From 70df5381789e17816c76e17c0a38c4bc68d3028a Mon Sep 17 00:00:00 2001 From: Florent Hivert Date: Sun, 5 Jul 2026 10:02:45 +0200 Subject: [PATCH] Release of Combi-1.1.0 --- combi/1.1.0/ALEA.Ccpo.html | 2381 ++ combi/1.1.0/ALEA.Misc.html | 316 + combi/1.1.0/ALEA.Qmeasure.html | 902 + combi/1.1.0/Combi.Basic.combclass.html | 350 + combi/1.1.0/Combi.Basic.congr.html | 590 + combi/1.1.0/Combi.Basic.ordtype.html | 578 + combi/1.1.0/Combi.Basic.unitriginv.html | 208 + combi/1.1.0/Combi.Combi.Dyckword.html | 741 + combi/1.1.0/Combi.Combi.Yamanouchi.html | 399 + combi/1.1.0/Combi.Combi.bintree.html | 1041 + combi/1.1.0/Combi.Combi.composition.html | 515 + combi/1.1.0/Combi.Combi.fibered_set.html | 141 + combi/1.1.0/Combi.Combi.multinomial.html | 118 + combi/1.1.0/Combi.Combi.ordtree.html | 270 + combi/1.1.0/Combi.Combi.partition.html | 1929 ++ combi/1.1.0/Combi.Combi.permuted.html | 294 + combi/1.1.0/Combi.Combi.setpartition.html | 405 + combi/1.1.0/Combi.Combi.skewpart.html | 955 + 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.../1.1.0/Combi.MPoly.MurnaghanNakayama.html | 510 + combi/1.1.0/Combi.MPoly.Schur_altdef.html | 833 + combi/1.1.0/Combi.MPoly.Schur_mpoly.html | 169 + combi/1.1.0/Combi.MPoly.antisym.html | 636 + combi/1.1.0/Combi.MPoly.homogsym.html | 870 + combi/1.1.0/Combi.MPoly.sympoly.html | 1521 ++ combi/1.1.0/Combi.SSRcomplements.ordcast.html | 96 + .../1.1.0/Combi.SSRcomplements.permcomp.html | 100 + combi/1.1.0/Combi.SSRcomplements.sorted.html | 178 + combi/1.1.0/Combi.SSRcomplements.tools.html | 357 + .../1.1.0/Combi.SymGroup.Frobenius_char.html | 499 + combi/1.1.0/Combi.SymGroup.cycles.html | 383 + combi/1.1.0/Combi.SymGroup.cycletype.html | 637 + combi/1.1.0/Combi.SymGroup.permcent.html | 396 + combi/1.1.0/Combi.SymGroup.presentSn.html | 1454 ++ combi/1.1.0/Combi.SymGroup.reprSn.html | 309 + combi/1.1.0/Combi.SymGroup.towerSn.html | 534 + combi/1.1.0/Combi.SymGroup.weak_order.html | 313 + combi/1.1.0/coqdoc.css | 338 + combi/1.1.0/depend.map | 59 + combi/1.1.0/depend.png | Bin 0 -> 323971 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00000000..f508be1b --- /dev/null +++ b/combi/1.1.0/ALEA.Ccpo.html @@ -0,0 +1,2381 @@ + + + + + +ALEA.Ccpo + + + + +
+ + + +
+ +

Library ALEA.Ccpo

+ +
+
+ +
+

Ccpo.v: Specification and properties of a cpo

+ +
+
+ +
+From Stdlib Require Export Arith.
+From Stdlib Require Import Lia.
+ +
+From Stdlib Require Export Classes.SetoidTactics.
+From Stdlib Require Export Classes.SetoidClass.
+From Stdlib Require Export Classes.Morphisms.
+ +
+Declare Scope signature_scope.
+#[local] Open Scope signature_scope.
+Declare Scope O_scope.
+Delimit Scope O_scope with O.
+#[local] Open Scope O_scope.
+ +
+
+ +
+

Ordered type

+ +
+
+ +
+Definition eq_rel {A} (E1 E2:relation A) := forall x y, E1 x y <-> E2 x y.
+ +
+Class Order {A} (E:relation A) (R:relation A) :=
+  {reflexive : Reflexive R;
+   order_eq : forall x y, R x y /\ R y x <-> E x y;
+   transitive : Transitive R }.
+ +
+Generalizable Variables A E R.
+ +
+#[export] Instance OrderEqRefl `{Order A E R} : Reflexive E.
+Qed.
+ +
+#[export] Instance OrderEqSym `{Order A E R} : Symmetric E.
+Qed.
+ +
+#[export] Instance OrderEqTrans `{Order A E R} : Transitive E.
+Qed.
+ +
+#[export] Instance OrderEquiv `{Order A E R} : Equivalence E.
+Qed.
+Opaque OrderEquiv.
+ +
+Class ord A :=
+   { Oeq : relation A;
+      Ole : relation A;
+      #[global] order_rel :: Order Oeq Ole }.
+ +
+Lemma OrdSetoid `(o:ord A) : Setoid A.
+ +
+ +
+Add Parametric Relation {A} {o:ord A} : A (@Oeq _ o)
+reflexivity proved by OrderEqRefl
+symmetry proved by OrderEqSym
+transitivity proved by OrderEqTrans
+as Oeq_setoid.
+ +
+ +
+Infix " ≤ " := Ole : O_scope.
+Infix " ≈ " := Oeq : type_scope.
+ +
+Definition Oge {O} {o:ord O} := fun (x y:O) => y ≤ x.
+Infix ">=" := Oge : O_scope.
+ +
+Lemma Ole_refl_eq : forall {O} {o:ord O} (x y:O), x ≈ y -> x ≤ y.
+ +
+#[export] Hint Immediate Ole_refl_eq : core.
+ +
+Lemma Ole_refl_eq_inv : forall {O} {o:ord O} (x y:O), x ≈ y -> y ≤ x.
+ +
+#[export] Hint Immediate Ole_refl_eq_inv : core.
+ +
+Lemma Ole_trans : forall {O} {o:ord O} (x y z:O), x ≤ y -> y ≤ z -> x ≤ z.
+ +
+Lemma Ole_refl : forall {O} {o:ord O} (x:O), x ≤ x.
+ +
+#[export] Hint Resolve Ole_refl : core.
+ +
+Add Parametric Relation {A} {o:ord A} : A (@Ole _ o)
+reflexivity proved by Ole_refl
+transitivity proved by Ole_trans
+as Ole_setoid.
+ +
+Lemma Ole_antisym : forall {O} {o:ord O} (x y:O), x ≤ y -> y ≤ x -> x ≈ y.
+#[export] Hint Immediate Ole_antisym : core.
+ +
+Lemma Oeq_refl : forall {O} {o:ord O} (x:O), x ≈ x.
+#[export] Hint Resolve Oeq_refl : core.
+ +
+Lemma Oeq_refl_eq : forall {O} {o:ord O} (x y:O), x = y -> x ≈ y.
+#[export] Hint Resolve Oeq_refl_eq : core.
+ +
+Lemma Oeq_sym : forall {O} {o:ord O} (x y:O), x ≈ y -> y ≈ x.
+ +
+Lemma Oeq_le : forall {O} {o:ord O} (x y:O), x ≈ y -> x ≤ y.
+ +
+Lemma Oeq_le_sym : forall {O} {o:ord O} (x y:O), x ≈ y -> y ≤ x.
+ +
+#[export] Hint Resolve Oeq_le : core.
+#[export] Hint Immediate Oeq_sym Oeq_le_sym : core.
+ +
+Lemma Oeq_trans
+   : forall {O} {o:ord O} (x y z:O), x ≈ y -> y ≈ z -> x ≈ z.
+#[export] Hint Resolve Oeq_trans : core.
+ +
+Add Parametric Morphism `(o:ord A): (Ole (ord:=o))
+with signature (Oeq (A:=A) ==> Oeq (A:=A) ==> iff) as Ole_eq_compat_iff.
+Qed.
+ +
+
+ +
+Equivalence of orders +
+
+ +
+Definition eq_ord {O} (o1 o2:ord O) := eq_rel (Ole (ord:=o1)) (Ole (ord:=o2)).
+ +
+Lemma eq_ord_equiv : forall {O} (o1 o2:ord O), eq_ord o1 o2 ->
+      eq_rel (Oeq (ord:=o1)) (Oeq (ord:=o2)).
+ +
+ +
+Lemma Ole_eq_compat :
+     forall {O} {o:ord O} (x1 x2 : O),
+       x1 ≈ x2 -> forall x3 x4 : O, x3 ≈ x4 -> x1 ≤ x3 -> x2 ≤ x4.
+ +
+Lemma Ole_eq_right : forall {O} {o:ord O} (x y z: O),
+             x ≤ y -> y ≈ z -> x ≤ z.
+ +
+Lemma Ole_eq_left : forall {O} {o:ord O} (x y z: O),
+             x ≈ y -> y ≤ z -> x ≤ z.
+ +
+Add Parametric Morphism `{o:ord A} : (Oeq (A:=A))
+       with signature Oeq ⇾ Oeq ⇾ iff as Oeq_iff_morphism.
+Qed.
+ +
+Add Parametric Morphism `{o:ord A} : (Ole (A:=A))
+       with signature Oeq ⇾ Oeq ⇾ iff as Ole_iff_morphism.
+Qed.
+ +
+Add Parametric Morphism `{o:ord A} : (Ole (A:=A))
+       with signature Ole --> Ole ⇾ Basics.impl as Ole_impl_morphism.
+Qed.
+ +
+
+ +
+

Definition and properties of x < y

+ +
+
+Definition Olt `{o:ord A} (r1 r2:A) : Prop := (r1 ≤ r2) /\ ~ (r1 ≈ r2).
+ +
+Infix "<" := Olt : O_scope.
+ +
+Lemma Olt_eq_compat `{o:ord A} :
+forall x1 x2 : A, x1 ≈ x2 -> forall x3 x4 : A, x3 ≈ x4 -> x1 < x3 -> x2 < x4.
+ +
+Add Parametric Morphism `{o:ord A} : (Olt (A:=A))
+with signature Oeq ⇾ Oeq ⇾ iff as Olt_iff_morphism.
+Qed.
+ +
+Lemma Olt_neq `{o:ord A} : forall x y:A, x < y -> ~ x ≈ y.
+ +
+Lemma Olt_neq_rev `{o:ord A} : forall x y:A, x < y -> ~ y ≈ x.
+ +
+Lemma Olt_le `{o:ord A} : forall x y, x < y -> x ≤ y.
+ +
+Lemma Olt_notle `{o:ord A} : forall x y, x < y -> ~ y ≤ x.
+ +
+Lemma Olt_trans `{o:ord A} : forall x y z:A, x < y -> y < z -> x < z.
+ +
+Lemma Ole_diff_lt `{o:ord A} : forall x y : A, x ≤ y -> ~ x ≈ y -> x < y.
+ +
+Lemma Ole_notle_lt `{o:ord A} : forall x y : A, x ≤ y -> ~ y ≤ x -> x < y.
+ +
+#[export] Hint Immediate Olt_neq Olt_neq_rev Olt_le Olt_notle : core.
+#[export] Hint Resolve Ole_diff_lt : core.
+ +
+Lemma Olt_antirefl `{o:ord A} : forall x:A, ~ x < x.
+ +
+Lemma Ole_lt_trans `{o:ord A} : forall x y z:A, x ≤ y -> y < z -> x < z.
+ +
+Lemma Olt_le_trans `{o:ord A} : forall x y z:A, x < y -> y ≤ z -> x < z.
+ +
+#[export] Hint Resolve Olt_antirefl : core.
+ +
+Lemma Ole_not_lt `{o:ord A} : forall x y:A, x ≤ y -> ~ y < x.
+#[export] Hint Resolve Ole_not_lt : core.
+ +
+Add Parametric Morphism `{o:ord A} : (Olt (A:=A))
+       with signature Ole --> Ole ⇾ Basics.impl as Olt_le_compat.
+Qed.
+ +
+
+ +
+

Dual order

+ +
+ +
    +
  • Iord x y = y ≤ x + +
    • +
+
+
+Definition Iord : forall O {o:ord O}, ord O.
+Defined.
+ +
+Arguments Iord O {o}.
+ +
+
+ +
+

Order on functions

+ +
+
+ +
+Definition fun_ext A B (R:relation B) : relation (A -> B) :=
+                fun f g => forall x, R (f x) (g x).
+Arguments fun_ext A [B] R.
+ +
+
+ +
+
    +
  • ford f g := forall x, f x ≤ g x + +
    • +
+
+
+#[export] Program Instance ford A O {o:ord O} : ord (A -> O) :=
+  {Oeq:=fun_ext A (Oeq (A:=O)); Ole:=fun_ext A (Ole (A:=O))}.
+ +
+Lemma ford_le_elim : forall A O (o:ord O) (f g:A -> O), f ≤ g -> forall n, f n ≤ g n.
+#[export] Hint Immediate ford_le_elim : core.
+ +
+Lemma ford_le_intro : forall A O (o:ord O) (f g:A -> O), ( forall n, f n ≤ g n ) -> f ≤ g.
+#[export] Hint Resolve ford_le_intro : core.
+ +
+Lemma ford_eq_elim : forall A O (o:ord O) (f g:A -> O), f ≈ g -> forall n, f n ≈ g n.
+#[export] Hint Immediate ford_eq_elim : core.
+ +
+Lemma ford_eq_intro : forall A O (o:ord O) (f g:A -> O), ( forall n, f n ≈ g n ) -> f ≈ g.
+#[export] Hint Resolve ford_eq_intro : core.
+ +
+
+ +
+

Monotonicity

+ +
+ +

Definition and properties

+ +
+
+ +
+Generalizable Variables Oa Ob Oc Od.
+ +
+Class monotonic `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob) :=
+      monotonic_def : forall x y, x ≤ y -> f x ≤ f y.
+ +
+Lemma monotonic_intro : forall `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob),
+  (forall x y, x ≤ y -> f x ≤ f y) -> monotonic f.
+#[export] Hint Resolve monotonic_intro : core.
+ +
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob) {m:monotonic f} : f
+with signature (Ole (A:=Oa) ⇾ Ole (A:=Ob))
+as monotonic_morphism.
+Qed.
+ +
+Class stable `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob) :=
+      stable_def : forall x y, x ≈ y -> f x ≈ f y.
+#[export] Hint Unfold stable : core.
+ +
+Lemma stable_intro : forall `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob),
+  (forall x y, x ≈ y -> f x ≈ f y) -> stable f.
+#[export] Hint Resolve stable_intro : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob) {s:stable f} : f
+with signature (Oeq (A:=Oa) ⇾ Oeq (A:=Ob))
+as stable_morphism.
+Qed.
+ +
+#[export] Typeclasses Opaque monotonic stable.
+ +
+#[export] Instance monotonic_stable `{o1:ord Oa} `{o2:ord Ob} (f : Oa -> Ob) {m:monotonic f}
+         : stable f.
+Qed.
+ +
+
+ +
+

Type of monotonic functions

+ +
+
+ +
+Record fmon `{o1:ord Oa} `{o2:ord Ob}:= mon
+          {fmont :> Oa -> Ob;
+           fmonotonic: monotonic fmont}.
+ +
+#[export]
+Existing Instance fmonotonic.
+ +
+Arguments mon [Oa o1 Ob o2] fmont {fmonotonic}.
+Arguments fmon Oa [o1] Ob {o2}.
+ +
+#[export] Hint Resolve fmonotonic : core.
+ +
+ +
+Notation "Oa -m> Ob" := (fmon Oa Ob)
+   (right associativity, at level 30) : O_scope.
+Notation "Oa --m> Ob" := (fmon Oa (o1:=Iord Oa) Ob )
+   (right associativity, at level 30) : O_scope.
+Notation "Oa --m-> Ob" := (fmon Oa (o1:=Iord Oa) Ob (o2:=Iord Ob))
+   (right associativity, at level 30) : O_scope.
+Notation "Oa -m-> Ob" := (fmon Oa Ob (o2:=Iord Ob))
+   (right associativity, at level 30) : O_scope.
+ +
+Open Scope O_scope.
+ +
+Lemma mon_simpl : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob){mf: monotonic f} x,
+      mon f x = f x.
+#[export] Hint Resolve mon_simpl : core.
+ +
+#[export] Instance fstable `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> Ob) : stable f.
+Qed.
+ +
+#[export] Hint Resolve fstable : core.
+ +
+Lemma fmon_le : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> Ob) x y,
+                x ≤ y -> f x ≤ f y.
+#[export] Hint Resolve fmon_le : core.
+ +
+Lemma fmon_eq : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> Ob) x y,
+                x ≈ y -> f x ≈ f y.
+#[export] Hint Resolve fmon_eq : core.
+ +
+#[export] Program Instance fmono Oa Ob {o1:ord Oa} {o2:ord Ob} : ord (Oa -m> Ob)
+   := {Oeq := fun (f g : Oa-m> Ob)=> forall x, f x ≈ g x;
+       Ole := fun (f g : Oa-m> Ob)=> forall x, f x ≤ g x}.
+ +
+Lemma mon_le_compat : forall `{o1:ord Oa} `{o2:ord Ob} (f g:Oa -> Ob)
+      {mf:monotonic f} {mg:monotonic g}, f ≤ g -> mon f ≤ mon g.
+#[export] Hint Resolve mon_le_compat : core.
+ +
+Lemma mon_eq_compat : forall `{o1:ord Oa} `{o2:ord Ob} (f g:Oa-> Ob)
+      {mf:monotonic f} {mg:monotonic g}, f ≈ g -> mon f ≈ mon g.
+#[export] Hint Resolve mon_eq_compat : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob}
+       : (fmont (Oa:=Oa) (Ob:=Ob))
+       with signature Oeq ⇾ Oeq ⇾ Oeq as fmont_eq_morphism.
+Qed.
+ +
+
+ +
+

Monotonicity and dual order

+ +
+
+ +
+Lemma Imonotonic `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) {m:monotonic f}
+         : monotonic (o1:=Iord Oa) (o2:=Iord Ob) f.
+ +
+#[export] Hint Extern 2 (@monotonic _ (Iord _) _ (Iord _) _) => apply @Imonotonic
+  : typeclass_instances.
+ +
+Definition imon `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) {m:monotonic f}
+   : Oa --m-> Ob := mon (o1:=Iord Oa) (o2:=Iord Ob) f.
+ +
+Lemma imon_simpl : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) {m:monotonic f} (x:Oa),
+     imon f x = f x.
+ +
+
+ +
+
    +
  • Iord (A -> U) corresponds to A -> Iord U + +
    • +
+
+
+ +
+Lemma Iord_app {A} `{o1:ord Oa} (x: A) : ((A -> Oa) --m-> Oa).
+ +
+
+ +
+
    +
  • Imon f uses f as monotonic function over the dual order. + +
    • +
+
+
+ +
+Definition Imon : forall `{o1:ord Oa} `{o2:ord Ob}, (Oa -m> Ob) -> (Oa --m-> Ob).
+Defined.
+ +
+Lemma Imon_simpl : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> Ob)(x:Oa),
+                   Imon f x = f x.
+ +
+
+ +
+

Monotonicity and equality

+ +
+
+ +
+Lemma mon_fun_eq_monotonic
+  : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) (g:Oa -m> Ob),
+            f ≈ g -> monotonic f.
+ +
+Definition mon_fun_subst `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) (g:Oa -m> Ob) (H:f ≈ g)
+   : Oa -m> Ob := mon f (fmonotonic:= mon_fun_eq_monotonic _ _ H).
+ +
+Lemma mon_fun_eq
+  : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -> Ob) (g:Oa -m> Ob)
+            (H:f ≈ g), g ≈ mon_fun_subst f g H.
+ +
+
+ +
+

Monotonic functions with 2 arguments

+ +
+
+ +
+Class monotonic2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc) :=
+    monotonic2_intro : forall (x y:Oa) (z t:Ob), x ≤ y -> z ≤ t -> f x z ≤ f y t.
+ +
+#[export] Instance mon2_intro `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+    {m1:monotonic f} {m2: forall x, monotonic (f x)} : monotonic2 f | 10.
+Qed.
+ +
+Lemma mon2_elim1 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+    {m:monotonic2 f} : monotonic f.
+ +
+Lemma mon2_elim2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+    {m:monotonic2 f} : forall x, monotonic (f x).
+#[export] Hint Immediate mon2_elim1 mon2_elim2: typeclass_instances.
+ +
+Definition mon_comp {A} `{o1: ord Oa} `{o2: ord Ob}
+         (f:A -> Oa -> Ob) {mf:forall x, monotonic (f x)} : A -> Oa -m> Ob
+         := fun x => mon (f x).
+ +
+#[export] Instance mon_fun_mon `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+    {m:monotonic2 f} : monotonic (fun x => mon (f x)).
+Qed.
+ +
+Class stable2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc) :=
+    stable2_intro : forall (x y:Oa) (z t:Ob), x ≈ y -> z ≈ t -> f x z ≈ f y t.
+ +
+#[export] Instance monotonic2_stable2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+    (f:Oa -> Ob -> Oc) {m:monotonic2 f} : stable2 f.
+Qed.
+ +
+#[export] Typeclasses Opaque monotonic2 stable2.
+ +
+Definition mon2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+     {mf:monotonic2 f} : Oa -m> Ob -m> Oc := mon (fun x => mon (f x)).
+ +
+Lemma mon2_simpl : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -> Oc)
+     {mf:monotonic2 f} x y, mon2 f x y = f x y.
+#[export] Hint Resolve mon2_simpl : core.
+ +
+Lemma mon2_le_compat : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+     (f g:Oa -> Ob -> Oc) {mf: monotonic2 f} {mg:monotonic2 g},
+     f ≤ g -> mon2 f ≤ mon2 g.
+ +
+Definition fun2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -m> Oc)
+     : Oa -> Ob -> Oc := fun x => f x.
+ +
+#[export] Instance fmon2_mon `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc} (f:Oa -> Ob -m> Oc) :
+       forall x:Oa, monotonic (fun2 f x).
+Qed.
+ +
+#[export] Instance fun2_monotonic `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+         (f:Oa -> Ob -m> Oc) {mf:monotonic f} : monotonic (fun2 f).
+Qed.
+#[export] Hint Resolve fun2_monotonic : core.
+ +
+#[export] Instance fmonotonic2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc)
+         : monotonic2 (fun2 f).
+Qed.
+#[export] Hint Resolve fmonotonic2 : core.
+ +
+Definition mfun2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc)
+   : Oa-m> (Ob -> Oc) := mon (fun2 f).
+ +
+Lemma mfun2_simpl : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc) x y,
+     mfun2 f x y = f x y.
+ +
+#[export] Instance mfun2_mon `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+         (f:Oa -m> Ob -m> Oc) x : monotonic (mfun2 f x).
+Qed.
+ +
+Lemma mon2_fun2 : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+     (f:Oa -m> Ob -m> Oc), mon2 (fun2 f) ≈ f.
+ +
+Lemma fun2_mon2 : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+      (f:Oa -> Ob -> Oc) {mf:monotonic2 f} , fun2 (mon2 f) ≈ f.
+#[export] Hint Resolve mon2_fun2 fun2_mon2 : core.
+ +
+#[export] Instance fstable2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc)
+                : stable2 (fun2 f).
+Qed.
+#[export] Hint Resolve fstable2 : core.
+ +
+Definition Imon2 : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc},
+     (Oa -m> Ob -m> Oc) -> (Oa --m> Ob --m-> Oc).
+Defined.
+ +
+Lemma Imon2_simpl : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+      (f:Oa -m> Ob -m> Oc) (x:Oa) (y: Ob),
+      Imon2 f x y = f x y.
+ +
+Lemma Imonotonic2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+      (f:Oa -> Ob -> Oc){mf : monotonic2 f}
+      : monotonic2 (o1:=Iord Oa) (o2:=Iord Ob) (o3:=Iord Oc) f.
+ +
+#[export] Hint Extern 2 (@monotonic2 _ (Iord _) _ (Iord _) _ (Iord _) _) => apply @Imonotonic2
+  : typeclass_instances.
+ +
+Definition imon2 `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+      (f:Oa -> Ob -> Oc){mf : monotonic2 f} : Oa --m> Ob --m-> Oc :=
+      mon2 (o1:=Iord Oa) (o2:=Iord Ob) (o3:=Iord Oc) f.
+ +
+Lemma imon2_simpl : forall `{o1: ord Oa} `{o2: ord Ob} `{o3:ord Oc}
+      (f:Oa -> Ob -> Oc){mf : monotonic2 f} (x:Oa) (y:Ob),
+      imon2 f x y = f x y.
+ +
+
+ +
+

Strict monotonicity

+ +
+
+ +
+Lemma inj_strict_mon : forall `{o1: ord Oa} `{o2: ord Ob} (f:Oa -> Ob) {mf:monotonic f},
+      (forall x y, f x ≈ f y -> x ≈ y) -> forall x y, x < y -> f x < f y.
+ +
+
+ +
+

Sequences

+

Usual order on natural numbers

+ +
+
+ +
+#[export] Program Instance natO : ord nat :=
+    { Oeq := fun n m : nat => n = m;
+       Ole := fun n m : nat => (n ≤ m)%nat}.
+ +
+Lemma le_Ole : forall n m, ((n ≤ m)%nat)-> n ≤ m.
+#[export] Hint Resolve le_Ole : core.
+ +
+Lemma nat_monotonic : forall {O} {o:ord O}
+               (f:nat -> O), (forall n, f n ≤ f (S n)) -> monotonic f.
+#[export] Hint Resolve nat_monotonic : core.
+ +
+Lemma nat_monotonic_inv : forall {O} {o:ord O}
+               (f:nat -> O), (forall n, f (S n) ≤ f n) -> monotonic (o2:=Iord O) f.
+#[export] Hint Resolve nat_monotonic_inv : core.
+ +
+Definition fnatO_intro : forall {O} {o:ord O} (f:nat -> O), (forall n, f n ≤ f (S n)) -> nat -m> O.
+Defined.
+ +
+Lemma fnatO_elim : forall {O} {o:ord O} (f:nat -m> O) (n:nat), f n ≤ f (S n).
+#[export] Hint Resolve fnatO_elim : core.
+ +
+
+ +
+
    +
  • (mseq_lift_left f n) k = f (n+k) + +
    • +
+
+
+ +
+Definition seq_lift_left {O} (f:nat -> O) n := fun k => f (n+k)%nat.
+ +
+#[export] Instance mon_seq_lift_left
+  : forall n {O} {o:ord O} (f:nat -> O) {m:monotonic f}, monotonic (seq_lift_left f n).
+Qed.
+ +
+Definition mseq_lift_left : forall {O} {o:ord O} (f:nat -m> O) (n:nat), nat -m> O.
+Defined.
+ +
+Lemma mseq_lift_left_simpl : forall {O} {o:ord O} (f:nat -m> O) (n k:nat),
+    mseq_lift_left f n k = f (n+k)%nat.
+ +
+Lemma mseq_lift_left_le_compat : forall {O} {o:ord O} (f g:nat -m> O) (n:nat),
+             f ≤ g -> mseq_lift_left f n ≤ mseq_lift_left g n.
+#[export] Hint Resolve mseq_lift_left_le_compat : core.
+ +
+Add Parametric Morphism {O} {o:ord O} : (@mseq_lift_left _ o)
+  with signature Oeq ⇾ eq ⇾ Oeq
+  as mseq_lift_left_eq_compat.
+Qed.
+#[export] Hint Resolve mseq_lift_left_eq_compat : core.
+ +
+Add Parametric Morphism {O} {o:ord O}: (@seq_lift_left O)
+  with signature Oeq ⇾ eq ⇾ Oeq
+  as seq_lift_left_eq_compat.
+Qed.
+#[export] Hint Resolve seq_lift_left_eq_compat : core.
+ +
+
+ +
+
    +
  • (mseq_lift_right f n) k = f (k+n) + +
    • +
+
+
+ +
+Definition seq_lift_right {O} (f:nat -> O) n := fun k => f (k+n)%nat.
+ +
+#[export] Instance mon_seq_lift_right
+   : forall n {O} {o:ord O} (f:nat -> O) {m:monotonic f}, monotonic (seq_lift_right f n).
+Qed.
+ +
+Definition mseq_lift_right : forall {O} {o:ord O} (f:nat -m> O) (n:nat), nat -m> O.
+Defined.
+ +
+Lemma mseq_lift_right_simpl : forall {O} {o:ord O} (f:nat -m> O) (n k:nat),
+    mseq_lift_right f n k = f (k+n)%nat.
+ +
+Lemma mseq_lift_right_le_compat : forall {O} {o:ord O} (f g:nat -m> O) (n:nat),
+             f ≤ g -> mseq_lift_right f n ≤ mseq_lift_right g n.
+#[export] Hint Resolve mseq_lift_right_le_compat : core.
+ +
+Add Parametric Morphism {O} {o:ord O} : (mseq_lift_right (o:=o))
+   with signature Oeq ⇾ eq ⇾ Oeq
+   as mseq_lift_right_eq_compat.
+Qed.
+ +
+Add Parametric Morphism {O} {o:ord O}: (@seq_lift_right O)
+  with signature Oeq ⇾ eq ⇾ Oeq
+  as seq_lift_right_eq_compat.
+Qed.
+#[export] Hint Resolve seq_lift_right_eq_compat : core.
+ +
+Lemma mseq_lift_right_left : forall {O} {o:ord O} (f:nat -m> O) n,
+       mseq_lift_left f n ≈ mseq_lift_right f n.
+ +
+
+ +
+

Monotonicity and functions

+
    +
  • (shift f x) n = f n x + +
    • +
+
+
+ +
+#[export] Instance shift_mon_fun {A} `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> (A -> Ob)) :
+       forall x:A, monotonic (fun (y:Oa) => f y x).
+Qed.
+ +
+Definition shift {A} `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> (A -> Ob)) : A -> Oa -m> Ob
+   := fun x => (mon (fun y => f y x)).
+ +
+Infix " ◊" := shift (at level 30, no associativity) : O_scope.
+ +
+Lemma shift_simpl : forall {A} `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> (A -> Ob)) x y,
+      (f <o> x) y = f y x.
+ +
+Lemma shift_le_compat : forall {A} `{o1:ord Oa} `{o2:ord Ob} (f g:Oa -m> (A -> Ob)),
+             f ≤ g -> shift f ≤ shift g.
+#[export] Hint Resolve shift_le_compat : core.
+ +
+Add Parametric Morphism {A} `{o1:ord Oa} `{o2:ord Ob}
+    : (shift (A:=A) (Oa:=Oa) (Ob:=Ob)) with signature Oeq ⇾ eq ⇾ Oeq
+as shift_eq_compat.
+Qed.
+ +
+#[export] Instance ishift_mon {A} `{o1:ord Oa} `{o2:ord Ob} (f:A -> (Oa -m> Ob)) :
+       monotonic (fun (y:Oa) (x:A) => f x y).
+Qed.
+ +
+Definition ishift {A} `{o1:ord Oa} `{o2:ord Ob} (f:A -> (Oa -m> Ob)) : Oa -m> (A -> Ob)
+   := mon (fun (y:Oa) (x:A) => f x y) (fmonotonic:=ishift_mon f).
+ +
+Lemma ishift_simpl : forall {A} `{o1:ord Oa} `{o2:ord Ob} (f:A -> (Oa -m> Ob)) x y,
+      ishift f x y = f y x.
+ +
+Lemma ishift_le_compat : forall {A} `{o1:ord Oa} `{o2:ord Ob} (f g:A -> (Oa -m> Ob)),
+             f ≤ g -> ishift f ≤ ishift g.
+#[export] Hint Resolve ishift_le_compat : core.
+ +
+Add Parametric Morphism {A} `{o1:ord Oa} `{o2:ord Ob}
+    : (ishift (A:=A) (Oa:=Oa) (Ob:=Ob)) with signature Oeq ⇾ eq ⇾ Oeq
+as ishift_eq_compat.
+Qed.
+ +
+ +
+#[export] Instance shift_fun_mon `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> (Ob -> Oc))
+     {m:forall x, monotonic (f x)} : monotonic (shift f).
+Qed.
+ +
+#[export] Instance shift_mon2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc)
+     : monotonic2 (fun x y => f y x).
+Qed.
+#[export] Hint Resolve shift_mon_fun shift_fun_mon shift_mon2 : core.
+ +
+Definition mshift `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Oa -m> Ob -m> Oc)
+    : Ob -m> Oa -m> Oc := mon2 (fun x y => f y x).
+ +
+
+ +
+
    +
  • id c = c + +
    • +
+
+
+ +
+Definition id O {o:ord O} : O -> O := fun x => x.
+ +
+#[export] Instance mon_id : forall {O:Type} {o:ord O}, monotonic (id O).
+Qed.
+ +
+
+ +
+
    +
  • (cte c) n = c + +
    • +
+
+
+ +
+Definition cte A `{o1:ord Oa} (c:Oa) : A -> Oa := fun x => c.
+ +
+#[export] Instance mon_cte : forall `{o1:ord Oa} `{o2:ord Ob} (c:Ob), monotonic (cte Oa c).
+Qed.
+ +
+Definition mseq_cte {O} {o:ord O} (c:O) : nat -m> O := mon (cte nat c).
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} : (@cte Oa Ob _)
+  with signature Ole ⇾ Ole as cte_le_compat.
+Qed.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} : (@cte Oa Ob _)
+  with signature Oeq ⇾ Oeq as cte_eq_compat.
+Qed.
+ +
+#[export] Instance mon_diag `{o1:ord Oa} `{o2:ord Ob}(f:Oa -m> (Oa -m> Ob))
+     : monotonic (fun x => f x x).
+Qed.
+#[export] Hint Resolve mon_diag : core.
+ +
+Definition diag `{o1:ord Oa} `{o2:ord Ob}(f:Oa -m> (Oa -m> Ob)) : Oa-m> Ob
+     := mon (fun x => f x x).
+ +
+Lemma fmon_diag_simpl : forall `{o1:ord Oa} `{o2:ord Ob} (f:Oa -m> (Oa -m> Ob)) (x:Oa),
+             diag f x = f x x.
+ +
+Lemma diag_le_compat : forall `{o1:ord Oa} `{o2:ord Ob} (f g:Oa -m> (Oa -m> Ob)),
+             f ≤ g -> diag f ≤ diag g.
+#[export] Hint Resolve diag_le_compat : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} : (diag (Oa:=Oa) (Ob:=Ob))
+   with signature Oeq ⇾ Oeq as diag_eq_compat.
+Qed.
+ +
+Lemma diag_shift : forall `{o1:ord Oa} `{o2:ord Ob} (f: Oa -m> Oa -m> Ob),
+                   diag f ≈ diag (mshift f).
+ +
+#[export] Hint Resolve diag_shift : core.
+ +
+Lemma mshift_simpl : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (h:Oa -m> Ob -m> Oc) (x : Ob) (y:Oa), mshift h x y = h y x.
+ +
+Lemma mshift_le_compat : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (f g:Oa -m> Ob -m> Oc), f ≤ g -> mshift f ≤ mshift g.
+#[export] Hint Resolve mshift_le_compat : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} : (@mshift Oa _ Ob _ Oc _)
+     with signature Oeq ⇾ Oeq as mshift_eq_compat.
+Qed.
+ +
+Lemma mshift2_eq : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (h : Oa -m> Ob -m> Oc),
+             mshift (mshift h) ≈ h.
+ +
+
+ +
+
    +
  • (f@g) x = f (g x) + +
    • +
+
+
+ +
+#[export] Instance monotonic_comp `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+   (f:Ob -> Oc){mf : monotonic f} (g:Oa -> Ob){mg:monotonic g} : monotonic (fun x => f (g x)).
+Qed.
+#[export] Hint Resolve monotonic_comp : core.
+ +
+#[export] Instance monotonic_comp_mon `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+   (f:Ob -m> Oc)(g:Oa -m> Ob) : monotonic (fun x => f (g x)).
+Qed.
+#[export] Hint Resolve monotonic_comp_mon : core.
+ +
+Definition comp `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f:Ob -m> Oc) (g:Oa -m> Ob)
+       : Oa -m> Oc := mon (fun x => f (g x)).
+ +
+Infix "@" := comp (at level 35) : O_scope.
+ +
+Lemma comp_simpl : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (f:Ob -m> Oc) (g:Oa -m> Ob) (x:Oa), (f@g) x = f (g x).
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}: (@comp Oa _ Ob _ Oc _)
+    with signature Ole ++> Ole ++> Ole
+    as comp_le_compat.
+Qed.
+ +
+#[export] Hint Immediate comp_le_compat : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} : (@comp Oa _ Ob _ Oc _)
+    with signature Oeq ⇾ Oeq ⇾ Oeq
+    as comp_eq_compat.
+Qed.
+ +
+#[export] Hint Immediate comp_eq_compat : core.
+ +
+
+ +
+
    +
  • (f@2 g) h x = f (g x) (h x) + +
    • +
+
+
+ +
+#[export] Instance mon_app2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}
+      (f:Ob -> Oc -> Od) (g:Oa -> Ob) (h:Oa -> Oc)
+      {mf:monotonic2 f}{mg:monotonic g} {mh:monotonic h}
+      : monotonic (fun x => f (g x) (h x)).
+Qed.
+ +
+#[export] Instance mon_app2_mon `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}
+      (f:Ob -m> Oc -m> Od) (g:Oa -m> Ob) (h:Oa -m> Oc)
+      : monotonic (fun x => f (g x) (h x)).
+Qed.
+ +
+Definition app2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}
+        (f:Ob -m> Oc -m> Od) (g:Oa -m> Ob) (h:Oa -m> Oc) : Oa -m> Od
+        := mon (fun x => f (g x) (h x)).
+ +
+Infix "@²" := app2 (at level 70) : O_scope.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}:
+        (@app2 Oa _ Ob _ Oc _ Od _)
+    with signature Ole ++> Ole ++> Ole ++> Ole
+    as app2_le_compat.
+Qed.
+ +
+#[export] Hint Immediate app2_le_compat : core.
+ +
+Add Parametric Morphism `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}:
+        (@app2 Oa _ Ob _ Oc _ Od _)
+    with signature Oeq ⇾ Oeq ⇾ Oeq ⇾ Oeq
+    as app2_eq_compat.
+Qed.
+ +
+#[export] Hint Immediate app2_eq_compat : core.
+ +
+Lemma app2_simpl :
+    forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}
+            (f:Ob -m> Oc -m> Od) (g:Oa -m> Ob) (h:Oa -m> Oc) (x:Oa),
+    (f@2 g) h x = f (g x) (h x).
+ +
+Lemma comp_monotonic_right :
+      forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f: Ob -m> Oc) (g1 g2:Oa -m> Ob),
+               g1 ≤ g2 -> f @ g1 ≤ f @ g2.
+#[export] Hint Resolve comp_monotonic_right : core.
+ +
+Lemma comp_monotonic_left :
+      forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} (f1 f2: Ob -m> Oc) (g:Oa -m> Ob),
+               f1 ≤ f2 -> f1 @ g ≤ f2 @ g.
+#[export] Hint Resolve comp_monotonic_left : core.
+ +
+#[export] Instance comp_monotonic2 : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc},
+             monotonic2 (@comp Oa _ Ob _ Oc _).
+Qed.
+#[export] Hint Resolve comp_monotonic2 : core.
+ +
+Definition fcomp `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} :
+   (Ob -m> Oc) -m> (Oa -m> Ob) -m> (Oa -m> Oc) := mon2 (@comp Oa _ Ob _ Oc _).
+ +
+Arguments fcomp Oa [o1] Ob [o2] Oc {o3}.
+ +
+Lemma fcomp_simpl : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (f:Ob -m> Oc) (g:Oa -m> Ob), fcomp _ _ _ f g = f@g.
+ +
+Definition fcomp2 `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od} :
+        (Oc -m> Od) -m> (Oa -m> Ob -m> Oc) -m> (Oa -m> Ob -m> Od):=
+        (fcomp Oa (Ob -m> Oc) (Ob -m> Od))@(fcomp Ob Oc Od).
+ +
+Arguments fcomp2 Oa [o1] Ob [o2] Oc [o3] Od {o4}.
+ +
+Lemma fcomp2_simpl : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4:ord Od}
+      (f:Oc -m> Od) (g:Oa -m> Ob -m> Oc) (x:Oa)(y:Ob), fcomp2 _ _ _ _ f g x y = f (g x y).
+ +
+Lemma fmon_le_compat2 : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (f: Oa -m> Ob -m> Oc) (x y:Oa) (z t:Ob), x ≤ y -> z ≤ t -> f x z ≤ f y t.
+#[export] Hint Resolve fmon_le_compat2 : core.
+ +
+Lemma fmon_cte_comp : forall `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc}
+      (c:Oc)(f:Oa -m> Ob), (mon (cte Ob c)) @ f ≈ mon (cte Oa c).
+ +
+
+ +
+

Abstract relational notion of lubs

+ +
+
+Record islub O (o:ord O) I (f:I -> O) (x:O) : Prop := mk_islub
+     { le_islub : forall i, f i ≤ x;
+       islub_le : forall y, (forall i, f i ≤ y) -> x ≤ y}.
+Arguments islub [O o I] f x.
+Arguments le_islub [O o I f x].
+Arguments islub_le [O o I f x].
+ +
+Definition isglb O (o:ord O) I (f:I -> O) (x:O) : Prop
+     := islub (o:=Iord O) f x.
+Arguments isglb [O o I].
+ +
+Lemma le_isglb O (o:ord O) I (f:I -> O) (x:O) :
+         isglb f x -> forall i, x ≤ f i.
+ +
+Lemma isglb_le O (o:ord O) I (f:I -> O) (x:O) :
+         isglb f x -> forall y, (forall i, y ≤ f i) -> y ≤ x.
+Arguments le_isglb [O o I f x].
+Arguments isglb_le [O o I f x].
+ +
+Lemma mk_isglb O (o:ord O) I (f:I -> O) (x:O) :
+      (forall i, x ≤ f i) -> (forall y, (forall i, y ≤ f i) -> y ≤ x)
+      -> isglb f x.
+ +
+Lemma islub_eq_compat O (o:ord O) I (f g:I -> O) (x y:O):
+      f ≈ g -> x ≈ y -> islub f x -> islub g y.
+ +
+Lemma islub_eq_compat_left O (o:ord O) I (f g:I -> O) (x:O):
+      f ≈ g -> islub f x -> islub g x.
+ +
+Lemma islub_eq_compat_right O (o:ord O) I (f:I -> O) (x y:O):
+       x ≈ y -> islub f x -> islub f y.
+ +
+Lemma isglb_eq_compat O (o:ord O) I (f g:I -> O) (x y:O):
+      f ≈ g -> x ≈ y -> isglb f x -> isglb g y.
+ +
+Lemma isglb_eq_compat_left O (o:ord O) I (f g:I -> O) (x:O):
+      f ≈ g -> isglb f x -> isglb g x.
+ +
+Lemma isglb_eq_compat_right O (o:ord O) I (f:I -> O) (x y:O):
+       x ≈ y -> isglb f x -> isglb f y.
+ +
+Add Parametric Morphism {O} {o:ord O} I : (@islub _ o I)
+with signature Oeq ⇾ Oeq ⇾ iff
+as islub_morphism.
+Qed.
+ +
+Add Parametric Morphism {O} {o:ord O} I : (@isglb _ o I)
+with signature Oeq ⇾ Oeq ⇾ iff
+as isglb_morphism.
+Qed.
+ +
+Add Parametric Morphism {O} {o:ord O} I : (@islub _ o I)
+with signature (@pointwise_relation I O (@Oeq _ _)) ⇾ Oeq ⇾ iff
+as islub_morphism_ext.
+Qed.
+ +
+Add Parametric Morphism {O} {o:ord O} I : (@isglb _ o I)
+with signature (@pointwise_relation I O (@Oeq _ _)) ⇾ Oeq ⇾ iff
+as isglb_morphism_ext.
+Qed.
+ +
+Lemma islub_incr_ext {O} {o:ord O} (f :nat -> O) (x:O) (n:nat):
+      (forall k, f k ≤ f (S k)) -> islub f x -> islub (fun k => f (n + k)) x.
+ +
+Lemma islub_incr_lift {O} {o:ord O} (f :nat -> O) (x:O) (n:nat):
+      (forall k, f k ≤ f (S k)) -> islub (fun k => f (n + k)) x -> islub f x.
+ +
+Lemma isglb_decr_ext {O} {o:ord O} (f :nat -> O) (x:O) (n:nat):
+      (forall k, f (S k) ≤ f k) -> isglb f x -> isglb (fun k => f (n + k)) x.
+ +
+Lemma isglb_decr_lift {O} {o:ord O} (f :nat -> O) (x:O) (n:nat):
+      (forall k, f (S k) ≤ f k) -> isglb (fun k => f (n + k)) x -> isglb f x.
+ +
+#[export] Hint Resolve islub_incr_ext isglb_decr_ext : core.
+ +
+Lemma islub_exch {O} {o:ord O} (F :nat -> nat -> O) (f g : nat -> O)(x:O) :
+      (forall m, islub (fun n => F n m) (f m))
+       -> (forall n, islub (F n) (g n)) -> islub f x -> islub g x.
+ +
+Lemma islub_decr {O} {o:ord O} {I} (f g : I -> O) (x y : O) :
+      (f ≤ g) -> islub f x -> islub g y -> x ≤ y.
+ +
+Lemma islub_unique_eq {O} {o:ord O} {I} (f g : I -> O) (x y : O) :
+      (f ≈ g) -> islub f x -> islub g y -> x ≈ y.
+ +
+Lemma islub_unique {O} {o:ord O} {I} (f : I -> O) (x y : O) :
+           islub f x -> islub f y -> x ≈ y.
+ +
+Lemma islub_fun_intro O (o:ord O) {I A} (F : I -> A -> O) (f : A -> O) :
+           (forall x, islub (fun i => F i x) (f x)) -> islub F f.
+ +
+
+ +
+

Basic operators of omega-cpos

+
    +
  • Constant : 0 +
      +
    • lub : limit of monotonic sequences + +
      • +
    + +
    • +
+ +
+
+ +
+Generalizable Variables D.
+ +
+
+ +
+

Definition of cpos

+ +
+
+Class cpo `{o:ord D} : Type := mk_cpo
+  {D0 : D; lub: forall (f:nat -m> D), D;
+   Dbot : forall x:D, D0 ≤ x;
+   le_lub : forall (f : nat -m> D) (n:nat), f n ≤ lub f;
+   lub_le : forall (f : nat -m> D) (x:D), (forall n, f n ≤ x) -> lub f ≤ x}.
+ +
+Arguments cpo D {o}.
+ +
+Notation "0" := D0 : O_scope.
+ +
+#[export] Hint Resolve Dbot le_lub lub_le : core.
+ +
+Definition mon_ord_equiv : forall `{o:ord D1} `{o1:ord D2} {o2:ord D2},
+      eq_ord o1 o2 -> fmon D1 D2 (o2:=o2) -> fmon D1 D2 (o2:=o1).
+Defined.
+ +
+Lemma mon_ord_equiv_simpl : forall `{o:ord D1} `{o1:ord D2} {o2:ord D2}
+      (H:eq_ord o1 o2) (f:fmon D1 D2 (o2:=o2)) (x:D1),
+      mon_ord_equiv H f x = f x.
+ +
+Definition cpo_ord_equiv `{o1:ord D} (o2:ord D)
+       : eq_ord o1 o2 -> cpo (o:=o1) D -> cpo (o:=o2) D.
+Defined.
+ +
+
+ +
+

Least upper bounds

+ +
+
+ +
+Add Parametric Morphism `{c:cpo D} : (lub (cpo:=c))
+             with signature Ole ++> Ole as lub_le_compat.
+Qed.
+#[export] Hint Resolve lub_le_compat : core.
+ +
+Add Parametric Morphism `{c:cpo D}: (lub (cpo:=c))
+      with signature Oeq ⇾ Oeq as lub_eq_compat.
+Qed.
+#[export] Hint Resolve lub_eq_compat : core.
+ +
+Notation "'mlub' f" := (lub (mon f)) (at level 60) : O_scope .
+ +
+Lemma mlub_le_compat : forall `{c:cpo D} (f g:nat -> D) {mf:monotonic f} {mg:monotonic g},
+                f ≤ g -> mlub f ≤ mlub g.
+#[export] Hint Resolve mlub_le_compat : core.
+ +
+Lemma mlub_eq_compat : forall `{c:cpo D} (f g:nat -> D) {mf:monotonic f} {mg:monotonic g},
+                f ≈ g -> mlub f ≈ mlub g.
+#[export] Hint Resolve mlub_eq_compat : core.
+ +
+Lemma le_mlub : forall `{c:cpo D} (f:nat -> D) {m:monotonic f} (n:nat), f n ≤ mlub f.
+ +
+Lemma mlub_le : forall `{c:cpo D}(f:nat -> D) {m:monotonic f}(x:D), (forall n, f n ≤ x) -> mlub f ≤ x.
+#[export] Hint Resolve le_mlub mlub_le : core.
+ +
+Lemma islub_mlub : forall `{c:cpo D}(f:nat -> D) {m:monotonic f},
+            islub f (mlub f).
+ +
+Lemma islub_lub : forall `{c:cpo D}(f:nat -m> D),
+            islub f (lub f).
+ +
+#[export] Hint Resolve islub_mlub islub_lub : core.
+ +
+#[export] Instance lub_mon `{c:cpo D} : monotonic lub.
+Qed.
+ +
+Definition Lub `{c:cpo D} : (nat -m> D) -m> D := mon lub.
+ +
+#[export] Instance monotonic_lub_comp {O} {o:ord O} `{c:cpo D} (f:O -> nat -> D){mf:monotonic2 f}:
+         monotonic (fun x => mlub (f x)).
+Qed.
+ +
+Lemma lub_cte : forall `{c:cpo D} (d:D), mlub (cte nat d) ≈ d.
+ +
+#[export] Hint Resolve lub_cte : core.
+ +
+Lemma mlub_lift_right : forall `{c:cpo D} (f:nat -m> D) n,
+      lub f ≈ mlub (seq_lift_right f n).
+#[export] Hint Resolve mlub_lift_right : core.
+ +
+Lemma mlub_lift_left : forall `{c:cpo D} (f:nat -m> D) n,
+      lub f ≈ mlub (seq_lift_left f n).
+#[export] Hint Resolve mlub_lift_left : core.
+ +
+Lemma lub_lift_right : forall `{c:cpo D} (f:nat -m> D) n,
+      lub f ≈ lub (mseq_lift_right f n).
+#[export] Hint Resolve lub_lift_right : core.
+ +
+Lemma lub_lift_left : forall `{c:cpo D} (f:nat -m> D) n,
+      lub f ≈ lub (mseq_lift_left f n).
+#[export] Hint Resolve lub_lift_left : core.
+ +
+Lemma lub_le_lift : forall `{c:cpo D} (f g:nat -m> D)
+      (n:nat), (forall k, n ≤ k -> f k ≤ g k) -> lub f ≤ lub g.
+ +
+Lemma lub_eq_lift : forall `{c:cpo D} (f g:nat -m> D) {m:monotonic f} {m':monotonic g}
+      (n:nat), (forall k, n ≤ k -> f k ≈ g k) -> lub f ≈ lub g.
+ +
+Lemma lub_seq_eq : forall `{c:cpo D} (f:nat -> D) (g: nat-m> D) (H:f ≈ g),
+      lub g ≈ lub (mon_fun_subst f g H).
+ +
+Lemma lub_Olt : forall `{c:cpo D} (f:nat -m> D) (k:D),
+      k < lub f -> ~ (forall n, f n ≤ k).
+ +
+
+ +
+
    +
  • (lub_fun h) x = lub_n (h n x) + +
    • +
+
+
+Definition lub_fun {A} `{c:cpo D} (h : nat -m> (A -> D)) : A -> D
+               := fun x => mlub (h <o> x).
+ +
+ +
+#[export] Instance lub_shift_mon {O} {o:ord O} `{c:cpo D} (h : nat -m> (O -m> D))
+          : monotonic (fun (x:O) => lub (mshift h x)).
+Qed.
+#[export] Hint Resolve lub_shift_mon : core.
+ +
+
+ +
+

Functional cpos

+ +
+
+ +
+#[export] Program Instance fcpo {A: Type} `(c:cpo D) : cpo (A -> D) :=
+  {D0 := fun x:A => (0:D); lub := fun f => lub_fun f}.
+ +
+Lemma fcpo_lub_simpl : forall {A} `{c:cpo D} (h:nat -m> (A -> D))(x:A),
+      (lub h) x = lub (h <o> x).
+ +
+Lemma lub_ishift : forall {A} `{c:cpo D} (h:A -> (nat -m> D)),
+       lub (ishift h) ≈ fun x => lub (h x).
+ +
+
+ +
+

Cpo of monotonic functions

+ +
+
+ +
+#[export] Program Instance fmon_cpo {O} {o:ord O} `{c:cpo D} : cpo (O -m> D) :=
+  { D0 := mon (cte O (0:D));
+    lub := fun h:nat -m> (O -m> D) => mon (fun (x:O) => lub (cpo:=c) (mshift h x))}.
+ +
+Lemma fmon_lub_simpl : forall {O} {o:ord O} `{c:cpo D}
+      (h:nat -m> (O -m> D))(x:O), (lub h) x = lub (mshift h x).
+#[export] Hint Resolve fmon_lub_simpl : core.
+ +
+#[export] Instance mon_fun_lub : forall {O} {o:ord O} `{c:cpo D}
+         (h:nat -m> (O -> D)) {mh:forall n, monotonic (h n)}, monotonic (lub h).
+Qed.
+ +
+ +
+
+ +
+Link between lubs on ordinary functions and monotonic functions +
+
+ +
+Lemma lub_mon_fcpo : forall {O} {o:ord O} `{c:cpo D} (h:nat -m> (O -m> D)),
+      lub h ≈ mon (lub (mfun2 h)).
+ +
+Lemma lub_fcpo_mon : forall {O} {o:ord O} `{c:cpo D} (h:nat -m> (O -> D))
+     {mh:forall x, monotonic (h x)}, lub h ≈ lub (mon2 h).
+ +
+Lemma double_lub_diag : forall `{c:cpo D} (h : nat -m> nat -m> D),
+        lub (lub h) ≈ lub (diag h).
+#[export] Hint Resolve double_lub_diag : core.
+ +
+Lemma double_lub_shift : forall `{c:cpo D} (h : nat -m> nat -m> D),
+        lub (lub h) ≈ lub (lub (mshift h)).
+#[export] Hint Resolve double_lub_shift : core.
+ +
+ +
+
+ +
+

Continuity

+ +
+
+ +
+Lemma lub_comp_le :
+    forall `{c1:cpo D1} `{c2:cpo D2} (f:D1 -m> D2) (h : nat -m> D1),
+                lub (f @ h) ≤ f (lub h).
+#[export] Hint Resolve lub_comp_le : core.
+ +
+Lemma lub_app2_le : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+        (F:D1 -m> D2 -m> D3) (f : nat -m> D1) (g: nat -m> D2),
+        lub ((F @² f) g) ≤ F (lub f) (lub g).
+#[export] Hint Resolve lub_app2_le : core.
+ +
+Class continuous `{c1:cpo D1} `{c2:cpo D2} (f:D1 -m> D2) :=
+    cont_intro : forall (h : nat -m> D1), f (lub h) ≤ lub (f @ h).
+ +
+#[export] Typeclasses Opaque continuous.
+ +
+Lemma continuous_eq_compat : forall `{c1:cpo D1} `{c2:cpo D2}(f g:D1 -m> D2),
+                  f ≈ g -> continuous f -> continuous g.
+ +
+Add Parametric Morphism `{c1:cpo D1} `{c2:cpo D2} : (@continuous D1 _ _ D2 _ _)
+     with signature Oeq ⇾ iff
+as continuous_eq_compat_iff.
+Qed.
+ +
+Lemma lub_comp_eq :
+    forall `{c1:cpo D1} `{c2:cpo D2} (f:D1 -m> D2) (h : nat -m> D1),
+             continuous f -> f (lub h) ≈ lub (f @ h).
+#[export] Hint Resolve lub_comp_eq : core.
+ +
+
+ +
+
    +
  • mon0 x == 0 + +
    • +
+
+
+#[export] Instance cont0 `{c1:cpo D1} `{c2:cpo D2} : continuous (mon (cte D1 (0:D2))).
+Qed.
+ +
+
+ +
+
    +
  • double_app f g n m = f m (g n) + +
    • +
+
+
+Definition double_app `{o1:ord Oa} `{o2:ord Ob} `{o3:ord Oc} `{o4: ord Od}
+      (f:Oa -m> Oc -m> Od) (g:Ob -m> Oc)
+        : Ob -m> (Oa -m> Od) := mon ((mshift f) @ g).
+ +
+ +
+
+ +
+

Continuity

+ +
+
+ +
+Class continuous2 `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (F:D1 -m> D2 -m> D3) :=
+continuous2_intro : forall (f : nat -m> D1) (g :nat -m> D2),
+                 F (lub f) (lub g) ≤ lub ((F @² f) g).
+ +
+Lemma continuous2_app : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+            (F : D1 -m> D2 -m> D3) {cF:continuous2 F} (k:D1), continuous (F k).
+ +
+#[export] Typeclasses Opaque continuous2.
+ +
+Lemma continuous2_eq_compat :
+   forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f g : D1 -m> D2 -m> D3),
+   f ≈ g -> continuous2 f -> continuous2 g.
+ +
+Lemma continuous2_continuous : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+           (F : D1 -m> D2 -m> D3), continuous2 F -> continuous F.
+#[export] Hint Immediate continuous2_continuous : core.
+ +
+Lemma continuous2_left : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+             (F : D1 -m> D2 -m> D3) (h:nat -m> D1) (x:D2),
+             continuous F -> F (lub h) x ≤ lub (mshift (F @ h) x).
+ +
+Lemma continuous2_right : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+             (F : D1 -m> D2 -m> D3) (x:D1)(h:nat -m> D2),
+             continuous2 F -> F x (lub h) ≤ lub (F x @ h).
+ +
+Lemma continuous_continuous2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+      (F : D1 -m> D2 -m> D3) (cFr: forall k:D1, continuous (F k)) (cF: continuous F),
+      continuous2 F.
+ +
+#[export] Hint Resolve continuous2_app continuous2_continuous continuous_continuous2 : core.
+ +
+Lemma lub_app2_eq : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+      (F : D1 -m> D2 -m> D3) {cFr:forall k:D1, continuous (F k)} {cF : continuous F},
+      forall (f:nat -m> D1) (g:nat -m> D2),
+      F (lub f) (lub g) ≈ lub ((F@2 f) g).
+ +
+Lemma lub_cont2_app2_eq : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+      (F : D1 -m> D2 -m> D3){cF : continuous2 F},
+      forall (f:nat -m> D1) (g:nat -m> D2),
+      F (lub f) (lub g) ≈ lub ((F@2 f) g).
+ +
+Lemma mshift_continuous2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+             (F : D1 -m> D2 -m> D3), continuous2 F -> continuous2 (mshift F).
+#[export] Hint Resolve mshift_continuous2 : core.
+ +
+Lemma monotonic_sym : forall `{o1:ord D1} `{o2:ord D2} (F : D1 -> D1 -> D2),
+      (forall x y, F x y ≈ F y x) -> (forall k:D1, monotonic (F k)) -> monotonic F.
+#[export] Hint Immediate monotonic_sym : core.
+ +
+Lemma monotonic2_sym : forall `{o1:ord D1} `{o2:ord D2} (F : D1 -> D1 -> D2),
+      (forall x y, F x y ≈ F y x) -> (forall k:D1, monotonic (F k)) -> monotonic2 F.
+#[export] Hint Immediate monotonic2_sym : core.
+ +
+Lemma continuous_sym : forall `{c1:cpo D1} `{c2:cpo D2} (F : D1 -m> D1 -m> D2),
+      (forall x y, F x y ≈ F y x) -> (forall k:D1, continuous (F k)) -> continuous F.
+ +
+Lemma continuous2_sym : forall `{c1:cpo D1} `{c2:cpo D2} (F : D1 -m>D1 -m>D2),
+      (forall x y, F x y ≈ F y x) -> (forall k, continuous (F k)) -> continuous2 F.
+#[export] Hint Resolve continuous2_sym : core.
+ +
+
+ +
+
    +
  • continuity is preserved by composition + +
    • +
+
+
+ +
+Lemma continuous_comp : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+   (f:D2 -m> D3)(g:D1 -m> D2), continuous f -> continuous g -> continuous (mon (f@g)).
+#[export] Hint Resolve continuous_comp : core.
+ +
+Lemma continuous2_comp : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} `{c4:cpo D4}
+  (f:D1 -m> D2)(g:D2 -m> D3 -m> D4),
+  continuous f -> continuous2 g -> continuous2 (g @ f).
+#[export] Hint Resolve continuous2_comp : core.
+ +
+Lemma continuous2_comp2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} `{c4:cpo D4}
+    (f:D3 -m> D4)(g:D1 -m> D2 -m> D3),
+    continuous f -> continuous2 g -> continuous2 (fcomp2 D1 D2 D3 D4 f g).
+#[export] Hint Resolve continuous2_comp2 : core.
+ +
+Lemma continuous2_app2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} `{c4:cpo D4}
+    (F : D1 -m> D2 -m> D3) (f:D4 -m> D1)(g:D4 -m> D2), continuous2 F ->
+    continuous f -> continuous g -> continuous ((F @² f) g).
+#[export] Hint Resolve continuous2_app2 : core.
+ +
+
+ +
+

Cpo of continuous functions

+ +
+
+ +
+#[export] Instance lub_continuous `{c1:cpo D1} `{c2:cpo D2}
+     (f:nat -m> (D1 -m> D2)) {cf:forall n, continuous (f n)}
+     : continuous (lub f).
+Qed.
+ +
+Record fcont `{c1:cpo D1} `{c2:cpo D2}: Type
+     := cont {fcontm :> D1 -m> D2; fcontinuous : continuous fcontm}.
+#[export] Existing Instance fcontinuous.
+ +
+#[export] Hint Resolve fcontinuous : core.
+Arguments fcont D1 [o][c1] D2 {o0}{c2}.
+Arguments cont [D1][o][c1] [D2][o0][c2] fcontm {fcontinuous}.
+ +
+ +
+Infix "->" := fcont (at level 30, right associativity) : O_scope.
+ +
+Definition fcont_fun `{c1:cpo D1} `{c2:cpo D2} (f:D1 -c> D2) : D1 -> D2 := fun x => f x.
+ +
+#[export] Program Instance fcont_ord `{c1:cpo D1} `{c2:cpo D2} : ord (D1 -c> D2)
+  := {Oeq := fun f g => forall x, f x ≈ g x; Ole := fun f g => forall x, f x ≤ g x}.
+ +
+Lemma fcont_le_intro : forall `{c1:cpo D1} `{c2:cpo D2} (f g : D1 -c> D2),
+    (forall x, f x ≤ g x) -> f ≤ g.
+ +
+Lemma fcont_le_elim : forall `{c1:cpo D1} `{c2:cpo D2} (f g : D1 -c> D2),
+     f ≤ g -> forall x, f x ≤ g x.
+ +
+Lemma fcont_eq_intro : forall `{c1:cpo D1} `{c2:cpo D2} (f g : D1 -c> D2),
+      (forall x, f x ≈ g x) -> f ≈ g.
+ +
+Lemma fcont_eq_elim : forall `{c1:cpo D1} `{c2:cpo D2} (f g : D1 -c> D2),
+       f ≈ g -> forall x, f x ≈ g x.
+ +
+Lemma fcont_le : forall `{c1:cpo D1} `{c2:cpo D2} (f : D1 -c> D2) (x y : D1),
+            x ≤ y -> f x ≤ f y.
+#[export] Hint Resolve fcont_le : core.
+ +
+Lemma fcont_eq : forall `{c1:cpo D1} `{c2:cpo D2} (f : D1 -c> D2) (x y : D1),
+            x ≈ y -> f x ≈ f y.
+#[export] Hint Resolve fcont_eq : core.
+ +
+Definition fcont0 D1 `{c1:cpo D1} D2 `{c2:cpo D2} : D1 -c> D2 := cont (mon (cte D1 (0:D2))).
+ +
+#[export] Instance fcontm_monotonic : forall `{c1:cpo D1} `{c2:cpo D2},
+         monotonic (fcontm (D1:=D1) (D2:=D2)).
+Qed.
+ +
+Definition Fcontm D1 `{c1:cpo D1} D2 `{c2:cpo D2} : (D1 -c> D2) -m> (D1 -m> D2) :=
+     mon (fcontm (D1:=D1) (D2:=D2)).
+ +
+#[export] Instance fcont_lub_continuous :
+    forall `{c1:cpo D1} `{c2:cpo D2} (f:nat -m> (D1 -c> D2)),
+    continuous (lub (D:=D1 -m> D2) (Fcontm D1 D2 @ f)).
+Qed.
+ +
+Definition fcont_lub `{c1:cpo D1} `{c2:cpo D2} : (nat -m> (D1 -c> D2)) -> D1 -c> D2 :=
+     fun f => cont (lub (D:=D1 -m> D2) (Fcontm D1 D2 @ f)).
+ +
+#[export] Program Instance fcont_cpo `{c1:cpo D1} `{c2:cpo D2} : cpo (D1-c> D2) :=
+  {D0:=fcont0 D1 D2; lub:=fcont_lub (D1:=D1) (D2:=D2)}.
+ +
+Definition fcont_app {O} {o:ord O} `{c1:cpo D1} `{c2:cpo D2} (f: O -m> D1 -c> D2) (x:D1) : O -m> D2
+         := mshift (Fcontm D1 D2 @ f) x.
+ +
+Infix "<_>" := fcont_app (at level 70) : O_scope.
+ +
+Lemma fcont_app_simpl : forall {O} {o:ord O} `{c1:cpo D1} `{c2:cpo D2} (f: O -m> D1 -c> D2)(x:D1)(y:O),
+            (f <_> x) y = f y x.
+ +
+#[export] Instance ishift_continuous :
+   forall {A:Type} `{c1:cpo D1} `{c2:cpo D2} (f: A -> (D1 -c> D2)),
+          continuous (ishift f).
+Qed.
+ +
+Definition fcont_ishift {A:Type} `{c1:cpo D1} `{c2:cpo D2} (f: A -> (D1 -c> D2))
+        : D1 -c> (A -> D2) := cont _ (fcontinuous:=ishift_continuous f).
+ +
+#[export] Instance mshift_continuous : forall {O} {o:ord O} `{c1:cpo D1} `{c2:cpo D2} (f: O -m> (D1 -c> D2)),
+         continuous (mshift (Fcontm D1 D2 @ f)).
+Qed.
+ +
+Definition fcont_mshift {O} {o:ord O} `{c1:cpo D1} `{c2:cpo D2} (f: O -m> (D1 -c> D2))
+   : D1 -c> O -m> D2 := cont (mshift (Fcontm D1 D2 @ f)).
+ +
+Lemma fcont_app_continuous :
+       forall {O} {o:ord O} `{c1:cpo D1} `{c2:cpo D2} (f: O -m> D1 -c> D2) (h:nat -m> D1),
+            f <_> (lub h) ≤ lub (D:=O -m> D2) ((fcont_mshift f) @ h).
+ +
+Lemma fcont_lub_simpl : forall `{c1:cpo D1} `{c2:cpo D2} (h:nat -m> D1 -c> D2)(x:D1),
+            lub h x = lub (h <_> x).
+ +
+#[export] Instance cont_app_monotonic : forall `{o1:ord D1} `{c2:cpo D2} `{c3:cpo D3} (f:D1 -m> D2 -m> D3)
+            (p:forall k, continuous (f k)),
+            monotonic (Ob:=D2 -c> D3) (fun (k:D1) => cont _ (fcontinuous:=p k)).
+Qed.
+ +
+Definition cont_app `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f:D1 -m> D2 -m> D3)
+            (p:forall k, continuous (f k)) : D1 -m> (D2 -c> D3)
+    := mon (fun k => cont (f k) (fcontinuous:=p k)).
+ +
+Lemma cont_app_simpl :
+forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}(f:D1 -m> D2 -m> D3)(p:forall k, continuous (f k))
+        (k:D1), cont_app f p k = cont (f k).
+ +
+#[export] Instance cont2_continuous `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f:D1 -m> D2 -m> D3)
+           (p:continuous2 f) : continuous (cont_app f (continuous2_app f)).
+Qed.
+ +
+Definition cont2 `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f:D1 -m> D2 -m> D3)
+           {p:continuous2 f} : D1 -c> (D2 -c> D3)
+:= cont (cont_app f (continuous2_app f)).
+ +
+#[export] Instance Fcontm_continuous `{c1:cpo D1} `{c2:cpo D2} : continuous (Fcontm D1 D2).
+Qed.
+#[export] Hint Resolve Fcontm_continuous : core.
+ +
+#[export] Instance fcont_comp_continuous : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+    (f:D2 -c> D3) (g:D1 -c> D2), continuous (f @ g).
+Qed.
+ +
+Definition fcont_comp `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f:D2 -c> D3) (g:D1 -c> D2)
+   : D1 -c> D3 := cont (f @ g).
+ +
+Infix "@_" := fcont_comp (at level 35) : O_scope.
+ +
+Lemma fcont_comp_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+       (f:D2 -c> D3)(g:D1 -c> D2) (x:D1), (f @_ g) x = f (g x).
+ +
+Lemma fcontm_fcont_comp_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+       (f:D2 -c> D3)(g:D1 -c> D2), fcontm (f @_ g) = f @ g.
+ +
+Lemma fcont_comp_le_compat : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+      (f g : D2 -c> D3) (k l :D1 -c> D2),
+      f ≤ g -> k ≤ l -> f @_ k ≤ g @_ l.
+#[export] Hint Resolve fcont_comp_le_compat : core.
+ +
+Add Parametric Morphism `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+    : (@fcont_comp _ _ c1 _ _ c2 _ _ c3)
+      with signature Ole ++> Ole ++> Ole as fcont_comp_le_morph.
+Qed.
+ +
+Add Parametric Morphism `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+    : (@fcont_comp _ _ c1 _ _ c2 _ _ c3)
+      with signature Oeq ⇾ Oeq ⇾ Oeq as fcont_comp_eq_compat.
+Qed.
+ +
+Definition fcont_Comp D1 `{c1:cpo D1} D2 `{c2:cpo D2} D3 `{c3:cpo D3}
+      : (D2 -c> D3) -m> (D1 -c> D2) -m> D1 -c> D3
+      := mon2 _ (mf:=fcont_comp_le_compat (D1:=D1) (D2:=D2) (D3:=D3)).
+ +
+Lemma fcont_Comp_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+                        (f:D2 -c> D3) (g:D1 -c> D2), fcont_Comp D1 D2 D3 f g = f @_ g.
+ +
+#[export] Instance fcont_Comp_continuous2
+   : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}, continuous2 (fcont_Comp D1 D2 D3).
+Qed.
+ +
+Definition fcont_COMP D1 `{c1:cpo D1} D2 `{c2:cpo D2} D3 `{c3:cpo D3}
+      : (D2 -c> D3) -c> (D1 -c> D2) -c> D1 -c> D3
+      := cont2 (fcont_Comp D1 D2 D3).
+ +
+Lemma fcont_COMP_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+        (f: D2 -c> D3) (g:D1 -c> D2),
+        fcont_COMP D1 D2 D3 f g = f @_ g.
+ +
+Definition fcont2_COMP D1 `{c1:cpo D1} D2 `{c2:cpo D2} D3 `{c3:cpo D3} D4 `{c4:cpo D4}
+   : (D3 -c> D4) -c> (D1 -c> D2 -c> D3) -c> D1 -c> D2 -c> D4 :=
+     (fcont_COMP D1 (D2 -c> D3) (D2 -c> D4)) @_ (fcont_COMP D2 D3 D4).
+ +
+Definition fcont2_comp `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} `{c4:cpo D4}
+           (f:D3 -c> D4)(F:D1 -c> D2 -c> D3) := fcont2_COMP D1 D2 D3 D4 f F.
+ +
+Infix "@@_" := fcont2_comp (at level 35) : O_scope.
+ +
+Lemma fcont2_comp_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} `{c4:cpo D4}
+       (f:D3 -c> D4)(F:D1 -c> D2 -c> D3)(x:D1)(y:D2), (f @@_ F) x y = f (F x y).
+ +
+Lemma fcont_le_compat2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f : D1 -c> D2 -c> D3)
+    (x y : D1) (z t : D2), x ≤ y -> z ≤ t -> f x z ≤ f y t.
+#[export] Hint Resolve fcont_le_compat2 : core.
+ +
+Lemma fcont_eq_compat2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f : D1 -c> D2 -c> D3)
+    (x y : D1) (z t : D2), x ≈ y -> z ≈ t -> f x z ≈ f y t.
+#[export] Hint Resolve fcont_eq_compat2 : core.
+ +
+Lemma fcont_continuous : forall `{c1:cpo D1} `{c2:cpo D2} (f:D1 -c> D2)(h:nat -m> D1),
+            f (lub h) ≤ lub (f @ h).
+#[export] Hint Resolve fcont_continuous : core.
+ +
+#[export] Instance fcont_continuous2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+         (f:D1 -c> D2 -c> D3), continuous2 (Fcontm D2 D3 @ f).
+Qed.
+#[export] Hint Resolve fcont_continuous2 : core.
+ +
+#[export] Instance cshift_continuous2 : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+         (f:D1 -c> D2 -c> D3), continuous2 (mshift (Fcontm D2 D3 @ f)).
+Qed.
+#[export] Hint Resolve cshift_continuous2 : core.
+ +
+Definition cshift `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3} (f:D1 -c> D2 -c> D3)
+   : D2 -c> D1 -c> D3 := cont2 (mshift (Fcontm D2 D3 @ f)).
+ +
+Lemma cshift_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+           (f:D1 -c> D2 -c> D3) (x:D2) (y:D1), cshift f x y = f y x.
+ +
+Definition fcont_SEQ D1 `{c1:cpo D1} D2 `{c2:cpo D2} D3 `{c3:cpo D3}
+   : (D1 -c> D2) -c> (D2 -c> D3) -c> D1 -c> D3 := cshift (fcont_COMP D1 D2 D3).
+ +
+Lemma fcont_SEQ_simpl : forall `{c1:cpo D1} `{c2:cpo D2} `{c3:cpo D3}
+       (f: D1 -c> D2) (g:D2 -c> D3), fcont_SEQ D1 D2 D3 f g = g @_ f.
+ +
+ +
+#[export] Instance Id_mon : forall `{o1:ord Oa}, monotonic (fun x:Oa => x).
+Qed.
+ +
+Definition Id Oa {o1:ord Oa} : Oa -m> Oa := mon (fun x => x).
+ +
+Lemma Id_simpl : forall `{o1:ord Oa} (x:Oa), Id Oa x = x.
+ +
+ +
+
+ +
+

Fixpoints

+ +
+
+ +
+Fixpoint iter_ {D} {o} `{c: @cpo D o} (f : D -m> D) n {struct n} : D
+    := match n with O => 0 | S m => f (iter_ f m) end.
+ +
+Lemma iter_incr : forall `{c: cpo D} (f : D -m> D) n, iter_ f n ≤ f (iter_ f n).
+#[export] Hint Resolve iter_incr : core.
+ +
+#[export] Instance iter_mon : forall `{c: cpo D} (f : D -m> D), monotonic (iter_ f).
+Qed.
+ +
+Definition iter `{c: cpo D} (f : D -m> D) : nat -m> D := mon (iter_ f).
+ +
+Definition fixp `{c: cpo D} (f : D -m> D) : D := mlub (iter_ f).
+ +
+Lemma fixp_le : forall `{c: cpo D} (f : D -m> D), fixp f ≤ f (fixp f).
+#[export] Hint Resolve fixp_le : core.
+ +
+Lemma fixp_eq : forall `{c: cpo D} (f : D -m> D) {mf:continuous f},
+      fixp f ≈ f (fixp f).
+ +
+Lemma fixp_inv : forall `{c: cpo D} (f : D -m> D) g, f g ≤ g -> fixp f ≤ g.
+ +
+Definition fixp_cte : forall `{c:cpo D} (d:D), fixp (mon (cte D d)) ≈ d.
+Qed.
+#[export] Hint Resolve fixp_cte : core.
+ +
+Lemma fixp_le_compat : forall `{c:cpo D} (f g : D -m> D),
+      f ≤ g -> fixp f ≤ fixp g.
+#[export] Hint Resolve fixp_le_compat : core.
+ +
+#[export] Instance fixp_monotonic `{c:cpo D} : monotonic fixp.
+Qed.
+ +
+Add Parametric Morphism `{c:cpo D} : (fixp (c:=c))
+    with signature Oeq ⇾ Oeq as fixp_eq_compat.
+Qed.
+#[export] Hint Resolve fixp_eq_compat : core.
+ +
+Definition Fixp D `{c:cpo D} : (D -m> D) -m> D := mon fixp.
+ +
+Lemma Fixp_simpl : forall `{c:cpo D} (f:D-m>D), Fixp D f = fixp f.
+ +
+#[export] Instance iter_monotonic `{c:cpo D} : monotonic iter.
+Qed.
+ +
+Definition Iter D `{c:cpo D} : (D -m> D) -m> (nat -m> D) := mon iter.
+ +
+Lemma IterS_simpl : forall `{c:cpo D} f n, Iter D f (S n) = f (Iter D f n).
+ +
+Lemma iterO_simpl : forall `{c:cpo D} (f: D-m> D), iter f O = (0:D).
+ +
+Lemma iterS_simpl : forall `{c:cpo D} f n, iter f (S n) = f (iter f n).
+ +
+Lemma iter_continuous : forall `{c:cpo D} (h : nat -m> (D -m> D)),
+       (forall n, continuous (h n)) -> iter (lub h) ≤ lub (mon iter @ h).
+ +
+#[export] Hint Resolve iter_continuous : core.
+ +
+Lemma iter_continuous_eq : forall `{c:cpo D} (h : nat -m> (D -m> D)),
+    (forall n, continuous (h n)) -> iter (lub h) ≈ lub (mon iter @ h).
+ +
+Lemma fixp_continuous : forall `{c:cpo D} (h : nat -m> (D -m> D)),
+       (forall n, continuous (h n)) -> fixp (lub h) ≤ lub (mon fixp @ h).
+#[export] Hint Resolve fixp_continuous : core.
+ +
+Lemma fixp_continuous_eq : forall `{c:cpo D} (h : nat -m> (D -m> D)),
+       (forall n, continuous (h n)) -> fixp (lub h) ≈ lub (mon fixp @ h).
+ +
+Definition Fixp_cont D `{c:cpo D} : (D -c> D) -m> D := Fixp D @ (Fcontm D D).
+ +
+Lemma Fixp_cont_simpl : forall `{c:cpo D} (f:D -c> D), Fixp_cont D f = fixp (fcontm f).
+ +
+#[export] Instance Fixp_cont_continuous : forall D `{c:cpo D}, continuous (Fixp_cont D).
+Qed.
+ +
+Definition FIXP D `{c:cpo D} : (D -c> D) -c> D := cont (Fixp_cont D).
+ +
+Lemma FIXP_simpl : forall `{c:cpo D} (f:D -c> D), FIXP D f = Fixp D (fcontm f).
+ +
+Lemma FIXP_le_compat : forall `{c:cpo D} (f g : D -c> D),
+            f ≤ g -> FIXP D f ≤ FIXP D g.
+#[export] Hint Resolve FIXP_le_compat : core.
+ +
+Lemma FIXP_eq_compat : forall `{c:cpo D} (f g : D -c> D),
+            f ≈ g -> FIXP D f ≈ FIXP D g.
+#[export] Hint Resolve FIXP_eq_compat : core.
+ +
+Lemma FIXP_eq : forall `{c:cpo D} (f:D -c> D), FIXP D f ≈ f (FIXP D f).
+#[export] Hint Resolve FIXP_eq : core.
+ +
+Lemma FIXP_inv : forall `{c:cpo D} (f:D -c> D) (g : D), f g ≤ g -> FIXP D f ≤ g.
+ +
+
+ +
+

Iteration of functional

+ +
+
+Lemma FIXP_comp_com : forall `{c:cpo D} (f g:D-c>D),
+       g @_ f ≤ f @_ g-> FIXP D g ≤ f (FIXP D g).
+ +
+Lemma FIXP_comp : forall `{c:cpo D} (f g:D-c>D),
+       g @_ f ≤ f @_ g -> f (FIXP D g) ≤ FIXP D g -> FIXP D (f @_ g) ≈ FIXP D g.
+ +
+Fixpoint fcont_compn {D} {o} `{c:@cpo D o}(f:D -c> D) (n:nat) {struct n} : D -c> D :=
+             match n with O => f | S p => fcont_compn f p @_ f end.
+ +
+Lemma fcont_compn_Sn_simpl :
+     forall `{c:cpo D}(f:D -c> D) (n:nat), fcont_compn f (S n) = fcont_compn f n @_ f.
+ +
+Lemma fcont_compn_com : forall `{c:cpo D}(f:D-c>D) (n:nat),
+            f @_ (fcont_compn f n) ≤ fcont_compn f n @_ f.
+ +
+Lemma FIXP_compn :
+     forall `{c:cpo D} (f:D-c>D) (n:nat), FIXP D (fcont_compn f n) ≈ FIXP D f.
+ +
+Lemma fixp_double : forall `{c:cpo D} (f:D-c>D), FIXP D (f @_ f) ≈ FIXP D f.
+ +
+ +
+
+ +
+

Induction principle

+ +
+
+Definition admissible `{c:cpo D}(P:D->Type) :=
+          forall f : nat -m> D, (forall n, P (f n)) -> P (lub f).
+ +
+Lemma fixp_ind : forall `{c:cpo D}(F:D -m> D)(P:D->Type),
+       admissible P -> P 0 -> (forall x, P x -> P (F x)) -> P (fixp F).
+ +
+Definition admissible2 `{c1:cpo D1}`{c2:cpo D2}(R:D1 -> D2 -> Type) :=
+    forall (f : nat -m> D1) (g:nat -m> D2), (forall n, R (f n) (g n)) -> R (lub f) (lub g).
+ +
+Lemma fixp_ind_rel : forall `{c1:cpo D1}`{c2:cpo D2}(F:D1 -m> D1) (G:D2-m> D2)
+       (R:D1 -> D2 -> Type),
+       admissible2 R -> R 0 0 -> (forall x y, R x y -> R (F x) (G y)) -> R (fixp F) (fixp G).
+ +
+Lemma lub_le_fixp : forall `{c1:cpo D1}`{c2:cpo D2} (f:D1-m>D2) (F:D1 -m> D1)
+                                         (s:nat-m> D2),
+          s O ≤ f 0 -> (forall x n, s n ≤ f x -> s (S n) ≤ f (F x))
+          -> lub s ≤ f (fixp F).
+ +
+Lemma fixp_le_lub : forall `{c1:cpo D1}`{c2:cpo D2} (f:D1-m>D2) (F:D1 -m> D1)
+                                         (s:nat-m> D2) {fc:continuous f},
+          f 0 ≤ s O -> (forall x n, f x ≤ s n -> f (F x) ≤ s (S n)) -> f (fixp F) ≤ lub s.
+ +
+ +
+Ltac continuity cont Cont Hcont:=
+  match goal with
+ | |- (Ole ?x1 (lub (mon (fun (n:nat) => cont (@?g n))))) =>
+      let f := fresh "f" in (
+           pose (f:=g); assert (monotonic f) ;
+                               [auto | (transitivity (lub (Cont@(mon f))); [rewrite <- Hcont | auto])]
+           )
+end.
+ +
+Ltac gen_monotonic :=
+match goal with |- context [(@mon _ _ _ _ ?f ?mf)] => generalize (mf:monotonic f)
+end.
+ +
+Ltac gen_monotonic1 f :=
+match goal with |- context [(@mon _ _ _ _ f ?mf)] => generalize (mf:monotonic f)
+end.
+ +
+
+ +
+

Function for conditionnal choice defined as a morphism

+ +
+
+ +
+Definition fif {A} (b:bool) : A -> A -> A := fun e1 e2 => if b then e1 else e2.
+ +
+#[export] Instance fif_mon2 `{o:ord A} (b:bool) : monotonic2 (@fif _ b).
+Qed.
+ +
+Definition Fif `{o:ord A} (b:bool) : A -m> A -m> A := mon2 (@fif _ b).
+ +
+Lemma Fif_simpl : forall `{o:ord A} (b:bool) (x y:A), Fif b x y = fif b x y.
+ +
+Lemma Fif_continuous_right `{c:cpo A} (b:bool) (e:A) : continuous (Fif b e).
+ +
+Lemma Fif_continuous_left `{c:cpo A} (b:bool) : continuous (Fif (A:=A) b).
+#[export] Hint Resolve Fif_continuous_right Fif_continuous_left : core.
+ +
+Lemma fif_continuous_left `{c:cpo A} (b:bool) (f:nat-m> A):
+    fif b (lub f) ≈ lub (Fif b@f).
+ +
+Lemma fif_continuous_left2 :
+forall (A : Type) (o : ord A) (c : cpo A) (b : bool) (f : nat -m> A) (g:A),
+fif b (lub f) g ≈ lub (Fif b @ f) g.
+ +
+Lemma fif_continuous_right `{c:cpo A} (b:bool) e (f:nat-m> A):
+    fif b e (lub f) ≈ lub (Fif b e@f).
+ +
+#[export] Hint Resolve fif_continuous_right fif_continuous_left fif_continuous_left2 : core.
+ +
+#[export] Instance Fif_continuous2 `{c:cpo A} (b:bool) : continuous2 (Fif (A:=A) b).
+Qed.
+ +
+Lemma fif_continuous2 `{c:cpo A} (b:bool) (f g : nat-m> A):
+      fif b (lub f) (lub g) ≈ lub ((Fif b@2 f) g).
+ +
+Add Parametric Morphism `{o:ord A} (b:bool) : (@fif A b)
+with signature Ole ⇾ Ole ⇾ Ole
+as fif_le_compat.
+Qed.
+ +
+Add Parametric Morphism `{o:ord A} (b:bool) : (@fif A b)
+with signature Oeq ⇾ Oeq ⇾ Oeq
+as fif_eq_compat.
+Qed.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/ALEA.Misc.html b/combi/1.1.0/ALEA.Misc.html new file mode 100644 index 00000000..92fb886d --- /dev/null +++ b/combi/1.1.0/ALEA.Misc.html @@ -0,0 +1,316 @@ + + + + + +ALEA.Misc + + + + +
+ + + +
+ +

Library ALEA.Misc

+ +
+ +
+
+ +
+

Misc.v: Preliminaries

+ +
+
+ +
+Set Implicit Arguments.
+From Stdlib Require Export Arith.
+From Stdlib Require Import Lia.
+ +
+From Stdlib Require Import Classes.SetoidTactics.
+From Stdlib Require Import Classes.SetoidClass.
+From Stdlib Require Import Classes.Morphisms.
+ +
+#[local] Open Scope signature_scope.
+ +
+Lemma beq_nat_neq: ∀ x y : nat, x <> y → false = Nat.eqb x y.
+ +
+Lemma if_beq_nat_nat_eq_dec : ∀ A (x y:nat) (a b:A),
+  (if Nat.eqb x y then a else b) = if eq_nat_dec x y then a else b.
+ +
+Definition ifte A (test:bool) (thn els:A) := if test then thn else els.
+ +
+Add Parametric Morphism (A:Type) : (@ifte A)
+  with signature (eq ⇒eq ⇒ eq ⇒ eq) as ifte_morphism1.
+ +
+Add Parametric Morphism (A:Type) x : (@ifte A x)
+  with signature (eq ⇒ eq ⇒ eq) as ifte_morphism2.
+ +
+Add Parametric Morphism (A:Type) x y : (@ifte A x y)
+  with signature (eq ⇒ eq) as ifte_morphism3.
+ +
+
+ +
+

Definition of iterator compn

+ + compn f u n x is defined as (f (u (n-1)).. (f (u 0) x)) +
+
+ +
+Fixpoint compn (A:Type)(f:A → A → A) (x:A) (u:nat → A) (n:nat) {struct n}: A :=
+   match n with O ⇒ x | (S p) ⇒ f (u p) (compn f x u p) end.
+ +
+Lemma comp0 : ∀ (A:Type) (f:A → A → A) (x:A) (u:nat → A), compn f x u 0 = x.
+ +
+Lemma compS : ∀ (A:Type) (f:A → A → A) (x:A) (u:nat → A) (n:nat),
+              compn f x u (S n) = f (u n) (compn f x u n).
+ +
+
+ +
+

Reducing if constructs

+ +
+
+ +
+Lemma if_then : ∀ (P:Prop) (b:{ P }+{ ¬ P })(A:Type)(p q:A),
+      P → (if b then p else q) = p.
+ +
+Lemma if_else : ∀ (P :Prop) (b:{ P }+{ ¬ P })(A:Type)(p q:A),
+      ¬P → (if b then p else q) = q.
+ +
+Lemma if_then_not : ∀ (P Q:Prop) (b:{ P }+{ Q })(A:Type)(p q:A),
+      ¬ Q → (if b then p else q) = p.
+ +
+Lemma if_else_not : ∀ (P Q:Prop) (b:{ P }+{ Q })(A:Type)(p q:A),
+      ¬P → (if b then p else q) = q.
+ +
+
+ +
+

Classical reasoning

+ +
+
+ +
+Definition class (A:Prop) := ¬ ¬ A → A.
+ +
+Lemma class_neg : ∀ A:Prop, class ( ¬ A).
+ +
+Lemma class_false : class False.
+#[export] Hint Resolve class_neg class_false : core.
+ +
+Definition orc (A B:Prop) := ∀ C:Prop, class C → (A → C) → (B → C) → C.
+ +
+Lemma orc_left : ∀ A B:Prop, A → orc A B.
+ +
+Lemma orc_right : ∀ A B:Prop, B → orc A B.
+ +
+#[export] Hint Resolve orc_left orc_right : core.
+ +
+Lemma class_orc : ∀ A B, class (orc A B).
+ +
+Arguments class_orc [A B].
+ +
+Lemma orc_intro : ∀ A B, ( ¬ A → ¬ B → False) → orc A B.
+ +
+Lemma class_and : ∀ A B, class A → class B → class (A ∧ B).
+ +
+Lemma excluded_middle : ∀ A, orc A ( ¬ A).
+ +
+Definition exc (A :Type)(P:A → Prop) :=
+   ∀ C:Prop, class C → (∀ x:A, P x → C) → C.
+ +
+Lemma exc_intro : ∀ (A :Type)(P:A → Prop) (x:A), P x → exc P.
+ +
+Lemma class_exc : ∀ (A :Type)(P:A → Prop), class (exc P).
+ +
+Lemma exc_intro_class : ∀ (A:Type) (P:A → Prop), ((∀ x, ¬ P x) → False) → exc P.
+ +
+Lemma not_and_elim_left : ∀ A B, ¬ (A ∧ B) → A → ¬B.
+ +
+Lemma not_and_elim_right : ∀ A B, ¬ (A ∧ B) → B → ¬A.
+ +
+#[export] Hint Resolve class_orc class_and class_exc excluded_middle : core.
+ +
+Lemma class_double_neg : ∀ P Q: Prop, class Q → (P → Q) → ¬ ¬ P → Q.
+ +
+
+ +
+

Extensional equality

+ +
+
+ +
+Definition feq A B (f g : A → B) := ∀ x, f x = g x.
+ +
+Lemma feq_refl : ∀ A B (f:A→B), feq f f.
+ +
+Lemma feq_sym : ∀ A B (f g : A → B), feq f g → feq g f.
+ +
+Lemma feq_trans : ∀ A B (f g h: A → B), feq f g → feq g h → feq f h.
+ +
+#[export] Hint Resolve feq_refl : core.
+#[export] Hint Immediate feq_sym : core.
+#[export] Hint Unfold feq : core.
+ +
+Add Parametric Relation (A B : Type) : (A → B) (feq (A:=A) (B:=B))
+  reflexivity proved by (feq_refl (A:=A) (B:=B))
+  symmetry proved by (feq_sym (A:=A) (B:=B))
+  transitivity proved by (feq_trans (A:=A) (B:=B))
+as feq_rel.
+ +
+
+ +
+Computational version of elimination on CompSpec +
+
+ +
+Lemma CompSpec_rect : ∀ (A : Type) (eq lt : A → A → Prop) (x y : A)
+       (P : comparison → Type),
+       (eq x y → P Eq) →
+       (lt x y → P Lt) →
+       (lt y x → P Gt)
+    → ∀ c : comparison, CompSpec eq lt x y c → P c.
+ +
+
+ +
+Decidability +
+
+Lemma dec_sig_lt : ∀ P : nat → Prop, (∀ x, {P x}+{ ¬ P x})
+  → ∀ n, {i | i < n ∧ P i}+{∀ i, i < n → ¬ P i}.
+ +
+Lemma dec_exists_lt : ∀ P : nat → Prop, (∀ x, {P x}+{ ¬ P x})
+  → ∀ n, {exists i, i < n ∧ P i}+{¬ exists i, i < n ∧ P i}.
+ +
+Definition eq_nat2_dec : ∀ p q : nat*nat, { p=q }+{¬ p=q }.
+Defined.
+ +
+Lemma nat_compare_specT
+   : ∀ x y : nat, CompareSpecT (x = y) (x < y)%nat (y < x)%nat (Nat.compare x y).
+ +
+
+ +
+

Preliminary lemmas relating the ordre on nat and N

+ +
+
+ +
+From Stdlib Require Export NArith.
+ +
+Lemma N2Nat_lt_mono : ∀ n m, (n < m)%N ↔ (N.to_nat n < N.to_nat m)%nat.
+ +
+Lemma N2Nat_le_mono : ∀ n m, (n <= m)%N ↔ (N.to_nat n <= N.to_nat m)%nat.
+ +
+Lemma N2Nat_inj_lt : ∀ n m, (n < m)%N → (N.to_nat n < N.to_nat m)%nat.
+ +
+Lemma N2Nat_inj_le : ∀ n m, (n <= m)%N → (N.to_nat n <= N.to_nat m)%nat.
+ +
+Lemma N2Nat_inj_pos : ∀ n, (0 < n)%N → (0 < N.to_nat n)%nat.
+#[export] Hint Resolve N2Nat_inj_lt N2Nat_inj_pos N2Nat_inj_le : core.
+ +
+Lemma Nsucc_pred_pos: ∀ n : N, (0 < n)%N → N.succ (N.pred n) = n.
+#[export] Hint Resolve Nsucc_pred_pos : core.
+ +
+Lemma Npos : ∀ n, (0 <= n)%N.
+#[export] Hint Resolve Npos : core.
+ +
+Lemma Neq_lt_0 : ∀ n, (n=0)%N ∨ (0<n)%N.
+#[export] Hint Resolve Neq_lt_0 : core.
+ +
+Lemma Nlt0_le1 : ∀ n, (0<n)%N → (1<=n)%N.
+#[export] Hint Immediate Nlt0_le1 : core.
+ +
+Ltac Nineq :=
+     match goal with
+      |- (N.le ?n ?m) ⇒ compute; discriminate
+    | |- (N.lt ?n ?m) ⇒ compute; reflexivity
+    | |- ¬ (eq (A:=N) ?n ?m) ⇒ rewrite <- N.compare_eq_iff; compute; discriminate
+    end.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/ALEA.Qmeasure.html b/combi/1.1.0/ALEA.Qmeasure.html new file mode 100644 index 00000000..17e3f7f5 --- /dev/null +++ b/combi/1.1.0/ALEA.Qmeasure.html @@ -0,0 +1,902 @@ + + + + + +ALEA.Qmeasure: Finite probabilities + + + + +
+ + + +
+ +

Library ALEA.Qmeasure: Finite probabilities

+ +
+
+ +
+ +
+
+
+ +
+

Definition of finite probabilities as measures with values in rational numbers

+ +
+
+ +
+Require Import Misc Ccpo.
+ +
+Set Implicit Arguments.
+ +
+From Stdlib Require Arith.
+From Stdlib Require Import Lia.
+ +
+From mathcomp Require Import ssreflect ssrfun eqtype choice.
+From mathcomp Require Import ssrbool ssrnat seq order fintype finfun.
+From mathcomp Require Import bigop ssralg ssrnum ssrint rat.
+Import GRing.
+Import Order.Theory.
+Import Num.Theory.
+ +
+ +
+Delimit Scope order_scope with Omc.
+#[local] Open Scope O_scope.
+Delimit Scope O_scope with O.
+#[local] Open Scope ring_scope.
+ +
+#[export] Program Instance ratO : ord rat :=
+     { Oeq := fun n m : rat => n = m;
+       Ole := fun n m : rat => (n <= m)%R}.
+ +
+
+ +
+Functions to be measured +
+
+ +
+Definition MF (A:Type) := A -> rat.
+ +
+#[export] Instance MFO (A:Type) : ord (MF A) := ford A rat.
+ +
+
+ +
+Type of measures on A +
+
+ +
+Definition M A := MF A -m> rat.
+ +
+#[export] Instance MO (A:Type) : ord (M A) := fmono (MF A) rat.
+ +
+#[export] Instance app_mon (A:Type) (x:A) : monotonic (fun (f:MF A) => f x).
+ #[export] Hint Resolve app_mon : core.
+ +
+
+ +
+Monadic operators on M A +
+
+ +
+Definition munit (A:Type) (x:A) : M A := mon (fun (f:MF A) => f x).
+ +
+Definition mstar (A B:Type) : M A -> (A -> M B) -> M B.
+Defined.
+ +
+Lemma star_simpl (A B:Type) (a:M A) (F:A -> M B) (f:MF B) :
+  mstar a F f = a (fun x => F x f).
+ +
+
+ +
+

Properties of monadic operators

+ +
+
+Lemma law1 (A B:Type) (x:A) (F:A -> M B) (f:MF B) :
+  mstar (munit x) F f == F x f.
+ +
+Lemma law2 (A:Type) (a:M A) (f:MF A) :
+  mstar a (fun x:A => munit x) f == a (fun x:A => f x).
+ +
+Lemma law3 (A B C:Type) (a:M A) (F:A -> M B) (G:B -> M C) (f:MF C) :
+  mstar (mstar a F) G f == mstar a (fun x:A => mstar (F x) G) f.
+ +
+
+ +
+

Properties of distributions

+

Expected properties of measures

+ +
+
+ +
+Definition stable_add (A:Type) (m:M A) : Prop :=
+  forall f g:MF A, m (f \+ g) = (m f) + (m g).
+ +
+Definition stable_sub (A:Type) (m:M A) : Prop :=
+  forall f g:MF A, m (f \- g) = (m f) - (m g).
+ +
+Definition stable_mull (A:Type) (m:M A) : Prop :=
+  forall (k:rat) (f:MF A), m (k \*o f) = k * (m f).
+ +
+Definition stable_opp (A:Type) (m:M A) : Prop :=
+  forall (f:MF A), m (fun x => - (f x)) = - (m f).
+ +
+
+ +
+Linearity on rational is deduced from stability with respect to substraction +
+
+ +
+Section StabilityProperties.
+ +
+Variable A : Type.
+Variable m : M A.
+Hypothesis Mstable_sub : stable_sub m.
+ +
+Implicit Types (f g : MF A).
+ +
+Lemma Mstable_eq : stable m.
+Hint Resolve Mstable_eq : core.
+ +
+Lemma Mstable0 : m \0 = 0.
+Hint Resolve Mstable0 : core.
+ +
+Lemma Mstable_opp : stable_opp m.
+ +
+Lemma Mstable_add : stable_add m.
+ +
+Lemma Mstable_addn f (n : nat) : m (fun x => f x *+ n) = (m f) *+ n.
+ +
+Lemma Mstable_subn f (n : nat) : m (fun x => f x *- n) = (m f) *- n.
+ +
+Lemma Mstable_divn f (n : nat) :
+  m (fun x => f x / n%:~R) = (m f) / (n%:~R).
+ +
+Lemma Mstable_addi f (n : int) : m (fun x => f x *~ n) = (m f) *~ n.
+ +
+Lemma Mstable_muli f (n : int) : m (fun x => n%:~R * f x) = (n%:~R) * (m f).
+ +
+Lemma Mstable_divi f (n : int) : m (fun x => f x / n%:~R) = (m f) / (n%:~R).
+ +
+Lemma Mstable_mull : stable_mull m.
+ +
+Lemma Mstable_linear (p q : rat) f g :
+  m ((p \*o f) \+ (q \*o g)) = p * (m f) + q * (m g).
+ +
+End StabilityProperties.
+#[export] Hint Resolve Mstable_eq : core.
+#[export] Hint Resolve Mstable0 : core.
+#[export] Hint Resolve Mstable_opp : core.
+#[export] Hint Resolve Mstable_add : core.
+ +
+Lemma unit_stable_eq (A:Type) (x:A) : stable (munit x).
+ +
+Lemma star_stable_eq (A B:Type) (m:M A) (F:A -> M B) : stable (mstar m F).
+ +
+Lemma unit_monotonic A (x:A) (f g : MF A) :
+  (f <= g)%O -> munit x f <= munit x g.
+ +
+Lemma star_monotonic A B (m:M A) (F:A -> M B) (f g : MF B) :
+  (f <= g)%O -> mstar m F f <= mstar m F g.
+ +
+Lemma star_le_compat A B (m1 m2:M A) (F1 F2:A -> M B) :
+  (m1 <= m2)%O -> (F1 <= F2)%O -> (mstar m1 F1 <= mstar m2 F2)%O.
+#[export] Hint Resolve star_le_compat : core.
+ +
+
+ +
+

Stability for substraction of unit and star

+ +
+
+Lemma unit_stable_sub (A:Type) (x:A) : stable_sub (munit x).
+ +
+Lemma star_stable_sub (A B:Type) (m:M A) (F:A -> M B) :
+  stable_sub m -> (forall a:A, stable_sub (F a)) -> stable_sub (mstar m F).
+ +
+
+ +
+

Definition of distribution

+ +Finite distributions are monotonic measure functions such that +
    +
  • mu (f - g) = mu f - mu g + +
    • +
  • mu 1 = 1 + +
    • +
+ +
+
+ +
+Record distr (A:Type) : Type := {
+       mu : M A;
+       mu_stable_sub : stable_sub mu;
+       mu_prob : mu (fun x => 1) = 1
+   }.
+ +
+#[export] Hint Resolve mu_stable_sub mu_prob : core.
+ +
+
+ +
+

Properties of measures

+ +
+
+Section MeasureProp.
+ +
+Context {A : Type} (m: distr A).
+Implicit Types (f g : MF A).
+ +
+Lemma mu_monotonic : monotonic (mu m).
+ Hint Resolve mu_monotonic : core.
+ +
+Lemma mu_stable_eq : stable (mu m).
+ Hint Resolve mu_stable_eq : core.
+ +
+Lemma mu_zero : mu m \0 = 0.
+ Hint Resolve mu_zero : core.
+ +
+Lemma mu_zero_eq f : (forall x, f x = 0) -> mu m f = 0.
+ +
+Lemma mu_one_eq f : (forall x, f x = 1) -> mu m f = 1.
+ +
+Lemma mu_stable_inv f : mu m (fun x => 1 - f x) = 1 - (mu m f).
+ +
+Lemma mu_stable_add f g : mu m (f \+ g) = mu m f + mu m g.
+ +
+Lemma mu_stable_mull (q : rat) f : mu m (q \*o f) = q * mu m f.
+ +
+Lemma mu_add_zero f g : mu m f = 0 -> mu m g = 0 -> mu m (f \+ g) = 0.
+ +
+Lemma mu_stable_pos f : (\0 <= f)%O -> 0 <= mu m f.
+ +
+Lemma mu_stable_le1 f : (forall x, f x <= 1) -> mu m f <= 1.
+ +
+Lemma mu_cte (c:rat) : mu m (fun x => c) = c.
+ +
+Lemma mu_stable_mulr (c:rat) f : mu m (c \o* f) = (mu m f) * c.
+ +
+Lemma mu_stable_inv_inv f : mu m f = 1 - mu m (fun x => 1 - f x).
+ +
+End MeasureProp.
+#[export] Hint Resolve mu_monotonic : core.
+#[export] Hint Resolve mu_stable_eq : core.
+#[export] Hint Resolve mu_zero : core.
+#[export] Hint Immediate mu_zero_eq : core.
+#[export] Hint Immediate mu_one_eq : core.
+#[export] Hint Resolve mu_stable_inv : core.
+#[export] Hint Resolve mu_stable_add : core.
+#[export] Hint Resolve mu_stable_mull : core.
+#[export] Hint Resolve mu_add_zero : core.
+#[export] Hint Resolve mu_cte : core.
+#[export] Hint Resolve mu_stable_inv_inv : core.
+ +
+#[export] Program Instance Odistr (A:Type) : ord (distr A) :=
+    {Ole := fun (f g : distr A) => (mu f <= mu g)%O;
+     Oeq := fun (f g : distr A) => Oeq (mu f) (mu g)}.
+ +
+
+ +
+

Monadic operators for distributions

+ +
+
+Section MonDistrib.
+ +
+Variables (A B : Type).
+ +
+Definition Munit : A -> distr A.
+ +
+Definition Mlet : distr A -> (A -> distr B) -> distr B.
+ +
+Lemma Munit_simpl (q : A -> rat) x : mu (Munit x) q = q x.
+ +
+Lemma Munit_simpl_eq (q : A -> rat) x : mu (Munit x) q == q x.
+ +
+Lemma Mlet_simpl (m : distr A) (M : A -> distr B) (f : B -> rat) :
+  mu (Mlet m M) f = mu m (fun x => (mu (M x) f)).
+ +
+Lemma Mlet_simpl_eq (m : distr A) (M : A -> distr B) (f : B -> rat) :
+mu (Mlet m M) f == mu m (fun x => (mu (M x) f)).
+ +
+End MonDistrib.
+ +
+
+ +
+

Operations on distributions

+ +
+
+Section OperDistr.
+ +
+Variables (A B C : Type).
+ +
+Lemma Munit_eq_compat (x y : A) : x = y -> Munit x == Munit y.
+ +
+Lemma Mlet_le_compat (m1 m2 : distr A) (M1 M2 : A -> distr B) :
+  (m1 <= m2 -> M1 <= M2 -> Mlet m1 M1 <= Mlet m2 M2)%O.
+Hint Resolve Mlet_le_compat : core.
+ +
+Add Parametric Morphism : (Mlet (A:=A) (B:=B))
+    with signature Ole ==> Ole ==> Ole
+      as Mlet_le_morphism.
+ +
+Add Parametric Morphism : (Mlet (A:=A) (B:=B))
+    with signature Ole ==> (@pointwise_relation A (distr B) (@Ole _ _)) ==> Ole
+      as Mlet_le_pointwise_morphism.
+ +
+#[export] Instance Mlet_mon2 : monotonic2 (@Mlet A B).
+ +
+Definition MLet : distr A -m> (A -> distr B) -m> distr B
+  := mon2 (@Mlet A B).
+ +
+Lemma MLet_simpl (m : distr A) (M : A -> distr B) (f : B -> rat) :
+  mu (MLet m M) f = mu m (fun x => mu (M x) f).
+ +
+Lemma Mlet_eq_compat (m1 m2 : distr A) (M1 M2 : A -> distr B) :
+  (m1 == m2 -> M1 == M2 -> Mlet m1 M1 == Mlet m2 M2)%type.
+ Hint Resolve Mlet_eq_compat : core.
+ +
+Add Parametric Morphism : (Mlet (A:=A) (B:=B))
+    with signature Oeq ==> Oeq ==> Oeq
+      as Mlet_eq_morphism.
+ +
+Add Parametric Morphism : (Mlet (A:=A) (B:=B))
+    with signature Oeq ==> (@pointwise_relation A (distr B) (@Oeq _ _)) ==> Oeq
+      as Mlet_Oeq_pointwise_morphism.
+ +
+Lemma mu_le_compat (m1 m2 : distr A) :
+  (m1 <= m2 -> forall f g : A -> rat, f <= g -> mu m1 f <= mu m2 g)%O.
+ +
+Lemma mu_eq_compat (m1 m2 : distr A) :
+  (m1 == m2 -> forall f g : A -> rat, f == g -> mu m1 f = mu m2 g)%type.
+Hint Immediate mu_le_compat mu_eq_compat : core.
+ +
+Add Parametric Morphism : (mu (A:=A))
+    with signature Ole ==> Ole
+      as mu_le_morphism.
+ +
+Add Parametric Morphism : (mu (A:=A))
+    with signature Oeq ==> Oeq
+      as mu_eq_morphism.
+ +
+Add Parametric Morphism (a : distr A) : (@mu A a)
+    with signature (@pointwise_relation A rat (@eq _) ==> Oeq)
+      as mu_distr_eq_morphism.
+ +
+Add Parametric Morphism (a : distr A) : (@mu A a)
+    with signature (@pointwise_relation A rat (@Oeq _ _) ==> Oeq)
+      as mu_distr_Oeq_morphism.
+ +
+Add Parametric Morphism (a : distr A) : (@mu A a)
+    with signature (@pointwise_relation _ _ (@Ole _ _) ==> Ole)
+      as mu_distr_le_morphism.
+ +
+Add Parametric Morphism : (@Mlet A B)
+    with signature (Ole ==> @pointwise_relation _ _ (@Ole _ _) ==> Ole)
+      as mlet_distr_le_morphism.
+ +
+Add Parametric Morphism : (@Mlet A B)
+    with signature (Oeq ==> @pointwise_relation _ _ (@Oeq _ _) ==> Oeq)
+      as mlet_distr_eq_morphism.
+ +
+
+ +
+

Properties of monadic operators

+ +
+
+Lemma Mlet_unit (x : A) (m : A -> distr B) : Mlet (Munit x) m == m x.
+ +
+Lemma Mlet_ext (m : distr A) : Mlet m (fun x => Munit x) == m.
+ +
+Lemma Mlet_assoc (m1 : distr A) (m2 : A -> distr B) (m3 : B -> distr C) :
+  Mlet (Mlet m1 m2) m3 == Mlet m1 (fun x:A => Mlet (m2 x) m3).
+ +
+Lemma let_indep (m1 : distr A) (m2 : distr B) (f : MF B) :
+  mu m1 (fun => mu m2 f) = mu m2 f.
+ +
+Lemma let_indep_distr (m1 : distr A) (m2 : distr B) :
+  Mlet m1 (fun => m2) == m2.
+ +
+Section MuBool.
+Context {m : distr A}.
+Implicit Types (f g : A -> bool).
+ +
+Lemma mu_bool_le1 {f} : mu m (fun x => (f x)%:~R) <= 1%:~R.
+Hint Resolve mu_bool_le1 : core.
+ +
+Lemma mu_bool_0le {f} : 0%:~R <= mu m (fun x => (f x)%:~R).
+ Hint Resolve mu_bool_0le : core.
+ +
+Lemma mu_bool_impl {f g} :
+  (forall x, (f x) ==> (g x)%B) ->
+  (mu m (fun x => (f x)%:~R) <= mu m (fun x => (g x)%:~R)).
+ +
+Lemma mu_bool_impl1 {f g} :
+  (forall x, (f x) ==> (g x)%B) ->
+  mu m (fun x => (f x)%:~R) = 1 -> mu m (fun x => (g x)%:~R) = 1.
+ +
+Lemma mu_bool_negb0 {f g} :
+  (forall x, (f x) ==> ~~ (g x)%B) ->
+  mu m (fun x => (f x)%:~R) = 1 ->
+  mu m (fun x => (g x)%:~R) = 0.
+ +
+Lemma mu_bool_negb {f} :
+  mu m (fun x => (~~ f x)%:~R) = 1 - mu m (fun x => (f x)%:~R).
+ +
+Lemma mu_bool_negb1 {f g} :
+  (forall x, (~~ (f x) ==> g x)%B) ->
+  mu m (fun x => (f x)%:~R) = 0 ->
+  mu m (fun x => (g x)%:~R) = 1.
+ +
+End MuBool.
+ +
+End OperDistr.
+ +
+#[export] Hint Resolve mu_bool_le1 : core.
+#[export] Hint Resolve mu_bool_0le : core.
+ +
+ +
+
+ +
+

Examples of distributions

+ +
+ +

Flipping a coin:

+ + +
+ +The distribution associated to flip () is + f --> [1/2] (f true) + [1/2] (f false) +
+
+ +
+Notation "[1/2]" := (2%:~R)^-1.
+ +
+#[export] Instance flip_mon :
+  monotonic (fun (f : bool -> rat) => [1/2] * (f true) + [1/2] * (f false)).
+ +
+Definition flip : M bool :=
+  mon (fun (f : bool -> rat) => [1/2] * (f true) + [1/2] * (f false)).
+ +
+Lemma flip_stable_sub : stable_sub flip.
+ +
+Lemma flip_prob : flip (fun x => 1) = 1.
+ +
+Lemma flip_true : flip (fun b => (b%:~R)) = [1/2].
+ +
+Lemma flip_false : flip (fun b => (~~b)%:~R) = [1/2].
+ +
+#[export] Hint Resolve flip_true flip_false : core.
+ +
+Definition Flip : distr bool.
+ +
+Lemma Flip_simpl f : mu Flip f = [1/2] * (f true) + [1/2] * (f false).
+ +
+
+ +
+

Finite distributions given by points and rational coefficients

+ +
+
+ +
+ +
+Section FiniteDistributions.
+ +
+Variable A : Type.
+Variable p : seq A.
+ +
+
+ +
+We use a finite sequent of points, give a rational coefficient + to each point but only consider positive ones +
+
+ +
+Definition weight (c : A -> rat) : rat := \sum_(i <- p | 0 < c i) (c i).
+ +
+Lemma weight_nonneg c : 0 <= weight c.
+Hint Resolve weight_nonneg : core.
+ +
+Lemma weight_case c : (weight c = 0) \/ 0 < (weight c)^-1.
+ +
+#[export] Instance finite_mon (c : A -> rat) :
+  monotonic (fun f => (\sum_(i <- p | 0 < c i) (c i * f i)) / weight c).
+ +
+Definition mfinite (c : A -> rat) : M A :=
+  mon (fun f => (\sum_(i <- p | 0 < c i) (c i * f i))/weight c).
+ +
+Lemma finite_simpl (c : A -> rat) f :
+  mfinite c f = (\sum_(i <- p | 0 < c i) (c i * f i))/weight c.
+ +
+Lemma finite_stable_sub (c : A -> rat) : stable_sub (mfinite c).
+ +
+End FiniteDistributions.
+ +
+
+ +
+We have a distribution when the total weight is positive +
+
+ +
+Record fin (A : Type) : Type :=
+  mkfin { points : seq A;
+          coeff : A -> rat;
+          weight_pos : 0 < weight points coeff
+        }.
+#[export] Hint Resolve weight_pos : core.
+ +
+Lemma inv_weight_pos A (d : fin A) : 0 < (weight (points d) (coeff d))^-1.
+ #[export] Hint Resolve inv_weight_pos : core.
+ +
+Lemma weight_is_unit A (d : fin A) : (weight (points d) (coeff d)) \is a unit.
+ #[export] Hint Resolve weight_is_unit : core.
+ +
+Definition fprob A (d : fin A) (a : A) : rat :=
+  coeff d a / weight (points d) (coeff d).
+ +
+Definition Finite A : fin A -> distr A.
+ +
+Lemma Finite_simpl A (d : fin A) :
+  mu (Finite d) = mfinite (points d) (coeff d).
+ +
+Lemma Finite_eq_in (A : eqType) (d : fin A) (a : A) :
+  uniq (points d) -> (a \in points d)%SEQ -> 0 < coeff d a ->
+  mu (Finite d) (fun x => (x==a)%:~R) = fprob d a.
+ +
+Lemma Finite_eq_out (A : eqType) (d : fin A) (a : A) :
+  (a \notin points d)%SEQ \/ (coeff d a <= 0)%R ->
+  mu (Finite d) (fun x => (x==a)%:~R) = 0.
+ +
+Lemma Finite_in_seq (A : eqType) (d : fin A) :
+  mu (Finite d) (fun x => (x \in points d)%:~R) = 1%R.
+ +
+
+ +
+

Uniform distribution on a non empty sequence of points

+ +
+
+ +
+Record unif (A : Type) : Type :=
+  mkunif { upoints :> seq A; _ : size upoints != O }.
+ +
+Definition usize A (p : unif A) : nat := size (upoints p).
+ +
+Lemma usize_pos A (p : unif A) : usize p != O.
+ +
+Definition unif2fin A (p:unif A) : fin A.
+ +
+Definition Uniform A (d : unif A) : distr A := Finite (unif2fin d).
+ +
+Lemma Uniform_simpl A (d : unif A) :
+  mu (Uniform d) = mfinite (upoints d) (fun A => 1%R).
+ +
+Lemma weight1_size A (d : seq A) : weight d (fun x => 1) = (size d)%:~R.
+ +
+Lemma mu_uniform_sum A (d : unif A) (f : A -> rat) :
+  mu (Uniform d) f = (\sum_(i <- d) f i) / (size d)%:~R.
+ +
+Lemma Uniform_eq_in (A : eqType) (d : unif A) (a : A) :
+  uniq d -> (a \in upoints d)%SEQ ->
+  mu (Uniform d) (fun x => (x==a)%:~R) = 1 / (usize d)%:~R.
+ +
+Lemma Uniform_eq_out (A : eqType) (d : unif A) (a : A) :
+  (a \notin upoints d)%SEQ ->
+  mu (Uniform d) (fun x => (x==a)%:~R) = 0%:~R.
+ +
+Lemma Uniform_in_seq (A : eqType) (d : unif A) :
+  mu (Uniform d) (fun x => (x \in upoints d)%:~R) = 1%:~R.
+ +
+Fact succ_neq0 (n m : nat) : (n==m.+1)%N -> (n!=0)%N.
+ +
+Lemma Uniform_unif_seq_eq (A : eqType) (d1 d2 : unif A) :
+  (d1 == d2 :> seq A) -> Uniform d1 == Uniform d2.
+ +
+
+ +
+Uniform distribution on a sequence with a default value +
+
+ +
+Definition unif_def A (d : A) (s : seq A) : unif A.
+ +
+Lemma Uniform_def_ne A (d : A) (s : seq A) :
+  forall (Hs : (size s != 0)%N), Uniform (unif_def d s) = Uniform (mkunif s Hs).
+ +
+
+ +
+

Uniform distribution between 0 and n included

+ +
+
+Section UnifNat.
+ +
+Implicit Types (n a : nat).
+ +
+Definition unifnat n : unif nat := mkunif (iota 0 (n.+1)) (eq_refl true).
+Definition Random n : distr nat := Uniform (unifnat n).
+ +
+Lemma Random_simpl n : mu (Random n) = mfinite (iota 0 (n.+1)) (fun x => 1).
+ +
+Lemma Random_eq_in n a :
+  (a <= n)%N -> mu (Random n) (fun x => (x==a)%:~R) = 1 / (n.+1)%:~R.
+ +
+Lemma Random_eq_out n a :
+  (n < a)%N -> mu (Random n) (fun x => (x==a)%:~R) = 0.
+ +
+Lemma mu_random_sum n (f : nat -> rat) :
+  mu (Random n) f = (\sum_(0 <= i < n.+1) f i) / (n.+1)%:~R.
+ +
+Lemma Random_in_range n :
+  mu (Random n) (fun x => (x <= n)%N%:~R) = 1%:~R.
+ +
+End UnifNat.
+ +
+
+ +
+

Distribution and big sums

+ +
+
+ +
+ +
+Lemma mu_stable_sum (A : Type) (m : distr A)
+      (I : Type) (s : seq I) (f : I -> A -> rat) :
+  mu m (fun a => \sum_(i <- s) f i a) = \sum_(i <- s) (mu m (f i)).
+ +
+Section Bigsums.
+ +
+Variable A : eqType.
+Implicit Types (x : A) (s : seq A) (m : distr A).
+ +
+Lemma in_seq_sum s x :
+  uniq s -> (x \in s)%:~R = \sum_(i <- s) (x == i)%:~R :> rat.
+ +
+Lemma mu_in_seq m s :
+  uniq s ->
+  mu m (fun x => (x \in s)%:~R) = \sum_(a <- s) mu m (fun x => (x == a)%:~R).
+ +
+Lemma mu_bool_cond m (f g : A -> bool) :
+  mu m (fun x => (f x)%:~R) = 1 ->
+  mu m (fun x => (g x)%:~R) = mu m (fun x => (f x && g x)%:~R).
+ +
+Lemma mu_pos_cond (m : distr A) (f : A -> bool) (g : A -> rat) :
+  (forall x, 0 <= g x <= 1) ->
+  mu m (fun x => (f x)%:~R) = 1 ->
+  mu m (fun x => (g x)) = mu m (fun x => ((f x)%:~R * g x)).
+ +
+End Bigsums.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Basic.combclass.html b/combi/1.1.0/Combi.Basic.combclass.html new file mode 100644 index 00000000..da7ee10d --- /dev/null +++ b/combi/1.1.0/Combi.Basic.combclass.html @@ -0,0 +1,350 @@ + + + + + +Combi.Basic.combclass: Fintypes for Combinatorics + + + + +
+ + + +
+ +

Library Combi.Basic.combclass: Fintypes for Combinatorics

+ +
+
+ +
+ +
+
+
+ +
+

Fintypes for Combinatorics

+ + +
+ +The goal of this file is to define various way to easily build finite +subtype of a countable type knowing a lists of its elements. We provide four +ways, three from a list (see sub_subFinType, sub_uniq_subFinType and +sub_undup_subFinType below) and one by taking the disjoint union of already +constructed subfintypes (see union_subFinType below). +
+
+ +
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+Require Import tools.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+Summing count_mem in a finType +
+
+Lemma sum_count_mem (T : finType) (P : pred T) l :
+  \sum_(i | P i) (count_mem i) l = count P l.
+ +
+
+ +
+

Building subtype from a sequence

+ + +
+ +Here is how to construct a fintype from a list: we are given +
    +
  • a type T which is a countType + +
    • +
  • a type TP which is subCountType of T for a predicate P. + +
    • +
  • a list lst of element from T whose element veryfies the predicate P. + +
    • +
+ +
+ +We define three possible ways to provide TP with a subFinType structure: +
    +
  • sub_subFinType which suppose that any element verifying P appears only + once in lst; + +
    • +
  • sub_uniq_subFinType which suppose that any element verifying P appears in + lst and that lst is duplicate free (uniq); + +
    • +
  • sub_undup_subFinType which suppose that any element verifying P appears in + lst and remove the duplicate elements. + + +
    • +
+
+
+ +
+Section EnumFintype.
+Context {T : countType} {P : pred T} (TP : subCountType P).
+Variable subenum : seq T.
+Hypothesis subenumP : all P subenum.
+Hypothesis subenum_countE : forall x : T, P x -> count_mem x subenum = 1.
+ +
+Lemma sub_enumE : subenum =i P.
+ +
+Definition subType_seq : seq TP := pmap insub subenum.
+Lemma subType_seqP : map val subType_seq = subenum.
+Lemma finite_subP : Finite.axiom subType_seq.
+Definition seq_finType : finType :=
+  HB.pack TP (isFinite.Build TP finite_subP).
+ +
+Lemma enum_subE : map val (enum (seq_finType)) = subenum.
+ Lemma card_subE : #|seq_finType| = size subenum.
+ +
+End EnumFintype.
+ +
+Module Example1.
+ +
+Definition is_one n := n == 1.
+Record isOne := IsOne { one :> nat; _ : is_one one}.
+Lemma all_isOne : all is_one [:: 1].
+ Lemma isOne_count_1 x : is_one x -> count_mem x [:: 1] = 1.
+ +
+Lemma enum_isOne : map val (enum (isOne : finType)) = [:: 1].
+ Lemma card_isOne : #|isOne : finType| = 1.
+ +
+End Example1.
+ +
+
+ +
+

Method 2 - Each element appears and the lists is uniq

+ +
+
+Section UniqFinType.
+Context {T : countType} {P : pred T} (TP : subCountType P).
+Variable subenum : seq T.
+Hypothesis subenumE : subenum =i P.
+Hypothesis subenum_uniq : uniq subenum.
+ +
+Lemma all_subenum : all P subenum.
+ Lemma subenum_countE x : P x -> count_mem x subenum = 1.
+Definition uniq_finType : finType :=
+  seq_finType TP all_subenum subenum_countE.
+End UniqFinType.
+ +
+Module Example2.
+ +
+Definition is_one n := n == 1.
+Record isOne := IsOne { one :> nat; _ : is_one one}.
+Lemma all_isoneE : [:: 1] =i is_one.
+ Lemma isOne_uniq : uniq [:: 1].
+ +
+Lemma enum_isOne : map val (enum (isOne : finType)) = [:: 1].
+ Lemma card_isOne : #|isOne : finType| = 1.
+ +
+End Example2.
+ +
+
+ +
+

Method 3 - Each element appears, we remove the duplicates

+ +
+
+Section SubUndup.
+Context {T : countType} {P : pred T} (TP : subCountType P).
+Variable subenum : seq T.
+Hypothesis subenumP : all P subenum.
+Hypothesis subenum_in : forall x : T, P x -> x \in subenum.
+ +
+Lemma finite_sub_undupP :
+  Finite.axiom (undup (subType_seq TP subenum)).
+Definition undup_finType : finType :=
+  HB.pack TP (isFinite.Build TP finite_sub_undupP).
+ +
+Lemma enum_sub_undupE : map val (enum undup_finType) = undup subenum.
+ +
+End SubUndup.
+ +
+Module Example3.
+ +
+Definition is_one n := n == 1.
+Record isOne := IsOne { one :> nat; _ : is_one one}.
+Lemma all_isOne : all is_one [:: 1; 1].
+ Lemma isOne_in n : is_one n -> n \in [:: 1; 1].
+ +
+Lemma enum_isOne : map val (enum (isOne : finType)) = [:: 1].
+ Lemma card_isOne : #|isOne : finType| = 1.
+ +
+End Example3.
+ +
+
+ +
+

Finite subtype obtained as a finite the dijoint union of finite subtypes

+ + +
+ +Here is how to construct a union of disjoint finite subtype of a countable +type. More precisely, we want to define a type for + +
+ + U := Union_(i : TI | Pi i) TPi i + +
+ +For the constructed type U, we need the following data: +
    +
  • a type T which is a countType. + +
    • +
  • a type TP which is subCountType of T for a predicate P. + +
    • +
+The index type must be also countable, it should be given by +
    +
  • a type TI which is a countType. + +
    • +
  • a type TPI which is subCountType of TI for a predicate PI. + +
    • +
+For all index i : TPI, there must be a finite type, given by +
    +
  • a type TPi i which is a subFinType (Pi (val i)) for a predicate Pi i. + +
    • +
+Finally the sets { { x | Pi i } | PI i } should define a partition +of { x | P x }. This is ensured by providing +
    +
  • a map FI : T -> TI which recover the index of an element x of T. + +
    • +
+Together with the two following requirements: +
    +
  • for all index i : TPi and x : T, the statement Pi i x must be + equivalent to P x && i == FI x. + +
    • +
  • forall x : T, such that P x the assertion PI (FI x) must holds. + +
    • +
+From all these data union_subFinType is a subFinType of T for the +predicate P that is a subFinType structure for TP. + +
+ +See stpn_subFinType and yamn_subFinType for example of usage. + +
+
+Section SubtypesDisjointUnion.
+ +
+Variable T : countType.
+Variable P : pred T.
+Variable TP : subCountType P.
+ +
+Variable TI : countType.
+Variable PI : pred TI.
+Variable TPI : subFinType PI.
+ +
+Variable Pi : TI -> pred T.
+Variable TPi : forall i : TPI, subFinType (Pi (val i)).
+ +
+Variable FI : T -> TI.
+Hypothesis HPTi : forall i : TPI, (predI P (pred1 (val i) \o FI)) =1 (Pi (val i)).
+Hypothesis Hpart : forall x : T, P x -> PI (FI x).
+ +
+Definition enum_union := flatten [seq map val (enum (TPi i)) | i : TPI].
+ +
+Lemma all_unionP : all P enum_union.
+ +
+Lemma count_unionP x : P x -> count_mem x enum_union = 1.
+ +
+Let union_enum := subType_seq TP enum_union.
+ +
+Lemma subType_unionE : map val union_enum = enum_union.
+Lemma finite_unionP : Finite.axiom union_enum.
+Definition union_finType : finType :=
+  HB.pack TP (isFinite.Build TP finite_unionP).
+ +
+Lemma enum_unionE :
+  map val (enum union_finType) = enum_union.
+ +
+Lemma card_unionE : #|union_finType| = \sum_(i : TPI) #|TPi i|.
+ +
+End SubtypesDisjointUnion.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Basic.congr.html b/combi/1.1.0/Combi.Basic.congr.html new file mode 100644 index 00000000..704abfc0 --- /dev/null +++ b/combi/1.1.0/Combi.Basic.congr.html @@ -0,0 +1,590 @@ + + + + + +Combi.Basic.congr: Rewriting rule and congruencies of words + + + + +
+ + + +
+ +

Library Combi.Basic.congr: Rewriting rule and congruencies of words

+ +
+
+ +
+ +
+
+
+ +
+

Equivalence and congruence closure of a rewriting rule on words

+ + +
+ +If what follows a relation on T is a term of type rel T and a rewriting rule +on T is a term of type T -> seq T. + +
+ +Recall that a word congruence is an equivalence relation compatible with the +concatenation product. This is the following definitions + +
+ +
    +
  • congruence_rel r == the relation r on seq T is a congruence + +
    • +
  • congruence_rule == the rule rule on seq T is a congruence rule + +
    • +
+ +
+ +Given a rewriting rule, the goal of this file is to compute its word +congruence closure as well as its congruence classes. We therefore suppose +that equivalence classes are finite. This also ensure that the generated +equivalence relation is decidable by bounding the length of the rewriting +paths. Concretely, this is done by requiring that rule is included in a +reflexive relation invar which is invariant by rewriting rule, that is: + +
+ + Hypothesis Hinvar : forall x0 x, invar x0 x -> all (invar x0) (rule x). + +
+ +We show that this is true for homogeneous rules (ie: that preserve size) for +words over a finite alphabet as well as multi-homogeneous rules (ie: that +permutes words). Precisely, under the hypothesis + +
+ + Hypothesis Hsym : forall x y, x \in rule y -> y \in rule x. + +
+ +1- for homogeneous rules (ie: that preserve size) for words over a finite +alphabet: + +
+ + Hypothesis Hhom : forall u : word, all (szinvar u) (rule u). + +
+ +We define + +
+ +
    +
  • gencongr_hom == the word congrugence generated by a rule + +
    • +
  • genclass_hom == the class for gencongr_multhom + +
    • +
+ +
+ +2- for multi-homogeneous rules (ie: that permutes words): + +
+ + Hypothesis Hmulthom : forall u : word, all (perm_eq u) (rule u). + +
+ +We define + +
+ +
    +
  • gencongr_multhom == the word congrugence generated by a rule + +
    • +
  • genclass_multhom == the class for gencongr_multhom + +
    • +
+ +
+ + +
+ +Assuming the two latter hypothesis, we define a relation gencongr which is +the congruence transitive closure of rule. The main results here are +
    +
  • gencongrP : gencongr is the smallest transitive congruence relation + contaning rule. + +
    • +
  • gencongr_unique : gencongr is uniquely characterized by the property above + +
    • +
  • gencongr_ind : induction principle on classes for gencongr, any property + preserved along the rewriting rule holds for classes. +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From Stdlib Require Import Recdef.
+Require Import permcomp permuted multinomial vectNK.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Transitive closure of a rule with finite classes

+ Since we only need to contruct it and not to compute with it, we use a simple but extremely inefficient algorithm +
+
+Section Transitive.
+ +
+Variable T : eqType.
+Variable rule : T -> seq T.
+Variable invar : pred T.
+Hypothesis Hinvar : forall x, invar x -> all invar (rule x).
+ +
+Section FullKnown.
+ +
+Variable full : seq T.
+Hypothesis Hfull : forall x, invar x -> x \in full.
+ +
+Theorem full_bound :
+  forall s : seq T, all invar s -> uniq s -> size s <= size full.
+ +
+End FullKnown.
+ +
+Variable bound : nat.
+Hypothesis Hbound: forall s : seq T, all invar s -> uniq s -> size s <= bound.
+ +
+Lemma invar_undupE s : all invar (undup s) = all invar s.
+ +
+Definition step s := undup (s ++ flatten (map rule s)).
+ +
+Lemma step_mem s x y : y \in rule x -> x \in s -> y \in step s.
+ +
+Lemma uniq_step s : uniq (step s).
+ +
+Lemma undup_step s : undup (step s) = step s.
+ +
+Lemma invar_step s : all invar s -> all invar (step s).
+ +
+Lemma subset_step s : {subset s <= step s}.
+ +
+Lemma subset_undup_step s : {subset undup s <= step s}.
+ +
+
+ +
+This is a very unefficient transitive closure algorithm +
+
+Function trans
+         (s : seq T) {measure (fun s => bound - size (undup s))} : seq T :=
+  let us := undup s in
+  if all invar us then
+    let news := step s in
+    if size news > size us then trans news
+    else us
+  else us.
+ +
+Lemma subset_s_trans_s s : {subset s <= trans s}.
+ +
+Variant rewrite_path x y : Prop :=
+  Rew : forall l, path (fun t => [mem (rule t)]) x l ->
+                  y = last x l -> rewrite_path x y.
+ +
+Arguments Rew {x y} (l).
+ +
+Lemma invar_rewrite_path x y : invar x -> rewrite_path x y -> invar y.
+ +
+Lemma step_closed x s y :
+  undup s =i step s -> x \in s -> rewrite_path x y -> y \in s.
+ +
+Lemma transP y l :
+  all invar l ->
+  reflect (exists x, x \in l /\ rewrite_path x y) (y \in trans l).
+ +
+Lemma rewrite_path_trans x y z :
+  rewrite_path x y -> rewrite_path y z -> rewrite_path x z.
+ +
+Lemma rewrite_path_sym x y :
+  (forall x y, x \in rule y -> y \in rule x) ->
+  rewrite_path x y -> rewrite_path y x.
+ +
+End Transitive.
+ +
+
+ +
+

Dealing with the dependance of invar on x

+ +
+
+Section Depend.
+ +
+Variable T : eqType.
+Variable rule : T -> seq T.
+ +
+Record invariant_context :=
+  InvariantContext {
+      invar : T -> pred T;
+      Hinvar_refl : forall x, invar x x;
+      Hinvar_all : forall x0 x, invar x0 x -> all (invar x0) (rule x);
+      bound : T -> nat;
+      Hbound: forall x s, all (invar x) s -> uniq s -> size s <= bound x
+    }.
+Variable inv : invariant_context.
+Hypothesis Hsym : forall x y, x \in rule y -> y \in rule x.
+ +
+Definition rclass x := trans (@Hinvar_all inv x) (@Hbound inv x) [:: x].
+Definition rtrans : rel T := fun x y => y \in rclass x.
+ +
+Lemma rtransP x y : reflect (rewrite_path rule y x) (rtrans y x).
+ +
+Theorem equiv_rtrans : equivalence_rel rtrans.
+ +
+Lemma rtrans_ind (P : T -> Prop) x :
+  P x -> (forall y z, P y -> z \in rule y -> P z) ->
+  forall t, rtrans x t -> P t.
+ +
+Lemma rule_rtrans x y : y \in rule x -> rtrans x y.
+ +
+Lemma rewrite_path_min (r : rel T) :
+  equivalence_rel r -> (forall x y, y \in rule x -> r x y) ->
+  forall x y, rewrite_path rule x y -> r x y.
+ +
+Lemma rtrans_min (r : rel T) :
+  equivalence_rel r -> (forall x y, y \in rule x -> r x y) ->
+  forall x y, rtrans x y -> r x y.
+ +
+End Depend.
+ +
+Definition congruence_rel (T : eqType) (r : rel (seq T)) :=
+  forall a b1 c b2, r b1 b2 -> r (a ++ b1 ++ c) (a ++ b2 ++ c).
+ +
+Definition congruence_rule (T : eqType) (rule : seq T -> seq (seq T)) :=
+  congruence_rel (fun a b => a \in rule b).
+ +
+
+ +
+

Basic facts on congruences:

+ equivalence of various definitions and immediate consequences +
+
+Section CongruenceFacts.
+ +
+Variable Alph : eqType.
+Notation word := (seq Alph).
+ +
+Variable r : rel word.
+Hypothesis Hcongr : congruence_rel r.
+Hypothesis Hequiv : equivalence_rel r.
+ +
+Lemma congr_cons w1 w2 a : r w1 w2 -> r (a :: w1) (a :: w2).
+ +
+Lemma congr_rcons w1 w2 a : r w1 w2 -> r (rcons w1 a) (rcons w2 a).
+ +
+Lemma congr_catl u1 u2 v : r u1 u2 -> r (u1 ++ v) (u2 ++ v).
+ +
+Lemma congr_catr u v1 v2 : r v1 v2 -> r (u ++ v1) (u ++ v2).
+ +
+Lemma congr_cat u1 u2 v1 v2 : r u1 u2 -> r v1 v2 -> r (u1 ++ v1) (u2 ++ v2).
+ +
+End CongruenceFacts.
+ +
+
+ +
+

Congruence closure of a bounded rule

+ +
+
+Section CongruenceClosure.
+ +
+Variable Alph : eqType.
+Notation word := (seq Alph).
+Implicit Types (a b c u v w : word).
+ +
+Variable rule : word -> seq word.
+ +
+Hypothesis Hsym : forall u v : word, v \in rule u -> u \in rule v.
+Variable inv : invariant_context rule.
+ +
+Definition congrrule s :=
+  flatten [seq
+             [seq (triple.1.1) ++ b1 ++ (triple.2) | b1 <- rule triple.1.2]
+          | triple <- cut3 s ].
+ +
+Lemma congrruleP u1 u2 :
+  reflect (exists a v1 b v2,
+              [/\ u1 = a ++ v1 ++ b, u2 = a ++ v2 ++ b & v1 \in rule v2])
+          (u1 \in congrrule u2).
+ +
+Lemma rule_congrrule u v : v \in rule u -> v \in congrrule u.
+ +
+Lemma congrrule_is_congr : congruence_rule congrrule.
+ +
+Hypothesis Hinvar_congr :
+  forall u a b1 b2 c,
+    invar inv b1 b2 -> invar inv u (a ++ b1 ++ c) -> invar inv u (a ++ b2 ++ c).
+ +
+Lemma congrrule_invar u v :
+ invar inv u v -> all (invar inv u) (congrrule v).
+ +
+Lemma congrrule_sym u v : v \in congrrule u -> u \in congrrule v.
+ +
+Definition invcont_congr :=
+  InvariantContext (Hinvar_refl inv) congrrule_invar (@Hbound _ _ inv).
+ +
+Definition gencongr := rtrans invcont_congr.
+Definition genclass := rclass invcont_congr.
+ +
+Lemma genclassE u v : (u \in genclass v) = (u \in gencongr v).
+ +
+Lemma gencongr_equiv : equivalence_rel gencongr.
+ +
+Lemma gencongr_is_congr : congruence_rel gencongr.
+ +
+Lemma rule_gencongr u v : v \in rule u -> v \in gencongr u.
+ +
+Lemma gencongr_min (r : rel word) :
+  equivalence_rel r -> congruence_rel r ->
+  (forall x y, y \in rule x -> r x y) ->
+  forall x y, gencongr x y -> r x y.
+ +
+Theorem gencongr_ind (P : word -> Prop) x :
+  P x ->
+  (forall a b1 c b2, P (a ++ b1 ++ c) -> b2 \in rule b1 -> P (a ++ b2 ++ c)) ->
+  forall y, gencongr x y -> P y.
+ +
+Lemma gencongr_invar u v : gencongr u v-> invar inv u v.
+ +
+Variant Generated_EquivCongruence (grel : rel word) :=
+  GenCongr : equivalence_rel grel ->
+             congruence_rel grel ->
+             ( forall u v, v \in rule u -> grel u v ) ->
+             ( forall r : rel word,
+                      equivalence_rel r -> congruence_rel r ->
+                      (forall x y, y \in rule x -> r x y) ->
+                      forall x y, grel x y -> r x y
+             ) -> Generated_EquivCongruence grel.
+ +
+Theorem gencongrP : Generated_EquivCongruence gencongr.
+ +
+Lemma gencongr_imply r1 r2 :
+  Generated_EquivCongruence r1 -> Generated_EquivCongruence r2 ->
+  forall x y, r1 x y -> r2 x y.
+ +
+Theorem gencongr_unique grel :
+  Generated_EquivCongruence grel -> grel =2 gencongr.
+ +
+Theorem gencongr_generic_ind grel (P : word -> Prop) x :
+  Generated_EquivCongruence grel ->
+  P x ->
+  (forall a b1 c b2, P (a ++ b1 ++ c) -> b2 \in rule b1 -> P (a ++ b2 ++ c)) ->
+  forall y, grel x y -> P y.
+ +
+End CongruenceClosure.
+ +
+
+ +
+

Multi-homogeneous congruence rules

+ +
+
+Section InvarContMultHom.
+ +
+Variable Alph : eqType.
+Notation word := (seq Alph).
+ +
+Variable rule : word -> seq word.
+Hypothesis Hmulthom : forall u : word, all (perm_eq u) (rule u).
+ +
+Lemma perm_bound (x : word) (s : seq word) :
+  all (perm_eq x) s -> uniq s -> size s <= (size x)`!.
+ +
+Lemma perm_invar (x0 x : word) : perm_eq x0 x -> all (perm_eq x0) (rule x).
+ +
+Definition invcont_perm :=
+  InvariantContext (@perm_refl _) perm_invar perm_bound.
+ +
+Hypothesis Hsym : forall u v : word, v \in rule u -> u \in rule v.
+Hypothesis Hcongr : congruence_rule rule.
+ +
+Lemma perm_invar_congr u (a b1 b2 c : word) :
+  invar invcont_perm b1 b2 ->
+  invar invcont_perm u (a ++ b1 ++ c) -> invar invcont_perm u (a ++ b2 ++ c).
+ +
+Definition gencongr_multhom := gencongr perm_invar_congr.
+Definition genclass_multhom := genclass perm_invar_congr.
+ +
+End InvarContMultHom.
+ +
+
+ +
+

Homogeneous congruence rules on a finite type

+ +
+
+Section InvarContHom.
+ +
+Variable Alph : finType.
+Notation word := (seq Alph).
+ +
+Variable rule : word -> seq word.
+Let szinvar (u : word) := [pred v : word | size v == size u].
+Hypothesis Hhom : forall u : word, all (szinvar u) (rule u).
+ +
+Lemma size_bound (x : word) (s : seq word) :
+  all (szinvar x) s -> uniq s -> size s <= #|Alph|^(size x).
+ +
+Lemma size_invar (x0 x : word) : szinvar x0 x -> all (szinvar x0) (rule x).
+ +
+Lemma size_invar_refl (x : word) : szinvar x x.
+ +
+Definition invcont_size :=
+  InvariantContext size_invar_refl size_invar size_bound.
+ +
+Hypothesis Hcongr : congruence_rule rule.
+ +
+Lemma size_invar_congr u (a b1 b2 c : word) :
+  invar invcont_size b1 b2 ->
+  invar invcont_size u (a ++ b1 ++ c) -> invar invcont_size u (a ++ b2 ++ c).
+ +
+Definition gencongr_hom := gencongr size_invar_congr.
+Definition genclass_hom := genclass size_invar_congr.
+ +
+End InvarContHom.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Basic.ordtype.html b/combi/1.1.0/Combi.Basic.ordtype.html new file mode 100644 index 00000000..69967258 --- /dev/null +++ b/combi/1.1.0/Combi.Basic.ordtype.html @@ -0,0 +1,578 @@ + + + + + +Combi.Basic.ordtype: Ordered Types + + + + +
+ + + +
+ +

Library Combi.Basic.ordtype: Ordered Types

+ +
+
+ +
+ +
+
+
+ +
+

Ordered type

+ + +
+ +Inhabited Types: + +
+ +
    +
  • inhType == interface for inhabited types + +
    • +
  • inhPordType == interface for partially ordered inhabited types + +
    • +
  • inhOrdType == interface for totally ordered inhabited types + +
    • +
  • inhOrdFinType == interface for totally ordered finite types + +
    • +
+ +
+ +Sequence on a totally ordered type: + +
+ +
    +
  • maxL a L == the maximum of a and the element of the sequence L + +
    • +
  • allLeq v a == a is smaller or equal than all the element of v + +
    • +
  • allLnt v a == a is strictly smaller than all the element of v + +
    + + +
    • +
  • rembig w == w minus last occurence of its largest letter + +
    • +
  • posbig w == the position of the last occurence of the largest letter of w + +
    + + +
    • +
  • shift_pos pos i == if i < pos then i else i.+1 + +
    • +
  • shiftinv_pos pos i == if i < pos then i else i.-1 + +
    • +
+ +
+ +Cover relation: + +
+ +
    +
  • covers x y == y covers x where x and y belongs to a common + finPOrderType. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import order.
+Require Import tools.
+ +
+Set Implicit Arguments.
+ +
+#[local] Open Scope order_scope.
+Import Order.Theory.
+ +
+
+ +
+

Induction on partially ordered types

+ +
+
+ +
+Lemma finord_wf disp (T : finPOrderType disp) (P : T -> Type) :
+  (forall x, (forall y, y < x -> P y) -> P x) -> forall x, P x.
+ +
+Lemma finord_wf_down disp (T : finPOrderType disp) (P : T -> Type) :
+  (forall x, (forall y, y > x -> P y) -> P x) -> forall x, P x.
+ +
+
+ +
+

Covering relation

+ +
+ + We only define covering relation for finite type, since it cannot be decided and it is not very useful for infinite orders. +
+
+Definition covers {disp} {T : finPOrderType disp} :=
+  [rel x y : T | (x < y) && [forall z, ~~(x < z < y)]].
+ +
+Section CoversFinPOrder.
+ +
+Context disp (T : finPOrderType disp).
+Implicit Type (x y : T).
+ +
+Lemma coversP x y : reflect (x < y /\ (forall z, ~(x < z < y))) (covers x y).
+ +
+Lemma ltcovers x y : covers x y -> x < y.
+ +
+Lemma coversEV x y : covers x y -> forall z, x <= z <= y -> z = x \/ z = y.
+ +
+Lemma covers_dual x y :
+  covers (T := T^d) y x = covers x y.
+ +
+Lemma covers_ind (P : T -> Type) :
+  (forall x y, covers x y -> P x -> P y) ->
+  forall x, P x -> forall y, x <= y -> P y.
+ +
+Lemma covers_connect x y : (connect covers x y) = (x <= y).
+ +
+Lemma covers_path x y :
+  reflect (exists2 s, path covers x s & y = last x s) (x <= y).
+ +
+End CoversFinPOrder.
+ +
+Lemma covers_rind disp (T : finPOrderType disp) (P : T -> Type) :
+  (forall x y, covers y x -> P x -> P y) ->
+  forall x, P x -> forall y, x >= y -> P y.
+ +
+
+ +
+

Inhabited types

+ +
+
+ +
+#[short(type=inhType)]
+HB.structure Definition Inhabited := {T of isInhabited T & Choice T}.
+ +
+#[warning="-HB.no-new-instance"]
+HB.instance Definition _ := isInhabited.Build T (ex_intro _ x is_true_true).
+ +
+ +
+ +
+Definition inh {T : inhType} := xchoose (@inh_ex T).
+ +
+Lemma inh_xchooseE (T : inhType) (exP : exists x0 : T, true) :
+  xchoose exP = @inh T.
+ +
+Lemma inh_chooseE (T : inhType) (x0 : T) :
+  choose xpredT x0 = @inh T.
+ +
+#[short(type=inhFinType)]
+HB.structure Definition InhFinite := { T of isInhabited T & Finite T }.
+ +
+
+ +
+

Inhabited ordered types

+ +
+
+#[short(type=inhPOrderType)]
+HB.structure Definition InhPOrder d :=
+  {T of isInhabited T & Order.POrder d T}.
+ +
+#[short(type=inhLatticeType)]
+HB.structure Definition InhLattice d :=
+  {T of isInhabited T & Order.Lattice d T}.
+ +
+#[short(type=inhTBLatticeType)]
+HB.structure Definition InhTBLattice d :=
+  {T of isInhabited T & Order.TBLattice d T}.
+ +
+#[short(type=inhOrderType)]
+HB.structure Definition InhOrder d :=
+  {T of isInhabited T & Order.Total d T}.
+ +
+
+ +
+

Inhabited finite partially ordered types

+ +
+
+#[short(type=inhFinPOrderType)]
+HB.structure Definition InhFinPOrder d :=
+  {T of isInhabited T & Order.POrder d T & Finite T}.
+ +
+#[short(type=inhFinLatticeType)]
+HB.structure Definition InhFinLattice d :=
+  {T of isInhabited T & Order.FinLattice d T}.
+ +
+#[short(type=inhFinOrderType)]
+HB.structure Definition InhFinOrder d :=
+  {T of isInhabited T & Order.FinTotal d T}.
+ +
+ +
+Section Dual.
+ +
+ +
+ +
+End Dual.
+ +
+
+ +
+

sequences over an ordered types

+

Maximum of a sequence

+ +
+
+Section MaxSeq.
+ +
+Context disp (T : orderType disp).
+Implicit Type a b c : T.
+Implicit Type u v : seq T.
+ +
+Definition maxL a := foldl Order.max a.
+ +
+Lemma maxLb a u : a <= maxL a u.
+ +
+Lemma in_maxL a u : (maxL a u) \in a :: u.
+ +
+Lemma maxXL a b u : Order.max a (maxL b u) = maxL (Order.max a b) u.
+ +
+Lemma maxL_cat a u b v : maxL a (u ++ b :: v) = Order.max (maxL a u) (maxL b v).
+ +
+End MaxSeq.
+ +
+
+ +
+

Comparison of the elements of a sequence to an element

+ +
+
+Section AllLeqLtn.
+ +
+Context disp (T : orderType disp).
+Implicit Type a b c : T.
+Implicit Type u v : seq T.
+ +
+Definition allLeq v a := all (<= a) v.
+Definition allLtn v a := all (< a) v.
+ +
+Lemma allLtn_notin s b : allLeq s b -> b \notin s -> allLtn s b.
+ +
+Lemma maxLPt a u : allLeq u (maxL a u).
+Lemma maxLP a u : allLeq (a :: u) (maxL a u).
+ +
+Lemma allLtnW v a : allLtn v a -> allLeq v a.
+ +
+Lemma allLeqE u a : allLeq u a -> maxL a u = a.
+Lemma allLeqP u a : reflect (maxL a u = a) (allLeq u a).
+ +
+Lemma allLeqCons b u a : b <= a -> allLeq u a -> allLeq (b :: u) a.
+Lemma allLtnCons b u a : b < a -> allLtn u a -> allLtn (b :: u) a.
+ +
+Lemma allLeqConsE u a b : allLeq (b :: u) a = (maxL b u <= a).
+ +
+Lemma allLtnConsE u a b : allLtn (b :: u) a = (maxL b u < a).
+ +
+Lemma allLeq_consK b u a : allLeq (b :: u) a -> allLeq u a.
+Lemma allLtn_consK b u a : allLtn (b :: u) a -> allLtn u a.
+ +
+Lemma allLeq_catE u v a : allLeq (u ++ v) a = allLeq u a && allLeq v a.
+Lemma allLtn_catE u v a : allLtn (u ++ v) a = allLtn u a && allLtn v a.
+ +
+Lemma maxL_perm a u b v : perm_eq (a :: u) (b :: v) -> maxL a u = maxL b v.
+ +
+Lemma perm_allLeq u v a : perm_eq u v -> allLeq u a -> allLeq v a.
+Lemma perm_allLeqE u v a : perm_eq u v -> allLeq u a = allLeq v a.
+Lemma perm_allLtn u v a : perm_eq u v -> allLtn u a -> allLtn v a.
+Lemma perm_allLtnE u v a : perm_eq u v -> allLtn u a = allLtn v a.
+ +
+Lemma allLeq_rev u a : allLeq (rev u) a = allLeq u a.
+Lemma allLtn_rev u a : allLtn (rev u) a = allLtn u a.
+ +
+Lemma allLeq_rconsK b u a : allLeq (rcons u b) a -> allLeq u a.
+Lemma allLtn_rconsK b u a : allLtn (rcons u b) a -> allLtn u a.
+ +
+Lemma allLeq_last b u a : allLeq (rcons u b) a -> b <= a.
+Lemma allLtn_last b u a : allLtn (rcons u b) a -> b < a.
+ +
+Lemma maxL_LbR a v L b R :
+  a :: v = L ++ b :: R -> allLeq L b -> allLeq R b -> maxL a v = b.
+ +
+End AllLeqLtn.
+ +
+
+ +
+

Removing the largest letter of a sequence

+ +
+
+Section RemoveBig.
+ +
+Context disp (T : orderType disp).
+Variable Z : T.
+Implicit Type a b c : T.
+Implicit Type u v w r : seq T.
+ +
+
+ +
+Remove the last occurence of the largest letter from w +
+
+Fixpoint rembig w :=
+  if w is a :: v then
+    if allLtn v a then v else a :: rembig v
+  else [::].
+ +
+
+ +
+Position of the last occurence of the largest letter of w +
+
+Fixpoint posbig w :=
+  if w is a :: v then
+    if allLtn v a then 0 else (posbig v).+1
+  else 0.
+ +
+Lemma size_rembig w : size (rembig w) = (size w).-1.
+ +
+Lemma rembig_catR a u b v :
+  maxL a u <= maxL b v -> rembig (a :: u ++ b :: v) = a :: u ++ rembig (b :: v).
+ +
+Lemma rembig_catL a u b v :
+  maxL a u > maxL b v -> rembig (a :: u ++ b :: v) = rembig (a :: u) ++ b :: v.
+ +
+Lemma rembig_cat u v :
+  rembig (u ++ v) = (rembig u) ++ v \/ rembig (u ++ v) = u ++ (rembig v).
+ +
+Lemma rembig_eq_permL u1 u2 v :
+  perm_eq u1 u2 ->
+  (rembig (u1 ++ v) = (rembig u1) ++ v /\
+   rembig (u2 ++ v) = (rembig u2) ++ v)
+  \/
+  (rembig (u1 ++ v) = u1 ++ (rembig v) /\
+   rembig (u2 ++ v) = u2 ++ (rembig v)).
+ +
+Lemma rembig_eq_permR u v1 v2 :
+  perm_eq v1 v2 ->
+  (rembig (u ++ v1) = (rembig u) ++ v1 /\
+   rembig (u ++ v2) = (rembig u) ++ v2)
+  \/
+  (rembig (u ++ v1) = u ++ (rembig v1) /\
+   rembig (u ++ v2) = u ++ (rembig v2)).
+ +
+Lemma rembigP w wb : wb != [::] ->
+  reflect
+    (exists u b v, [/\ w = u ++ v, wb = u ++ b :: v, allLeq u b & allLtn v b])
+    (w == rembig wb).
+ +
+Lemma perm_rembig u v :
+  perm_eq u v -> perm_eq (rembig u) (rembig v).
+ +
+Lemma rembig_rev_uniq s : uniq s -> rev (rembig s) = rembig (rev s).
+ +
+Lemma rembig_subseq s : subseq (rembig s) s.
+ +
+Lemma rembig_uniq s : uniq s -> uniq (rembig s).
+ +
+Open Scope nat_scope.
+ +
+Lemma posbig_size_cons l s : posbig (l :: s) < size (l :: s).
+ +
+Lemma posbig_size s : s != [::] -> posbig s < size s.
+ +
+Lemma posbigE u b v :
+  (allLeq u b && allLtn v b) = (posbig (u ++ b :: v) == size u).
+ +
+Lemma posbig_take_dropE l s :
+  take (posbig (l :: s)) (rembig (l :: s)) ++
+     maxL l s
+     :: drop (posbig (l :: s)) (rembig (l :: s)) = l :: s.
+ +
+Lemma nth_posbig l s : nth Z (l :: s) (posbig (l :: s)) = maxL l s.
+ +
+Lemma allLeq_posbig l s :
+  allLeq (take (posbig (l :: s)) (l :: s)) (maxL l s).
+ +
+Lemma allLtn_posbig l s :
+  allLtn (drop (posbig (l :: s)).+1 (l :: s)) (maxL l s).
+ +
+Lemma rembigE l s :
+  take (posbig (l :: s)) (l :: s) ++
+       drop (posbig (l :: s)).+1 (l :: s) = rembig (l :: s).
+ +
+Lemma nth_lt_posbig i s : i < posbig s -> nth Z (rembig s) i = nth Z s i.
+ +
+Definition shift_pos pos i := if i < pos then i else i.+1.
+Definition shiftinv_pos pos i := if i < pos then i else i.-1.
+ +
+Lemma shift_posK pos i : shiftinv_pos pos (shift_pos pos i) = i.
+ +
+Lemma shiftinv_posK pos i : i != pos -> shift_pos pos (shiftinv_pos pos i) = i.
+ +
+Lemma nth_rembig s i :
+  nth Z s (shift_pos (posbig s) i) = nth Z (rembig s) i.
+ +
+Lemma nth_inspos s pos i n :
+  pos <= size s ->
+  nth Z ((take pos s) ++ n :: (drop pos s)) i =
+  if i == pos then n else nth Z s (shiftinv_pos pos i).
+ +
+Lemma shift_pos_mono pos : {mono shift_pos pos : i j / i <= j}.
+ +
+Lemma shiftinv_pos_homo pos : {homo shiftinv_pos pos : i j / i <= j}.
+ +
+End RemoveBig.
+ +
+ +
+Lemma maxL_iota n i : maxL i (iota i.+1 n) = i + n.
+ +
+Lemma maxL_iota_n n : maxL 0%N (iota 1 n) = n.
+ +
+Lemma rembig_iota n i : rembig (iota i n.+1) = iota i n.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Basic.unitriginv.html b/combi/1.1.0/Combi.Basic.unitriginv.html new file mode 100644 index 00000000..3f04e41a --- /dev/null +++ b/combi/1.1.0/Combi.Basic.unitriginv.html @@ -0,0 +1,208 @@ + + + + + +Combi.Basic.unitriginv: Uni-triangular Matrices + + + + +
+ + + +
+ +

Library Combi.Basic.unitriginv: Uni-triangular Matrices

+ +
+
+ +
+ +
+
+
+ +
+

Triangular matrix with 1 on the diagonal

+ + +
+ +We deal with "matrices" which are triangular for a possibly partial order +with 1 on the diagonal. The goal is to show that such a matrix is invertible +on any ring and to give formulas for the inverse. The matrices are given as a +function M : T -> T -> R for a finite partially ordered type T and a +commutative unit ring R. + +
+ +
    +
  • unitrig M == M is unitriangular where M. + +
    • +
  • Mat M == transform M to a usual mathcomp square matrix of order #|T| + +
    • +
  • Minv M == the inverse of the matrix M. + +
    • +
+ +
+ +We show that such a matrix has determinant 1 (Lemma det_unitrig) and is +therefore invertible. Moreover Lemma Minv_unitrig says that the inverse +is unitriangular too. + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat order.
+From mathcomp Require Import fintype bigop ssralg.
+From mathcomp Require Import finset fingroup perm matrix.
+ +
+Require ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+Import GRing.Theory.
+#[local] Open Scope ring_scope.
+ +
+Section UniTriangular.
+ +
+Variable R : comUnitRingType.
+Context disp (T : finPOrderType disp).
+Implicit Type M : T -> T -> R.
+Implicit Types t u v : T.
+ +
+Definition unitrig M :=
+  [forall t, M t t == 1] && [forall t, [forall (u | M u t != 0), (t <= u)%O]].
+Lemma unitrigP M :
+  reflect ((forall t, M t t = 1) /\ (forall t u, M u t != 0 -> (t <= u)%O))
+          (unitrig M).
+ +
+Lemma unitrig1 : unitrig (fun x y => (x == y)%:R).
+ +
+
+ +
+TODO : construct the group of unitriangular matrix +
+
+ +
+Lemma unitrig_suml M (Mod : lmodType R) (F : T -> Mod) u :
+  unitrig M ->
+  \sum_(t : T) M u t *: F t = \sum_(t | (t <= u)%O) M u t *: F t.
+ +
+Lemma unitrig_sum1l M (Mod : lmodType R) (F : T -> Mod) u :
+  unitrig M ->
+  \sum_(t : T) M u t *: F t = F u + \sum_(t | (t < u)%O) M u t *: F t.
+ +
+Lemma unitrig_sumr M (Mod : lmodType R) (F : T -> Mod) t :
+  unitrig M ->
+  \sum_(u : T) M u t *: F u = \sum_(u | (t <= u)%O) M u t *: F u.
+ +
+Lemma unitrig_sum1r M (Mod : lmodType R) (F : T -> Mod) t :
+  unitrig M ->
+  \sum_(u : T) M u t *: F u = F t + \sum_(u | (t < u)%O) M u t *: F u.
+ +
+Lemma unitrig_sumlV M (Mod : lmodType R) (F : T -> Mod) u :
+  unitrig M ->
+  \sum_(t : T) M t u *: F t = \sum_(t | (u <= t)%O) M t u *: F t.
+ +
+Lemma unitrig_sum1lV M (Mod : lmodType R) (F : T -> Mod) u :
+  unitrig M ->
+  \sum_(t : T) M t u *: F t = F u + \sum_(t | (u < t)%O) M t u *: F t.
+ +
+Lemma unitrig_sumrV M (Mod : lmodType R) (F : T -> Mod) t :
+  unitrig M ->
+  \sum_(u : T) M t u *: F u = \sum_(u | (u <= t)%O) M t u *: F u.
+ +
+Lemma unitrig_sum1rV M (Mod : lmodType R) (F : T -> Mod) t :
+  unitrig M ->
+  \sum_(u : T) M t u *: F u = F t + \sum_(u | (u < t)%O) M t u *: F u.
+ +
+End UniTriangular.
+ +
+Section TriangularInv.
+ +
+Context (R : comUnitRingType) disp (T : finPOrderType disp).
+Variable M : T -> T -> R.
+Implicit Types t u v : T.
+ +
+Hypothesis Munitrig : unitrig M.
+ +
+#[local] Notation n := #|{: T}|.
+Definition Mat : 'M[R]_n := \matrix_(i, j < n) M (enum_val i) (enum_val j).
+ +
+Lemma det_unitrig : \det Mat = 1.
+ +
+Definition Minv t u : R := invmx Mat (enum_rank t) (enum_rank u).
+ +
+Lemma Minvl t u : \sum_(v : T) (Minv t v) * (M v u) = (u == t)%:R.
+ +
+Lemma Minvr t u : \sum_(v : T) (M t v) * (Minv v u) = (u == t)%:R.
+ +
+Lemma Minv_trig t u : Minv u t != 0 -> (t <= u)%O.
+ +
+Lemma Minv_uni t : Minv t t = 1.
+ +
+Lemma Minv_unitrig : unitrig Minv.
+ +
+Lemma Minv_lincombl (Mod : lmodType R) (F G : T -> Mod) :
+  (forall t, F t = \sum_u M t u *: G u) ->
+  (forall t, G t = \sum_u Minv t u *: F u).
+ +
+Lemma Minv_lincombr (Mod : lmodType R) (F G : T -> Mod) :
+  (forall t, F t = \sum_u M u t *: G u) ->
+  (forall t, G t = \sum_u Minv u t *: F u).
+ +
+End TriangularInv.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.Dyckword.html b/combi/1.1.0/Combi.Combi.Dyckword.html new file mode 100644 index 00000000..b13d8c6b --- /dev/null +++ b/combi/1.1.0/Combi.Combi.Dyckword.html @@ -0,0 +1,741 @@ + + + + + +Combi.Combi.Dyckword: Dyck Words + + + + +
+ + + +
+ +

Library Combi.Combi.Dyckword: Dyck Words

+ +
+
+ +
+ +
+
+
+ +
+

Dyck Words

+ + +
+ +A Dyck word is a word on the two letter alphabet "(" and ")" which is well +parentesized. More formally + +
+ +
    +
  • for all prefix, the number of "(" is larger that the number of ")"; + +
    • +
  • the total number of "(" is equal to those of ")". + +
    • +
+ +
+ +We encode those words using: + +
+ +
    +
  • brace == a type with two values Open and Close and locally use + the notations "{{" and "}}" + +
    • +
  • height w == the difference between the number of "{{" and "}}" + +
    • +
  • prefixes w == the list of all the prefix of w + +
    + + +
    • +
  • w is a Dyck_prefix <=> for all prefix, the number of "{{" is larger than + the number of "}}" + +
    • +
  • w is a Dyck_word <=> w is a Dyck word, that is a Dyck_prefix of height 0 + +
    + + +
    • +
  • Dyck == a Sigma-type for Dyck words. It is canonically a countType + +
    • +
  • [Dyck of s] == a Dyck word for the sequence s; the proof is canonically + infered. + +
    • +
  • [Dyck of s by pf] == a Dyck word for the sequence s using the proof pf + +
    • +
  • [Dyck {{ D1 }} D2] == the term of type Dyck corresponding to the word + "(D1)D2", assuming that D1 and D2 are of type Dyck + +
    • +
+ +
+ +Dyck words are in bijection with binary trees: + +
+ +
    +
  • Dyck_of_bintree t == the image of the binary tree t by the standard + bijection from trees to Dyck words. + +
    • +
  • bintree_of_Dyck D == the converse bijection + +
    • +
+ +
+ +The main result of this file is to show that the number of Dyck word is the +so-called Catalan numbers defined below: + +
+ +
    +
  • Catalan n == the n-th Catalan number 'C(n.*2, n) %/ n.+1 + +
    • +
  • Dyck_hsz n == the set of tuple of half size n which are Dyck words + +
    • +
  • bal_hsz n == the set of tuple of half size n which are balanced words + +
    • +
+ +
+ +We prove bijectively the equality with catalan binomial formula, using the +rotation trick: there is a (n+1) to 1 map from balanced words to Dyck words. + +
+ +
    +
  • minh w == the minimal height of prefixes of w + +
    • +
  • pfminh w == the first position where the minimal height is reached + +
    • +
  • Dyck_of_bal w == given a balanced word w returns the unique Dyck word + D such that D) is a rotation of w) + +
    • +
  • bal_of_Dyck rt w == the word obtained by rotating w) by rt and + removing the last letter. + +
    + + +
    • +
  • bal_part n == the partition of balanced words given by the fiber of the + Dyck_of_bal map. The main argument is to show that all the fibers + have the same cardinality n+1. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import div ssralg ssrint ssrnum binomial.
+Require Import tools combclass bintree.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory GRing.Theory Num.Theory.
+ +
+Section PreimPartition.
+ +
+Variables (T rT : finType) (f : T -> rT) (D : {set T}).
+ +
+Lemma mem_preim_partition (B : {set T}) :
+  B \in preim_partition f D ->
+  exists x0 : T, (x0 \in B) /\ (B = D :&: f @^-1: [set f x0]).
+ +
+Lemma imset_transversal_preim :
+  f @: (transversal (preim_partition f D) D) = f @: D.
+ +
+Lemma card_preim_partition : #|preim_partition f D| = #|f @: D|.
+ +
+End PreimPartition.
+ +
+Lemma card_preim_nth (m : nat) (T : eqType) (s : seq T) (P : pred T) (u : T):
+  size s = m -> #|[set i : 'I_m | preim (nth u s) P i]| = count P s.
+ +
+
+ +
+

Braces and Dyck words

+ +
+
+Inductive brace : Set := | Open : brace | Close : brace.
+ +
+#[local] Notation "{{" := Open.
+#[local] Notation "}}" := Close.
+ +
+Definition bool_of_brace b := if b is {{ then true else false.
+Definition brace_of_bool b := if b then {{ else }}.
+ +
+Lemma bool_of_braceK : cancel bool_of_brace brace_of_bool.
+Lemma brace_of_boolK : cancel brace_of_bool bool_of_brace.
+ +
+ +
+Lemma size_count_braceE (w : seq brace) :
+  size w = count_mem {{ w + count_mem }} w.
+ +
+Import GRing.Theory Num.Theory.
+Open Scope int_scope.
+Open Scope ring_scope.
+ +
+
+ +
+

Height of a word

+ +
+
+Section Defs.
+ +
+Implicit Type n : nat.
+Implicit Type u v w : seq brace.
+ +
+Definition height w : int := Posz (count_mem {{ w) - Posz (count_mem }} w).
+ +
+Lemma height_nil : height [::] = 0.
+ +
+Lemma height_cons l v :
+  height (l :: v) = (if l == {{ then 1 else -1) + height v.
+ +
+Lemma height_rcons v l :
+  height (rcons v l) = height v + (if l == {{ then 1 else -1).
+ +
+Lemma height_cat u v : height (u ++ v) = height u + height v.
+ +
+Definition height_simpl := (height_cons, height_rcons, height_cat, height_nil).
+ +
+Lemma height_drop n u : height (drop n u) = height u - height (take n u).
+ +
+Lemma height_rev u : height (rev u) = height u.
+ +
+Lemma height_nseq n b :
+  height (nseq n b) = Posz n * (if b == {{ then 1 else -1).
+ +
+Definition prefixes w := [seq take n w | n <- iota 0 (size w).+1].
+ +
+Lemma take_prefixes w i : take i w \in prefixes w.
+Lemma mem_prefixesP w p : reflect (exists i, p = take i w) (p \in prefixes w).
+ +
+
+ +
+

Definitions

+ +
+
+Definition Dyck_prefix :=
+  [qualify a w : seq brace |
+   all (fun p : seq brace => height p >= 0) (prefixes w)].
+Definition Dyck_word :=
+  [qualify a w : seq brace | (w \is a Dyck_prefix) && (height w == 0)].
+ +
+Lemma Dyck_prefixP w :
+  reflect (forall n, height (take n w) >= 0) (w \is a Dyck_prefix).
+ +
+Lemma Dyck_wordP w :
+  reflect
+    ((forall n, height (take n w) >= 0) /\ (height w = 0))
+    (w \is a Dyck_word).
+ +
+Lemma height_take_leq (h : int) w :
+  (forall n : nat, h <= height (take n w)) <->
+  (forall n : nat, (n <= size w)%N -> h <= height (take n w)).
+ +
+Lemma Dyck_word_OwC w :
+  w \is a Dyck_word -> {{ :: w ++ [:: }}] \is a Dyck_word.
+ +
+Lemma Dyck_word_cat w1 w2 :
+  w1 \is a Dyck_word -> w2 \is a Dyck_word -> w1 ++ w2 \is a Dyck_word.
+Lemma Dyck_word_flatten l :
+  all (mem Dyck_word) l -> flatten l \is a Dyck_word.
+ +
+Lemma Dyck_word_OwCw w1 w2 :
+  w1 \is a Dyck_word -> w2 \is a Dyck_word ->
+  {{ :: w1 ++ }} :: w2 \is a Dyck_word.
+ +
+End Defs.
+ +
+
+ +
+

Sigma type for Dyck words

+ +
+
+Section DyckType.
+ +
+Record Dyck := DyckWord {dyckword :> seq brace;
+                         is_dyckword :> dyckword \is a Dyck_word}.
+ +
+Implicit Type D : Dyck.
+ +
+Lemma DyckP D : (dyckword D) \is a Dyck_word.
+ Hint Resolve DyckP : core.
+ +
+Definition dyck D mkD : Dyck :=
+  mkD (let: DyckWord _ DP := D return dyckword D \is a Dyck_word in DP).
+ +
+Lemma dyckE D : dyck (fun sP => @DyckWord D sP) = D.
+ +
+Canonical nil_Dyck := (@DyckWord [::] is_true_true).
+Canonical cat_Dyck D1 D2 := DyckWord (Dyck_word_cat D1 D2).
+Canonical join_Dyck D1 D2 := DyckWord (Dyck_word_OwCw D1 D2).
+ +
+End DyckType.
+ +
+#[warning="-notation-incompatible-prefix"]
+Notation "'[' 'Dyck' 'of' s ]" := (dyck (fun sP => @DyckWord s sP))
+  (at level 9, format "[ 'Dyck' 'of' s ]") : form_scope.
+ +
+#[warning="-notation-incompatible-prefix"]
+Notation "'[' 'Dyck' 'of' s 'by' pf ]" := (@DyckWord s pf)
+  (at level 9, format "[ 'Dyck' 'of' s 'by' pf ]") : form_scope.
+ +
+#[warning="-notation-incompatible-prefix"]
+Notation "'[' 'Dyck' '{{' D1 '}}' D2 ]" := (join_Dyck D1 D2)
+  (at level 8, format "[ 'Dyck' '{{' D1 '}}' D2 ]",
+   D1 at next level) : form_scope.
+ +
+
+ +
+

Standard factorization of Dyck words

+ +
+
+Section DyckFactor.
+ +
+Implicit Type D : Dyck.
+ +
+Lemma Dyck_cut_ex D :
+  D != [Dyck of [::]] -> exists i, (i != 0)%N && (height (take i D) == 0).
+ +
+Theorem factor_Dyck D :
+  D != [Dyck of [::]] -> { DD : Dyck * Dyck | D = [Dyck {{ DD.1 }} DD.2] }.
+ +
+Theorem join_Dyck_inj D1 D2 E1 E2 :
+  [Dyck {{D1}}D2] = [Dyck {{E1}}E2] -> (D1, D2) = (E1, E2).
+ +
+End DyckFactor.
+ +
+
+ +
+

Induction on Dyck words following the standard factorization

+ +
+
+Section DyckSetInd.
+ +
+Implicit Type D : Dyck.
+ +
+Variable P : Dyck -> Type.
+Hypotheses (Pnil : P nil_Dyck)
+           (Pcons : forall D1 D2, P D1 -> P D2 -> P [Dyck {{ D1 }} D2]).
+ +
+Theorem Dyck_ind D : P D.
+ +
+End DyckSetInd.
+ +
+
+ +
+

Examples of application of the induction principle

+ +
+
+Lemma Dyck_size_even (D : Dyck) : ~~ odd (size D).
+ +
+Lemma factor_Dyck_seq D :
+  { DS : seq Dyck | D = foldr join_Dyck nil_Dyck DS }.
+ +
+Lemma foldr_join_Dyck_inj : injective (foldr join_Dyck nil_Dyck).
+ +
+Lemma size_foldr_join_Dyck DS :
+  (size (foldr join_Dyck nil_Dyck DS) =
+   sumn (map (size \o dyckword) DS) + 2 * size DS)%N.
+ +
+
+ +
+

Bijection with binary trees

+ +
+
+Section BijBinTrees.
+ +
+Fixpoint Dyck_of_bintree t :=
+  if t is BinNode l r then
+    [Dyck {{ (Dyck_of_bintree l) }} Dyck_of_bintree r]
+  else nil_Dyck.
+ +
+Lemma bintree_of_Dyck_spec D :
+  { t : bintree | Dyck_of_bintree t = D /\
+                  forall t', Dyck_of_bintree t' = D -> t = t' }.
+Definition bintree_of_Dyck D := proj1_sig (bintree_of_Dyck_spec D).
+ +
+Lemma bintree_of_DyckK D : Dyck_of_bintree (bintree_of_Dyck D) = D.
+ +
+Lemma bintree_of_nil_Dyck : bintree_of_Dyck nil_Dyck = BinLeaf.
+ +
+Lemma bintree_of_join_Dyck D1 D2 :
+  bintree_of_Dyck ([Dyck {{ D1 }} D2]) =
+  BinNode (bintree_of_Dyck D1) (bintree_of_Dyck D2).
+ +
+Theorem Dyck_of_bintreeK t : bintree_of_Dyck (Dyck_of_bintree t) = t.
+ +
+Lemma size_Dyck_of_bintree t :
+  size (Dyck_of_bintree t) = (size_tree t).*2.
+ +
+Lemma size_bintree_of_Dyck D :
+  (size_tree (bintree_of_Dyck D)).*2 = size D.
+ +
+End BijBinTrees.
+ +
+
+ +
+

Dyck and balanced words

+

Rotation map from balanced words to a Dyck words

+ +
+
+Section BalToDyck.
+ +
+Variable w : seq brace.
+ +
+Definition minh :=
+  - Posz (\max_(s <- prefixes w) (count_mem }} s - count_mem {{ s)).
+ +
+Lemma minhE :
+  minh = - Posz (\max_(i < (size w).+1)
+                  (count_mem }} (take i w) - count_mem {{ (take i w))).
+ +
+Lemma minhP : forall i : nat, height (take i w) >= minh.
+ +
+Fact exists_minh : exists i : nat, height (take i w) == minh.
+Definition pfminh := ex_minn exists_minh.
+ +
+Lemma pfminhP : height (take pfminh w) = minh.
+ +
+Lemma pfminh_size : (pfminh <= size w)%N.
+ +
+Lemma pfminh_min :
+  forall i : nat, (i < pfminh)%N -> height (take i w) > minh.
+ +
+Lemma pfminhE n :
+  (n <= size w)%N ->
+  (forall i : nat, height (take i w) >= height (take n w)) ->
+  (forall i : nat, (i < n)%N -> height (take i w) > height (take n w)) ->
+  n = pfminh.
+ +
+Hypothesis Hbal1 : height w = -1.
+ +
+Lemma minh_neg : minh < 0.
+ +
+Lemma pfminh_pos : (pfminh > 0)%N.
+ +
+Lemma nth_pfminh : nth {{ w pfminh.-1 = }}.
+ +
+Lemma last_rot_pfminh : last {{ (rot pfminh w) = }}.
+ +
+Lemma rot_pfminhE :
+  rcons (take (size w).-1 (rot pfminh w)) }} = rot pfminh w.
+ +
+Theorem rot_is_Dyck : take (size w).-1 (rot pfminh w) \is a Dyck_word.
+ +
+End BalToDyck.
+ +
+Section DyckToBal.
+ +
+Variables (w : seq brace) (rt : nat).
+Hypothesis HDyck : w \is a Dyck_word.
+Hypothesis Hrt: (rt <= (size w))%N.
+ +
+Lemma rrw_bal1 : height (rot rt (rcons w }})) = -1.
+ +
+Lemma pfminh_rrw :
+  pfminh (rot rt (rcons w }})) = ((size w).+1 - rt)%N.
+ +
+Lemma minh_rrw : minh (rot rt (rcons w }})) = - height (take rt w) - 1.
+ +
+End DyckToBal.
+ +
+Close Scope ring_scope.
+ +
+
+ +
+

Catalan numbers

+ +
+
+Definition Catalan n := 'C(n.*2, n) %/ n.+1.
+ +
+Definition Dyck_hsz n : {set n.*2.-tuple brace} :=
+  [set w : n.*2.-tuple brace | tval w \is a Dyck_word].
+Definition bal_hsz n : {set n.*2.-tuple brace} :=
+  [set w : n.*2.-tuple brace | height w == 0].
+ +
+Theorem card_bintreesz_dyck n : #|bintreesz n| = #|Dyck_hsz n|.
+ +
+Lemma count_mem_height0 n (w : n.*2.-tuple brace) b :
+  height w = 0 -> count_mem b w = n.
+ +
+Lemma card_bal_hsz n : #|bal_hsz n| = 'C(n.*2, n).
+ +
+
+ +
+

The rotation bijection

+ +
+
+Section DyckWordRotationBijection.
+ +
+Variable n : nat.
+Implicit Types u v w D : n.*2.-tuple brace.
+ +
+Fact size_UnDn : size (nseq n {{ ++ nseq n }}) == n.*2.
+ Definition UnDn := Tuple size_UnDn.
+ +
+Lemma UnDn_Dyck : tval UnDn \is a Dyck_word.
+ +
+Fact size_UDn : size (flatten (nseq n [:: {{; }}])) == n.*2.
+ Definition UDn := Tuple size_UDn.
+ +
+Lemma UDn_Dyck : tval UDn \is a Dyck_word.
+ +
+Fact size_Dyck_of_bal w :
+  size (take (size w) (rot (pfminh (rcons w }})) (rcons w }}))) == n.*2.
+ Definition Dyck_of_bal w := Tuple (size_Dyck_of_bal w).
+ +
+Lemma Dyck_of_balP w : height w = 0 -> tval (Dyck_of_bal w) \is a Dyck_word.
+ +
+Lemma Dyck_of_Dyck_hsz D : tval D \is a Dyck_word -> Dyck_of_bal D = D.
+ +
+Fact size_bal_of_Dyck rt w :
+  size (take (size w) (rotr rt (rcons w }}))) == n.*2.
+ Definition bal_of_Dyck rt w := Tuple (size_bal_of_Dyck rt w).
+ +
+Lemma bal_of_DyckP rt w :
+  (rt <= size w)%N ->
+  nth {{ (rcons w }}) (size w - rt) = }} ->
+  height w = 0 -> height (bal_of_Dyck rt w) = 0.
+ +
+Lemma rcons_bal_of_Dyck w k :
+  (k <= size w)%N ->
+  nth {{ (rcons w }}) (size w - k) = }} ->
+  rcons (bal_of_Dyck k w) }} = rotr k (rcons w }}).
+ +
+Lemma bal_of_DyckK D rt :
+  (rt <= size D)%N ->
+  nth {{ (rcons D }}) (size D - rt) = }} ->
+  tval D \is a Dyck_word ->
+  Dyck_of_bal (bal_of_Dyck rt D) = D.
+ +
+Lemma Dyck_of_balK w :
+  height w = 0 -> w = bal_of_Dyck (pfminh (rcons w }})) (Dyck_of_bal w).
+ +
+Lemma preim_Dyck_of_balE D :
+  tval D \is a Dyck_word ->
+  (bal_hsz n) :&: (Dyck_of_bal @^-1: [set D]) =
+  [ set bal_of_Dyck (nat_of_ord rt) D |
+    rt : 'I_((size D).+1) & nth {{ (rcons D }}) (size D - rt) == }} ].
+ +
+Lemma card_preim_Dyck_of_bal D :
+  tval D \is a Dyck_word ->
+  #|(bal_hsz n) :&: (Dyck_of_bal @^-1: [set D])| = n.+1.
+ +
+End DyckWordRotationBijection.
+ +
+Arguments Dyck_of_bal {n}.
+Arguments bal_of_Dyck {n}.
+ +
+
+ +
+

Main theorem for catalan numbers

+ +
+
+Theorem card_bal_Dyck_hsz n : #|bal_hsz n| = #|Dyck_hsz n| * n.+1.
+ +
+Lemma div_central_binomial n : n.+1 %| 'C(n.*2, n).
+ +
+Theorem card_Dyck_hsz n : #|Dyck_hsz n| = Catalan n.
+ +
+Theorem Catalan_binE n : Catalan_bin n = Catalan n.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.Yamanouchi.html b/combi/1.1.0/Combi.Combi.Yamanouchi.html new file mode 100644 index 00000000..7939cd0b --- /dev/null +++ b/combi/1.1.0/Combi.Combi.Yamanouchi.html @@ -0,0 +1,399 @@ + + + + + +Combi.Combi.Yamanouchi: Yamanouchi Words + + + + +
+ + + +
+ +

Library Combi.Combi.Yamanouchi: Yamanouchi Words

+ +
+
+ +
+ +
+
+
+ +
+

Yamanouchi words.

+ + +
+ +A Yamanouchi word w is a seq nat such as for all i in any suffix of w, +the number of occurence of i is larger than the number of occurence of i+1. +Yamanouchi words are in bijection with standard tableaux. + +
+ +We define the following notions and notations: + +
+ +
    +
  • evalseq s == the evaluation of the sequence s stored as a sequence, + that is the sequence whose i-th entry is the number + of occurences of i in s; the final zeroes are not stored + so that the sequence ends with a non zero entry. + +
    • +
  • evalseq_count == an alternate definition of the previous sequence + +
    • +
  • is_yam s == s is a Yamanouchi word + +
    • +
  • is_yam_of_eval ev s == s is a Yamanouchi word of evaluation ev. + +
    • +
  • decr_yam s == the Yamanouchi word obtained by removing the zero and + decrassing all the other entries by 1 + +
    • +
  • hyper_yam ev == the hyperstandard (decreasing) Yamanouchi word of + evaluation ev such as 33 2222 11111 0000000 + +
    • +
+ +
+ +Enumeration of Yamanouchi words: + +
+ +
    +
  • is_yam_of_eval ev y == y is a yamanouchi word of evalutation ev + +
    • +
  • is_yam_of_size n y == y is a yamanouchi word of size n + +
    • +
  • enum_yameval ev == the lists of all yamanouchi word of evalutation ev + +
    • +
+ +
+ + +
+ +Sigma types for Yamanouchi words: + +
+ +
    +
  • yameval ev == a type for seq nat which are Yamanouchi words of + evaluation ev; it is canonically a finType + +
    • +
  • yamn n == a type for seq nat which are Yamanouchi words of + size n; it is canonically a finType + +
    • +
  • hyper_yameval ev == the the hyperstandard (decreasing) Yamanouchi word of + evaluation ev such as 33 2222 11111 0000000 as a yameval + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+Require Import tools combclass partition.
+ +
+Set Implicit Arguments.
+ +
+Section Yama.
+ +
+Implicit Type s : seq nat.
+ +
+
+ +
+

Evaluation of a sequence of integers (mostly Yamanouchi word)

+ +
+
+Fixpoint evalseq s :=
+  if s is s0 :: s'
+  then incr_nth (evalseq s') s0
+  else [::].
+ +
+
+ +
+equivalent definition +
+
+Definition evalseq_count :=
+  [fun s => [seq (count_mem i) s | i <- iota 0 (foldr maxn 0 (map S s))]].
+ +
+Lemma foldr_maxn s : foldr maxn 0 [seq i.+1 | i <- s] = \max_(i <- s) S i.
+ +
+Lemma evalseq_countE : evalseq_count =1 evalseq.
+ +
+Lemma nth_evalseq s i : nth 0 (evalseq s) i = count_mem i s.
+ +
+Lemma perm_evalseq s t : perm_eq s t = (evalseq s == evalseq t).
+ +
+Lemma evalseq0 y : evalseq y = [::] -> y = [::].
+ +
+Lemma evalseq_cons l s : evalseq (l :: s) = incr_nth (evalseq s) l.
+ +
+Lemma evalseq_eq_size y : sumn (evalseq y) = size y.
+ +
+
+ +
+

Yamanouchi words:

+ Sequence of rows of the corners for an increasing sequence of partitions. They are in bijection with standard tableaux +
+
+Fixpoint is_yam s :=
+  if s is s0 :: s'
+  then is_part (evalseq s) && is_yam s'
+  else true.
+Definition is_yam_of_eval ev y := (is_yam y) && (evalseq y == ev).
+ +
+Lemma is_yamP s :
+  reflect
+    (forall i n, count_mem n (drop i s) >= count_mem n.+1 (drop i s))
+    (is_yam s).
+ +
+Lemma is_yam_ijP s :
+  reflect
+    (forall d i j, i <= j -> count_mem i (drop d s) >= count_mem j (drop d s))
+    (is_yam s).
+ +
+Lemma is_part_eval_yam s : is_yam s -> is_part (evalseq s).
+ +
+Lemma is_yam_tl l0 s : is_yam (l0 :: s) -> is_yam s.
+ +
+Lemma is_yam_catr s t : is_yam (s ++ t) -> is_yam t.
+ +
+Lemma last_yam y : is_yam y -> last 0 y = 0.
+ +
+
+ +
+Remove the zeroes from a yam and decrease all the other entries by 1 +
+
+Fixpoint decr_yam s :=
+  if s is s0 :: s'
+  then if s0 is s0'.+1
+       then s0' :: (decr_yam s')
+       else (decr_yam s')
+  else [::].
+ +
+Lemma evalseq_decr_yam s : evalseq (decr_yam s) = behead (evalseq s).
+ +
+Lemma is_yam_decr s : is_yam s -> is_yam (decr_yam s).
+ +
+Lemma is_rem_corner_yam l0 s :
+  is_yam (l0 :: s) -> is_rem_corner (evalseq (l0 :: s)) l0.
+ +
+Lemma is_add_corner_yam l0 s :
+  is_yam (l0 :: s) -> is_add_corner (evalseq s) l0.
+ +
+
+ +
+

Hyperstandard Yamanouchi word : 33 2222 11111 0000000

+ +
+
+Fixpoint hyper_yam_rev ev :=
+  if ev is s0 :: s then
+    nseq s0 (size s) ++ hyper_yam_rev s
+  else [::].
+Definition hyper_yam ev := hyper_yam_rev (rev ev).
+ +
+Lemma size_hyper_yam ev : size (hyper_yam ev) = sumn ev.
+ +
+Lemma incr_nth_size s : incr_nth s (size s) = rcons s 1.
+ +
+Lemma part_rcons_ind s sn : is_part (rcons s sn.+2) -> is_part (rcons s sn.+1).
+ +
+
+ +
+"is_part ev" is just here to ensure that sh doesn't ends with a 0 +
+
+Lemma evalseq_hyper_yam ev : is_part ev -> evalseq (hyper_yam ev) = ev.
+ +
+Lemma hyper_yamP ev : is_part ev -> is_yam (hyper_yam ev).
+ +
+Lemma hyper_yam_of_eval ev : is_part ev -> is_yam_of_eval ev (hyper_yam ev).
+ +
+End Yama.
+ +
+Lemma is_yam_cat_any y0 y1 z :
+  is_yam y0 -> is_yam y1 -> evalseq y0 = evalseq y1 ->
+  is_yam (z ++ y0) -> is_yam (z ++ y1).
+ +
+
+ +
+

Enumeration of Yamanouchi words

+ +
+
+Fixpoint enum_yamevaln n ev :=
+  if n is n'.+1 then
+    flatten [seq [seq i :: y | y <- enum_yamevaln n' (decr_nth ev i)] |
+                  i <- iota 0 (size ev) & is_rem_corner ev i]
+  else [:: [::]].
+Definition enum_yameval ev := enum_yamevaln (sumn ev) ev.
+Definition is_yam_of_size n y := (is_yam y) && (size y == n).
+ +
+Lemma enum_yamevalP ev:
+  is_part ev -> all (is_yam_of_eval ev) (enum_yameval ev).
+ +
+Lemma enum_yameval_countE ev :
+  is_part ev ->
+  forall y, is_yam_of_eval ev y -> count_mem y (enum_yameval ev) = 1.
+ +
+
+ +
+

Sigma types for Yamanouchi words

+ +
+
+Section YamOfEval.
+ +
+Variable ev : intpart.
+ +
+Structure yameval : Set :=
+  YamEval {yamevalval :> seq nat; _ : is_yam_of_eval ev yamevalval}.
+ +
+Lemma yamevalP (y : yameval) : is_yam y.
+ +
+Lemma eval_yameval (y : yameval) : evalseq y = ev.
+ +
+Lemma size_yameval (y : yameval) : size y = sumn ev.
+ +
+Lemma enum_yamevalE : map val (enum {: yameval}) = enum_yameval ev.
+ +
+Definition hyper_yameval := YamEval (hyper_yam_of_eval (intpartP ev)).
+ +
+End YamOfEval.
+#[export] Hint Resolve yamevalP : core.
+ +
+Section YamOfSize.
+ +
+Variable n : nat.
+ +
+Lemma yamn_PredEq (ev : intpartn n) :
+  predI (is_yam_of_size n) (pred1 (val ev) \o evalseq)
+  =1 is_yam_of_eval (val ev).
+ +
+Lemma yamn_partition_evalseq yam :
+  is_yam_of_size n yam -> (is_part_of_n n) (evalseq yam).
+ +
+Structure yamn : Set :=
+  Yamn {yamnval :> seq nat; _ : is_yam_of_size n yamnval}.
+ +
+Lemma yamnP (y : yamn) : is_yam y.
+ +
+Lemma size_yamn (y : yamn) : size y = n.
+ +
+
+ +
+Check of disjoint union enumeration +
+
+Lemma enum_yamnE :
+  map val (enum {: yamn}) = flatten [seq enum_yameval p | p <- enum_partn n].
+ +
+End YamOfSize.
+ +
+#[export] Hint Resolve yamnP yamevalP : core.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.bintree.html b/combi/1.1.0/Combi.Combi.bintree.html new file mode 100644 index 00000000..56b4a78b --- /dev/null +++ b/combi/1.1.0/Combi.Combi.bintree.html @@ -0,0 +1,1041 @@ + + + + + +Combi.Combi.bintree: Binary Trees + + + + +
+ + + +
+ +

Library Combi.Combi.bintree: Binary Trees

+ +
+
+ +
+ +
+
+
+ +
+

Binary Trees

+ + +
+ +A binary tree is a rooted tree such that each node has two distiguished, +children the left one and the right one. The main goal of this file is to +construct the Tamari lattice which is the transitive closure of the directed +graph of left rotation. We show that it is indeed a lattice. + +
+ +Basic definitions: + +
+ +
    +
  • bintree == the type of binary trees. This is canonically a countType + +
    • +
  • BinLeaf == the leaf for binary trees + +
    • +
  • BinNode left right == the binary tree with subtrees left right + +
    • +
+ +
+ +Binary trees of size n: + +
+ +
    +
  • size_tree t == the number of node of the tree t + +
    • +
  • enum_bintreesz n == the list of a trees of size n + +
    • +
  • bintreesz n == the Sigma type for trees of size n. This is + canonically a finType with enumeration enum_bintreesz n + +
    • +
  • Catalan_bin n == Catalan number defined using the binary tree recursion: + C_0 = 1 and C_{n+1} = sum__(i+j = n) C_i C_j. We show that they + count the number of binary trees of size n. The binomial formula + will be proved using Dyck word. + +
    • +
+ +
+ +Left branch surgery: + +
+ +
    +
  • left_branch t == the sequence of the right subtrees of the nodes + starting for the root of t and going only left + +
    • +
  • from_left s == the tree whose left branch is s + +
    • +
  • cat_left t1 t2 == the tree whose left branch is the concatenation of + the left branches of t1 and t2 + +
    + + +
    • +
  • rightcomb n == the right comb binary tree of size n as a bintree + +
    • +
  • leftcomb n == the left comb binary tree of size n as a bintree + +
    • +
  • flip t == the left/right mirror of t + +
    • +
  • rightcombsz n == the right comb binary tree of size n as a bintreesz n + +
    • +
  • leftcombsz n == the left comb binary tree of size n as a bintreesz n + +
    • +
  • flipsz t == the left/right mirror of t as a bintreesz n + +
    • +
+ +
+ +Rotations and Tamari order: + +
+ +
    +
  • rotations t == the list of right rotations of the tree t + +
    • +
  • t1 <=T t2 == t1 is smaller than t2 in the Tamari order + +
    • +
+ +
+ +Tamari bracketing vectors: + +
+ +
    +
  • right_sizes t == the sequence in infix order reading of the node of t + of the the sizes of the right subtrees + +
    • +
  • from_vct vct == recovers the tree t from its right sizes vector. + +
    • +
  • v \is a TamariVector == v is the right sizes vector of some tree t. + The characterization of a Tamari vector of size n is that + for all i < n then v_i + 1 < n and + for all j such that i < j <= v_i + i then v_j + j <= v_i + i. + +
    • +
+The function right_sizes and from_vct are two inverse bijections from +binary trees to Tamari vectors as stated in theorems right_sizesK, +right_sizesP and from_vctK. + +
+ +
    +
  • vctleq v1 v2 = v1 <=V v2 == v1 and v2 are of the same length + and v1 is smaller than v2 componentwise (ie. for all i then + v1_i <= v2_i + +
    • +
  • vctmin v1 v2 == the componentwise min of v1 and v2 + +
    • +
+ +
+ +Tamari Lattice: + +
+ +
    +
  • t1 \/T t2 == the meet of t1 and t2 in the Tamari lattice. + +
    • +
  • t1 /\T t2 == the join of t1 and t2 in the Tamari lattice. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+Require Import tools combclass ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import Order.Theory.
+ +
+Reserved Notation "x '<=T' y" (at level 70, y at next level).
+Reserved Notation "x '<T' y" (at level 70, y at next level).
+Reserved Notation "x '/\T' y" (at level 70, y at next level).
+Reserved Notation "x '\/T' y" (at level 70, y at next level).
+ +
+
+ +
+

Inductive type for binary trees

+ +
+
+Inductive bintree : Set :=
+  | BinLeaf : bintree
+  | BinNode : bintree -> bintree -> bintree.
+ +
+Fixpoint eq_bintree (b1 b2 : bintree) : bool :=
+  match b1, b2 with
+  | BinLeaf, BinLeaf => true
+  | BinNode l1 r1, BinNode l2 r2 => eq_bintree l1 l2 && eq_bintree r1 r2
+  | _, _ => false
+  end.
+ +
+Lemma eq_bintreeP : Equality.axiom eq_bintree.
+ +
+Fixpoint tree_encode (t : bintree) : GenTree.tree unit :=
+  match t with
+  | BinLeaf => GenTree.Leaf tt
+  | BinNode tl tr => GenTree.Node 2 [:: tree_encode tl; tree_encode tr]
+  end.
+Fixpoint tree_decode (gt : GenTree.tree unit) :=
+  match gt with
+  | GenTree.Leaf tt => BinLeaf
+  | GenTree.Node 2 [:: tl; tr] => BinNode (tree_decode tl) (tree_decode tr)
+  | _ => BinLeaf
+ +
+unused case +
+
+  end.
+ +
+Lemma tree_encodeK : cancel tree_encode tree_decode.
+ +
+ +
+Fixpoint size_tree t :=
+  match t with
+  | BinLeaf => 0
+  | BinNode l r => 1 + (size_tree l) + (size_tree r)
+  end.
+ +
+Lemma LeafP t : reflect (t = BinLeaf) (size_tree t == 0).
+ Lemma size_tree_eq0 t : (size_tree t == 0) = (t == BinLeaf).
+ +
+Section Size.
+ +
+Variable (n : nat).
+ +
+Structure bintreesz : predArgType :=
+  BinTreeSZ {trval :> bintree; _ : size_tree trval == n}.
+ +
+Lemma bintreeszP (t : bintreesz) : size_tree t = n.
+ +
+End Size.
+ +
+
+ +
+

Catalan numbers

+ +
+ + The intent of the two following recursive definition is the recursion of lemmas + CatalanS and enum_bintreeszS: +
+Fixpoint Catalan n :=
+  if n is n'.+1 then
+    sumn [Catalan i * Catalan (n - i) | i <- iota 0 n.+1]
+  else 1. +
+ +however i and n - i are not structurally smaller than n.+1 so the +definition is refused by coq as not well founded. So we write the following +functions which returns a cache containing the list of the results of +Catalan i and enum_bintreesz i for i = 0 ... n. Otherwise said, to define +the enum_bintreesz function we need a strong nat induction where Coq only +allows simple nat induction. +
+ + A seq of size n.+1 whose i-th element is Catalan i +
+
+Fixpoint Catalan_bin_leq n :=
+  if n is n.+1 then
+    let cr := Catalan_bin_leq n in
+    let new := sumn [seq nth 0 cr i * nth 0 cr (n - i) | i <- iota 0 n.+1]
+    in rcons cr new
+  else [:: 1].
+Definition Catalan_bin n := last 0 (Catalan_bin_leq n).
+ +
+
+ +
+A seq of size n.+1 whose i-th element contains the list of all binary trees + of size i +
+
+Fixpoint enum_bintreesz_leq n :=
+  if n is n'.+1 then
+    let rec := enum_bintreesz_leq n' in
+    let listn :=
+        flatten [seq [seq BinNode tl tr |
+                      tl <- nth [::] rec i,
+                      tr <- nth [::] rec (n' - i)] | i <- iota 0 n'.+1]
+    in rcons rec listn
+  else [:: [:: BinLeaf]].
+Definition enum_bintreesz n := last [::] (enum_bintreesz_leq n).
+ +
+
+ +
+Some tests : +
+
+Goal [seq Catalan_bin n | n <- iota 0 10] =
+     [:: 1; 1; 2; 5; 14; 42; 132; 429; 1430; 4862].
+ Goal [seq size (enum_bintreesz n) | n <- iota 0 10] =
+     [:: 1; 1; 2; 5; 14; 42; 132; 429; 1430; 4862].
+ +
+Lemma size_Catalan_bin_leq n : size (Catalan_bin_leq n) = n.+1.
+ +
+Lemma size_enum_bintreesz n : size (enum_bintreesz_leq n) = n.+1.
+ +
+Lemma size_enum_bintreeszE n : size (enum_bintreesz n) = Catalan_bin n.
+ +
+Lemma Catalan_bin_leqE i m :
+  i <= m -> nth 0 (Catalan_bin_leq m) i = Catalan_bin i.
+ +
+Lemma enum_bintreesz_leq_leqE i n m :
+  i <= n <= m ->
+  nth [::] (enum_bintreesz_leq n) i = nth [::] (enum_bintreesz_leq m) i.
+ +
+Lemma enum_bintreesz_leqE n m :
+  n <= m -> enum_bintreesz n = nth [::] (enum_bintreesz_leq m) n.
+ +
+Lemma Catalan_bin0 : Catalan_bin 0 = 1.
+ +
+Lemma enum_bintreesz0 : enum_bintreesz 0 = [:: BinLeaf].
+ +
+Lemma Catalan_binS n :
+  Catalan_bin n.+1 = \sum_(0 <= i < n.+1) Catalan_bin i * Catalan_bin (n - i).
+ +
+Lemma enum_bintreeszE n :
+  enum_bintreesz n.+1 =
+  flatten [seq [seq BinNode tl tr |
+                tl <- enum_bintreesz i,
+                tr <- enum_bintreesz (n - i)] | i <- iota 0 n.+1].
+ +
+Lemma size_mem_enum_bintreeszP n t :
+  t \in enum_bintreesz n -> size_tree t = n.
+ +
+Lemma enum_bintreeszP n :
+  all (fun t => size_tree t == n) (enum_bintreesz n).
+ +
+Lemma enum_bintreesz_uniq n : uniq (enum_bintreesz n).
+ +
+Lemma mem_enum_bintreesz n t :
+  size_tree t == n -> t \in enum_bintreesz n.
+ +
+Lemma enum_bintreesz_countE n t :
+  size_tree t == n -> count_mem t (enum_bintreesz n) = 1.
+ +
+ +
+Theorem card_bintreesz n : #|bintreesz n| = Catalan_bin n.
+ +
+
+ +
+

Left branch surgery in binary trees

+ +
+
+Fixpoint left_branch t :=
+  if t is BinNode l r then r :: left_branch l else [::].
+Fixpoint from_left l :=
+  if l is l0 :: l then BinNode (from_left l) l0 else BinLeaf.
+Definition cat_left t1 t2 := from_left (left_branch t1 ++ left_branch t2).
+ +
+Lemma left_branchK : cancel left_branch from_left.
+ Lemma from_leftK : cancel from_left left_branch.
+ +
+Lemma cat_left0t t : cat_left BinLeaf t = t.
+ Lemma cat_leftt0 t : cat_left t BinLeaf = t.
+ Lemma cat_left_Node l r :
+  cat_left (BinNode BinLeaf r) l = BinNode l r.
+ Lemma cat_leftA t u v :
+  cat_left (cat_left t u) v = cat_left t (cat_left u v).
+ Lemma size_from_left s :
+  size_tree (from_left s) = sumn [seq (size_tree t).+1 | t <- s].
+ +
+Lemma size_cat_left t u :
+  size_tree (cat_left t u) = size_tree t + size_tree u.
+ +
+Lemma from_left_cat s1 s2 :
+  from_left (s1 ++ s2) = cat_left (from_left s1) (from_left s2).
+ +
+Lemma size_left_branch t :
+  all (fun l => size_tree l < size_tree t) (left_branch t).
+ +
+
+ +
+

Various particular binary trees

+ +
+
+Fixpoint rightcomb n :=
+  if n is n'.+1 then
+    BinNode BinLeaf (rightcomb n')
+  else BinLeaf.
+Fixpoint leftcomb n :=
+  if n is n'.+1 then
+    BinNode (leftcomb n') BinLeaf
+  else BinLeaf.
+Fixpoint flip t :=
+  if t is BinNode l r then
+    BinNode (flip r) (flip l)
+  else BinLeaf.
+ +
+Section SizeN.
+ +
+Variable (n : nat).
+ +
+Fact size_rightcomb : size_tree (rightcomb n) == n.
+ Canonical rightcombsz := BinTreeSZ size_rightcomb.
+ +
+Lemma size_leftcomb : size_tree (leftcomb n) == n.
+ Canonical leftcombsz := BinTreeSZ size_leftcomb.
+ +
+Lemma size_flip t : size_tree (flip t) = size_tree t.
+Lemma flipK : involutive flip.
+ Lemma flip_rightcomb : flip (rightcomb n) = leftcomb n.
+ Lemma flip_leftcomb : flip (leftcomb n) = rightcomb n.
+ +
+Fact flipsz_subproof (t : bintreesz n) : size_tree (flip t) == n.
+ Canonical flipsz t := BinTreeSZ (flipsz_subproof t).
+ +
+Lemma flipszK : involutive flipsz.
+ Lemma flipsz_rightcomb : flipsz rightcombsz = leftcombsz.
+ Lemma flipsz_leftcomb : flipsz leftcombsz = rightcombsz.
+ +
+End SizeN.
+ +
+
+ +
+

Tamari's lattice

+

Rotations in binary trees

+ +
+
+Fixpoint rotations t :=
+  if t is BinNode l r then
+    let rec := [seq BinNode lrot r | lrot <- rotations l] ++
+               [seq BinNode l rrot | rrot <- rotations r]
+    in if l is BinNode ll lr then
+         BinNode ll (BinNode lr r) :: rec
+       else rec
+  else [::].
+ +
+Lemma rotations_left_sub l1 l2 r :
+  l1 \in rotations l2 -> BinNode l1 r \in rotations (BinNode l2 r).
+ +
+Lemma rotations_right_sub l r1 r2 :
+  r1 \in rotations r2 -> BinNode l r1 \in rotations (BinNode l r2).
+ +
+Lemma rotationP t1 t2 :
+  reflect
+    (exists a b c,
+        [\/ [/\ t1 = BinNode (BinNode a b) c & t2 = BinNode a (BinNode b c)],
+            [/\ t1 = BinNode a c, t2 = BinNode b c & b \in rotations a] |
+            [/\ t1 = BinNode a b, t2 = BinNode a c & c \in rotations b]])
+    (t2 \in rotations t1).
+ +
+Lemma size_rotations t t' :
+  t' \in rotations t -> size_tree t' = size_tree t.
+ +
+Lemma rotations_flip_impl t1 t2:
+  (t1 \in rotations t2) -> (flip t2 \in rotations (flip t1)).
+ +
+Lemma rotations_flip t1 t2:
+  (t1 \in rotations t2) = (flip t2 \in rotations (flip t1)).
+ +
+Lemma catleft_rotations t t1 t2 :
+  t1 \in rotations t2 -> (cat_left t t1) \in rotations (cat_left t t2).
+ +
+Lemma rightcomb_rotations n : rotations (rightcomb n) = [::].
+ Lemma rightcomb_rotationsE t :
+  (rotations t == [::]) = (t == rightcomb (size_tree t)).
+Lemma leftcomb_rotations t : leftcomb (size_tree t) \notin rotations t.
+ +
+
+ +
+

Tamari vectors

+ +
+
+Fixpoint right_sizes t :=
+  if t is BinNode l r then
+        right_sizes l ++ size_tree r :: right_sizes r
+  else [::].
+Fixpoint is_Tamari v :=
+  if v is v0 :: v' then
+    [&& v0 <= size v',
+     is_Tamari v' &
+     all (fun i => nth 0 v' i + i < v0) (iota 0 v0)]
+  else true.
+Definition TamariVector := [qualify a v : seq nat | is_Tamari v].
+ +
+Lemma sumn_right_sizes_gt t t' :
+  t' \in rotations t -> sumn (right_sizes t) < sumn (right_sizes t').
+ +
+Lemma rotations_neq t t' : t' \in rotations t -> t != t'.
+ +
+Lemma size_right_sizes t : size (right_sizes t) = size_tree t.
+ +
+Lemma right_sizes_left_comb n :
+  right_sizes (leftcomb n) = nseq n 0.
+ +
+Lemma right_sizes_from_left s :
+  right_sizes (from_left s) =
+  flatten [seq size_tree t :: right_sizes t | t <- rev s].
+ +
+Lemma right_sizes_cat_left t1 t2 :
+  right_sizes (cat_left t1 t2) = right_sizes t2 ++ right_sizes t1.
+ +
+Lemma Tamari_consP v0 v :
+  reflect
+    [/\ v0 <= size v,
+     v \is a TamariVector &
+     forall i, i < v0 -> nth 0 v i + i < v0]
+    (v0 :: v \is a TamariVector).
+ +
+Lemma TamariP v :
+  reflect
+    ((forall i, i < size v -> nth 0 v i + i < size v) /\
+     (forall i j, i < j <= nth 0 v i + i -> nth 0 v j + j <= nth 0 v i + i))
+    (v \is a TamariVector).
+ +
+Lemma Tamari_drop n v :
+  v \is a TamariVector -> drop n v \is a TamariVector.
+ +
+Lemma Tamari_catr u v :
+  u ++ v \is a TamariVector -> v \is a TamariVector.
+ +
+Lemma Tamari_take v0 v :
+  v0 :: v \is a TamariVector -> take v0 v \is a TamariVector.
+ +
+Lemma Tamari_cat v1 v2 :
+  v1 \is a TamariVector ->
+  v2 \is a TamariVector ->
+  v1 ++ v2 \is a TamariVector.
+ +
+Lemma cons_TamariP v : v \is a TamariVector -> size v :: v \is a TamariVector.
+ +
+Theorem right_sizesP t : right_sizes t \is a TamariVector.
+ +
+#[local] Hint Resolve right_sizesP : core.
+ +
+
+ +
+

Bijection with Tamari vectors

+ +
+
+Fixpoint from_vct_rec fuel lft vct :=
+  if fuel is fuel.+1 then
+    if vct is v0 :: vct' then
+      from_vct_rec fuel
+                   (BinNode lft (from_vct_rec fuel BinLeaf (take v0 vct')))
+                   (drop v0 vct')
+    else lft
+  else BinLeaf.
+Definition from_vct_acc lft vct := from_vct_rec (size vct).+1 lft vct.
+Definition from_vct := from_vct_acc BinLeaf.
+ +
+Section Tests.
+ +
+Goal [seq right_sizes t | t <- enum_bintreesz 3] =
+[:: [:: 2; 1; 0]; [:: 2; 0; 0]; [:: 0; 1; 0]; [:: 1; 0; 0]; [:: 0; 0; 0]].
+ +
+Goal [seq right_sizes t | t <- enum_bintreesz 4] =
+[:: [:: 3; 2; 1; 0]; [:: 3; 2; 0; 0]; [:: 3; 0; 1; 0]; [:: 3; 1; 0; 0];
+    [:: 3; 0; 0; 0]; [:: 0; 2; 1; 0]; [:: 0; 2; 0; 0]; [:: 1; 0; 1; 0];
+    [:: 0; 0; 1; 0]; [:: 2; 1; 0; 0]; [:: 2; 0; 0; 0]; [:: 0; 1; 0; 0];
+    [:: 1; 0; 0; 0]; [:: 0; 0; 0; 0] ].
+ +
+Let bla := Eval hnf in nth BinLeaf (enum_bintreesz 5) 21.
+Goal right_sizes bla = [:: 0; 0; 2; 1; 0].
+ +
+Goal [seq right_sizes rot | rot <- rotations bla] =
+[:: [:: 0; 3; 2; 1; 0]; [:: 1; 0; 2; 1; 0]].
+ +
+Goal all
+     (fun i => all (fun t => t == from_vct (right_sizes t)) (enum_bintreesz i))
+     (iota 0 7).
+ +
+Goal all
+     (fun i => all (fun t => right_sizes t \is a TamariVector) (enum_bintreesz i))
+     (iota 0 7).
+ +
+Goal [:: 1; 1; 0] \isn't a TamariVector.
+ Goal [:: 2; 0; 1; 0] \isn't a TamariVector.
+ +
+Goal [:: 2; 0; 0; 1; 0] \is a TamariVector.
+ +
+End Tests.
+ +
+Lemma from_vct_acc_nil lft : from_vct_acc lft [::] = lft.
+ +
+Lemma from_vct_fuelE fuel lft vct :
+  size vct < fuel -> from_vct_rec fuel lft vct = from_vct_acc lft vct.
+ +
+Lemma from_vct_accE lft v0 vct :
+  from_vct_acc lft (v0 :: vct) =
+  from_vct_acc (BinNode lft (from_vct_acc BinLeaf (take v0 vct)))
+               (drop v0 vct).
+ +
+Lemma from_vct_cat_leftE lft vct :
+  from_vct_acc lft vct = cat_left (from_vct_acc BinLeaf vct) lft.
+ +
+Lemma size_from_vct_acc s :
+  size_tree (from_vct_acc BinLeaf s) = size s.
+ +
+Lemma size_from_vct s :
+  size_tree (from_vct s) = size s.
+ +
+Theorem right_sizesK : cancel right_sizes from_vct.
+ +
+Theorem from_vctK : {in TamariVector, cancel from_vct right_sizes}.
+ +
+Lemma from_vct_cat u v t :
+  u \is a TamariVector ->
+  from_vct_acc t (u ++ v) = from_vct_acc (from_vct_acc t u) v.
+ +
+Lemma from_vct0 n : from_vct (nseq n 0) = leftcomb n.
+ +
+
+ +
+

Comparison of Tamari vectors

+ +
+
+ +
+Definition vctleq v1 v2 :=
+  (size v1 == size v2) && (all (fun p => p.1 <= p.2) (zip v1 v2)).
+Definition vctmin v1 v2 := [seq minn p.1 p.2 | p <- zip v1 v2].
+ +
+#[local] Notation "x '<=V' y" := (vctleq x y) (at level 70, y at next level).
+ +
+Section TestsComp.
+ +
+Goal all (fun t => all
+                     (vctleq (right_sizes t))
+                     [seq right_sizes rot | rot <- rotations t])
+     (enum_bintreesz 6).
+ +
+End TestsComp.
+ +
+Lemma vctleqP v1 v2 :
+  reflect (size v1 = size v2 /\ forall i, nth 0 v1 i <= nth 0 v2 i)
+          (v1 <=V v2).
+ +
+Lemma vctleq_refl : reflexive vctleq.
+ Lemma vctleq_trans : transitive vctleq.
+Lemma vctleq_anti : antisymmetric vctleq.
+ +
+Lemma vctleq_sumn_right_sizes t1 t2 :
+  right_sizes t1 <=V right_sizes t2 ->
+  sumn (right_sizes t1) <= sumn (right_sizes t2) ?= iff (t1 == t2).
+ +
+Lemma all_leqzip_refl l : all (fun p => p.1 <= p.2) (zip l l).
+ +
+Lemma size_vctmin v1 v2 : size (vctmin v1 v2) = minn (size v1) (size v2).
+ Lemma nth_vctmin v1 v2 i :
+  nth 0 (vctmin v1 v2) i = minn (nth 0 v1 i) (nth 0 v2 i).
+ +
+Lemma vctminC: commutative vctmin.
+ +
+Lemma vctminPr v1 v2 :
+  size v1 = size v2 -> vctmin v1 v2 <=V v2.
+ +
+Lemma vctminPl v1 v2 :
+  size v1 = size v2 -> vctmin v1 v2 <=V v1.
+ +
+Lemma vctminP v1 v2 v :
+  v <=V v1 -> v <=V v2 -> v <=V (vctmin v1 v2).
+ +
+Lemma vctmin_Tamari v1 v2 :
+  size v1 = size v2 ->
+  v1 \is a TamariVector ->
+  v2 \is a TamariVector ->
+  vctmin v1 v2 \is a TamariVector.
+ +
+
+ +
+

Correspondence between rotation on binary trees and Tamari vectors

+ +
+
+ +
+Lemma rotations_vctleq_impl t1 t2 :
+  t1 \in rotations t2 -> right_sizes t2 <=V right_sizes t1.
+ +
+Lemma Tamari_add_head v0 v :
+  v0 :: v \is a TamariVector ->
+  v0 < size v ->
+  (nth 0 v v0 + v0).+1 :: v \is a TamariVector.
+ +
+Lemma rotations_add_head v0 v t :
+  v0 :: v \is a TamariVector ->
+  (nth 0 v v0 + v0).+1 :: v \is a TamariVector ->
+  from_vct_acc t ((nth 0 v v0 + v0).+1 :: v) \in
+    rotations (from_vct_acc t (v0 :: v)).
+ +
+Lemma Tamari_add_min u v0 v w0 w :
+  u ++ w0 :: w \is a TamariVector ->
+  u ++ v0 :: v <=V u ++ w0 :: w ->
+  v0 < w0 ->
+  nth 0 v v0 + v0 < w0.
+ +
+Lemma Tamari_add_bounded u v0 v w0 w :
+  u ++ v0 :: v \is a TamariVector ->
+  u ++ w0 :: w \is a TamariVector ->
+  u ++ v0 :: v <=V u ++ w0 :: w ->
+  v0 < w0 ->
+  u ++ (nth 0 v v0 + v0).+1 :: v \is a TamariVector.
+ +
+Lemma rotations_add u v0 v :
+  u ++ v0 :: v \is a TamariVector ->
+  u ++ (nth 0 v v0 + v0).+1 :: v \is a TamariVector ->
+  from_vct (u ++ (nth 0 v v0 + v0).+1 :: v)
+           \in rotations (from_vct (u ++ v0 :: v)).
+ +
+Lemma rotations_add_bounded u v0 v w0 w :
+  u ++ v0 :: v \is a TamariVector ->
+  u ++ w0 :: w \is a TamariVector ->
+  u ++ v0 :: v <=V u ++ w0 :: w ->
+  v0 < w0 ->
+  from_vct (u ++ (nth 0 v v0 + v0).+1 :: v)
+           \in rotations (from_vct (u ++ v0 :: v)).
+ +
+Lemma vctleq_rotation t1 t2 :
+  right_sizes t1 <=V right_sizes t2 ->
+  t1 != t2 ->
+  exists t, t \in rotations t1 /\ right_sizes t <=V right_sizes t2.
+ +
+Theorem rotations_right_sizesP t1 t2 :
+  reflect
+    (exists u v0 v,
+        right_sizes t1 = u ++ v0 :: v /\
+        right_sizes t2 = u ++ (nth 0 v v0 + v0).+1 :: v)
+    (t2 \in rotations t1).
+ +
+Lemma vct_succ u v0 v w :
+  w \is a TamariVector ->
+  u ++ v0 :: v <=V w ->
+  w <=V u ++ (nth 0 v v0 + v0).+1 :: v ->
+  w = u ++ v0 :: v \/ w = u ++ (nth 0 v v0 + v0).+1 :: v.
+ +
+Fact Tamari_display : Order.disp_t.
+Notation "x <=T y" := (@Order.le Tamari_display _ x y).
+Notation "x <T y" := (@Order.lt Tamari_display _ x y).
+Notation "x /\T y" := (@Order.meet Tamari_display _ x y).
+Notation "x \/T y" := (@Order.join Tamari_display _ x y).
+ +
+
+ +
+

Definition of Tamari order

+ +
+
+Module TamariLattice.
+Section TamariLattice.
+ +
+Variable n : nat.
+Implicit Type t : bintreesz n.
+ +
+Definition Tamari := connect (fun t1 t2 : bintreesz n => grel rotations t1 t2).
+ +
+Lemma Tamari_sumn_right_sizes t1 t2 :
+  Tamari t1 t2 ->
+  sumn (right_sizes t1) <= sumn (right_sizes t2) ?= iff (t1 == t2).
+ +
+Fact Tamari_refl : reflexive Tamari.
+ Fact Tamari_trans : transitive Tamari.
+ Fact Tamari_anti : antisymmetric Tamari.
+ +
+#[export] HB.instance Definition _ :=
+  Order.Le_isPOrder.Build Tamari_display (bintreesz n)
+    Tamari_refl Tamari_anti Tamari_trans.
+ +
+Lemma TamariE t1 t2 :
+  (t1 <=T t2) = connect (fun t t' => grel rotations t t') t1 t2.
+ +
+Lemma rotations_Tamari t t' : trval t' \in rotations t -> t <T t'.
+ +
+Lemma Tamari_flip t1 t2 : (flipsz t2 <=T flipsz t1) = (t1 <=T t2).
+ +
+Theorem Tamari_vctleq t1 t2 :
+  (right_sizes t1 <=V right_sizes t2) = (t1 <=T t2).
+ +
+Lemma Tmeet_proof t1 t2 :
+  size_tree (from_vct (vctmin (right_sizes t1) (right_sizes t2))) == n.
+Definition Tmeet t1 t2 := BinTreeSZ (Tmeet_proof t1 t2).
+Definition Tjoin t1 t2 := flipsz (Tmeet (flipsz t1) (flipsz t2)).
+ +
+Lemma TmeetC t1 t2 : Tmeet t1 t2 = Tmeet t2 t1.
+ +
+Lemma TmeetPr t1 t2 : Tmeet t1 t2 <=T t2.
+ +
+Fact TmeetP t t1 t2 : (t <=T Tmeet t1 t2) = (t <=T t1) && (t <=T t2).
+Fact TjoinP t1 t2 t : (Tjoin t1 t2 <=T t) = (t1 <=T t) && (t2 <=T t).
+ #[export] HB.instance Definition _ :=
+  Order.POrder_MeetJoin_isLattice.Build Tamari_display
+    (bintreesz n) TmeetP TjoinP.
+ +
+Lemma flipsz_meet t1 t2 : flipsz (t1 \/T t2) = (flipsz t1 /\T flipsz t2).
+ Lemma flipsz_join t1 t2 : flipsz (t1 /\T t2) = (flipsz t1 \/T flipsz t2).
+ +
+Lemma right_sizes_meet t1 t2 :
+  right_sizes (t1 /\T t2) = vctmin (right_sizes t1) (right_sizes t2).
+ +
+Fact leftcomb_bottom t : leftcombsz n <=T t.
+Fact rightcomb_top t : t <=T rightcombsz n.
+ #[export] HB.instance Definition _ :=
+  Order.hasBottom.Build Tamari_display (bintreesz n) leftcomb_bottom.
+#[export] HB.instance Definition _ :=
+  Order.hasTop.Build Tamari_display (bintreesz n) rightcomb_top.
+ +
+Lemma botETamari : \bot%O = leftcombsz n.
+ Lemma topETamari : \top%O = rightcombsz n.
+ +
+End TamariLattice.
+ +
+Module Exports.
+ +
+Definition TamariE := TamariE.
+Definition rotations_Tamari := rotations_Tamari.
+Definition Tamari_flip := Tamari_flip.
+Definition Tamari_vctleq := Tamari_vctleq.
+Definition flipsz_meet := flipsz_meet.
+Definition flipsz_join := flipsz_join.
+Definition right_sizes_meet := right_sizes_meet.
+Definition botETamari := botETamari.
+Definition topETamari := topETamari.
+ +
+End Exports.
+End TamariLattice.
+ +
+Section TamariCover.
+ +
+Variable n : nat.
+Implicit Type t : bintreesz n.
+ +
+Lemma Tamari_succ t1 t2 t :
+  trval t2 \in rotations t1 -> t1 <=T t -> t <=T t2 -> t = t1 \/ t = t2.
+ +
+Lemma covers_Tamari t1 t2 : (trval t2 \in rotations t1) = (covers t1 t2).
+ +
+End TamariCover.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.composition.html b/combi/1.1.0/Combi.Combi.composition.html new file mode 100644 index 00000000..5d943e45 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.composition.html @@ -0,0 +1,515 @@ + + + + + +Combi.Combi.composition: Integer Composition + + + + +
+ + + +
+ +

Library Combi.Combi.composition: Integer Composition

+ +
+
+ +
+ +
+
+
+ +
+

Integer Compositions.

+ + +
+ +Integer Composition are stored as seq nat. We define the following: + +
+ +
    +
  • is_comp s == s is a composition, ie. s doesn't contains any 0 + +
    • +
  • is_comp_of_n sm s == s is a composition of sum sm + +
    • +
  • intcomp == a sigma type for is_comp + +
    + + +
    • +
  • intcompn sm == a sigma type for the predicate is_comp_of_n sm. + this is canonically a subFinType + +
    + + +
    • +
  • rowcomp n == the trivial composition + +
    • +
  • rowcompn n == the trivial composition as a intcompn n + +
    + + +
    • +
  • colcomp n == the composition [:: 1; 1; ...] + +
    • +
  • colcompn n == the composition [:: 1; 1; ...] as a intcompn n + +
    + + +
    • +
  • rev_intcompn c == the reverse composition inside intcompn n. + +
    • +
+ +
+ +Bijection with subsets: Consistently, with permutation starting at 0, +descents are starting at zero and therefore of type 'I_n.-1. +In the following we assume. + +
+ +
    +
  • partsums s == sorted sequence of partial sums (excluding the trivial + and full sum) + +
    • +
  • descset c == the descent set of c : intcompn n + +
    • +
  • from_descset d == the composition (of type c : intcompn n) whose + descent set is d : {set 'I_n.-1}. + +
    • +
+ +
+ +Compositions and partitions: + +
+ +
    +
  • partn_of_compn n c == the partition in 'P_n obtained by sorting + c : intcompn n + +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+Require Import tools combclass sorted partition subseq ordtype.
+ +
+Set Implicit Arguments.
+ +
+ +
+ +
+Definition bottom := f' \bot%O.
+Lemma isobottom x : (bottom <= x)%O.
+ +
+ +
+ +
+ +
+Definition top := f' \top%O.
+Lemma isotop x : (x <= top)%O.
+ +
+ +
+
+ +
+

Definitions and basic properties

+ +
+
+Section Defs.
+ +
+Implicit Type s : seq nat.
+ +
+Definition is_comp s := 0 \notin s.
+ +
+Lemma is_compP s : reflect (forall i, i \in s -> i != 0) (is_comp s).
+ +
+Lemma is_comp1 i : is_comp [:: i] = (i != 0).
+ +
+Lemma is_comp_cons i s : is_comp (i :: s) = (i != 0) && (is_comp s).
+ +
+Lemma is_comp_rcons s sn : is_comp (rcons s sn) = (sn != 0) && (is_comp s).
+ +
+Lemma is_comp_cat s1 s2 : is_comp (s1 ++ s2) = (is_comp s1) && (is_comp s2).
+ +
+Lemma part_is_comp s : is_part s -> is_comp s.
+ +
+
+ +
+Compositions and sumn +
+
+Lemma comp0 s : is_comp s -> sumn s = 0 -> s = [::].
+ +
+Lemma size_comp s : is_comp s -> size s <= sumn s.
+ +
+End Defs.
+ +
+
+ +
+

Sigma Types for Compositions

+ +
+
+Structure intcomp : Type := IntComp {cval :> seq nat; _ : is_comp cval}.
+ +
+Lemma intcompP (p : intcomp) : is_comp p.
+ +
+#[export] Hint Resolve intcompP : core.
+ +
+Fixpoint enum_compn_rec aux n : (seq (seq nat)) :=
+  if aux is aux'.+1 then
+    if n == 0 then [:: [::]]
+ else
+      flatten [seq [seq i :: p | p <- enum_compn_rec aux' (n - i) ] |
+               i <- iota 1 n]
+  else [:: [::]].
+Definition enum_compn n := enum_compn_rec n n.
+ +
+Definition is_comp_of_n sm := [pred s | (sumn s == sm) & is_comp s].
+ +
+Lemma enum_compn_rec_any aux1 aux2 n :
+  n <= aux1 -> n <= aux2 -> enum_compn_rec aux1 n = enum_compn_rec aux2 n.
+ +
+Lemma enum_compn_any aux n :
+  n <= aux -> enum_compn_rec aux n = enum_compn n.
+ +
+Lemma enum_compnE n :
+  n != 0 ->
+  enum_compn n = flatten [seq [seq i :: p | p <- enum_compn (n - i) ] |
+                          i <- iota 1 n].
+ +
+Lemma enum_compn_allP n : all (is_comp_of_n n) (enum_compn n).
+ +
+Lemma enum_compn_countE n :
+  forall s, is_comp_of_n n s -> count_mem s (enum_compn n) = 1.
+ +
+Lemma enum_compnP n s : (is_comp_of_n n s) = (s \in enum_compn n).
+ +
+Section CompOfn.
+ +
+Variable n : nat.
+ +
+Structure intcompn : Set :=
+  IntCompN {cnval :> seq nat; _ : is_comp_of_n n cnval}.
+ +
+Implicit Type (c : intcompn) (d : {set 'I_n.-1}).
+ +
+Lemma intcompnP c : is_comp c.
+ +
+Hint Resolve intcompnP : core.
+ +
+Definition intcomp_of_intcompn c := IntComp (intcompnP c).
+Coercion intcomp_of_intcompn : intcompn >-> intcomp.
+ +
+Lemma intcompn_sumn c : sumn c = n.
+ +
+Lemma enum_intcompnE : map val (enum {: intcompn}) = enum_compn n.
+ +
+Lemma card_intcompn : #|{: intcompn}| = 2 ^ n.-1.
+ +
+Lemma rev_intcompn_spec c : is_comp_of_n n (rev c).
+ Definition rev_intcompn c := IntCompN (rev_intcompn_spec c).
+Lemma rev_intcompnK : involutive rev_intcompn.
+ +
+
+ +
+

Bijection with subsets

+ +
+
+Definition partsums s := [seq sumn (take i s) | i <- iota 1 (size s).-1].
+Definition descset c : {set 'I_n.-1} :=
+  [set i in pmap insub [seq i.-1 | i <- partsums c]].
+ +
+Lemma size_partsums s : size (partsums s) = (size s).-1.
+ +
+Lemma partsums_cat s1 s2 :
+  s1 != [::] -> s2 != [::] ->
+  partsums (s1 ++ s2) =
+  partsums s1 ++ sumn s1 :: [seq sumn s1 + i | i <- partsums s2].
+ +
+Lemma partsums_cons i s :
+  s != [::] -> partsums (i :: s) = i :: [seq i + j | j <- partsums s].
+ +
+Lemma all_partsums c : all (fun i => 0 < i < n) (partsums c).
+ +
+Lemma from_descset_spec d :
+  is_comp_of_n n (if n is 0 then [::]
+                  else pairmap (fun a b => b - a) 0
+                               (rcons [seq (val i).+1 | i in d] n)).
+Definition from_descset d := IntCompN (from_descset_spec d).
+ +
+Lemma diff_nth_sumn_take s i m :
+  m <= size s -> i.+1 < m ->
+  nth 0 [seq sumn (take i0 s) | i0 <- iota 1 m] i.+1 -
+  nth 0 [seq sumn (take i0 s) | i0 <- iota 1 m] i = nth 0 s i.+1.
+ +
+Lemma sorted_ltn_partsums c : sorted ltn (partsums c).
+ +
+Lemma val_descset c : [seq (val i).+1 | i in descset c] = partsums c.
+ +
+Lemma card_descset c : #|descset c| = (size c).-1.
+ +
+Lemma descsetK : cancel descset from_descset.
+ +
+Lemma descset_inj : injective descset.
+ +
+Lemma descset_bij : bijective descset.
+ +
+Lemma from_descsetK : cancel from_descset descset.
+ +
+End CompOfn.
+ +
+Lemma intcompn0 (sh : intcompn 0) : sh = [::] :> seq nat.
+ +
+Lemma intcompn1 (sh : intcompn 1) : sh = [:: 1] :> seq nat.
+ +
+Lemma intcompn2 (sh : intcompn 2) :
+  sh = [:: 2] :> seq nat \/ sh = [:: 1; 1] :> seq nat.
+ +
+Definition intcompn_cast m n (eq_mn : m = n) p :=
+  let: erefl in _ = n := eq_mn return intcompn n in p.
+ +
+Lemma intcompn_castE m n (eq_mn : m = n) p :
+  val (intcompn_cast eq_mn p) = val p.
+ +
+Definition rowcomp d := if d is _.+1 then [:: d] else [::].
+Fact rowcompnP d : is_comp_of_n d (rowcomp d).
+ Canonical rowcompn d : intcompn d := IntCompN (rowcompnP d).
+ +
+Definition colcomp d := nseq d 1%N.
+Fact colcompnP d : is_comp_of_n d (colcomp d).
+ Canonical colcompn d : intcompn d := IntCompN (colcompnP d).
+ +
+Section DescSet.
+ +
+Variable (n : nat).
+Implicit Types (c : intcompn n) (d : {set 'I_n.-1}).
+ +
+Lemma mem_partsum_non0 u0 u i : i \in partsums (u0 :: u) -> u != [::].
+ +
+Lemma mem_partsum_gt x v0 v1 v :
+  v0 < x -> x \in partsums (v0 :: v1 :: v) -> x \in partsums (v0 + v1 :: v).
+ +
+Lemma subdescset_partsumP c1 c2 :
+  reflect {subset partsums c1 <= partsums c2} (descset c1 \subset descset c2).
+ +
+Lemma subseq_partsumE c1 c2 :
+  (subseq (partsums c1) (partsums c2)) = (descset c1 \subset descset c2).
+ +
+Lemma subdescsetP c1 c2 :
+  reflect (exists2 c : seq (seq nat), c1 = map sumn c :> seq nat &
+                                      c2 = flatten c :> seq nat)
+          (descset c1 \subset descset c2).
+ +
+End DescSet.
+ +
+Module RefinementOrder.
+Section RefinementOrder.
+Import Order.DefaultSetSubsetOrder.
+ +
+Variable (n : nat).
+Definition type := intcompn n.
+#[local] Notation "'CRef" := type.
+Implicit Types (c : 'CRef) (d : {set 'I_n.-1}).
+#[local] Notation SetIn := ({set ('I_n.-1 : finType)}).
+ +
+#[export] HB.instance Definition _ := SubType.copy 'CRef (intcompn n).
+#[export] HB.instance Definition _ := Finite.copy 'CRef (intcompn n).
+Fact compnref_display : Order.disp_t.
+#[export] HB.instance Definition _ : Order.isPOrder compnref_display 'CRef :=
+  Order.CanIsPartial compnref_display (@descsetK n).
+#[export] HB.instance Definition _ :=
+  isInhabitedType.Build 'CRef (rowcompn n).
+ +
+Lemma leEcompnref c1 c2 :
+  (c1 <= c2)%O = (descset c1 \subset descset c2).
+ +
+Lemma descset_mono :
+  {mono (@descset n) : A B / (A <= B :> 'CRef)%O >-> (A <= B :> SetIn)%O}.
+ +
+#[export] HB.instance Definition _ :=
+  Order.IsoLattice.Build compnref_display 'CRef
+    (@descsetK n) (@from_descsetK n) descset_mono.
+ +
+Lemma descset_meet c1 c2 :
+  descset (c1 `&` c2)%O = descset c1 :&: descset c2.
+ Lemma descset_join c1 c2 :
+  descset (c1 `|` c2)%O = descset c1 :|: descset c2.
+ +
+#[export] HB.instance Definition _ :=
+  Order.IsoDistrLattice.Build compnref_display 'CRef
+    (@descsetK n) (@from_descsetK n) descset_mono.
+ +
+Lemma descset_rowcompn : descset (rowcompn n) = set0.
+ Lemma descset_colcompn : descset (colcompn n) = setT.
+ +
+#[export] HB.instance Definition _ :=
+  IsoBottom.Build compnref_display 'CRef
+                 (@descsetK n) (@from_descsetK n) descset_mono.
+#[export] HB.instance Definition _ :=
+  IsoTop.Build compnref_display 'CRef
+                 (@descsetK n) (@from_descsetK n) descset_mono.
+ +
+Lemma topEcompnref : \top%O = colcompn n :> 'CRef.
+ +
+Lemma botEcompnref : \bot%O = rowcompn n :> 'CRef.
+ +
+Lemma compnref_rev c1 c2 :
+  (rev_intcompn c1 <= rev_intcompn c2 :> 'CRef)%O = (c1 <= c2)%O.
+ +
+End RefinementOrder.
+ +
+Module Exports.
+ +
+Notation intcompnref := type.
+Definition leEcompnref := leEcompnref.
+Definition descset_mono := descset_mono.
+Definition descset_meet := descset_meet.
+Definition descset_join := descset_join.
+Definition topEcompnref := topEcompnref.
+Definition botEcompnref := botEcompnref.
+Definition compnref_rev := compnref_rev.
+ +
+End Exports.
+End RefinementOrder.
+ +
+#[export] Hint Resolve intcompP intcompnP : core.
+ +
+Lemma part_of_comp_subproof n (c : intcompn n) :
+  is_part_of_n n (sort geq c).
+Canonical partn_of_compn n (c : intcompn n) :=
+  IntPartN (part_of_comp_subproof c).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.fibered_set.html b/combi/1.1.0/Combi.Combi.fibered_set.html new file mode 100644 index 00000000..e9fcd50c --- /dev/null +++ b/combi/1.1.0/Combi.Combi.fibered_set.html @@ -0,0 +1,141 @@ + + + + + +Combi.Combi.fibered_set: Bijection beetween fibered sets + + + + +
+ + + +
+ +

Library Combi.Combi.fibered_set: Bijection beetween fibered sets

+ +
+
+ +
+ +
+
+
+ +
+

Bijection beetween fibered sets

+ + +
+ +Given a type with equality I a fiberedset is a set fbset in an ihnabited +fintype T equiped with a function fbfun : fbset -> I. The preimage of some +i under fbfun is called the fiber of i. The goal of this file is to show +that if two fibered set S1 S2 have their fiber of i of the same +cardinality for all i, then there there exists a bijection fbbij from S1 +to S2 which sends the fiber of i to the fiber of i. That is fbbij +commute with the fbfun : forall x, fbfun2 (fbbij x) = fbfun1 x. + +
+ +
    +
  • fibered_set == record for fibered sets on a carrier finType + +
    • +
  • fiber S i == the i-fiber of S + +
    + + +
    • +
  • fbbij U V == a bijection from U to V provided the two fibered sets + U and V have their fiber of i of the same cardinality: + +
    + + Hypothesis HcardEq : forall i, #|fiber U i| = #|fiber V i|. + +
    • +
+ +
+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq fintype finset.
+ +
+Set Implicit Arguments.
+ +
+Section BijFiberedSet.
+ +
+Variable I : eqType.
+ +
+Record fibered_set := FiberedSet {
+                          carrier : finType;
+                          elt : carrier;
+                          fbset :> {set carrier};
+                          fbfun : carrier -> I }.
+ +
+Definition fiber (S : fibered_set) v := [set x | x in S & fbfun x == v].
+ +
+Lemma mem_fiber (S : fibered_set) x : x \in S -> x \in fiber S (fbfun x).
+ +
+Definition fbbij (U V : fibered_set) (x : carrier U) : carrier V :=
+  nth (elt V) (enum (fiber V (fbfun x))) (index x (enum (fiber U (fbfun x)))).
+ +
+Section Defs.
+ +
+Variable U V : fibered_set.
+Hypothesis HcardEq : forall i, #|fiber U i| = #|fiber V i|.
+ +
+Lemma fbbij_in_fiber x : x \in U -> fbbij V x \in fiber V (fbfun x).
+ +
+Lemma fbset_fbbij x : x \in U -> fbbij V x \in V.
+ +
+Lemma equi_fbbij x : x \in U -> fbfun (fbbij V x) = fbfun x.
+ +
+Lemma fbbijK : {in U, cancel (fbbij V) (fbbij U)}.
+ +
+End Defs.
+ +
+Lemma fbbijP (U V : fibered_set) :
+  (forall i, #|fiber U i| = #|fiber V i|) ->
+  [/\ {in U &, injective (fbbij V)},
+   [set fbbij V x | x in U] = V &
+   {in U, forall x, fbfun (fbbij V x) = fbfun x} ].
+ +
+End BijFiberedSet.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.multinomial.html b/combi/1.1.0/Combi.Combi.multinomial.html new file mode 100644 index 00000000..30329ac6 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.multinomial.html @@ -0,0 +1,118 @@ + + + + + +Combi.Combi.multinomial: Multinomial Coefficients + + + + +
+ + + +
+ +

Library Combi.Combi.multinomial: Multinomial Coefficients

+ +
+
+ +
+ +
+
+
+ +
+

Multinomial Coefficients

+ + +
+ +We define: +
    +
  • 'Cs == the multinomial coefficient associated to the sequence s. + +
    • +
+ +
+ +The main expression is Lemma multinomial_factd: + +
+ + 'C[s] = (sumn s)`! %/ \prod_(i <- s) i`!. + +
+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype fintype choice seq.
+From mathcomp Require Import bigop div binomial.
+ +
+Require Import tools.
+ +
+Set Implicit Arguments.
+ +
+Implicit Type (i a b : nat) (s t : seq nat).
+ +
+Fixpoint multinomial_rec s :=
+  if s is i :: s' then 'C(sumn s, i) * (multinomial_rec s') else 1.
+Arguments multinomial_rec : simpl nomatch.
+Definition multinomial := multinomial_rec.
+Arguments multinomial : simpl never.
+Notation "''C' [ s ]" := (multinomial s)
+  (at level 0, format "''C' [ s ]") : nat_scope.
+ +
+Lemma multinomial0 : 'C[[::]] = 1.
+ Lemma multinomial1 i : 'C[[:: i]] = 1.
+ Lemma multinomialE i s : 'C[i :: s] = 'C(i + sumn s, i) * 'C[s].
+ Lemma multinomial2 a b : 'C[[:: a; b]] = 'C(a + b, a).
+ +
+Lemma multinomial_fact s : 'C[s] * \prod_(i <- s) i`! = (sumn s)`!.
+ +
+Lemma multinomial_nseq n a : 'C[nseq n a] * (a`! ^ n) = (a * n)`!.
+ +
+Lemma multinomial_nseq1 n : 'C[nseq n 1] = n`!.
+ +
+Lemma dvdn_prodfact s : \prod_(i <- s) i`! %| (sumn s)`! .
+ +
+Lemma multinomial_factd s : 'C[s] = (sumn s)`! %/ \prod_(i <- s) i`!.
+ +
+Lemma multinomial_cat s t :
+  'C[s ++ t] = 'C(sumn s + sumn t, sumn s) * 'C[s] * 'C[t].
+ +
+Lemma perm_multinomial s t : perm_eq s t -> 'C[s] = 'C[t].
+ +
+Lemma multinomial_filter_neq0 s : 'C[[seq i <- s | i != 0]] = 'C[s].
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.ordtree.html b/combi/1.1.0/Combi.Combi.ordtree.html new file mode 100644 index 00000000..5f251133 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.ordtree.html @@ -0,0 +1,270 @@ + + + + + +Combi.Combi.ordtree + + + + +
+ + + +
+ +

Library Combi.Combi.ordtree

+ +
+
+ +
+

Combi.Combi.ordree : Ordered Trees

+ +
+
+
+ +
+

Ordered Trees

+ + +
+ +An ordered tree is a rooted tree such that each node has a possibly empty +list of child that are ordered trees. + +
+ +Basic definitions: + +
+ +
    +
  • ordtree == the type of ordered trees. This is canonically a countType + +
    • +
  • forest == the type of forest, that is sequence of ordered trees + +
    • +
  • OrdNode f == the ordered tree with subtrees from the forest f + +
    • +
+ +
+ +Ordered trees of size n: + +
+ +
    +
  • size_ordtree t == the number of node of the ordered tree t + +
    • +
  • enum_ordtreesz n == the list of a ordered trees of size n + +
    • +
  • ordtreesz n == the Sigma type for ordered trees of size n. + This is canonically a finType with enumeration enum_ordtreesz n + +
    • +
  • depth_ordtree t == the depth of the ordered tree t, that is the + maximum number of node on a branch. + +
    + + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+Require Import tools combclass bintree.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Inductive type for ordered trees

+ +
+
+Inductive ordtree : Set := OrdNode : seq ordtree -> ordtree.
+Notation forest := (seq ordtree).
+ +
+Lemma OrdNode_inj : injective OrdNode.
+ +
+
+ +
+Induction scheme for ordtrees +
+
+Section Recursion.
+ +
+Variables (P : ordtree -> Type) (PF : seq ordtree -> Type).
+Hypothesis HPnil : PF [::].
+Hypothesis IHforest : forall tr f, P tr -> PF f -> PF (tr :: f).
+Hypothesis IHtree : forall f, PF f -> P (OrdNode f).
+ +
+Fixpoint recforest rt f : PF f :=
+  if f is tr :: tlf then IHforest (rt tr) (recforest rt tlf)
+  else HPnil.
+Fixpoint rectree t : P t :=
+  let: OrdNode f := t in IHtree (recforest rectree f).
+ +
+End Recursion.
+Definition indtreeforest
+  (P : ordtree -> Prop) (PF : forest -> Prop) := @rectree P PF.
+ +
+Fixpoint eq_forest (eqtr : ordtree -> ordtree -> bool) (f1 f2 : seq ordtree) :=
+  match f1, f2 with
+  | [::], [::] => true
+  | tr1 :: tl1, tr2 :: tl2 => eqtr tr1 tr2 && eq_forest eqtr tl1 tl2
+  | _, _ => false
+  end.
+Fixpoint eq_ordtree tr1 tr2 :=
+  match tr1, tr2 with
+    OrdNode f1, OrdNode f2 => eq_forest eq_ordtree f1 f2
+  end.
+Fact eq_ordtreeP : Equality.axiom eq_ordtree.
+ +
+Section SimpleRecursion.
+ +
+Variables (P : ordtree -> Type).
+Hypothesis IHtree :
+  forall f : forest, (forall t : ordtree, t \in f -> P t) -> P (OrdNode f).
+Lemma rec_tree t : P t.
+ +
+End SimpleRecursion.
+Definition indtree (P : ordtree -> Prop) := @rec_tree P.
+ +
+Fixpoint ord_to_bintree (t : ordtree) : bintree :=
+  let fix f_to_bin t_to_bin (f : forest) : bintree :=
+    match f with
+    | [::] => BinLeaf
+    | t :: ftl => BinNode (t_to_bin t) (f_to_bin t_to_bin ftl)
+    end
+  in let: OrdNode f := t in f_to_bin ord_to_bintree f.
+Definition forest_to_bintree f := ord_to_bintree (OrdNode f).
+ +
+Fixpoint bin_to_forest (t : bintree) : forest :=
+  if t is BinNode l r then OrdNode (bin_to_forest l) :: bin_to_forest r
+  else [::].
+Definition bin_to_ordtree t := OrdNode (bin_to_forest t).
+ +
+Lemma bin_to_forestK : cancel bin_to_forest forest_to_bintree.
+ Lemma bin_to_ordtreeK : cancel bin_to_ordtree ord_to_bintree.
+ +
+Lemma ord_to_bintreeK : cancel ord_to_bintree bin_to_ordtree.
+Lemma forest_to_bintreeK : cancel forest_to_bintree bin_to_forest.
+ +
+ +
+Fixpoint size_ordtree t :=
+  let: OrdNode f := t in (sumn [seq size_ordtree t | t <- f]).+1.
+Lemma size_ordtreeE f :
+  size_ordtree (OrdNode f) = (sumn [seq size_ordtree t | t <- f]).+1.
+ Lemma size_ordtree_pos t : size_ordtree t > 0.
+ Lemma size_tree_eq1 t : (size_ordtree t == 1) = (t == OrdNode [::]).
+Lemma size_bin_to_ordtree bt :
+  size_ordtree (bin_to_ordtree bt) = (size_tree bt).+1.
+Lemma size_ord_to_bintree t :
+  size_ordtree t = (size_tree (ord_to_bintree t)).+1.
+ +
+Section OfSize.
+ +
+Variable (n : nat).
+ +
+Structure ordtreesz : predArgType :=
+  OrdTreeSZ {trval :> ordtree; _ : size_ordtree trval == n}.
+ +
+Lemma ordtreeszP (t : ordtreesz) : size_ordtree t = n.
+ +
+End OfSize.
+ +
+Section FinType.
+ +
+Implicit Type (t : ordtree).
+ +
+Definition enum_ordtreesz n :=
+  if n is n'.+1 then [seq bin_to_ordtree b | b <- enum_bintreesz n']
+  else [::].
+ +
+Lemma size_mem_enum_ordtreeszP n t :
+  t \in enum_ordtreesz n -> size_ordtree t = n.
+ +
+Lemma enum_ordtreeszP n :
+  all (fun t => size_ordtree t == n) (enum_ordtreesz n).
+ +
+Lemma enum_ordtreesz_uniq n : uniq (enum_ordtreesz n).
+ +
+Lemma mem_enum_ordtreesz n t :
+  size_ordtree t == n -> t \in enum_ordtreesz n.
+ +
+Lemma enum_ordtreesz_countE n t :
+  size_ordtree t == n -> count_mem t (enum_ordtreesz n) = 1.
+ +
+Theorem card_ordtreesz n : #|ordtreesz n.+1| = Catalan_bin n.
+ +
+End FinType.
+ +
+Fixpoint depth_ordtree t :=
+  let: OrdNode f := t in (foldr maxn 0 [seq depth_ordtree t | t <- f]).+1.
+Lemma depth_ordtreeE f :
+  depth_ordtree (OrdNode f) = (foldr maxn 0 [seq depth_ordtree t | t <- f]).+1.
+ Lemma depth_ordtree_pos t : depth_ordtree t > 0.
+ Lemma depth_tree_eq1 t : (depth_ordtree t == 1) = (t == OrdNode [::]).
+Lemma depth_tree_eq2P t :
+  reflect (exists n, t = OrdNode (nseq n.+1 (OrdNode [::])))
+    (depth_ordtree t == 2).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.partition.html b/combi/1.1.0/Combi.Combi.partition.html new file mode 100644 index 00000000..4924e7d2 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.partition.html @@ -0,0 +1,1929 @@ + + + + + +Combi.Combi.partition: Integer Partitions + + + + +
+ + + +
+ +

Library Combi.Combi.partition: Integer Partitions

+ +
+
+ +
+ +
+
+
+ +
+

Shapes and Integer Partitions

+ + +
+ +Partitions (and more generally shapes) are stored by terms of type seq nat. +We define the following predicates and operations on seq nat: + +
+ +
    +
  • in_shape sh (r, c) == the box with coordinate (r, c) belongs to the shape + sh, that is: c < sh[r]. + +
    • +
  • in_skew inn out (r, c) == the box with coordinate (r, c) belongs to the + shape out but not inn + +
    • +
  • box_in sh == a sigma type for boxes in sh : { b | in_shape sh b }; + it is canonically a subFinType. + +
    • +
  • box_skew inn out == a sigma type for boxes in the skew shape out / inn + +
    • +
  • enum_box_in sh == a full duplicate free list of the boxes in sh. + +
    • +
  • enum_box_skew inn out == a full duplicate free list of the boxes in + the skew shape out / inn + +
    • +
+ +
+ +Integer Partitions: + +
+ +
    +
  • is_part sh == sh is a partition + +
    • +
  • rem_trail0 sh == remove the trailing zeroes of a shape + +
    • +
  • is_add_corner sh i == i is the row of an addable corner of sh + +
    • +
  • is_rem_corner sh i == i is the row of a removable corner of sh + +
    • +
  • incr_nth sh i == the shape obtained by adding a box at the end of the + i-th row. This gives a partition if i is an addable + corner of sh (Lemma is_part_incr_nth) + +
    • +
  • decr_nth sh i == the shape obtained by removing a box at the end of the + i-th row. This gives a partition if i is an removable + corner of sh (Lemma is_part_decr_nth) + +
    • +
  • rem_corners sh == the list of the rows of the removable corners of sh. + +
    • +
  • incr_first_n sh n == adding 1 to the n'th first part of sh, + always gives a partition + +
    • +
  • conj_part sh == the conjugate of a partition + +
    • +
  • included s t == the Ferrers diagram of s is included in the + Ferrers diagram of t. This is an order. + +
    • +
  • s / t = diff_shape s t == the difference of the shapes s and t + +
    • +
  • outer_shape s t == add t to the shape s + +
    • +
+ +
+ + +
+ +Enumeration of integer partitions: + +
+ +
    +
  • is_part_of_n sm sh == sh is a partition of n + +
    • +
  • is_part_of_ns sm sz sh == sh is a partition of n of size sz + ("size" means "length") + +
    • +
  • is_part_of_nsk sm sz mx sh == sh is a partition of n of size sz + in parts at most mx + +
    • +
  • enum_partn sm == the list of all partitions of n + +
    • +
  • enum_partns sm sz == the list of all partitions of n of size sz + +
    • +
  • enum_partnsk sm sz mx == the list of all partitions of n of size sz + in parts at most mx + +
    • +
  • intpartn_nb sm == the number of partitions of n + +
    • +
  • intpartns_nb sm sz == the number of partitions of n of size sz + +
    • +
  • intpartnsk_nb sm sz mx == the number of partitions of n of size sz + in parts at most mx + +
    • +
+ +
+ + +
+ +Sigma types for integer partitions: + +
+ +
    +
  • intpart == a type for seq nat which are partitions; + canonically a subCountType of seq nat + +
    • +
  • conj_intpart == the conjugate of a intpart as a intpart + +
    • +
  • empty_intpart == the empty intpart + +
    + + +
    • +
  • 'P_n == a type for seq nat which are partitions of n; + canonically a finType + +
    • +
  • cast_intpartn m n eq_mn == the cast from 'P_m to 'P_n + +
    • +
  • rowpartn n == the one-row partition of sum n as a 'P_n + +
    • +
  • colpartn n == the one-column partition of sum n as a 'P_n + +
    • +
  • conj_intpartn == the conjugate of a 'P_n as a 'P_n + +
    • +
  • hookpartm n k == the hook shape partition of sum n.+1 as a 'P_n.+1 + whose arm length is k (not counting the corner), + that is, (k+1, 1, ..., 1). + +
    + + +
    • +
  • decr_nth_intpart p i == the shape obtained by removing a box at the end + of the i-th row if the result is a partition, else p + +
    • +
+ +
+ +Operations on partitions: + +
+ +
    +
  • union_intpart l k == the partition of type intpart obtained by + gathering the parts of l and k + +
    • +
  • l +|+ k = union_intpartn l k == the partition of type 'P_(m + n) + obtained by gathering the parts of l : 'P_m and k : 'P_n + +
    • +
+ +
+ +Comparison of partitions: + +
+ +
    +
  • 'YL == a type convertible to intpart which is canonically + a lattice partially ordered by included + ("Young's lattice"). + +
    • +
  • partdom s t == s is dominated by t, that is, the partial sums of s are + less-or-equal to the partial sums of t. + +
    • +
  • 'PDom_d == a type convertible to 'P_d which is canonically + a finite lattice partially ordered by partdom. + +
    • +
  • 'PLexi_d == a type convertible to 'P_d which is canonically + finite and totally ordered by the lexicographic order. + +
    • +
+ +
+ +Relation with set partitions: + +
+ +
    +
  • setpart_shape P == the shape of a set partition, i.e. + the sorted list of the cardinalities of the parts + +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import div ssralg ssrint ssrnum binomial.
+Require Import tools combclass sorted ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import LeqGeqOrder.
+Import Order.TTheory.
+ +
+Open Scope nat_scope.
+Declare Scope Combi_scope.
+Delimit Scope Combi_scope with Combi.
+Open Scope Combi_scope.
+ +
+Reserved Notation "''P_' n"
+         (at level 8, n at level 2, format "''P_' n").
+Reserved Notation "''PDom_' n"
+         (at level 8, n at level 2, format "''PDom_' n").
+Reserved Notation "''PLexi_' n"
+         (at level 8, n at level 2, format "''PLexi_' n").
+ +
+
+ +
+

Shapes

+ +
+
+Definition in_shape sh b := b.2 < nth 0 sh b.1.
+Definition in_skew inn out b := nth 0 inn b.1 <= b.2 < nth 0 out b.1.
+ +
+Lemma in_skewE inn out b :
+  in_skew inn out b = ~~ in_shape inn b && in_shape out b.
+ Lemma in_skew_out inn out b : in_skew inn out b -> in_shape out b.
+ Lemma in_skew_in inn out b : in_skew inn out b -> ~~ in_shape inn b.
+ +
+Lemma in_shape_nil : in_shape [::] =1 pred0.
+ Lemma in_skew_nil : in_skew [::] =2 in_shape.
+ +
+Lemma in_shape_size sh r c : in_shape sh (r, c) -> r < size sh.
+ +
+
+ +
+

Integer Partitions

+

Definitions and basic properties

+ +
+
+Section Defs.
+ +
+Implicit Types (s sh : seq nat) (i j r c : nat).
+ +
+Fixpoint is_part sh :=
+  if sh is sh0 :: sh'
+  then (sh0 >= head 1 sh') && (is_part sh')
+  else true.
+ +
+
+ +
+Partitions don't have 0 parts +
+
+Lemma part_head0F sh : head 1 sh == 0 -> is_part sh = false.
+ +
+Lemma part_head_non0 sh : is_part sh -> head 1 sh != 0.
+ +
+Lemma notin0_part sh : is_part sh -> 0 \notin sh.
+ +
+Lemma in_part_non0 sh i : is_part sh -> i \in sh -> i != 0.
+ +
+Lemma nth_part_non0 sh i : is_part sh -> i < size sh -> nth 0 sh i != 0.
+ +
+Lemma leq_head_sumn sh : head 0 sh <= sumn sh.
+ +
+Lemma part_leq_head sh i : is_part sh -> i \in sh -> i <= head 0 sh.
+ +
+
+ +
+Three equivalent definitions +
+
+Lemma is_partP sh :
+  reflect
+    (last 1 sh != 0 /\ forall i, (nth 0 sh i) >= nth 0 sh i.+1)
+    (is_part sh).
+ +
+Lemma is_part_ijP sh :
+  reflect
+    (last 1 sh != 0 /\ forall i j, i <= j -> (nth 0 sh i) >= nth 0 sh j)
+    (is_part sh).
+ +
+Lemma is_part_sortedE sh : (is_part sh) = (sorted geq sh) && (0 \notin sh).
+ +
+
+ +
+Sub-partitions +
+
+ +
+Lemma is_part_consK l0 sh : is_part (l0 :: sh) -> is_part sh.
+ +
+Lemma is_part_behead sh : is_part sh -> is_part (behead sh).
+ +
+Lemma is_part_subseq sh1 sh2 : subseq sh1 sh2 -> is_part sh2 -> is_part sh1.
+ +
+Lemma is_part_rconsK sh sn : is_part (rcons sh sn) -> is_part sh.
+ +
+Lemma is_part_catl sh1 sh2 : is_part (sh1 ++ sh2) -> is_part sh2.
+ Lemma is_part_catr sh1 sh2 : is_part (sh1 ++ sh2) -> is_part sh1.
+ +
+
+ +
+Boxes in a partitions +
+
+ +
+Lemma in_part_le sh r c j k :
+  is_part sh -> in_shape sh (r, c) -> j <= r -> k <= c -> in_shape sh (j, k).
+ +
+Lemma in_part_is_part sh :
+  (forall r c j k, j <= r -> k <= c ->
+                   in_shape sh (r, c) -> in_shape sh (j, k)) ->
+  last 1 sh != 0 -> is_part sh.
+ +
+
+ +
+Equality of partitions +
+
+Lemma part_eqP p q :
+  is_part p -> is_part q -> reflect (forall i, nth 0 p i = nth 0 q i) (p == q).
+ +
+
+ +
+Partitions and sumn +
+
+Lemma part0 sh : is_part sh -> sumn sh = 0 -> sh = [::].
+ +
+Lemma size_part sh : is_part sh -> size sh <= sumn sh.
+ +
+Lemma part_sumn_rectangle sh :
+  is_part sh -> sumn sh <= (head 0 sh) * (size sh).
+ +
+
+ +
+Removing trailing zeroes +
+
+Fixpoint rem_trail0 s :=
+  if s is s0 :: s' then
+    if (rem_trail0 s') is t1 :: t' then s0 :: t1 :: t'
+    else if s0 == 0 then [::] else [:: s0]
+  else [::].
+ +
+Lemma size_rem_trail0 s : size (rem_trail0 s) <= size s.
+ +
+Lemma nth_rem_trail0 s i : nth 0 (rem_trail0 s) i = nth 0 s i.
+ +
+Lemma sumn_rem_trail0 s : sumn (rem_trail0 s) = sumn s.
+ +
+Lemma is_part_rem_trail0 s : sorted geq s -> is_part (rem_trail0 s).
+ +
+
+ +
+

Corners, adding and removing corners

+ +
+
+Definition is_rem_corner sh i := nth 0 sh i > nth 0 sh i.+1.
+Definition is_add_corner sh i := (i == 0) || (nth 0 sh i < nth 0 sh i.-1).
+ +
+Lemma last_incr_nth_non0 sh i : last 1 sh != 0 -> last 1 (incr_nth sh i) != 0.
+ +
+Lemma is_part_incr_nth_size sh i :
+  is_part sh -> is_part (incr_nth sh i) -> i <= size sh.
+ +
+Lemma is_part_incr_nth sh i :
+  is_part sh -> is_add_corner sh i -> is_part (incr_nth sh i).
+ +
+Lemma del_rem_corner sh i :
+  last 1 sh != 0 -> is_part (incr_nth sh i) ->
+  is_rem_corner (incr_nth sh i) i = is_part sh.
+ +
+Lemma rem_corner_incr_nth sh i :
+  is_part sh -> is_add_corner sh i -> is_rem_corner (incr_nth sh i) i.
+ +
+Lemma is_rem_cornerP sh i : is_part sh ->
+  (i < size sh) && (~~ in_shape sh (i.+1, (nth 0 sh i).-1)) =
+  (is_rem_corner sh i).
+ +
+Fixpoint decr_nth v i {struct i} :=
+  if v is n :: v' then
+    if i is i'.+1 then n :: decr_nth v' i'
+    else if n is n'.+1 then
+           if n' is _.+1 then
+             n' :: v'
+           else [::]
+         else [::]
+  else [::].
+ +
+Lemma incr_nthK sh i :
+  is_part sh -> is_part (incr_nth sh i) -> decr_nth (incr_nth sh i) i = sh.
+ +
+Lemma decr_nthK sh i :
+  is_part sh -> is_rem_corner sh i -> incr_nth (decr_nth sh i) i = sh.
+ +
+Lemma sumn_incr_nth s i : sumn (incr_nth s i) = (sumn s).+1.
+ +
+Lemma nth_decr_nth sh i :
+  nth 0 (decr_nth sh i) i = (nth 0 sh i).-1.
+ +
+Lemma nth_decr_nth_neq sh i j :
+  is_part sh -> is_rem_corner sh i -> i != j ->
+  nth 0 (decr_nth sh i) j = nth 0 sh j.
+ +
+Lemma sumn_decr_nth sh i :
+  is_part sh -> is_rem_corner sh i -> (sumn (decr_nth sh i)) = (sumn sh).-1.
+ +
+Lemma is_part_decr_nth sh i :
+  is_part sh -> is_rem_corner sh i -> is_part (decr_nth sh i).
+ +
+Lemma add_corner_decr_nth sh i :
+  is_part sh -> is_rem_corner sh i -> is_add_corner (decr_nth sh i) i.
+ +
+Definition rem_corners sh := filter (is_rem_corner sh) (iota 0 (size sh)).
+ +
+Lemma rem_corners_uniq sh : uniq (rem_corners sh).
+ +
+
+ +
+

Conjugate of a partition

+ +
+
+ +
+Fixpoint incr_first_n sh n :=
+  if sh is s0 :: s then
+    if n is n'.+1 then s0.+1 :: incr_first_n s n'
+    else sh
+  else nseq n 1.
+Fixpoint conj_part sh :=
+  if sh is s0 :: sh then incr_first_n (conj_part sh) s0
+  else [::].
+ +
+Lemma is_part_nseq1 n : is_part (nseq n 1).
+ +
+Lemma nth_incr_first_n sh n i :
+  nth 0 (incr_first_n sh n) i = if i < n then (nth 0 sh i).+1 else nth 0 sh i.
+ +
+Lemma incr_first_n_nthC sh i j :
+  incr_first_n (incr_nth sh i) j = incr_nth (incr_first_n sh j) i.
+ +
+Lemma is_part_incr_first_n sh n :
+  is_part sh -> is_part (incr_first_n sh n).
+ +
+Lemma is_part_conj sh : is_part sh -> is_part (conj_part sh).
+ +
+Lemma conj_nseq n : 0 < n -> conj_part (nseq n 1) = [:: n].
+ +
+Lemma size_incr_first_n sh n :
+  size sh <= n -> size (incr_first_n sh n) = n.
+ +
+Lemma size_conj_part sh : is_part sh -> size (conj_part sh) = head 0 sh.
+ +
+Lemma sumn_incr_first_n sh n : sumn (incr_first_n sh n) = sumn sh + n.
+ +
+Lemma sumn_conj_part sh : sumn (conj_part sh) = sumn sh.
+ +
+Lemma conj_part_ind sh l :
+  sh != [::] -> is_part sh -> l >= size sh ->
+  conj_part (incr_first_n sh l) = l :: conj_part sh.
+ +
+Lemma conj_partK sh : is_part sh -> conj_part (conj_part sh) = sh.
+ +
+Lemma conj_part_incr_nth sh i :
+  is_part sh -> is_add_corner sh i ->
+  conj_part (incr_nth sh i) = incr_nth (conj_part sh) (nth 0 sh i).
+ +
+Lemma in_conj_part_impl sh : is_part sh ->
+  forall r c, in_shape sh (r, c) -> in_shape (conj_part sh) (c, r).
+ +
+Lemma in_conj_part sh : is_part sh ->
+  forall r c, in_shape sh (r, c) = in_shape (conj_part sh) (c, r).
+ +
+Lemma conj_ltnE sh :
+  is_part sh -> forall i j, nth 0 sh i > j = (nth 0 (conj_part sh) j > i).
+ +
+Lemma conj_leqE sh :
+  is_part sh -> forall i j, (nth 0 sh i <= j) = (nth 0 (conj_part sh) j <= i).
+ +
+Lemma nth_conjE sh r c :
+  is_part sh -> c != 0 ->
+  (nth 0 (conj_part sh) r == c) = (nth 0 sh c <= r < nth 0 sh c.-1).
+ +
+Lemma rem_corner_incr_first_n sh i :
+  is_part sh -> is_rem_corner (incr_first_n sh i.+1) i.
+ +
+Lemma rem_corner_incr_first_nE sh n i :
+  is_part sh -> is_rem_corner sh i -> is_rem_corner (incr_first_n sh n) i.
+ +
+Lemma is_add_corner_conj_part sh r :
+  is_part sh -> is_add_corner sh r -> is_add_corner (conj_part sh) (nth 0 sh r).
+ +
+Lemma rem_corner_conj_part sh i :
+  is_part sh -> is_rem_corner sh i ->
+  is_rem_corner (conj_part sh) (nth 0 sh i).-1.
+ +
+
+ +
+

Partial sum of partitions

+ +
+
+ +
+Lemma sumn_take_leq s k1 k2 :
+  k1 <= k2 -> sumn (take k1 s) <= sumn (take k2 s).
+ +
+Lemma sum_conj sh k : \sum_(l <- sh) minn l k = sumn (take k (conj_part sh)).
+ +
+Lemma sumn_take_inj s t :
+  is_part s -> is_part t ->
+  (forall k, sumn (take k s) = sumn (take k t)) -> s = t.
+ +
+
+ +
+

Inclusion of Partitions and Skew Partitions

+ +
+
+ +
+Fixpoint included inner outer :=
+  if inner is inn0 :: inn then
+    if outer is out0 :: out then
+      (inn0 <= out0) && (included inn out)
+    else false
+  else true.
+ +
+Lemma includedP inner outer :
+  reflect (size inner <= size outer /\ forall i, nth 0 inner i <= nth 0 outer i)
+          (included inner outer).
+ +
+Lemma part_includedP inner outer :
+  is_part inner ->
+  reflect (forall i, nth 0 inner i <= nth 0 outer i) (included inner outer).
+ +
+Lemma included_behead p1 p2 :
+  included p1 p2 -> included (behead p1) (behead p2).
+ +
+Lemma included_refl sh : included sh sh.
+ +
+Lemma included_trans : transitive included.
+ +
+Lemma included_incr_nth sh i : included sh (incr_nth sh i).
+ +
+Lemma included_decr_nth u i : included (decr_nth u i) u.
+ +
+Lemma included_incr_nth_inner inner outer i :
+  nth 0 inner i < nth 0 outer i ->
+  included inner outer -> included (incr_nth inner i) outer.
+ +
+Lemma size_included inner outer :
+  included inner outer -> size inner <= size outer.
+ +
+Lemma sumn_included inner outer :
+  included inner outer -> sumn inner <= sumn outer.
+ +
+Lemma included_sumnE inner outer :
+  is_part outer ->
+  included inner outer ->
+  sumn inner = sumn outer ->
+  inner = outer.
+ +
+Lemma included_anti sh1 sh2 :
+  is_part sh1 -> is_part sh2 ->
+  included sh1 sh2 -> included sh2 sh1 ->
+  sh1 = sh2.
+ +
+Lemma included_conj_part inner outer :
+  is_part inner -> is_part outer ->
+  included inner outer -> included (conj_part inner) (conj_part outer).
+ +
+Lemma included_conj_partE inner outer :
+  is_part inner -> is_part outer ->
+  included inner outer = included (conj_part inner) (conj_part outer).
+ +
+Fixpoint diff_shape inner outer :=
+  if inner is inn0 :: inn then
+    if outer is out0 :: out then
+      out0 - inn0 :: diff_shape inn out
+    else [::]
+  else outer.
+ +
+Notation "outer / inner" := (diff_shape inner outer) : Combi_scope.
+ +
+Definition pad (T : Type) (x : T) sz := [fun s => s ++ nseq (sz - size s) x].
+ +
+Lemma nth_pad (T : Type) n (p : T) (s : seq T) i :
+  nth p (pad p n s) i = nth p s i.
+ +
+Lemma head_pad (T : Type) n (p : T) (s : seq T) :
+  head p (pad p n s) = head p s.
+ +
+Definition outer_shape inner size_seq :=
+  [seq p.1 + p.2 | p <- zip (pad 0 (size (size_seq)) inner) size_seq].
+ +
+Lemma diff_shape_eq s : s / s = nseq (size s) 0.
+ +
+Lemma sumn_diff_shape_eq s : sumn (s / s) = 0.
+ +
+Lemma size_diff_shape inner outer : size (outer / inner) = size outer.
+ +
+Lemma nth_diff_shape inn out i :
+  nth 0 (out / inn) i = nth 0 out i - nth 0 inn i.
+ +
+Lemma sumn_diff_shape inner outer :
+  included inner outer -> sumn (outer / inner) = sumn outer - sumn inner.
+ +
+Lemma diff_shape_pad0 inner outer :
+  outer / ((pad 0 (size outer)) inner) = outer / inner.
+ +
+Lemma diff_shapeK inner outer :
+  included inner outer -> outer_shape inner (outer / inner) = outer.
+ +
+Lemma outer_shapeK inner diff :
+  size inner <= size diff -> (outer_shape inner diff) / inner = diff.
+ +
+Lemma outer_shape_pad0 inner sz :
+  outer_shape (pad 0 (size sz) inner) sz = outer_shape inner sz.
+ +
+Lemma included_pad0 inner outer :
+  included inner outer = included (pad 0 (size outer) inner) outer.
+ +
+End Defs.
+Notation "outer / inner" := (diff_shape inner outer) : Combi_scope.
+ +
+
+ +
+

Sigma Types for Partitions

+ +
+
+ +
+Structure intpart : Set := IntPart {pval :> seq nat; _ : is_part pval}.
+ +
+Lemma intpartP (p : intpart) : is_part p.
+ +
+Lemma intpart_sorted (p : intpart) : sorted geq p.
+ +
+#[export] Hint Resolve intpartP intpart_sorted : core.
+ +
+Lemma intpart_eqP (p q : intpart) :
+  reflect (forall i, nth 0 p i = nth 0 q i) (p == q).
+ +
+Canonical conj_intpart p := IntPart (is_part_conj (intpartP p)).
+ +
+Lemma conj_intpartK : involutive conj_intpart.
+ +
+Lemma intpart_sumn_take_inj (s t : intpart) :
+  (forall k, sumn (take k s) = sumn (take k t)) -> s = t.
+ +
+Canonical empty_intpart := IntPart (pval := [::]) is_true_true.
+ +
+Lemma empty_intpartP (p : intpart) : sumn p = 0 -> p = empty_intpart.
+ +
+Fixpoint enum_partnsk sm sz mx : (seq (seq nat)) :=
+  if sz is sz.+1 then
+    flatten [seq [seq i :: p | p <- enum_partnsk (sm - i) sz i] |
+             i <- iota 1 (minn sm mx)]
+  else if sm is sm.+1 then [::] else [:: [::]].
+ +
+Definition enum_partns sm sz := enum_partnsk sm sz sm.
+Definition enum_partn sm := flatten [seq enum_partns sm sz | sz <- iota 0 sm.+1 ].
+ +
+Definition is_part_of_n sm :=
+  [pred p | (sumn p == sm) & is_part p ].
+Definition is_part_of_ns sm sz :=
+  [pred p | (size p == sz) & is_part_of_n sm p].
+Definition is_part_of_nsk sm sz mx :=
+  [pred p | (head 1 p <= mx) & is_part_of_ns sm sz p].
+ +
+Lemma enum_partnsk_allP sm sz mx :
+  mx >= 1 -> all (is_part_of_nsk sm sz mx) (enum_partnsk sm sz mx).
+ +
+Lemma enum_partnsk_countE sm sz mx :
+  mx >= 1 ->
+  forall p, is_part_of_nsk sm sz mx p ->
+            count_mem p (enum_partnsk sm sz mx) = 1.
+ +
+Lemma enum_partnskE sm sz mx :
+  mx >= 1 ->
+  forall p, count_mem p (enum_partnsk sm sz mx) = is_part_of_nsk sm sz mx p.
+ +
+Lemma enum_partns_allP sm sz : all (is_part_of_ns sm sz) (enum_partns sm sz).
+ +
+Lemma enum_partns_countE sm sz p :
+  is_part_of_ns sm sz p -> count_mem p (enum_partns sm sz) = 1.
+ +
+Lemma enum_partnsE sm sz p :
+  count_mem p (enum_partns sm sz) = is_part_of_ns sm sz p.
+ +
+Lemma enum_partn_allP sm : all (is_part_of_n sm) (enum_partn sm).
+ +
+Lemma enum_partn_countE sm p :
+  is_part_of_n sm p -> count_mem p (enum_partn sm) = 1.
+ +
+Lemma enum_partnP n p : (is_part_of_n n p) = (p \in enum_partn n).
+ +
+Section PartOfn.
+ +
+Variable n : nat.
+ +
+Structure intpartn : Set :=
+  IntPartN {pnval :> seq nat; _ : is_part_of_n n pnval}.
+ +
+#[local] Notation "''P'" := intpartn.
+ +
+Implicit Type (p : 'P).
+Lemma intpartnP p : is_part p.
+ +
+Lemma intpartn_sorted p : sorted geq p.
+ +
+Hint Resolve intpartnP intpartn_sorted : core.
+ +
+Definition intpart_of_intpartn p := IntPart (intpartnP p).
+Coercion intpart_of_intpartn : intpartn >-> intpart.
+ +
+Lemma sumn_intpartn p : sumn p = n.
+ +
+Lemma intpartn_leq_head p i : i \in pnval p -> i <= head 0 p.
+ +
+Lemma intpartn_leq p i : i \in pnval p -> i <= n.
+ +
+Lemma enum_intpartnE : map val (enum {: 'P}) = enum_partn n.
+ +
+Fact conj_intpartnP p : is_part_of_n n (conj_part p).
+Canonical conj_intpartn p := IntPartN (conj_intpartnP p).
+ +
+Lemma conj_intpartnK : involutive conj_intpartn.
+ +
+End PartOfn.
+ +
+Notation "''P_' n" := (intpartn n).
+ +
+Lemma val_intpartn0 (sh : 'P_0) : sh = [::] :> seq nat.
+ +
+Lemma val_intpartn1 (sh : 'P_1) : sh = [:: 1] :> seq nat.
+ +
+Lemma val_intpartn2 (sh : 'P_2) :
+  sh = [:: 2] :> seq nat \/ sh = [:: 1; 1] :> seq nat.
+ +
+Lemma val_intpartn3 (sh : 'P_3) :
+  [\/ sh = [:: 3] :> seq nat,
+      sh = [:: 2; 1] :> seq nat |
+      sh = [:: 1; 1; 1] :> seq nat].
+ +
+Definition cast_intpartn m n (eq_mn : m = n) p :=
+  let: erefl in _ = n := eq_mn return 'P_n in p.
+ +
+Lemma cast_intpartnE m n (eq_mn : m = n) p :
+  val (cast_intpartn eq_mn p) = val p.
+ +
+Lemma cast_intpartn_id n eq_n (s : 'P_n) : cast_intpartn eq_n s = s.
+ +
+Lemma cast_intpartnK m n eq_m_n :
+  cancel (@cast_intpartn m n eq_m_n) (cast_intpartn (esym eq_m_n)).
+ +
+Lemma cast_intpartnKV m n eq_m_n :
+  cancel (cast_intpartn (esym eq_m_n)) (@cast_intpartn m n eq_m_n).
+ +
+Lemma cast_intpartn_inj m n eq_m_n : injective (@cast_intpartn m n eq_m_n).
+ +
+Lemma cast_intpartn_bij m n eq_m_n : bijective (@cast_intpartn m n eq_m_n).
+ +
+Lemma cast_conj_inpart m n eq_m_n (s : 'P_m) :
+  (@cast_intpartn m n eq_m_n) (conj_intpartn s) =
+  conj_intpartn (@cast_intpartn m n eq_m_n s).
+ +
+Fact rowpartn_subproof d : is_part_of_n d (if d is 0 then [::] else [:: d]).
+ Definition rowpartn d : 'P_d := locked (IntPartN (rowpartn_subproof d)).
+ +
+Fact colpartn_subproof d : is_part_of_n d (nseq d 1%N).
+Definition colpartn d : 'P_d := locked (IntPartN (colpartn_subproof d)).
+ +
+ +
+Lemma rowpartnE d : rowpartn d = (if d is 0 then [::] else [:: d]) :> seq nat.
+ Lemma colpartnE d : colpartn d = nseq d 1%N :> seq nat.
+ +
+Lemma rowpartn0E : rowpartn 0 = [::] :> seq nat.
+ Lemma rowpartnSE d : rowpartn d.+1 = [:: d.+1] :> seq nat.
+ +
+Lemma size_colpartn d : size (colpartn d) = d.
+ Lemma size_rowpartn d : size (rowpartn d) = (d != 0).
+ +
+Lemma part_nseq1P x0 sh :
+  is_part sh -> head x0 sh <= 1 -> sh = nseq (sumn sh) 1.
+Lemma colpartnP x0 d (la : 'P_d) : head x0 la <= 1 -> la = colpartn d.
+ +
+Lemma conj_rowpartn d : conj_intpartn (rowpartn d) = colpartn d.
+ Lemma conj_colpartn d : conj_intpartn (colpartn d) = rowpartn d.
+ +
+
+ +
+

Hook shaped partitions

+ +
+
+ +
+Definition hookpart d k :=
+  if d is d'.+1 then minn d'.+1 k.+1 :: nseq (d' - k) 1%N else [::].
+Fact hookpartn_subproof d k : is_part_of_n d (hookpart d k).
+Definition hookpartn d k : 'P_d :=
+  locked (IntPartN (hookpartn_subproof d k)).
+ +
+Lemma hookpartnE d k :
+  k < d -> (hookpartn d k) = k.+1 :: nseq (d.-1 - k) 1%N :> seq nat.
+ +
+Lemma size_hookpartn d k : k < d -> size (hookpartn d k) = d - k.
+ +
+Lemma behead_hookpartn d k : behead (hookpartn d k) = nseq (d.-1 - k) 1%N.
+ +
+Lemma hookpartn_col d : hookpartn d 0 = colpartn d.
+Lemma hookpartn_row d : hookpartn d d.-1 = rowpartn d.
+Lemma conj_hookpartn d k :
+  k < d -> conj_intpartn (hookpartn d k) = hookpartn d (d.-1 - k).
+ +
+Lemma sorted_geq_count_leq2E (s : seq nat) :
+  sorted geq s -> (count (leq 2) s <= 1) = (nth 0 s 1 <= 1).
+ +
+Lemma sorted_geq_nth0E (s : seq nat) :
+  sorted geq s -> nth 0 s 0 = \max_(i <- s) i.
+ +
+Lemma intpartn_count_leq2E d (la : 'P_d) :
+  (count (leq 2) la <= 1) = (nth 0 la 1 <= 1).
+Lemma intpartn_nth0 d (la : 'P_d) :
+  nth 0 la 0 = \max_(i <- la) i.
+ +
+Lemma hookpartnPE x0 x1 d (la : 'P_d) :
+  nth x0 la 1 <= 1 -> la = hookpartn d (nth x1 la 0).-1.
+ +
+Lemma hookpartnP d (la : 'P_d.+1) :
+  reflect (exists k, k < d.+1 /\ la = hookpartn d.+1 k) (nth 0 la 1 <= 1).
+ +
+Lemma intpartn0 : all_equal_to (rowpartn 0).
+ +
+Lemma intpartn1 : all_equal_to (rowpartn 1).
+ +
+Lemma intpartn2 (sh : 'P_2) : sh = rowpartn 2 \/ sh = colpartn 2.
+ +
+Lemma intpartn3 (sh : 'P_3) :
+  [\/ sh = rowpartn 3, sh = hookpartn 3 1 | sh = colpartn 3].
+ +
+
+ +
+

Removing a corner from a partition

+ +
+
+ +
+Lemma is_part_decr_nth_part (p : intpart) i :
+  is_part (if is_rem_corner p i then decr_nth p i else p).
+ +
+Definition decr_nth_intpart (p : intpart) i : intpart :=
+  IntPart (is_part_decr_nth_part p i).
+ +
+Lemma decr_nth_intpartE (p : intpart) i :
+  is_rem_corner p i -> decr_nth_intpart p i = decr_nth p i :> seq nat.
+ +
+Lemma intpart_rem_corner_ind (F : intpart -> Type) :
+  F empty_intpart ->
+  (forall p : intpart,
+      (forall i, is_rem_corner p i -> F (decr_nth_intpart p i)) -> F p) ->
+  forall p : intpart, F p.
+ +
+Lemma part_rem_corner_ind (F : seq nat -> Type) :
+  F [::] ->
+  (forall p, is_part p ->
+             (forall i, is_rem_corner p i -> F (decr_nth p i)) -> F p) ->
+  forall p, is_part p -> F p.
+ +
+
+ +
+

Lexicographic order on partitions of a fixed sum

+ +
+
+ +
+Module IntPartNLexi.
+Section IntPartNLexi.
+Import DefaultSeqLexiOrder.
+ +
+Variable d : nat.
+Definition intpartnlexi := 'P_d.
+#[local] Notation "'PLexi" := intpartnlexi.
+Implicit Type (sh : 'PLexi).
+ +
+#[export] HB.instance Definition _ := SubType.copy 'PLexi 'P_d.
+#[export] HB.instance Definition _ := Finite.copy 'PLexi 'P_d.
+#[export] HB.instance Definition _ := [Order of 'PLexi by <:].
+ +
+Lemma leEintpartnlexi sh1 sh2 :
+  (sh1 <= sh2)%O = (sh1 <= sh2 :> seqlexi nat)%O.
+ Lemma ltEintpartnlexi sh1 sh2 :
+  (sh1 < sh2)%O = (sh1 < sh2 :> seqlexi nat)%O.
+ +
+Lemma rowpartn_top sh : (sh <= rowpartn d :> 'PLexi)%O.
+Lemma colpartn_bot sh : (colpartn d <= sh :> 'PLexi)%O.
+ +
+#[export] HB.instance Definition _ :=
+  Order.hasBottom.Build _ 'PLexi colpartn_bot.
+#[export] HB.instance Definition _ :=
+  Order.hasTop.Build _ 'PLexi rowpartn_top.
+#[export] HB.instance Definition _ :=
+  isInhabitedType.Build 'PLexi (rowpartn d).
+ +
+Lemma botEintpartnlexi : \bot%O = colpartn d :> 'PLexi.
+ Lemma topEintpartnlexi : \top%O = rowpartn d :> 'PLexi.
+ +
+End IntPartNLexi.
+ +
+Module Exports.
+ +
+Notation "''PLexi_' n" := (intpartnlexi n).
+ +
+Definition leEintpartnlexi := leEintpartnlexi.
+Definition ltEintpartnlexi := ltEintpartnlexi.
+Definition botEintpartnlexi := botEintpartnlexi.
+Definition topEintpartnlexi := topEintpartnlexi.
+ +
+End Exports.
+End IntPartNLexi.
+ +
+
+ +
+

Counting functions

+ +
+
+ +
+Fixpoint intpartnsk_nb sm sz mx : nat :=
+  if sz is sz.+1 then
+    
+    iteri (minn sm mx) (fun i n => n + intpartnsk_nb (sm - i.+1) sz i.+1) 0
+  else if sm is sm.+1 then 0 else 1.
+Definition intpartns_nb sm sz := intpartnsk_nb sm sz sm.
+Definition intpartn_nb sm :=
+  iteri (sm.+1) (fun sz n => n + intpartns_nb sm sz) 0.
+ +
+Lemma size_enum_partnsk sm sz mx :
+  size (enum_partnsk sm sz mx) = (intpartnsk_nb sm sz mx).
+ +
+Lemma size_enum_partns sm sz : size (enum_partns sm sz) = (intpartns_nb sm sz).
+ +
+Lemma size_enum_partn sm : size (enum_partn sm) = intpartn_nb sm.
+ +
+Lemma card_intpartn sm : #|{: 'P_sm}| = intpartn_nb sm.
+ +
+#[export] Hint Resolve intpartP intpart_sorted intpartnP intpartn_sorted : core.
+ +
+
+ +
+

A finite type finType for coordinate of boxes inside a shape

+ +
+
+Section BoxInSkew.
+ +
+Variable inner outer : seq nat.
+ +
+Structure box_skew : Set :=
+  BoxSkew {box_skewval :> nat * nat; _ : in_skew inner outer box_skewval}.
+ +
+ +
+Lemma box_skewP (rc : box_skew) : in_skew inner outer rc.
+ +
+Definition enum_box_skew :=
+  [seq (r, c) | r <- iota 0 (size outer),
+                c <- iota (nth 0 inner r) (nth 0 outer r - nth 0 inner r)].
+ +
+Lemma enum_box_skew_uniq : uniq enum_box_skew.
+ +
+Lemma mem_enum_box_skew : enum_box_skew =i in_skew inner outer.
+ +
+ +
+Lemma enum_box_skewE : map val (enum {: box_skew}) = enum_box_skew.
+ +
+Lemma card_box_skew : #|{: box_skew}| = sumn (outer / inner).
+ +
+
+ +
+

Rewriting bigops running along the boxes of a shape

+ +
+
+Lemma big_box_skew R (idx : R) (op : Monoid.com_law idx) (f : nat * nat -> R):
+  \big[op/idx]_(b : box_skew) f b = \big[op/idx]_(b <- enum_box_skew) f b.
+ +
+Lemma big_box_skew2 R (idx : R) (op : Monoid.com_law idx) (f : nat -> nat -> R):
+  \big[op/idx]_(b : box_skew) f b.1 b.2 =
+  \big[op/idx]_(b <- enum_box_skew) f b.1 b.2.
+ +
+Lemma big_enum_box_skew
+      (R : Type) (idx : R) (op : Monoid.law idx) (f : nat -> nat -> R):
+  \big[op/idx]_(b <- enum_box_skew) f b.1 b.2 =
+  \big[op/idx]_(0 <= r < size outer)
+   \big[op/idx]_(nth 0 inner r <= c < nth 0 outer r) f r c.
+ +
+End BoxInSkew.
+#[export] Hint Resolve box_skewP : core.
+ +
+Notation box_in := (box_skew [::]).
+Notation enum_box_in := (enum_box_skew [::]).
+ +
+Lemma BoxIn_subproof sh rc : in_shape sh rc -> in_skew [::] sh rc.
+ Definition BoxIn sh rc (rc_sh : in_shape sh rc) : box_in sh :=
+  BoxSkew (BoxIn_subproof rc_sh).
+ +
+Lemma box_inP sh (rc : box_in sh) : in_shape sh rc.
+ +
+Lemma big_enum_box_in sh
+      (R : Type) (idx : R) (op : Monoid.law idx) (f : nat -> nat -> R):
+  \big[op/idx]_(b <- enum_box_in sh ) f b.1 b.2 =
+  \big[op/idx]_(0 <= r < size sh)
+   \big[op/idx]_(0 <= c < nth 0 sh r) f r c.
+ +
+Lemma card_box_in sh : #|{: box_in sh}| = sumn sh.
+ +
+Lemma enum_box_in_uniq sh : uniq (enum_box_in sh).
+ +
+Lemma mem_enum_box_in sh : enum_box_in sh =i in_shape sh.
+ +
+
+ +
+

Adding a box to a shape

+ +
+
+Lemma box_in_incr_nth sh i :
+  perm_eq ((i, nth 0 sh i) :: enum_box_in sh) (enum_box_in (incr_nth sh i)).
+ +
+
+ +
+

The union of two integer partitions

+ +
+
+ +
+Lemma merge_is_part l k :
+  is_part l -> is_part k -> is_part (merge geq l k).
+ +
+Lemma merge_sortedE l k :
+  is_part l -> is_part k -> merge geq l k = sort geq (l ++ k).
+ +
+Lemma sumn_union_part l k : sumn (merge geq l k) = sumn l + sumn k.
+ +
+Fact union_intpart_subproof (l : intpart) (k : intpart) :
+  is_part (merge geq l k).
+ Definition union_intpart (l : intpart) (k : intpart) :=
+  IntPart (union_intpart_subproof l k).
+ +
+Lemma union_intpartE l k : val (union_intpart l k) = sort geq (l ++ k).
+ +
+Lemma perm_union_intpart l k : perm_eq (union_intpart l k) (l ++ k).
+ +
+Lemma union_intpartC l k : union_intpart l k = union_intpart k l.
+ +
+Lemma union_intpartA l k j :
+  union_intpart l (union_intpart k j) = union_intpart (union_intpart k l) j.
+ +
+Section UnionPart.
+ +
+Variables (m n : nat) (l : 'P_m) (k : 'P_n).
+ +
+Lemma union_intpartn_subproof : is_part_of_n (m + n) (merge geq l k).
+Definition union_intpartn := IntPartN union_intpartn_subproof.
+ +
+Lemma union_intpartnE : val union_intpartn = sort geq (l ++ k).
+ +
+Lemma perm_union_intpartn : perm_eq union_intpartn (l ++ k).
+ +
+End UnionPart.
+ +
+Notation "a +|+ b" := (union_intpartn a b) (at level 50) : Combi_scope.
+Bind Scope Combi_scope with intpartn.
+ +
+Section IntpartnCons.
+ +
+Import LeqGeqOrder.
+ +
+Variables (d l0 : nat) (la : seq nat).
+Hypotheses (Hla : is_part_of_n d la)
+           (Hlla : is_part_of_n (l0 + d)%N (l0 :: la)).
+ +
+Lemma intpartn_cons : IntPartN Hlla = rowpartn l0 +|+ IntPartN Hla.
+ +
+End IntpartnCons.
+ +
+
+ +
+

Young lattice on partition

+ +
+
+Module YoungLattice.
+Section YoungLattice.
+ +
+Definition intpartYoung := intpart.
+#[local] Notation "'YL" := intpartYoung.
+Implicit Type (p q sh : 'YL).
+ +
+Definition le_Young p q := included p q.
+Fact le_Young_refl : reflexive le_Young.
+ Fact le_Young_trans : transitive le_Young.
+ Fact le_Young_anti : antisymmetric le_Young.
+#[export] HB.instance Definition _ := Countable.copy 'YL intpart.
+#[export] HB.instance Definition _ := Inhabited.copy 'YL intpart.
+ +
+Fact Young_display : Order.disp_t.
+#[export] HB.instance Definition _ :=
+  Order.Le_isPOrder.Build
+    Young_display 'YL le_Young_refl le_Young_anti le_Young_trans.
+ +
+Lemma le_YoungE sh1 sh2 : (sh1 <= sh2)%O = (included sh1 sh2).
+ Lemma le_YoungP sh1 sh2 :
+  reflect (forall i, nth 0 sh1 i <= nth 0 sh2 i) (sh1 <= sh2)%O.
+ +
+Lemma le_Young_sumn: {homo (fun x : 'YL => sumn x) : x y / (x <= y)%O }.
+ Lemma lt_Young_sumn: {homo (fun x : 'YL => sumn x) : x y / (x < y)%O }.
+ +
+Definition meet_Young_fun sh1 sh2 := [seq minn a.1 a.2 | a <- zip sh1 sh2].
+Lemma size_meet_Young sh1 sh2 :
+  size (meet_Young_fun sh1 sh2) = minn (size sh1) (size sh2).
+ Lemma nth_meet_Young sh1 sh2 i :
+  nth 0 (meet_Young_fun sh1 sh2) i = minn (nth 0 sh1 i) (nth 0 sh2 i).
+ +
+Fact meet_Young_subproof sh1 sh2 : is_part (meet_Young_fun sh1 sh2).
+Definition meet_Young sh1 sh2 : 'YL := IntPart (meet_Young_subproof sh1 sh2).
+ +
+Lemma meet_YoungC : commutative meet_Young.
+ +
+Lemma meet_Young_le sh1 sh2 : (meet_Young sh1 sh2 <= sh1)%O.
+ +
+Fact meet_YoungP sh sh1 sh2 :
+  (sh <= meet_Young sh1 sh2)%O = (sh <= sh1)%O && (sh <= sh2)%O.
+ +
+Definition join_Young_fun sh1 sh2 :=
+  [seq maxn a.1 a.2 | a <- zip (pad 0 (maxn (size sh1) (size sh2)) sh1)
+                               (pad 0 (maxn (size sh1) (size sh2)) sh2)].
+ +
+Lemma size_join_Young sh1 sh2 :
+  size (join_Young_fun sh1 sh2) = maxn (size sh1) (size sh2).
+ +
+Lemma nth_join_Young sh1 sh2 i :
+  nth 0 (join_Young_fun sh1 sh2) i = maxn (nth 0 sh1 i) (nth 0 sh2 i).
+ +
+Fact join_Young_subproof sh1 sh2 : is_part (join_Young_fun sh1 sh2).
+Definition join_Young sh1 sh2 : 'YL := IntPart (join_Young_subproof sh1 sh2).
+ +
+Lemma join_Young_le sh1 sh2 : (sh1 <= join_Young sh1 sh2)%O.
+ +
+Lemma join_YoungC : commutative join_Young.
+ +
+Fact join_YoungP sh1 sh2 sh :
+  (join_Young sh1 sh2 <= sh)%O = (sh1 <= sh)%O && (sh2 <= sh)%O.
+#[export] HB.instance Definition _ :=
+  Order.POrder_MeetJoin_isLattice.Build
+    Young_display 'YL meet_YoungP join_YoungP.
+ +
+Fact emptypart_bottom sh : (empty_intpart <= sh :> 'YL)%O.
+ +
+Lemma bottom_YoungE : \bot%O = empty_intpart :> 'YL.
+ +
+Fact Young_meetUl : @left_distributive 'YL 'YL Order.meet Order.join.
+#[export] HB.instance Definition _ :=
+  Order.Lattice_Meet_isDistrLattice.Build Young_display 'YL Young_meetUl.
+ +
+End YoungLattice.
+ +
+Module Exports.
+ +
+Implicit Type (p q : intpartYoung).
+ +
+Notation intpartYoung := intpartYoung.
+Definition le_YoungE := le_YoungE.
+Definition le_YoungP := le_YoungP.
+Definition le_Young_sumn := le_Young_sumn.
+Definition lt_Young_sumn := lt_Young_sumn.
+ +
+Lemma meet_YoungE p q :
+  (p `&` q)%O = [seq minn a.1 a.2 | a <- zip p q] :> seq nat.
+ Lemma nth_meet_Young p q i :
+  nth 0 (p `&` q)%O i = minn (nth 0 p i) (nth 0 q i).
+ Lemma size_meet_Young p q : size (p `&` q)%O = minn (size p) (size q).
+ +
+Lemma join_YoungE p q :
+  (p `|` q)%O =
+    [seq maxn a.1 a.2 | a <- zip (pad 0 (maxn (size p) (size q)) p)
+                                 (pad 0 (maxn (size p) (size q)) q)] :> seq nat.
+ Lemma nth_join_Young p q i :
+  nth 0 (p `|` q)%O i = maxn (nth 0 p i) (nth 0 q i).
+ Lemma size_join_Young p q : size (p `|` q)%O = maxn (size p) (size q).
+ +
+Definition bottom_YoungE := bottom_YoungE.
+ +
+End Exports.
+End YoungLattice.
+ +
+
+ +
+

Dominance order on partition

+ +
+
+Definition partdomsh n1 n2 (s1 s2 : seq nat) :=
+  all
+    (fun i => n1 + sumn (take i s1) <= n2 + sumn (take i s2))
+    (iota 0 (size s1).+1).
+Definition partdom := partdomsh 0 0.
+ +
+Lemma partdomshP {n1 n2 s1 s2} :
+  reflect (forall i, n1 + sumn (take i s1) <= n2 + sumn (take i s2))
+          (partdomsh n1 n2 s1 s2).
+ +
+Lemma partdomP s1 s2 :
+  reflect (forall i, sumn (take i s1) <= sumn (take i s2)) (partdom s1 s2).
+ +
+Lemma partdomsh_add y n1 n2 s1 s2 :
+  partdomsh (y + n1) (y + n2) s1 s2 = partdomsh n1 n2 s1 s2.
+ +
+Lemma partdom_nil s : partdom [::] s.
+ +
+Lemma partdom_refl : reflexive partdom.
+ +
+#[export] Hint Resolve partdom_nil partdom_refl : core.
+ +
+Lemma partdom_trans : transitive partdom.
+ +
+Lemma partdom_anti s1 s2 :
+  is_part s1 -> is_part s2 -> partdom s1 s2 -> partdom s2 s1 -> s1 = s2.
+ +
+Lemma partdomsh_cons2 y1 y2 s t n1 n2 :
+  partdomsh n1 n2 (y1 :: s) (y2 :: t) =
+  (n1 <= n2) && (partdomsh (n1 + y1) (n2 + y2) s t).
+ +
+Lemma partdomsh_cons2E y s t n1 n2 :
+  partdomsh n1 n2 (y :: s) (y :: t) = partdomsh n1 n2 s t.
+ +
+Lemma partdom_consK x s1 s2 : partdom (x :: s1) (x :: s2) -> partdom s1 s2.
+ +
+Lemma sumn_take_merge t x i :
+  is_part t ->
+  sumn (take i.+1 (merge geq [:: x] t)) =
+  maxn (sumn (take i.+1 t)) (x + sumn (take i t)).
+ +
+Lemma merge_cons x s t :
+  is_part (x :: s) -> is_part t ->
+  merge geq (x :: s) t = merge geq [:: x] (merge geq s t).
+ +
+Lemma partdomsh_merge1 n1 n2 x t1 t2 :
+  is_part t1 -> is_part t2 ->
+  partdomsh n1 n2 t1 t2 ->
+  partdomsh n1 n2 (merge geq [:: x] t1) (merge geq [:: x] t2).
+ +
+Lemma partdomsh_merge n1 n2 s t1 t2 :
+  is_part s -> is_part t1 -> is_part t2 ->
+  partdomsh n1 n2 t1 t2 ->
+  partdomsh n1 n2 (merge geq s t1) (merge geq s t2).
+ +
+Lemma partdom_union_intpartl (s t1 t2 : intpart) :
+  partdom t1 t2 -> partdom (union_intpart s t1) (union_intpart s t2).
+ +
+Lemma partdom_union_intpartr (s t1 t2 : intpart) :
+  partdom t1 t2 -> partdom (union_intpart t1 s) (union_intpart t2 s).
+ +
+Lemma partdom_union_intpart (s1 s2 t1 t2 : intpart) :
+  partdom s1 s2 -> partdom t1 t2 ->
+  partdom (union_intpart s1 t1) (union_intpart s2 t2).
+ +
+Module IntPartNDom.
+Section IntPartNDom.
+ +
+Variable d : nat.
+Definition intpartndom := 'P_d.
+#[local] Notation "'PDom" := intpartndom.
+Implicit Type (sh : 'PDom).
+ +
+#[export] HB.instance Definition _ := Finite.copy 'PDom 'P_d.
+#[export] HB.instance Definition _ := Inhabited.copy 'PDom 'P_d.
+ +
+Fact partdom_antisym : antisymmetric (fun x y : 'P_d => partdom x y).
+ +
+Lemma partdom_display : Order.disp_t.
+#[export] HB.instance Definition _ :=
+  Order.Le_isPOrder.Build partdom_display 'PDom
+    partdom_refl partdom_antisym partdom_trans.
+ +
+Lemma leEpartdom : @Order.le partdom_display 'PDom = partdom.
+ +
+#[local] Notation "sh '^#'" := (conj_intpartn sh : 'PDom)
+                              (at level 10, format "sh '^#'").
+ +
+Lemma sum_diff sh i :
+  \sum_(l <- sh) (l - i) =
+    \sum_(l <- take (nth 0 (conj_part sh) i) sh) (l - i).
+ +
+Lemma partdom_conj_intpartn sh1 sh2 : (sh2^# <= sh1^#)%O = (sh1 <= sh2)%O.
+ +
+Lemma take_intpartn_over sh i : d <= i -> take i sh = sh.
+ +
+Implicit Type t : d.+1.-tuple nat.
+ +
+Definition parttuple sh := [tuple sumn (take i sh) | i < d.+1].
+Definition from_parttuple t :=
+  rem_trail0 [seq nth 0 t i.+1 - nth 0 t i | i <- iota 0 d].
+Definition is_parttuple t :=
+  [&& tnth t ord0 == 0, tnth t ord_max == d, sorted leq t &
+      all (fun i => (nth 0 t i).*2 >= nth 0 t i.-1 + nth 0 t i.+1)
+          (iota 1 d.-1)].
+ +
+Lemma is_parttupleP t :
+  reflect
+    [/\ tnth t ord0 = 0, tnth t ord_max = d, sorted leq t &
+      forall i, 0 < i < d -> (nth 0 t i).*2 >= nth 0 t i.-1 + nth 0 t i.+1]
+    (is_parttuple t).
+ +
+Lemma nth_parttuple i sh :
+  i < d.+1 -> nth 0 (parttuple sh) i = sumn (take i sh).
+ +
+Lemma parttupleP sh : is_parttuple (parttuple sh).
+ +
+Lemma sum_diff_tuple t :
+  sorted leq t ->
+  forall i, i < d.+1 ->
+    \sum_(0 <= j < i) (nth 0 t j.+1 - nth 0 t j) = nth 0 t i - nth 0 t 0.
+ +
+Lemma from_parttupleP t : is_parttuple t -> is_part_of_n d (from_parttuple t).
+Definition part_fromtuple t (H : is_parttuple t) :=
+  IntPartN (from_parttupleP H).
+ +
+Lemma sumn_take_part_fromtuple t (H : is_parttuple t) i :
+  i < d.+1 -> sumn (take i (part_fromtuple H)) = nth 0 t i.
+Lemma from_parttupleK t (H : is_parttuple t) :
+  parttuple (part_fromtuple H) = t.
+ +
+Lemma parttupleK sh : from_parttuple (parttuple sh) = sh.
+Lemma parttuplePK sh : part_fromtuple (parttupleP sh) = sh.
+ +
+Definition parttuple_minn sh1 sh2 :=
+  [tuple minn (tnth (parttuple sh1) i) (tnth (parttuple sh2) i) | i < d.+1].
+ +
+Lemma parttuple_minnC sh1 sh2 :
+  parttuple_minn sh1 sh2 = parttuple_minn sh2 sh1.
+ +
+Lemma nth_parttuple_minn i sh1 sh2 :
+  i < d.+1 -> nth 0 (parttuple_minn sh1 sh2) i =
+              minn (sumn (take i sh1)) (sumn (take i sh2)).
+ +
+Lemma double_minn m n : (minn m n).*2 = minn m.*2 n.*2.
+ +
+Lemma parttuple_minnP sh1 sh2 : is_parttuple (parttuple_minn sh1 sh2).
+Definition meet_intpartn sh1 sh2 : 'PDom :=
+  IntPartN (from_parttupleP (parttuple_minnP sh1 sh2)).
+Definition join_intpartn sh1 sh2 := (meet_intpartn (sh1^#) (sh2^#))^#.
+ +
+Lemma meet_intpartnC sh1 sh2 : meet_intpartn sh1 sh2 = meet_intpartn sh2 sh1.
+ +
+Lemma le_meet_intpartn sh1 sh2 : (meet_intpartn sh1 sh2 <= sh1)%O.
+ +
+Fact meet_intpartnP sh sh1 sh2 :
+  (sh <= meet_intpartn sh1 sh2)%O = (sh <= sh1)%O && (sh <= sh2)%O.
+Fact join_intpartnP sh1 sh2 sh :
+  (join_intpartn sh1 sh2 <= sh)%O = (sh1 <= sh)%O && (sh2 <= sh)%O.
+#[export] HB.instance Definition _ :=
+  Order.POrder_MeetJoin_isLattice.Build partdom_display
+    'PDom meet_intpartnP join_intpartnP.
+ +
+Lemma sumn_take_pardom_meet i sh1 sh2 :
+  sumn (take i (sh1 `&` sh2)%O) = minn (sumn (take i sh1)) (sumn (take i sh2)).
+ +
+Lemma join_intpartnE sh1 sh2 : (sh1 `|` sh2)%O = (sh1^# `&` sh2^#)%O^#.
+ +
+End IntPartNDom.
+Section IntPartNTopBottom.
+ +
+Variable d : nat.
+#[local] Notation "'PDom" := (intpartndom d).
+Implicit Type (sh : 'PDom).
+ +
+Lemma partdom_rowpartn sh : (sh <= rowpartn d)%O.
+ +
+Lemma partdom_colpartn sh : (colpartn d <= sh :> 'PDom)%O.
+ +
+#[export] HB.instance Definition _ :=
+  Order.hasBottom.Build partdom_display 'PDom partdom_colpartn.
+#[export] HB.instance Definition _ :=
+  Order.hasTop.Build partdom_display 'PDom partdom_rowpartn.
+ +
+Lemma botEintpartndom : \bot%O = colpartn d :> 'PDom.
+ Lemma topEintpartndom : \top%O = rowpartn d :> 'PDom.
+ +
+End IntPartNTopBottom.
+ +
+Module Exports.
+ +
+Notation "''PDom_' n" := (intpartndom n).
+ +
+Definition leEpartdom := leEpartdom.
+Definition partdom_conj_intpartn := partdom_conj_intpartn.
+Definition sumn_take_pardom_meet := sumn_take_pardom_meet.
+Definition join_intpartnE := join_intpartnE.
+Definition botEintpartndom := botEintpartndom.
+Definition topEintpartndom := topEintpartndom.
+ +
+End Exports.
+End IntPartNDom.
+ +
+Lemma le_intpartndomlexi d :
+  {homo (id : 'PDom_d -> 'PLexi_d) : x y / (x <= y)%O}.
+Lemma lt_intpartndomlexi d :
+  {homo (id : 'PDom_d -> 'PLexi_d) : x y / (x < y)%O}.
+ +
+
+ +
+

Shape of set partitions and integer partitions

+ +
+
+Section SetPartitionShape.
+ +
+Variable T : finType.
+Implicit Types (A B X : {set T}) (P Q : {set {set T}}).
+ +
+Lemma count_set_of_card (p : pred nat) P :
+  count p [seq #{x} | x in P] = #|P :&: [set x | p #{x}]|.
+ +
+Definition setpart_shape P := sort geq [seq #{X} | X in P].
+ +
+Lemma setpart_shapeP P D :
+  partition P D -> is_part_of_n #|D| (setpart_shape P).
+ +
+Lemma ex_subset_card B k :
+  k <= #{B} -> exists2 A : {set T}, A \subset B & #{A} == k.
+ +
+Lemma ex_setpart_shape (s : seq nat) (A : {set T}) :
+  sumn s = #|A| -> 0 \notin s ->
+  exists P : seq {set T},
+    [/\ uniq P, partition [set X in P] A & [seq #{X} | X <- P] = s].
+ +
+Lemma ex_set_setpart_shape A (sh : 'P_#|A|) :
+  exists2 P, partition P A & setpart_shape P = sh.
+ +
+Lemma setpart_shape_union P Q :
+  [disjoint P & Q] ->
+  setpart_shape (P :|: Q) = sort geq (setpart_shape P ++ setpart_shape Q).
+ +
+End SetPartitionShape.
+ +
+Lemma setpart_shape_imset
+      (T1 T2 : finType) (f : T1 -> T2) (A : {set {set T1}}) :
+  injective f ->
+  setpart_shape [set f @: x | x : {set T1} in A] = setpart_shape A.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.permuted.html b/combi/1.1.0/Combi.Combi.permuted.html new file mode 100644 index 00000000..9d47298a --- /dev/null +++ b/combi/1.1.0/Combi.Combi.permuted.html @@ -0,0 +1,294 @@ + + + + + +Combi.Combi.permuted: Listing the Permutations of a tuple or seq + + + + +
+ + + +
+ +

Library Combi.Combi.permuted: Listing the Permutations of a tuple or seq

+ +
+
+ +
+ +
+
+
+ +
+

The list of the permuted tuple of a given tuple

+ + +
+ +The main goal is to show that, given a sequence s over an eqType there +are only finitely many sequences s' which are a permutation of s (that is +perm_eq s s') and to show that the number is a multinomial coefficient. + +
+ +
    +
  • permuted_tuple t == a sequence of tuples containing (with duplicates) all + tuple t' such that perm_eq t t' + +
    • +
  • permuted_seq s == a sequence of seqs containing (with duplicates) all seqs + s' such that perm_eq s s' + +
    • +
  • permuted t == sigma typle for tuple t' such that perm_eq t t'. this + is canonically a fintype, provided the type of the elements of + t is a countType. + +
    + + +
    • +
  • permutedact t s == the n tuple t permuted by the permutation s + +
    • +
  • permuted_action == the corresponding action of the symmetric group 'S_n + +
    • +
+ +
+ +The main result is the cardinality of the set of permuted of a tuple expressed +as a multinomial card_permuted_multinomial. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import perm fingroup action gproduct.
+ +
+Require Import tools combclass sorted partition composition multinomial.
+Require Import permcomp cycles.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Enumeration of the permutation of a tuple

+ +
+
+Section Permuted.
+ +
+Variable T : eqType.
+ +
+Section SizeN.
+ +
+Variable n : nat.
+ +
+Definition permuted_tuple (t : n.-tuple T) :=
+  [seq [tuple tnth t (aperm i p) | i < n] | p : 'S_n ].
+ +
+Lemma size_permuted_tuple (t : n.-tuple T) : size (permuted_tuple t) = n`!.
+ +
+Lemma perm_eq_permuted_tuple (s : seq T) (H : size s == n) :
+  forall s1, perm_eq s s1 ->
+             s1 \in [seq tval t | t <- permuted_tuple (Tuple H)].
+ +
+Lemma mem_enum_permuted (s t : n.-tuple T) :
+  perm_eq s t -> t \in permuted_tuple s.
+ +
+Lemma all_permuted (s : n.-tuple T) :
+  all (fun x : n.-tuple T => perm_eq s x) (permuted_tuple s).
+ +
+End SizeN.
+ +
+Definition permuted_seq s :=
+  [seq tval t | t <- permuted_tuple (Tuple (eq_refl (size s)))].
+ +
+Lemma size_permuted_seq s : size (permuted_seq s) = (size s)`!.
+ +
+Lemma eq_seqE s s1 : perm_eq s s1 -> s1 \in permuted_seq s.
+ +
+End Permuted.
+ +
+
+ +
+

Permutation of a tuple as a fintype

+ +
+
+Section FinType.
+ +
+Variable T : countType.
+Variable n : nat.
+Variable w : n.-tuple T.
+ +
+Structure permuted : predArgType :=
+  Permuted { tpval :> n.-tuple T; _ : perm_eq w tpval }.
+ +
+ +
+Lemma permutedP (p : permuted) : perm_eq w p.
+ +
+End FinType.
+ +
+#[export] Hint Resolve permutedP : core.
+ +
+Import GroupScope.
+ +
+
+ +
+

Action of 'S_n on permuted for and n.-tuple T.

+ + +
+ +There is no use defining an action on general tuple because most of the lemmas +on actions assume that the type acted upon is a finType. We could require +moreover that the underlying type is a fintype so that the set of tuple is a +fintype too, but the use we have in mind is T = nat allowing to deal with +the action on monomials. Instead of that, we decide to act only on the sigma +type permuted. + +
+
+ +
+Section ActOnTuple.
+ +
+Variables (T : countType) (n : nat) (w : n.-tuple T).
+Implicit Type (t : permuted w).
+ +
+#[local] Notation wp := (Permuted (perm_refl w)).
+ +
+Lemma permutedact_subproof t (s : 'S_n) :
+  perm_eq w [tuple tnth t (s^-1 i) | i < n].
+Definition permutedact t s := Permuted (permutedact_subproof t s).
+ +
+#[local] Notation "t # s" := (permutedact t s)
+  (at level 40, left associativity, format "t # s").
+ +
+Lemma permutedact_is_action : is_action [set: 'S_n] permutedact.
+Canonical permuted_action := Action permutedact_is_action.
+#[local] Notation pact := permuted_action.
+ +
+Lemma permuted_action_trans :
+  [transitive [set: 'S_n], on [set: permuted w] | pact].
+ +
+
+ +
+

The stabilizer of a tuple under permutation

+ +
+
+Lemma stab_tuple_prod :
+  'C[wp | pact] =
+  (\prod_(x : seq_sub w) Sym_group [set i | tnth w i == val x])%G.
+ +
+Lemma stab_tuple_dprod :
+  'C[wp | pact] =
+  \big[dprod/1]_(x : seq_sub w) Sym [set i | tnth w i == val x].
+ +
+Close Scope group_scope.
+ +
+Lemma card_stab_tuple :
+  #|('C[wp | pact])%G| = \prod_(x : seq_sub w) (count_mem (val x) w)`!.
+ +
+Lemma card_permuted_prod :
+  #|[set: permuted w]| * \prod_(x : seq_sub w) (count_mem (val x) w)`! = n`!.
+ +
+Lemma dvdn_card_permuted :
+  \prod_(x : seq_sub w) (count_mem (val x) w)`! %| n`!.
+ +
+Lemma card_permuted_seq_sub :
+  #|[set: permuted w]| = n`! %/ \prod_(x : seq_sub w) (count_mem (val x) w)`!.
+ +
+Lemma card_permuted :
+  #|[set: permuted w]| = n`! %/ \prod_(x <- undup w) (count_mem x w)`!.
+ +
+Lemma size_count_mem_undup (s : seq T) :
+  size s = \sum_(j <- undup s) count_mem j s.
+ +
+Theorem card_permuted_multinomial :
+  #|[set: permuted w]| = 'C[[seq count_mem x w | x <- undup w]].
+ +
+Corollary card_permuted_multinomial_subset (s : seq T) :
+  {subset w <= s} -> uniq s ->
+  #|[set: permuted w]| = 'C[[seq count_mem x w | x <- s]].
+ +
+End ActOnTuple.
+ +
+Require Import sorted.
+Import LeqGeqOrder.
+ +
+Lemma card_preim_part_of_compn n (sh : 'P_n) :
+  #|[set c | partn_of_compn c == sh]| =
+  (size sh)`! %/ \prod_(i <- iota 1 n) (count_mem i sh)`!.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.setpartition.html b/combi/1.1.0/Combi.Combi.setpartition.html new file mode 100644 index 00000000..88d5db3f --- /dev/null +++ b/combi/1.1.0/Combi.Combi.setpartition.html @@ -0,0 +1,405 @@ + + + + + +Combi.Combi.setpartition: Set Partitions + + + + +
+ + + +
+ +

Library Combi.Combi.setpartition: Set Partitions

+ +
+
+ +
+ +
+
+
+ +
+

Set Partitions and refinment lattice

+ + +
+ +In what follows T is a finType and S : {set T}. + +
+ +
    +
  • setpart S == a sigma type for partition of S. setpart S bears a + canonical structure of an inhabited finite lattice type. + +
    • +
  • setpart1 S == the partition of S made only of singleton + +
    • +
  • trivsetpart S == the partition whose only block is S itself + if S is non empty + +
    • +
  • setpart_set0 T == the trivial empty partition of set0 in setpart set0 + +
    • +
  • setpart_set1 x == the trivial partition of [set x] in setpart [set x] + +
    + + +
    • +
  • join_finer_eq P Q == the equivalence relation obtained as the join of + the one of P and Q. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import div ssralg ssrint ssrnum binomial.
+Require Import tools combclass sorted ordtype partition.
+ +
+Set Implicit Arguments.
+ +
+Import LeqGeqOrder.
+Import Order.TTheory.
+ +
+Lemma pblock_in (T : finType) (P : {set {set T}}) (x : T) :
+  set0 \notin P -> (pblock P x \in P) = (x \in cover P).
+ +
+Lemma mem_pblockC (T : finType) (P : {set {set T}}) :
+  trivIset P -> forall x y : T, (x \in pblock P y) = (y \in pblock P x).
+ +
+Lemma equivalence_partitionE (T : finType) (R S : rel T) (D : {set T}) :
+  R =2 S -> equivalence_partition R D = equivalence_partition S D.
+ +
+Section Defs.
+ +
+Variable T : finType.
+Variable S : {set T}.
+ +
+Structure setpart : predArgType :=
+  SetPart {setpartval :> {set {set T}}; _ : partition setpartval S}.
+ +
+Implicit Type (P Q : setpart) (A B : {set T}).
+ +
+Lemma setpartP P : partition P S.
+ +
+Lemma trivIsetpart P : trivIset P.
+ +
+Lemma cover_setpart P : cover P = S.
+ +
+Lemma setpart_non0 P : set0 \notin P.
+ +
+Hint Resolve setpartP trivIsetpart setpart_non0 : core.
+ +
+Lemma setpart_subset P A : A \in P -> A \subset S.
+ +
+Lemma setpart_eq P x A B :
+  A \in P -> B \in P -> x \in A -> x \in B -> A = B.
+ +
+Lemma mem_eq_pblock P A : A \in P -> forall x, x \in A -> pblock P x = A.
+ +
+Lemma mem_setpart_pblock P A : A \in P -> exists x, x \in A.
+ +
+Lemma mem_pblock_setpart P x y : x \in pblock P y -> x \in S.
+ +
+Lemma pblock_notin P x : (pblock P x != set0) = (x \in S).
+ +
+Fact setpart1_subproof : partition [set [set x] | x in S] S.
+Canonical setpart1 := SetPart setpart1_subproof.
+ +
+Lemma pblock_setpart1 x : x \in S -> pblock setpart1 x = [set x].
+ +
+Fact trivsetpart_subproof :
+  partition (T := T) (if S == set0 then set0 else [set S]) S.
+Canonical trivsetpart := SetPart trivsetpart_subproof.
+ +
+Lemma pblock_trivsetpart x : x \in S -> pblock trivsetpart x = S.
+ +
+End Defs.
+#[export] Hint Resolve setpartP trivIsetpart setpart_non0 : core.
+ +
+Section Empty.
+ +
+Variable T : finType.
+ +
+Fact part_ordinal0 : partition (T := T) set0 set0.
+Definition setpart_set0 := SetPart part_ordinal0.
+ +
+Lemma setpart_set0_eq_set0 (p : setpart set0) : p = setpart_set0.
+ +
+Lemma enum_setpart_set0 : enum (setpart (@set0 T)) = [:: setpart_set0 ].
+ +
+End Empty.
+ +
+Section Singleton.
+ +
+Variable T : finType.
+Variable x : T.
+ +
+Implicit Type (A B : {set T}).
+ +
+Fact part_ordinal1 : partition [set [set x]] [set x].
+Definition setpart_set1 := SetPart part_ordinal1.
+ +
+Lemma subset_set1 A :
+  A \subset [set x] -> (A = set0) \/ (A = [set x]).
+ +
+Lemma part_set1_eq (P : {set {set T}}) :
+  partition P [set x] -> P = [set [set x]].
+ +
+Lemma setpart_set1_eq_set1 (p : setpart [set x]) : p = setpart_set1.
+ +
+Lemma enum_setpart_set1 : enum (setpart [set x]) = [:: setpart_set1].
+ +
+End Singleton.
+ +
+Module RefinmentOrder.
+Section RefinmentOrder.
+ +
+Import LeqGeqOrder.
+ +
+Context {T : finType} (S : {set T}).
+ +
+Implicit Type (P Q R : setpart S) (A B X Y : {set T}).
+ +
+Definition is_finer P Q :=
+  [forall (x | x \in S), pblock P x \subset pblock Q x].
+ +
+Lemma is_finer_pblockP P Q :
+  reflect (forall x, x \in S -> pblock P x \subset pblock Q x) (is_finer P Q).
+ +
+Lemma is_finer_refl : reflexive is_finer.
+ +
+Lemma is_finer_trans : transitive is_finer.
+ +
+Fact is_finer_setpart_anti : antisymmetric is_finer.
+Lemma setpartfiner_display : Order.disp_t.
+#[export]
+HB.instance Definition _ :=
+  Order.Le_isPOrder.Build setpartfiner_display (setpart S)
+    is_finer_refl is_finer_setpart_anti is_finer_trans.
+ +
+Lemma is_finerP P Q :
+  reflect
+    (forall A, A \in P -> exists2 B : {set T}, B \in Q & A \subset B)
+    (P <= Q)%O.
+ +
+Fact setpart1_bottom P : (setpart1 S <= P)%O.
+Fact trivsetpart_top P : (P <= trivsetpart S)%O.
+#[export]
+HB.instance Definition _ :=
+  Order.hasBottom.Build setpartfiner_display (setpart S) setpart1_bottom.
+#[export]
+HB.instance Definition _ :=
+  Order.hasTop.Build setpartfiner_display (setpart S) trivsetpart_top.
+ +
+Fact meet_finer_subproof P Q :
+  partition [set A :&: B | A in P, B in Q & A :&: B != set0] S.
+Definition meet_finer P Q := SetPart (meet_finer_subproof P Q).
+ +
+Variant meet_spec P Q X : Prop :=
+  MeetSpec A B of A \in P & B \in Q & A :&: B != set0 & X = A :&: B.
+Lemma mem_meet_finerP P Q X :
+  reflect (meet_spec P Q X) (X \in meet_finer P Q).
+ +
+Lemma meet_finerC : commutative meet_finer.
+ +
+Lemma le_meet_finer P Q : (meet_finer P Q <= P)%O.
+ +
+Fact meet_finerP R P Q : (R <= meet_finer P Q)%O = (R <= P)%O && (R <= Q)%O.
+ +
+Lemma setpart_conn P : connect_sym (fun x y : T => y \in (pblock P x)).
+ +
+Definition join_finer_eq P Q :=
+  connect (relU (fun x y => y \in pblock P x) (fun x y => y \in pblock Q x)).
+ +
+Lemma join_finer_equivalence P Q :
+  {in S & &, equivalence_rel (join_finer_eq P Q)}.
+ +
+Definition join_finer P Q :=
+  SetPart (equivalence_partitionP (join_finer_equivalence P Q)).
+ +
+Lemma join_finerC : commutative join_finer.
+ +
+Lemma le_join_finer P Q : (P <= join_finer P Q)%O.
+ +
+Fact join_finerP P Q R : (join_finer P Q <= R)%O = (P <= R)%O && (Q <= R)%O.
+#[export]
+HB.instance Definition _ :=
+  Order.POrder_MeetJoin_isLattice.Build setpartfiner_display (setpart S)
+    meet_finerP join_finerP.
+ +
+End RefinmentOrder.
+ +
+Module Exports.
+ +
+Section Finer.
+ +
+Context {T : finType} (S : {set T}).
+Implicit Type (P Q R : setpart S) (A B X Y : {set T}).
+ +
+Definition is_finerP P := is_finerP P.
+Definition is_finer_pblockP P Q :
+  reflect (forall x, x \in S -> pblock P x \subset pblock Q x) (P <= Q)%O
+  := is_finer_pblockP P Q.
+ +
+Lemma is_finer_subpartP P Q :
+  reflect
+    (exists2 subpart : {set T} -> {set {set T}},
+        forall B, B \in Q -> partition (subpart B) B
+        & \bigcup_(B in Q) subpart B = P)
+    (P <= Q)%O.
+ +
+Lemma setpart_bottomE : \bot%O = setpart1 S.
+ Lemma setpart_topE : \top%O = trivsetpart S.
+ +
+Definition mem_meet_finerP P Q X : reflect (meet_spec P Q X) (X \in P `&` Q)%O
+  := mem_meet_finerP P Q X.
+ +
+Lemma join_finer_eq_in_S P Q x y :
+  x \in S -> join_finer_eq P Q x y -> y \in S.
+ +
+Lemma join_finerE P Q x y :
+  x \in S -> y \in pblock (P `|` Q)%O x = join_finer_eq P Q x y.
+ +
+End Finer.
+ +
+Notation join_finer_eq := join_finer_eq.
+ +
+End Exports.
+End RefinmentOrder.
+ +
+Section FinerCard.
+ +
+Context {T : finType} (S : {set T})
+  (P Q : setpart S) (finPQ : (P <= Q)%O).
+ +
+Implicit Types (A B : {set T}).
+ +
+Definition map_finer A : {set T} :=
+    if (boolP (A \in P)) is AltTrue pin then
+      let: exist2 B _ _ := sig2W (is_finerP P Q finPQ A pin) in B
+    else set0.
+ +
+Lemma map_finer_subset A : A \in P -> A \subset map_finer A.
+ +
+Lemma map_finer_in A : A \in P -> map_finer A \in Q.
+ +
+Lemma map_finer_pblock x : map_finer (pblock P x) = pblock Q x.
+ +
+Lemma image_map_finer : [set map_finer x | x in P] = Q.
+ +
+Lemma is_finer_card : #|Q| <= #|P| ?= iff (P == Q).
+ +
+End FinerCard.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.skewpart.html b/combi/1.1.0/Combi.Combi.skewpart.html new file mode 100644 index 00000000..5288756f --- /dev/null +++ b/combi/1.1.0/Combi.Combi.skewpart.html @@ -0,0 +1,955 @@ + + + + + +Combi.Combi.skewpart: Skew Partitions + + + + +
+ + + +
+ +

Library Combi.Combi.skewpart: Skew Partitions

+ +
+
+ +
+ +
+
+
+ +
+

Skew partitions:

+ + +
+ +Horizontal and vertical border strips: + +
+ +
    +
  • hb_strip inn out == out/inn is an horizontal border strip. That is inn + is included in out and each column contain at most one box + +
    • +
  • vb_strip inn out == out/inn is a vertical border strip. That is inn + is included in out and each row contain at most one box + +
    • +
+ +
+ +Ribbon border strips: + +
+ +According to the textbook definition, a ribbon is a non empty skew shape +which is connected and contains no 2x2 square. Unfortunately this definition +is totally inoperative. We use a different definition and prove the +equivalence with the textbook one. + +
+ +
    +
  • mindropeq s1 s2 == the minimal d such that drop d s1 == drop d s2 + +
    • +
  • ribbon_from inn out == out/inn is a ribbon shape starting at row 0 + +
    • +
  • ribbon inn out == out/inn is a ribbon shape + +
    • +
  • ribbon_on start stop inn out <-> out/inn is a ribbon shape starting and + ending at rows start and stop. + +
    • +
  • ribbon_height inn out == the height of the shape out/inn + +
    • +
  • add_ribbon_on sh start stop rem == adds a ribbon to sh from + start to stop with rem boxes at row start. The result + may not be a partition if there is an extra box. + +
    • +
  • add_ribbon sh nbox pos == tries to add a ribbon of size nbox.+1 + ending at row pos to the shape sh. Returns Some (res, h) + where res is the outer shape partition and h its height + on success, or None if not. Compared to add_ribbon_on, the + result is always a partition on success. + +
    • +
  • add_ribbon_intpartn sh nbox pos == sigma type version of add_ribbon. + Takes sh : 'P_d for some d and returns None or + Some (res, hgt) where res is of type 'P_(nbox.+1 + m). + +
    • +
+ +
+ +Textbook definition of ribbon: + +
+ +
    +
  • neig4 (r, c) == the 4 neighbors of r, c. + +
    • +
  • neig4box b == idem as a box_in_skew inner outer where b is itself a + box in out/inn + +
    • +
  • conn4_skew inner outer == the skew shape out/inn is 4-connected + +
    • +
  • has_no_square inner outer == the shape out/inn has no 2x2 square + +
    • +
  • ribbon_textbook inner outer == the textbook definition. + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+Require Import tools sorted partition.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Horizontal and vertical border strips

+ +
+
+Fixpoint hb_strip inner outer :=
+  if inner is inn0 :: inn then
+    if outer is out0 :: out then
+      (head 0 out <= inn0 <= out0) && (hb_strip inn out)
+    else false
+  else if outer is out0 :: out then out == [::]
+       else true.
+ +
+Fixpoint vb_strip inner outer :=
+  if outer is out0 :: out then
+    if inner is inn0 :: inn then
+      (inn0 <= out0 <= inn0.+1) && (vb_strip inn out)
+    else (out0 == 1) && (vb_strip [::] out)
+  else inner == [::].
+ +
+Lemma hb_strip_included inner outer :
+  hb_strip inner outer -> included inner outer.
+ +
+Lemma hb_strip_size inner outer :
+  hb_strip inner outer -> size inner <= size outer <= (size inner).+1.
+ +
+Lemma vb_strip_included inner outer :
+  vb_strip inner outer -> included inner outer.
+ +
+Lemma hb_stripP inner outer :
+  is_part inner -> is_part outer ->
+  reflect
+    (forall i, nth 0 outer i.+1 <= nth 0 inner i <= nth 0 outer i)
+    (hb_strip inner outer).
+ +
+Lemma vb_stripP inner outer :
+  is_part inner -> is_part outer ->
+  reflect
+    (forall i, nth 0 inner i <= nth 0 outer i <= (nth 0 inner i).+1)
+    (vb_strip inner outer).
+ +
+
+ +
+Usual definition : at most one box on each row +
+
+Lemma vb_strip_diffP inner outer :
+  is_part inner -> is_part outer ->
+  reflect
+    (included inner outer /\ all (geq 1) (outer / inner))
+    (vb_strip inner outer).
+ +
+Lemma vb_strip_conj inner outer :
+  is_part inner -> is_part outer ->
+  vb_strip inner outer -> hb_strip (conj_part inner) (conj_part outer).
+ +
+Lemma hb_strip_conj inner outer :
+  is_part inner -> is_part outer ->
+  hb_strip inner outer -> vb_strip (conj_part inner) (conj_part outer).
+ +
+Lemma hb_strip_conjE inner outer :
+  is_part inner -> is_part outer ->
+  hb_strip (conj_part inner) (conj_part outer) = vb_strip inner outer.
+ +
+Lemma vb_strip_conjE inner outer :
+  is_part inner -> is_part outer ->
+  vb_strip (conj_part inner) (conj_part outer) = hb_strip inner outer.
+ +
+
+ +
+

Ribbon border strip

+ +
+ +

The first common drop of two sequences

+ +
+
+Section MinDropEq.
+ +
+Variable (T : eqType).
+Implicit Types (s : seq T).
+ +
+Fact ex_dropeq s1 s2 : exists n, drop n s1 == drop n s2.
+Definition mindropeq s1 s2 := ex_minn (ex_dropeq s1 s2).
+ +
+Lemma mindropeqC s1 s2 : mindropeq s1 s2 = mindropeq s2 s1.
+ +
+Lemma mindropeq0 s1 s2 : mindropeq s1 s2 = 0 -> s1 = s2.
+ +
+Lemma mindropeq_eq s : mindropeq s s = 0.
+ +
+Lemma mindropeq_leq s1 s2 : mindropeq s1 s2 <= maxn (size s1) (size s2).
+ +
+Lemma mindropeq_nil s : mindropeq [::] s = size s.
+ +
+Lemma mindropeq_cons_eq a b s : a != b -> mindropeq (a :: s) (b :: s) = 1.
+ +
+Lemma mindropeq_cons_neq a b s1 s2 :
+  s1 != s2 -> mindropeq (a :: s1) (b :: s2) = (mindropeq s1 s2).+1.
+ +
+Lemma mindropeq_nthP x0 s t p :
+  x0 \notin s -> x0 \notin t ->
+  reflect
+    (nth x0 s p != nth x0 t p /\ (forall i, i > p -> nth x0 s i = nth x0 t i))
+    (mindropeq s t == p.+1).
+ +
+End MinDropEq.
+ +
+
+ +
+

An operative definition for ribbon border strips

+ +
+
+Fixpoint ribbon_from inner outer :=
+  if inner is inn0 :: inn then
+    if outer is out0 :: out then
+      (inn0 < out0) &&
+      ((out == inn) || ((head 0 out == inn0.+1) && (ribbon_from inn out)))
+    else false
+  else if outer is out0 :: out then head 0 out <= 1
+       else false.
+Fixpoint ribbon inner outer :=
+  ribbon_from inner outer ||
+  if (inner, outer) is (inn0 :: inn, out0 :: out) then
+    (out0 == inn0) && (ribbon inn out)
+  else false.
+ +
+Definition ribbon_on start stop inner outer :=
+  [/\ forall i, i > stop -> nth 0 outer i = nth 0 inner i,
+     forall i, start <= i < stop -> nth 0 outer i.+1 = (nth 0 inner i).+1,
+     nth 0 inner start < nth 0 outer start &
+     forall i, i < start -> nth 0 outer i = nth 0 inner i].
+Definition ribbon_height inner outer := count (ltn 0) (outer / inner).
+ +
+Section Test.
+ +
+Goal ~~ ribbon_from [::] [::].
+ Goal ~~ ribbon_from [:: 2] [:: 2].
+ Goal ribbon_from [::] [:: 4].
+ Goal ribbon_from [:: 2] [:: 4].
+ Goal ~~ ribbon_from [:: 2] [:: 4; 2].
+ Goal ribbon_from [:: 2] [:: 3].
+ Goal ribbon_from [:: 2] [:: 4; 3].
+ Goal ribbon_from [:: 3; 2] [:: 4; 4].
+ Goal ~~ ribbon_from [:: 3; 2] [:: 4; 4; 1].
+ Goal ~~ ribbon_from [:: 3; 2] [:: 4; 4; 2].
+ Goal ribbon_from [:: 3; 2; 2] [:: 4; 4; 2].
+ Goal ~~ ribbon_from [:: 2; 2] [:: 4; 4].
+ +
+Goal ribbon [:: 2] [:: 3].
+ Goal ribbon [:: 2; 2] [:: 3; 2].
+ Goal ribbon [:: 2; 2] [:: 3; 3].
+ Goal ribbon [:: 5; 3; 2; 2] [:: 5; 4; 4; 2].
+ Goal ~~ ribbon [:: 5; 3; 2; 2] [:: 5; 4; 4; 2; 1].
+ Goal ~~ ribbon [::] [::].
+ Goal ~~ ribbon [:: 2; 1] [:: 2; 1].
+ +
+End Test.
+ +
+Lemma ribbon_from_impl inn out : ribbon_from inn out -> ribbon inn out.
+ +
+Lemma ribbon_consK inn0 inn out0 out :
+  ribbon (inn0 :: inn) (out0 :: out) -> (out == inn) || ribbon inn out.
+Lemma ribbonE inn0 inn out0 out :
+  inn0 < out0 ->
+  ribbon (inn0 :: inn) (out0 :: out) = ribbon_from (inn0 :: inn) (out0 :: out).
+ +
+Lemma ribbon_from_noneq inner outer : ribbon_from inner outer -> inner != outer.
+Lemma ribbon_noneq inner outer : ribbon inner outer -> outer != inner.
+ +
+Lemma ribbon_onSS start stop inn0 inn out0 out :
+  ribbon_on start.+1 stop.+1 (inn0 :: inn) (out0 :: out)
+  <-> inn0 == out0 /\ ribbon_on start stop inn out.
+ +
+Section RibbonOn.
+ +
+Variables (start stop : nat) (inner outer : seq nat).
+Hypotheses (partinn : is_part inner)
+           (partout : is_part outer)
+           (Hrib : ribbon_on start stop inner outer).
+ +
+Lemma ribbon_on_start_stop : start <= stop.
+ +
+Lemma ribbon_on_nth_leq i :
+  (start <= i <= stop) = (nth 0 inner i < nth 0 outer i).
+ +
+Lemma ribbon_on_is_skew r c :
+  in_skew inner outer (r, c) -> start <= r <= stop.
+ +
+Lemma ribbon_on_included : included inner outer.
+ +
+Lemma ribbon_on_start_le : start <= size inner.
+ +
+Lemma ribbon_on_stop_lt : stop < size outer.
+ +
+Lemma ribbon_on_height : (stop - start).+1 = ribbon_height inner outer.
+ +
+Lemma ribbon_on_mindropeq : mindropeq inner outer = stop.+1.
+ +
+Lemma ribbon_on_stopE : stop = (mindropeq inner outer).-1.
+ +
+Lemma ribbon_on_startE :
+  start = mindropeq inner outer - ribbon_height inner outer.
+ +
+Lemma ribbon_on_sumn :
+  sumn (outer / inner) = (stop - start) + (nth 0 outer start - nth 0 inner stop).
+ +
+End RibbonOn.
+ +
+Lemma ribbon_on_inj inner outer start1 stop1 start2 stop2 :
+  is_part inner ->
+  ribbon_on start1 stop1 inner outer ->
+  ribbon_on start2 stop2 inner outer ->
+  (start1, stop1) = (start2, stop2).
+ +
+Lemma ribbon_fromP inner outer :
+  is_part inner -> is_part outer ->
+  reflect (exists stop, ribbon_on 0 stop inner outer)
+          (ribbon_from inner outer).
+ +
+Lemma ribbon_from_mindropeq inner outer :
+  is_part inner -> is_part outer -> ribbon_from inner outer ->
+  ribbon_on 0 (mindropeq inner outer).-1 inner outer.
+ +
+Lemma ribbonP inner outer :
+  is_part inner -> is_part outer ->
+  reflect (exists start stop, ribbon_on start stop inner outer)
+          (ribbon inner outer).
+ +
+Lemma ribbon_mindropeq inner outer :
+  is_part inner -> is_part outer -> ribbon inner outer ->
+  let min := mindropeq inner outer in
+  let height := ribbon_height inner outer in
+  ribbon_on (min - height) min.-1 inner outer.
+ +
+Lemma ribbon_from_included inner outer :
+  ribbon_from inner outer -> included inner outer.
+Lemma ribbon_included inner outer :
+  ribbon inner outer -> included inner outer.
+ +
+Lemma ribbon_sumn_lt inner outer :
+  is_part outer -> ribbon inner outer -> sumn inner < sumn outer.
+ +
+Lemma ribbon_sumn_diffE inner outer :
+  is_part outer -> ribbon inner outer ->
+  (sumn (outer / inner)).-1.+1 = (sumn (outer / inner)).
+ +
+Fixpoint startrem acc sh nbox pos :=
+  if (pos, sh) is (p.+1, s0 :: s) then
+    let c := nth 0 s p + nbox in
+    let cpos := s0 + pos in
+    if c >= cpos then (acc, c - cpos)
+    else startrem acc.+1 s nbox p
+  else (acc + (pos - nbox), nbox - pos). +
+ +
+Lemma startrem_acc_geq acc sh nbox pos : acc <= (startrem acc sh nbox pos).1.
+ +
+Lemma startrem_leq_pos sh nbox pos : (startrem 0 sh nbox pos).1 <= pos.
+Lemma startrem_leq_size sh nbox pos :
+  let (start, rem) := startrem 0 sh nbox pos in
+  0 < rem -> start <= size sh.
+Lemma startrem_leq sh nbox pos :
+  let (start, rem) := startrem 0 sh nbox pos in
+  0 < rem -> start <= minn pos (size sh).
+ +
+Lemma startrem_accE acc sh nbox pos :
+  let (start, rem) := startrem acc sh nbox pos in
+  nth 0 sh (start - acc) + pos + rem = nth 0 sh pos + (start - acc) + nbox.
+Lemma startremE sh nbox pos :
+  let (start, rem) := startrem 0 sh nbox pos in
+  nth 0 sh start + pos + rem = nth 0 sh pos + start + nbox.
+ +
+Lemma eq_interv_part sh st1 st2 v :
+  is_part sh ->
+  nth 0 sh st1.+1 <= st1 + v <= nth 0 sh st1 ->
+  nth 0 sh st2.+1 <= st2 + v <= nth 0 sh st2 ->
+  st1 = st2.
+ +
+Lemma startrem_accP acc sh nbox pos :
+  is_part sh ->
+  let (start, rem) := startrem acc sh nbox pos in
+  0 < rem ->
+  (start == acc) || (nth 0 sh (start - acc) + rem <= nth 0 sh (start - acc).-1).
+ +
+Lemma startrem0P acc sh nbox pos :
+  nth 0 sh 0 + pos <= nth 0 sh pos + nbox ->
+  startrem acc sh nbox pos = (acc, nth 0 sh pos + nbox - (nth 0 sh 0 + pos)).
+ +
+Lemma ribbon_on0_startrem stop inner outer acc :
+  is_part inner ->
+  is_part outer ->
+  ribbon_on 0 stop inner outer ->
+  startrem acc inner (sumn (outer / inner)) stop =
+  (acc, nth 0 outer 0 - nth 0 inner 0).
+ +
+Lemma ribbon_on_startrem_acc start stop inner outer acc :
+  is_part inner ->
+  is_part outer ->
+  ribbon_on start stop inner outer ->
+  startrem acc inner (sumn (outer / inner)) stop =
+  (acc + start, nth 0 outer start - nth 0 inner start).
+ +
+Lemma ribbon_on_startrem start stop inner outer :
+  is_part inner ->
+  is_part outer ->
+  ribbon_on start stop inner outer ->
+  startrem 0 inner (sumn (outer / inner)) stop =
+  (start, nth 0 outer start - nth 0 inner start).
+ +
+Lemma ribbon_startrem inner outer :
+  is_part inner -> is_part outer -> ribbon inner outer ->
+  let min := mindropeq inner outer in
+  let height := ribbon_height inner outer in
+  startrem 0 inner (sumn (outer / inner)) (mindropeq inner outer).-1 =
+  (min - height, nth 0 outer (min - height) - nth 0 inner (min - height)).
+ +
+Definition add_ribbon_on sh start stop rem :=
+  (take start sh)
+    ++ (nth 0 sh start + rem :: map S (drop start (take stop sh)))
+            ++ drop stop.+1 sh ++ nseq (stop - size sh) 1.
+Definition add_ribbon sh nbox pos :=
+  let: (start, rem) := startrem 0 sh nbox.+1 pos in
+  if rem > 0 then
+    Some (add_ribbon_on sh start pos rem, (pos - start).+1)
+  else None.
+ +
+Section NThAddRibbon.
+ +
+Variable (sh : seq nat) (start stop rem : nat).
+Hypothesis (lesmin : start <= minn stop (size sh)).
+ +
+#[local] Notation res := (add_ribbon_on sh start stop rem).
+ +
+
+ +
+Put some local result in the environment +
+
+#[local] Lemma less : start <= stop.
+ Let less := less.
+#[local] Lemma lessz : start <= size sh.
+ Let lessz := lessz.
+#[local] Lemma sztd :
+  size (drop start (take stop sh)) = minn stop (size sh) - start.
+ Let sztd := sztd.
+ +
+Lemma nth_add_ribbon_lt_start i : i < start -> nth 0 res i = nth 0 sh i.
+ +
+Lemma nth_add_ribbon_start : nth 0 res start = nth 0 sh start + rem.
+ +
+Lemma nth_add_ribbon_in i :
+  start < i.+1 <= stop -> nth 0 res i.+1 = (nth 0 sh i).+1.
+ +
+Lemma nth_add_ribbon_stop_lt i : stop < i -> nth 0 res i = nth 0 sh i.
+ +
+Lemma add_ribbon_on_remP : rem > 0 -> ribbon_on start stop sh res.
+ +
+Lemma is_part_add_ribbon_on nbox :
+  is_part sh -> rem > 0 ->
+  startrem 0 sh nbox stop = (start, rem) -> is_part res.
+ +
+End NThAddRibbon.
+ +
+Section Tests.
+ +
+Goal add_ribbon_on [:: 2; 2; 1; 1] 0 0 2 = [:: 4; 2; 1; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 1 1 2 = [:: 2; 4; 1; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 0 1 2 = [:: 4; 3; 1; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 0 2 2 = [:: 4; 3; 3; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 2 2 1 = [:: 2; 2; 2; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 2 3 1 = [:: 2; 2; 2; 2].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 2 4 1 = [:: 2; 2; 2; 2; 2].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 2 5 1 = [:: 2; 2; 2; 2; 2; 1].
+ Goal add_ribbon_on [:: 2; 2; 1; 1] 2 6 1 = [:: 2; 2; 2; 2; 2; 1; 1].
+ +
+Goal startrem 0 [:: 2; 2; 1; 1] 2 0 = (0, 2).
+ Goal add_ribbon [:: 2; 2; 1; 1] 1 0 = Some ([:: 4; 2; 1; 1], 1).
+ Goal startrem 0 [:: 2; 2; 1; 1] 2 1 = (0, 1).
+ Goal add_ribbon [:: 2; 2; 1; 1] 1 1 = Some ([:: 3; 3; 1; 1], 2).
+ +
+
+ +
+Tests : +
+sage: s[2, 2] * p[1]
+s[2, 2, 1] + s[3, 2]
+
+ +
+
+Goal pmap (add_ribbon [:: 2; 2] 0) (iota 0 10) =
+  [:: ([:: 3; 2], 1); ([:: 2; 2; 1], 1)].
+
+ +
+Tests : +
+sage: s[2, 2] * p[2]
+-s[2, 2, 1, 1] + s[2, 2, 2] - s[3, 3] + s[4, 2]
+
+ +
+
+Goal pmap (add_ribbon [:: 2; 2] 1) (iota 0 10) =
+  [:: ([:: 4; 2], 1); ([:: 3; 3], 2); ([:: 2; 2; 2], 1); ([:: 2; 2; 1; 1], 2)].
+
+ +
+Tests : +
+sage: s[3, 2, 1, 1] * p[4]
+-s[3, 2, 1, 1, 1, 1, 1, 1] + s[3, 2, 2, 2, 2] + s[3, 3, 3, 2] - s[5, 4, 1, 1]
+  + s[7, 2, 1, 1]
+
+ +
+
+Goal pmap (add_ribbon [:: 3; 2; 1; 1] 3) (iota 0 10) =
+  [:: ([:: 7; 2; 1; 1], 1);
+      ([:: 5; 4; 1; 1], 2);
+      ([:: 3; 3; 3; 2], 3);
+      ([:: 3; 2; 2; 2; 2], 3);
+      ([:: 3; 2; 1; 1; 1; 1; 1; 1], 4)].
+
+ +
+Tests : +
+sage: s[2, 2] * p[5]
+s[2, 2, 1, 1, 1, 1, 1] - s[2, 2, 2, 1, 1, 1] + s[3, 3, 3] - s[6, 3] + s[7, 2]
+
+ +
+
+Goal pmap (add_ribbon [:: 2; 2] 4) (iota 0 8) =
+  [:: ([:: 7; 2], 1); ([:: 6; 3], 2); ([:: 3; 3; 3], 3);
+      ([:: 2; 2; 2; 1; 1; 1], 4); ([:: 2; 2; 1; 1; 1; 1; 1], 5)].
+
+ +
+Tests : +
+sage: s[2, 2, 1] * p[5]
+s[2, 2, 1, 1, 1, 1, 1, 1] - s[2, 2, 2, 2, 1, 1] + s[4, 3, 3] - s[6, 3, 1] + s[7, 2, 1]
+
+ +
+
+Goal pmap (add_ribbon [:: 2; 2; 1] 4) (iota 0 8) =
+  [:: ([:: 7; 2; 1], 1); ([:: 6; 3; 1], 2); ([:: 4; 3; 3], 3);
+      ([:: 2; 2; 2; 2; 1; 1], 4); ([:: 2; 2; 1; 1; 1; 1; 1; 1], 5)].
+
+ +
+Tests : +
+s[5, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1] - s[5, 5, 2, 2, 2, 2, 1, 1, 1] + s[5, 5, 3, 3, 2, 2, 1]
+ - s[5, 5, 4, 3, 2, 2] + s[7, 6, 6, 1, 1] - s[11, 6, 2, 1, 1] + s[12, 5, 2, 1, 1]
+
+ +
+
+Goal pmap (add_ribbon [:: 5; 5; 2; 1; 1] 6) (iota 0 15) =
+  [:: ([:: 12; 5; 2; 1; 1], 1); ([:: 11; 6; 2; 1; 1], 2);
+     ([:: 7; 6; 6; 1; 1], 3); ([:: 5; 5; 4; 3; 2; 2], 4);
+      ([:: 5; 5; 3; 3; 2; 2; 1], 5); ([:: 5; 5; 2; 2; 2; 2; 1; 1; 1], 6);
+      ([:: 5; 5; 2; 1; 1; 1; 1; 1; 1; 1; 1; 1], 7)].
+ +
+Goal let sh := [:: 2; 2] in
+     all (fun p => ribbon sh p.1) (pmap (add_ribbon sh 0) (iota 0 8)).
+ Goal let sh := [:: 2; 2] in
+     all (fun p => ribbon sh p.1) (pmap (add_ribbon sh 4) (iota 0 8)).
+ Goal let sh := [:: 2; 2; 1] in
+     all (fun p => ribbon sh p.1) (pmap (add_ribbon sh 4) (iota 0 8)).
+ Goal let sh := [:: 5; 5; 2; 1; 1] in
+     all (fun p => ribbon sh p.1) (pmap (add_ribbon sh 6) (iota 0 15)).
+ +
+End Tests.
+ +
+Section Spec.
+ +
+Variables (sh : seq nat).
+Hypothesis (partsh : is_part sh).
+Variable (nbox pos : nat).
+Variables (res : seq nat) (hgt : nat).
+Hypothesis (Hret : add_ribbon sh nbox pos = Some (res, hgt)).
+ +
+Lemma sumn_mapS s : sumn [seq i.+1 | i <- s] = sumn s + size s.
+ +
+Lemma sumn_add_ribbon : sumn res = nbox.+1 + sumn sh.
+Lemma is_part_add_ribbon : is_part res.
+Lemma is_part_of_add_ribbon : is_part_of_n (nbox.+1 + sumn sh) res.
+ +
+Lemma size_add_ribbon : size res = maxn (size sh) pos.+1.
+ +
+Lemma add_ribbon_height : hgt = ribbon_height sh res.
+ +
+Lemma add_ribbon_onP : ribbon_on (pos.+1 - hgt) pos sh res.
+ +
+Lemma add_ribbonP : ribbon sh res.
+ +
+Lemma included_add_ribbon : included sh res.
+ +
+End Spec.
+ +
+Lemma ribbon_on_addE start stop inner outer :
+  is_part inner -> is_part outer -> ribbon_on start stop inner outer ->
+  add_ribbon_on inner start stop (nth 0 outer start - nth 0 inner start) =
+  outer.
+ +
+Lemma ribbon_addE inner outer :
+  is_part inner -> is_part outer -> ribbon inner outer ->
+  add_ribbon inner (sumn (outer / inner)).-1 (mindropeq inner outer).-1 =
+  Some (outer, ribbon_height inner outer).
+ +
+Section IntPartN.
+ +
+Variable (m : nat) (la : 'P_m).
+Variable nbox : nat.
+#[local] Notation "'Pr" := 'P_(nbox.+1 + m).
+ +
+Let val_id := fun p : ('Pr * nat) => let: (sh, h) := p in (val sh, h).
+ +
+Fact add_ribbon_intpartn_spec pos :
+  { res : option ('Pr * nat) | omap val_id res = add_ribbon la nbox pos }.
+Definition add_ribbon_intpartn (pos : nat) : option ('Pr * nat) :=
+  let: exist res _ := add_ribbon_intpartn_spec pos in res.
+ +
+Lemma add_ribbon_intpartnE pos :
+  add_ribbon la nbox pos =omap val_id (add_ribbon_intpartn pos).
+ Lemma add_ribbon_intpartnP pos res h :
+  add_ribbon_intpartn pos = Some (res, h) ->
+  add_ribbon la nbox pos = Some (val res, h).
+ +
+End IntPartN.
+ +
+
+ +
+

Equivalence with the textbook definition

+

Some complement on connect

+ +
+
+Section ConnectCompl.
+ +
+Variables (T : finType) (e : rel T).
+ +
+Lemma connect_rev : (connect (fun x : T => e^~ x)) =2 (fun x => (connect e)^~x).
+ +
+Lemma connect_from_sym : symmetric e -> connect_sym e.
+ +
+Lemma same_connect_rev : connect_sym e -> connect e =2 connect (fun x y => e y x).
+ +
+End ConnectCompl.
+ +
+
+ +
+

4 neigborhood

+ +
+
+Definition neig4 rc :=
+  let: (r, c) := rc in [:: (r, c.-1); (r, c.+1); (r.-1, c); (r.+1, c)].
+Lemma neig4_sym rc1 rc2 : rc1 \in neig4 rc2 -> rc2 \in neig4 rc1.
+Lemma grel_neig4_sym : symmetric (grel neig4).
+ +
+Section Connected4.
+ +
+Variable (inner outer : seq nat).
+#[local] Notation box := (box_skew inner outer).
+ +
+Definition neig4box (b : box) : seq box := pmap insub (neig4 b).
+Lemma neig4boxE :
+  relpre val (grel neig4) =2 grel (fun b : box => pmap insub (neig4 b)).
+ +
+Lemma neig4box_sym : symmetric (grel neig4box).
+ +
+End Connected4.
+ +
+
+ +
+

The textbook definition

+ +
+
+Definition conn4_skew inner outer :=
+  n_comp (grel (@neig4box inner outer)) predT == 1.
+Definition has_no_square inner outer :=
+  [forall b : box_skew inner outer, ~~ in_skew inner outer (b.1.+1, b.2.+1)].
+Definition ribbon_textbook inner outer :=
+  [&& inner != outer,
+   included inner outer,
+   conn4_skew inner outer &
+   has_no_square inner outer].
+ +
+Section TextBookDefStartStop.
+ +
+Variable (start stop : nat) (inner outer : seq nat).
+Hypotheses (partinn : is_part inner) (partout : is_part outer).
+Hypothesis (Hrib : ribbon_on start stop inner outer).
+#[local] Notation box := (box_skew inner outer).
+Implicit Type (b : box).
+ +
+#[local] Notation neig4 := (grel (@neig4box inner outer)).
+ +
+Lemma conn4_sym : symmetric (connect neig4).
+ +
+#[local] Definition ribbon_box_ex inner outer :=
+  let start := (mindropeq inner outer) - ribbon_height inner outer in
+  (start, nth 0 inner start).
+ +
+Lemma ribbon_on_box_exE :
+  ribbon_box_ex inner outer = (start, nth 0 inner start).
+ +
+Lemma ribbon_on_box_exP : in_skew inner outer (ribbon_box_ex inner outer).
+Definition ribbon_on_box_ex : box := BoxSkew ribbon_on_box_exP.
+#[local] Notation boxex := ribbon_on_box_ex.
+ +
+Lemma ribbon_on_conn4_box_ex b : connect neig4 boxex b.
+ +
+Lemma ribbon_on_conn4 b1 b2 : connect neig4 b1 b2.
+ +
+Lemma ribbon_on_conn4_skew : conn4_skew inner outer.
+ +
+Lemma ribbon_on_no_square r c :
+  in_skew inner outer (r, c) -> ~~ in_skew inner outer (r.+1, c.+1).
+ +
+End TextBookDefStartStop.
+ +
+Section TextBookImplDef.
+ +
+Variable (inner outer : seq nat).
+Hypotheses (partinn : is_part inner) (partout : is_part outer).
+Hypothesis (Htb : ribbon_textbook inner outer).
+#[local] Notation box := (box_skew inner outer).
+#[local] Notation stop := (mindropeq inner outer).-1.
+#[local] Notation neig4 := (grel (@neig4box inner outer)).
+ +
+Implicit Type (b : box).
+ +
+#[local] Lemma incl : included inner outer.
+ +
+Lemma mindropeq_non0 : mindropeq inner outer != 0.
+ +
+Lemma ribbontb_stop_ltn : nth 0 inner stop < nth 0 outer stop.
+ +
+Lemma ribbontb_start_subproof : exists i, nth 0 inner i < nth 0 outer i.
+ Definition ribbontb_start := ex_minn ribbontb_start_subproof.
+#[local] Notation start := ribbontb_start.
+ +
+Lemma ribbon_textbook_on : ribbon_on start stop inner outer.
+ +
+End TextBookImplDef.
+ +
+Theorem ribbon_textbookE inner outer :
+  is_part inner -> is_part outer ->
+  ribbon inner outer = ribbon_textbook inner outer.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.skewtab.html b/combi/1.1.0/Combi.Combi.skewtab.html new file mode 100644 index 00000000..413ed97b --- /dev/null +++ b/combi/1.1.0/Combi.Combi.skewtab.html @@ -0,0 +1,411 @@ + + + + + +Combi.Combi.skewtab: Skew Tableaux + + + + +
+ + + +
+ +

Library Combi.Combi.skewtab: Skew Tableaux

+ +
+
+ +
+ +
+
+
+ +
+

Skew tableau and skew yamanouchi words:

+ + +
+ +
    +
  • is_skew_yam inn out y == y ++ y0 is Yamanouchi of evaluation out for + any y0 of evaluation inn. + +
    • +
  • skew_dominate s u v == the row u dominate the row v when shifted by s. + +
    • +
  • is_skew_tableau inn t == t is a skew tableau with inner shape t. + +
    • +
  • skew_reshape inn out s == reshape the sequence s by the skew shape out/inn. + +
    • +
  • filter_leqX_tab n t == keeps only the entries greater than n in t. + +
    • +
  • join_tab t st == join the tableau t with the skew tableau st. + this gives a tableau if the inner shape of st is the shape of t and + the entries of t are smaller than the entries of st. + +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+ +
+Require Import tools partition skewpart Yamanouchi ordtype tableau std stdtab.
+ +
+Set Implicit Arguments.
+ +
+Import Order.Theory.
+ +
+
+ +
+

Skew Yamanouchi words

+ +
+
+Open Scope N.
+Open Scope Combi.
+ +
+Definition is_skew_yam innev outev sy :=
+  (forall y, is_yam_of_eval innev y -> is_yam_of_eval outev (sy ++ y)).
+ +
+Lemma skew_yam_nil sh : is_skew_yam sh sh [::].
+ +
+Lemma skew_nil_yamE eval y : is_yam_of_eval eval y -> is_skew_yam [::] eval y.
+ +
+Lemma skew_yam_cat sha shb shc y z :
+  is_skew_yam sha shb y -> is_skew_yam shb shc z -> is_skew_yam sha shc (z ++ y).
+ +
+Lemma is_skew_yamE innev outev z y0 :
+  is_yam_of_eval innev y0 ->
+  is_yam_of_eval outev (z ++ y0) ->
+  is_skew_yam innev outev z.
+ +
+Lemma is_part_skew_yam sha shb y :
+  is_part sha -> is_skew_yam sha shb y -> is_part shb.
+ +
+Lemma skew_yam_catrK sha shb shc y z :
+  is_part sha ->
+  is_skew_yam sha shb y ->
+  is_skew_yam sha shc (z ++ y) ->
+  is_skew_yam shb shc z.
+ +
+Lemma skew_yam_consK sha shc i y :
+  is_part sha -> is_skew_yam sha shc (i :: y) ->
+  is_skew_yam sha (decr_nth shc i) y.
+ +
+Lemma skew_yam_catK sha shc y z :
+  is_part sha -> is_skew_yam sha shc (y ++ z) ->
+  { shb | is_skew_yam sha shb z & is_skew_yam shb shc y }.
+ +
+Lemma skew_yam_included sha shb y :
+  is_part sha -> is_skew_yam sha shb y -> included sha shb.
+ +
+
+ +
+

Skew tableaux

+ +
+
+Section Dominate.
+ +
+Context disp (T : inhOrderType disp).
+Implicit Type u v : seq T.
+ +
+Definition skew_dominate sh u v := dominate (drop sh u) v.
+ +
+Lemma skew_dominate0 : skew_dominate 0 =2 (@dominate _ T).
+ +
+Lemma skew_dominate_take n sh u v :
+  skew_dominate sh u (take n v) -> skew_dominate sh u v.
+ +
+Lemma skew_dominate_no_overlap sh u v :
+  size u <= sh -> skew_dominate sh u v.
+ +
+Lemma skew_dominate_consl sh l u v :
+  skew_dominate sh u v -> skew_dominate sh.+1 (l :: u) v.
+ +
+Lemma skew_dominate_cut sh u v :
+  skew_dominate sh u v = skew_dominate sh u (take (size u - sh) v).
+ +
+Fixpoint is_skew_tableau inner t :=
+  if t is t0 :: t'
+  then [&& head 0 inner + size t0 != 0,
+        is_row t0,
+        skew_dominate ((head 0 inner) - (head 0 (behead inner)))
+                      (head [::] t') t0 & is_skew_tableau (behead inner) t']
+  else inner == [::].
+ +
+Lemma is_skew_tableauP inner t :
+  reflect
+    [/\ size inner <= size t,
+     forall i, i < size t -> nth 0 inner i + size (nth [::] t i) != 0,
+     forall i, is_row (nth [::] t i) &
+     forall i, skew_dominate ((nth 0 inner i) - (nth 0 inner i.+1))
+                             (nth [::] t i.+1) (nth [::] t i)]
+    (is_skew_tableau inner t).
+ +
+Lemma is_skew_tableau0 : is_skew_tableau [::] =1 is_tableau.
+ +
+Lemma is_skew_tableau_pad0 inner t :
+  is_skew_tableau inner t = is_skew_tableau (pad 0 (size t) inner) t.
+ +
+Definition skew_reshape (inner outer : seq nat) (s : seq T) :=
+  rev (reshape (rev (outer / inner)) s).
+ +
+Lemma size_skew_reshape inner outer s :
+  size (skew_reshape inner outer s) = size outer.
+ +
+Lemma shape_skew_reshape inner outer s :
+  included inner outer ->
+  size s = sumn (outer / inner) ->
+  shape (skew_reshape inner outer s) = outer / inner.
+ +
+Lemma to_word_skew_reshape inner outer s :
+  included inner outer ->
+  size s = sumn (outer / inner) ->
+  to_word (skew_reshape inner outer s) = s.
+ +
+Lemma skew_reshapeK inner t :
+  size inner <= size t ->
+  skew_reshape inner (outer_shape inner (shape t)) (to_word t) = t.
+ +
+Lemma row_hb_strip inner t :
+  is_part inner ->
+  is_skew_tableau inner t -> is_row (to_word t) ->
+  hb_strip inner (outer_shape inner (shape t)).
+ +
+Lemma hb_strip_rowE inner outer u :
+  is_part inner -> is_part outer ->
+  is_row u -> size u = sumn (outer / inner) ->
+  included inner outer &&
+           is_skew_tableau inner (skew_reshape inner outer u) =
+  hb_strip inner outer.
+ +
+End Dominate.
+ +
+
+ +
+

Skewing and joining tableaux

+ +
+
+Section FilterLeqGeq.
+ +
+Context disp (T : inhOrderType disp).
+Implicit Type l : T.
+Implicit Type r w : seq T.
+Implicit Type t : seq (seq T).
+ +
+Lemma filter_le_dominate n r1 r0 :
+  is_row r0 -> is_row r1 -> dominate r1 r0 ->
+  skew_dominate ((count (>%O n) r0) - (count (>%O n) r1))
+                (filter (<=%O n) r1) (filter (<=%O n) r0).
+ +
+Definition filter_le_tab n :=
+  [fun t : (seq (seq T)) => [seq [seq x <- i | (n <= x)%O] | i <- t]].
+ +
+Lemma is_skew_tableau_filter_le_tmp n t :
+  is_tableau t -> is_skew_tableau
+                    (shape ([seq [seq x <- i | (x < n)%O] | i <- t]))
+                    (filter_le_tab n t).
+ +
+Lemma filter_gt_first_row0 n r t :
+  is_tableau t ->
+  dominate (head [::] t) r ->
+  [seq x <- r | (x < n)%O] = [::] ->
+  [seq [seq x <- i | (n <= x)%O] | i <- t] = t.
+ +
+Lemma filter_le_first_row0 n r t :
+  is_tableau t ->
+  dominate (head [::] t) r ->
+  [seq x <- r | (x < n)%O] = [::] ->
+  [seq [seq x <- i | (x < n)%O] | i <- t] = nseq (size t) [::].
+ +
+Lemma included_shape_filter_gt c (t : seq (seq T)) :
+  is_tableau t -> included (shape (filter_gt_tab c t)) (shape t).
+ +
+Lemma shape_inner_filter_le n t :
+  is_tableau t ->
+  shape ([seq [seq x <- i | (x < n)%O] | i <- t]) =
+  pad 0 (size t) (shape (filter_gt_tab n t)).
+ +
+Lemma is_skew_tableau_filter_le n t:
+  is_tableau t ->
+  is_skew_tableau (shape (filter_gt_tab n t)) (filter_le_tab n t).
+ +
+Definition join_tab s t :=
+  [seq pr.1 ++ pr.2 | pr <- zip (pad [::] (size t) s) t].
+ +
+Lemma size_join_tab s t :
+  size s <= size t ->
+  size_tab (join_tab s t) = size_tab s + size_tab t.
+ +
+Lemma shape_join_tab s t :
+  shape (join_tab s t) =
+  ([seq pr.1 + pr.2 | pr <- zip (pad 0 (size t) (shape s)) (shape t)])%N.
+ +
+Lemma perm_join_tab s t :
+  size s <= size t ->
+  perm_eq (to_word (join_tab s t)) (to_word s ++ to_word t).
+ +
+Lemma join_tab_filter n t :
+  is_tableau t -> join_tab (filter_gt_tab n t) (filter_le_tab n t) = t.
+ +
+Lemma all_allLtn_cat (s0 s1 s : seq T) :
+  all (allLtn (s0 ++ s1)) s -> all (allLtn s0) s /\ all (allLtn s1) s.
+ +
+Lemma shape_join_tab_skew_reshape t sh w :
+  included (shape t) sh ->
+  size w = sumn (sh / (shape t)) ->
+  shape (join_tab t (skew_reshape (shape t) sh w)) = sh.
+ +
+Lemma join_tab_skew s t :
+  all (allLtn (to_word s)) (to_word t) ->
+  is_tableau s -> is_skew_tableau (shape s) t ->
+  is_tableau (join_tab s t).
+ +
+End FilterLeqGeq.
+ +
+
+ +
+

Standardisation of a tableau

+ +
+
+Section EqInvSkewTab.
+ +
+Lemma eq_inv_skew_dominate
+      d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+      (u1 v1 : seq T1) (u2 v2 : seq T2) s :
+  eq_inv (u1 ++ v1) (u2 ++ v2) ->
+  size u1 = size u2 ->
+  skew_dominate s u1 v1 -> skew_dominate s u2 v2.
+ +
+Lemma eq_inv_is_skew_tableau_reshape_size
+      inner outer d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+      (u1 : seq T1) (u2 : seq T2) :
+  size inner = size outer ->
+  eq_inv u1 u2 -> size u1 = sumn (outer / inner) ->
+  is_skew_tableau inner (skew_reshape inner outer u1) ->
+  is_skew_tableau inner (skew_reshape inner outer u2).
+ +
+Lemma is_skew_tableau_skew_reshape_pad0 inner outer
+      d (T : inhOrderType d) (u : seq T) :
+  is_skew_tableau inner (skew_reshape inner outer u) =
+  is_skew_tableau ((pad 0 (size outer)) inner)
+                  (skew_reshape ((pad 0 (size outer)) inner) outer u).
+ +
+Theorem eq_inv_is_skew_tableau_reshape
+        inner outer d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+        (u1 : seq T1) (u2 : seq T2) :
+  size inner <= size outer ->
+  eq_inv u1 u2 ->
+  size u1 = sumn (outer / inner) ->
+  is_skew_tableau inner (skew_reshape inner outer u1) ->
+  is_skew_tableau inner (skew_reshape inner outer u2).
+ +
+Theorem is_skew_tableau_reshape_std inner outer
+        d (T : inhOrderType d) (u : seq T) :
+  size inner <= size outer ->
+  size u = sumn (outer / inner) ->
+  is_skew_tableau inner (skew_reshape inner outer u) =
+  is_skew_tableau inner (skew_reshape inner outer (std u)).
+ +
+Theorem is_tableau_reshape_std sh d (T : inhOrderType d) (u : seq T) :
+  size u = sumn sh ->
+  is_tableau (skew_reshape [::] sh u) =
+  is_tableau (skew_reshape [::] sh (std u)).
+ +
+Theorem is_tableau_std d (T : inhOrderType d) (t : seq (seq T)) :
+  is_tableau t = is_tableau (skew_reshape [::] (shape t) (std (to_word t))).
+ +
+End EqInvSkewTab.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.std.html b/combi/1.1.0/Combi.Combi.std.html new file mode 100644 index 00000000..99d681c3 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.std.html @@ -0,0 +1,663 @@ + + + + + +Combi.Combi.std: Standard Words, i.e. Permutation as Words + + + + +
+ + + +
+ +

Library Combi.Combi.std: Standard Words, i.e. Permutation as Words

+ +
+
+ +
+ +
+
+
+ +
+

Standard words

+ + +
+ +That is words wich are permutations of 0, 1, ..., n-1: + +
+ +
    +
  • is_std s == s is a standard word + +
    • +
  • is_std_of_n n s == s is a standard word of size n + +
    • +
  • wordperm p == the standard word associated with a permutation p of 'S_n + +
    • +
+ +
+ +Sigma types for standard word + +
+ +
    +
  • stdwordn n == a type for seq nat which standard words size n; + it is canonically a finType + +
    • +
+ +
+ +Inversions and standardisation of a word over a totally ordered alphabet + +
+ +
    +
  • std s == given a word w over an ordType returns the standard word with + the same set of inversion (lemmas stdP and std_eq_invP) + +
    • +
  • versions w i j == i j is an non-inversion of w that is + i <= j < size w and w_i <= w_j + +
    + + +
    • +
  • eq_inv w1 w2 == w1 and w2 have the same versions which is equivalent + to the same set of inversions; in particular w1 and w2 + have the same size. + +
    + + +
    • +
  • std_spec s p == p is a standard word with the same versions as s + +
    • +
+ +
+ +The main result here is std_eq_invP which says that having the same inversions +is the same as having the same standardized. + +
+ +
    +
  • linvseq s t == t is the left inverse of s + +
    • +
  • invseq s t == s and t are inverse on of each other + +
    • +
  • invstd w == the inverse of the standardized of w that is the permutation + which sorts w in a stable way. + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm fingroup.
+ +
+Require Import tools combclass ordtype permcomp.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+ +
+Open Scope nat_scope.
+ +
+
+ +
+

Standard words and permutations

+ +
+
+Section StandardWords.
+ +
+Implicit Type n : nat.
+Definition is_std (s : seq nat) := perm_eq s (iota 0 (size s)).
+ +
+Lemma perm_std u v : is_std u -> perm_eq v u -> is_std v.
+ +
+Lemma std_perm u v : is_std u -> is_std v -> size u = size v -> perm_eq v u.
+ +
+Lemma mem_std p i : is_std p -> (i \in p) = (i < size p).
+ +
+Lemma std_uniq u : is_std u -> uniq u.
+ +
+Lemma std_max s0 s : is_std (s0 :: s) -> maxL s0 s = size s.
+ +
+Lemma is_stdP s : reflect (forall n, n < size s -> n \in s) (is_std s).
+ +
+Definition wordperm n (p : 'S_n) := [seq (p i : nat) | i : 'I_n].
+ +
+Open Scope group_scope.
+ +
+Lemma wordperm_iota n (p : 'S_n) : (wordperm p) =i iota 0 n.
+ +
+Lemma uniq_wordperm n (p : 'S_n) : uniq (wordperm p).
+ +
+Lemma wordperm_std n (p : 'S_n) : is_std (wordperm p).
+ +
+Lemma perm_of_std s : is_std s -> { p : 'S_(size s) | s = wordperm p }.
+ +
+Lemma is_std_wordpermP s :
+  reflect (exists p : 'S_(size s), s = wordperm p) (is_std s).
+ +
+Lemma wordperm_invP n (p : 'S_n) i (j : 'I_n) :
+  nth n (wordperm p) i = j <-> i = (p^-1) j.
+ +
+End StandardWords.
+ +
+
+ +
+

Standard words as a finType

+ +
+
+Section StdCombClass.
+ +
+Variable n : nat.
+ +
+Definition is_std_of_n := [pred w | (is_std w) && (size w == n) ].
+ +
+Structure stdwordn : Set :=
+  StdWordN {stdwordnval :> seq nat; _ : is_std_of_n stdwordnval}.
+ +
+ +
+Lemma stdwordnP (s : stdwordn) : is_std (val s).
+ +
+Lemma size_sdtn (s : stdwordn) : size (val s) = n.
+ +
+Definition enum_stdwordn := [seq wordperm p | p : 'S_n].
+ +
+Lemma enum_stdwordnE : enum_stdwordn =i is_std_of_n.
+ +
+Lemma wordperm_inj : injective (@wordperm n).
+ +
+Lemma enum_stdwordn_uniq : uniq enum_stdwordn.
+ +
+ +
+Lemma card_stdwordn : #|{: stdwordn}| = n`!.
+ +
+End StdCombClass.
+ +
+
+ +
+

Standardisation of a word over a totally ordered alphabet

+ +
+
+Section Standardisation.
+ +
+Context {disp} {Alph : orderType disp}.
+Implicit Type s u v w : seq Alph.
+ +
+Fixpoint std_rec n s :=
+  if n is n'.+1 then
+    let rec := std_rec n' (rembig s) in
+    let pos := posbig s in
+    take pos rec ++ n' :: drop pos rec
+  else [::].
+Definition std s := std_rec (size s) s.
+ +
+Lemma size_std_rec n s : size (std_rec n s) = n.
+ +
+Lemma size_std s : size (std s) = size s.
+ +
+Lemma std_is_std s : is_std (std s).
+ +
+Lemma in_std_ltn_size s i : i \in std s = (i < size s).
+ +
+Lemma allLtn_std_rec s : allLtn (std s) (size s).
+ +
+Lemma rembig_ins_std s pos :
+  rembig (take pos (std s) ++ size s :: drop pos (std s)) = std s.
+ +
+Lemma std_rembig s : std (rembig s) = rembig (std s).
+ +
+Lemma std_posbig s : posbig (std s) = posbig s.
+ +
+End Standardisation.
+ +
+Lemma std_std s : is_std s -> std s = s.
+ +
+Lemma std_stdE disp (T : orderType disp) (s : seq T) : std (std s) = std s.
+ +
+
+ +
+

Inversion sets and standardization

+ +
+
+Section EqInvDef.
+ +
+Definition versions d (T : inhOrderType d) (w : seq T) : rel nat :=
+  fun i j => (i <= j < size w) && (nth inh w i <= nth inh w j)%O.
+ +
+Definition eq_inv d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+           (w1 : seq T1) (w2 : seq T2) :=
+  (versions w1) =2 (versions w2).
+ +
+Context disp1 disp2 disp3
+   (S : inhOrderType disp1)
+   (T : inhOrderType disp2)
+   (U : inhOrderType disp3).
+ +
+Lemma eq_inv_refl (w : seq T) : eq_inv w w.
+ +
+Lemma eq_inv_nil : eq_inv (@nil S) (@nil T).
+ +
+Lemma eq_inv_sym (w1 : seq S) (w2 : seq T) :
+  eq_inv w1 w2 -> eq_inv w2 w1.
+ +
+Lemma eq_inv_trans (w1 : seq S) (w2 : seq T) (w3 : seq U) :
+  eq_inv w1 w2 -> eq_inv w2 w3 -> eq_inv w1 w3.
+ +
+End EqInvDef.
+ +
+Lemma eq_inv_size d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+      (w1 : seq T1) (w2 : seq T2) :
+  eq_inv w1 w2 -> size w1 = size w2.
+ +
+Section EqInvAltDef.
+ +
+Context disp1 disp2 disp3
+          (S : inhOrderType disp1)
+          (T : inhOrderType disp2)
+          (U : inhOrderType disp3).
+ +
+Lemma eq_invP (w1 : seq S) (w2 : seq T) :
+  (size w1 = size w2 /\
+   forall i j, i <= j < size w1 ->
+               ((nth inh w1 i <= nth inh w1 j) =
+                (nth inh w2 i <= nth inh w2 j))%O)
+    <-> (eq_inv w1 w2).
+ +
+Lemma eq_inv_inversionP (w1 : seq S) (w2 : seq T) :
+  (size w1 = size w2 /\
+   forall i j, i <= j < size w1 ->
+               (nth inh w1 i > nth inh w1 j) =
+               (nth inh w2 i > nth inh w2 j))%O
+    <-> (eq_inv w1 w2).
+ +
+Lemma eq_inv_consK (a : S) u (b : T) v :
+  eq_inv (a :: u) (b :: v) -> eq_inv u v.
+ +
+Lemma eq_inv_rconsK (a : S) u (b : T) v :
+  eq_inv (rcons u a) (rcons v b) -> eq_inv u v.
+ +
+End EqInvAltDef.
+ +
+Lemma eq_inv_allgt_imply
+      d1 d2 (S : inhOrderType d1) (T : inhOrderType d2) (a : S) u (b : T) v :
+  eq_inv (a :: u) (b :: v) -> (allLtn u a) -> (allLtn v b).
+ +
+Lemma eq_inv_allgtnX
+      d1 d2 (S : inhOrderType d1) (T : inhOrderType d2) (a : S) u (b : T) v :
+  eq_inv (a :: u) (b :: v) -> (allLtn u a) = (allLtn v b).
+ +
+Section EqInvPosRemBig.
+ +
+Context disp1 disp2 (S : inhOrderType disp1) (T : inhOrderType disp2).
+ +
+Lemma eq_inv_posbig (u : seq S) (v : seq T) :
+  eq_inv u v -> posbig u = posbig v.
+ +
+Lemma eq_inv_rembig (u : seq S) (v : seq T) :
+  eq_inv u v -> eq_inv (rembig u) (rembig v).
+ +
+Lemma std_eq_inv (u : seq S) (v : seq T) :
+  eq_inv u v -> std u = std v.
+ +
+Lemma eq_inv_std (u : seq T) : eq_inv u (std u).
+ +
+End EqInvPosRemBig.
+ +
+Section Spec.
+ +
+Context disp1 disp2 (S : inhOrderType disp1) (T : inhOrderType disp2).
+ +
+Variant std_spec (s : seq T) (p : seq nat) : Prop :=
+  | StdSpec : is_std p -> eq_inv s p -> std_spec s p.
+ +
+Lemma std_spec_uniq (u : seq T) p q :
+  std_spec u p -> std_spec u q -> p = q.
+ +
+Lemma std_specP (s : seq T) : std_spec s (std s).
+ +
+Lemma stdP (s : seq T) p :
+  reflect (std_spec s p) (std s == p).
+ +
+Lemma std_eq_invP (u : seq S) (v : seq T) :
+  reflect (eq_inv u v) (std u == std v).
+ +
+Lemma std_rconsK (u : seq S) (v : seq T) a b :
+  std (rcons u a) = std (rcons v b) -> std u = std v.
+ +
+Lemma eq_inv_catl (u1 v1 : seq S) (u2 v2 : seq T) :
+  eq_inv (u1 ++ v1) (u2 ++ v2) -> size u1 = size u2 -> eq_inv u1 u2.
+ +
+Lemma eq_inv_catr (u1 v1 : seq S) (u2 v2 : seq T) :
+  eq_inv (u1 ++ v1) (u2 ++ v2) -> size u1 = size u2 -> eq_inv v1 v2.
+ +
+End Spec.
+ +
+Section StdTakeDrop.
+ +
+Context disp1 disp2 (S : inhOrderType disp1) (T : inhOrderType disp2).
+Implicit Type u v : seq T.
+ +
+Lemma std_take_std u v : std (take (size u) (std (u ++ v))) = std u.
+ +
+Lemma std_drop_std u v : std (drop (size u) (std (u ++ v))) = std v.
+ +
+End StdTakeDrop.
+ +
+Section PermEq.
+ +
+Context disp (Alph : orderType disp).
+Implicit Type u v : seq Alph.
+ +
+Theorem perm_stdE u v : perm_eq u v -> std u = std v -> u = v.
+ +
+End PermEq.
+ +
+
+ +
+

Standardization and elementary transpositions of a word

+ +
+
+Section Transp.
+ +
+Context disp (Alph : inhOrderType disp).
+Implicit Type u v : seq Alph.
+ +
+Lemma nth_transp u v a b i :
+  i != size u -> i != (size u).+1 ->
+  nth inh (u ++ [:: a; b] ++ v) i = nth inh (u ++ [:: b; a] ++ v) i.
+ +
+Lemma nth_sizeu u v a b :
+  nth inh (u ++ [:: a; b] ++ v) (size u) = a.
+ +
+Lemma nth_sizeu1 u v a b :
+  nth inh (u ++ [:: a; b] ++ v) (size u).+1 = b.
+ +
+Lemma nth_sizeu2 u v a b c :
+  nth inh (u ++ [:: a; b; c] ++ v) (size u).+2 = c.
+ +
+End Transp.
+ +
+Lemma eq_inv_transp d1 d2 (S : inhOrderType d1) (T : inhOrderType d2)
+      (u v : seq S) a b (U V : seq T) A B :
+  (a < b)%O -> (A < B)%O -> size u = size U ->
+  eq_inv (u ++ [:: a; b] ++ v) (U ++ [:: A; B] ++ V) ->
+  eq_inv (u ++ [:: b; a] ++ v) (U ++ [:: B; A] ++ V).
+ +
+Lemma std_transp d (T : inhOrderType d)
+      (u v : seq T) a b
+      (U V : seq nat) A B :
+  (a < b)%O -> size u = size U ->
+  std (u ++ [:: a; b] ++ v) = (U ++ [:: A; B] ++ V) ->
+  std (u ++ [:: b; a] ++ v) = (U ++ [:: B; A] ++ V).
+ +
+Lemma std_cutabc d (T : orderType d) (u v : seq T) a b c :
+  exists U V A B C, size u = size U /\
+  std (u ++ [:: a; b; c] ++ v) = (U ++ [:: A; B; C] ++ V).
+ +
+
+ +
+

Inverse of a standard word

+ +
+
+Lemma perm_size_uniq (T : eqType) (s1 s2 : seq T) :
+  uniq s1 -> {subset s1 <= s2} -> size s2 <= size s1 -> perm_eq s1 s2.
+ +
+Section InvSeq.
+ +
+Implicit Type n : nat.
+ +
+Definition linvseq s :=
+  [fun t => all (fun i => nth (size s) t (nth (size t) s i) == i)
+              (iota 0 (size s))].
+Definition invseq s t := linvseq s t && linvseq t s.
+ +
+Lemma linvseqP s t :
+  reflect
+    (forall i, i < size s -> nth (size s) t (nth (size t) s i) = i)
+    (linvseq s t).
+ +
+Lemma invseq_sym s t : invseq s t -> invseq t s.
+ +
+Lemma size_all_leq n t : (forall i, i < n -> i \in t) -> n <= size t.
+ +
+Lemma linvseq_ltn_szt s t :
+  linvseq s t -> forall i, i < size s -> nth (size t) s i < size t.
+ +
+Lemma size_leq_invseq s t : linvseq s t -> size s <= size t.
+ +
+Lemma size_invseq s t : invseq s t -> size s = size t.
+ +
+Lemma linvseq_subset_iota s t : linvseq s t -> {subset iota 0 (size s) <= t}.
+ +
+Lemma invseq_is_std s t : invseq s t -> is_std s.
+ +
+Lemma linvseq_sizeP s t :
+  linvseq s t -> size s = size t -> invseq s t.
+ +
+Lemma invseq_nthE s t :
+  invseq s t ->
+  forall i j, i < size s -> j < size t ->
+              (i = nth (size s) t j <-> nth (size t) s i = j).
+ +
+
+ +
+

inverse of the standardized of w

+ +
+
+ +
+Definition invstd s := mkseq (fun i => index i s) (size s).
+ +
+Lemma invseq_invstd s : is_std s -> invseq s (invstd s).
+ +
+Lemma size_invstd p : size (invstd p) = size p.
+ +
+Lemma invstd_is_std p : is_std p -> is_std (invstd p).
+ +
+Lemma invseqE s t1 t2 : invseq s t1 -> invseq s t2 -> t1 = t2.
+ +
+Lemma invstdK s : is_std s -> invstd (invstd s) = s.
+ +
+Lemma invstd_inj s t :
+  is_std s -> is_std t -> (invstd s) = (invstd t) -> s = t.
+ +
+End InvSeq.
+ +
+Section Examples.
+ +
+Let u := [:: 4; 1; 2; 2; 5; 3].
+Let v := [:: 0; 4; 3; 3].
+ +
+Goal std u = [:: 4; 0; 1; 2; 5; 3].
+ Goal invstd (std u) = filter (gtn (size u)) (invstd (std (u ++ v))).
+ +
+End Examples.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.stdtab.html b/combi/1.1.0/Combi.Combi.stdtab.html new file mode 100644 index 00000000..2b35d6d8 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.stdtab.html @@ -0,0 +1,654 @@ + + + + + +Combi.Combi.stdtab: Standard Tableaux + + + + +
+ + + +
+ +

Library Combi.Combi.stdtab: Standard Tableaux

+ +
+
+ +
+ +
+
+
+ +
+

Standard Tableaux

+ + +
+ +We define the following notions: + +
+ +
    +
  • append_nth t b i == t with b appended to its i-th row + +
    + + +
    • +
  • is_stdtab t == t is a *standard tableau* that is a tableau whose + row reading is a standard word + +
    • +
  • last_big t b == the index of the first row of t which ends with b + +
    • +
  • remn t == remove the largest entry ie n from a standard tableau of size n + +
    • +
  • conj_tab t == the conjugate standard tableau of t (this is indeed a tableau + when t is itself a standard tableau. + +
    • +
+ +
+ +Bijection between Yamanouchi words and standard tableau + +
+ +
    +
  • stdtab_of_yam y == the standard tableau associated to y + +
    • +
  • yam_of_stdtab t == the Yamanouchi words associated to t + +
    • +
+ +
+ +Sigma type for standard tableaux: + +
+ +
    +
  • is_stdtab_of_shape sh == a predicate for standard tableau of shape sh. + +
    • +
  • stdtabsh sh == a sigma type for is_stdtab_of_shape sh where sh is an + integer partition (of type intpart). This is canonically a + finType. + +
    + + +
    • +
  • is_stdtab_of_n n == a predicate for standard tableau of size n + +
    • +
  • stdtabn n == a sigma type for is_stdtab_of_n n. This is canonically a + finType. + +
    + + +
    • +
  • shape_deg t == if t is of type stdtabn n, the shape of t in the + sigma type 'P_n + +
    + + +
    • +
  • hyper_stdtabsh sh == the hyperstandard tableau of shape sh : intpart, + that is the tableau obtained by filling the rows with consecutive + numbers, from bottom to top (in French conventions) + +
    + + +
    • +
  • conj_stdtabn t == the conjugate of t : stdtabn n in type stdtabn n + +
    • +
  • conj_stdtabsh t == the conjugate of t : stdtabsh sh + in type stdtabsh (conj_intpart sh) + +
    • +
+ +
+ + +
+ +Among the main results are the fact that stdtab_of_yam and yam_of_stdtab +are two converse bijections. That is: + +
+ +
    +
  • Lemma stdtab_of_yamP y : is_yam y -> is_stdtab (stdtab_of_yam y). + +
    + + +
    • +
  • Theorem stdtab_of_yamK y : is_yam y -> yam_of_stdtab (stdtab_of_yam y) = y. + +
    + + +
    • +
  • Lemma yam_of_stdtabP t : is_stdtab t -> is_yam (yam_of_stdtab t). + +
    + + +
    • +
  • Theorem yam_of_stdtabK t : is_stdtab t -> stdtab_of_yam (yam_of_stdtab t) = +t. + +
    • +
+ +
+ +Moreover, these bijections preserve the shape and therefore the size: + +
+ +
    +
  • Lemma shape_stdtab_of_yam y : shape (stdtab_of_yam y) = evalseq y. + +
    + + +
    • +
  • Lemma shape_yam_of_stdtab t : is_stdtab t -> evalseq (yam_of_stdtab t) = +shape t. + +
    + + +
    • +
  • Lemma size_stdtab_of_yam y : size_tab (stdtab_of_yam y) = size y. + +
    + + +
    • +
  • Lemma size_yam_of_stdtab t : is_stdtab t -> size (yam_of_stdtab t) = size_tab +t. + +
    + + +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm fingroup.
+Require Import tools combclass partition Yamanouchi ordtype std tableau.
+ +
+Set Implicit Arguments.
+ +
+Import Order.Theory.
+ +
+
+ +
+

Appending on the n-th row

+ +
+
+Section AppendNth.
+ +
+Context disp (T : inhOrderType disp).
+Implicit Type b : T.
+Implicit Type t : seq (seq T).
+ +
+Definition append_nth t b i := set_nth [::] t i (rcons (nth [::] t i) b).
+ +
+Lemma perm_append_nth t x pos :
+  perm_eq (to_word (append_nth t x pos)) (x :: to_word t).
+ +
+Lemma shape_append_nth t b i : shape (append_nth t b i) = incr_nth (shape t) i.
+ +
+Lemma size_append_nth t b i : size_tab (append_nth t b i) = (size_tab t).+1.
+ +
+Lemma get_tab_append_nth (t : seq (seq T)) l i r c :
+  get_tab (append_nth t l i) (r, c) =
+  if (r == i) && (c == nth 0 (shape t) i) then l else get_tab t (r, c).
+ +
+
+ +
+

Finding the largest entry

+ +
+
+Fixpoint last_big t b :=
+  if t is t0 :: t' then
+    if last b t0 == b then 0
+    else (last_big t' b).+1
+  else 0.
+ +
+Lemma allLeq_to_word_hd r t b : allLeq (to_word (r :: t)) b -> allLeq r b.
+Lemma allLeq_to_word_tl r t b : allLeq (to_word (r :: t)) b -> allLeq (to_word t) b.
+ +
+Lemma last_bigP t b i :
+  is_tableau t -> allLeq (to_word t) b ->
+  reflect (last b (nth [::] t i) = b /\
+           forall j, j < i -> (last b (nth [::] t j) < b)%O)
+          (i == last_big t b).
+ +
+Lemma last_big_append_nth t b lb :
+  (forall j : nat, j < lb -> (last b (nth [::] t j) < b)%O) ->
+  last_big (append_nth t b lb) b = lb.
+ +
+End AppendNth.
+ +
+
+ +
+

Bijection standard tableau <-> Yamanouchi words

+ +
+
+Section Bijection.
+ +
+Implicit Type y : seq nat.
+Implicit Types t : seq (seq nat).
+ +
+Definition is_stdtab t := is_tableau t && is_std (to_word t).
+ +
+Lemma stdtabP t : is_stdtab t -> is_tableau t.
+ +
+Fixpoint stdtab_of_yam y :=
+  if y is y0 :: y' then
+    append_nth (stdtab_of_yam y') (size y') y0
+  else [::].
+ +
+Lemma shape_stdtab_of_yam y : shape (stdtab_of_yam y) = evalseq y.
+ +
+Lemma size_stdtab_of_yam y : size_tab (stdtab_of_yam y) = size y.
+ +
+Lemma std_of_yam y : is_std (to_word (stdtab_of_yam y)).
+ +
+
+ +
+The following proof is an experiment to see if proof using get_tab are + easier than proof by induction on the structure of tableaux +
+
+Lemma is_tab_append_nth_size_alternative_proof r T n :
+   all (gtn n) (to_word T) ->
+   is_part (incr_nth (shape T) r) ->
+   is_tableau T -> is_tableau (append_nth T n r).
+ +
+Lemma is_tab_append_nth_size r T n :
+   all (gtn n) (to_word T) ->
+   is_part (incr_nth (shape T) r) ->
+   is_tableau T -> is_tableau (append_nth T n r).
+ +
+Lemma stdtab_of_yamP y : is_yam y -> is_stdtab (stdtab_of_yam y).
+ +
+Section StdTabInd.
+ +
+Fixpoint remn_rec t n :=
+  if t is t0 :: t' then
+    match t0 with
+      | [::] => [::]
+      | l0 :: t0' => if last l0 t0' == n then
+                       if t0' == [::] then [::]
+                       else (belast l0 t0') :: t'
+                     else t0 :: remn_rec t' n
+    end
+  else [::] .
+Definition remn t := remn_rec t (size_tab t).-1.
+ +
+Lemma remnP t :
+  is_stdtab t -> t != [::] ->
+  is_tableau (remn t) /\
+  append_nth (remn t) (size_tab t).-1 (last_big t (size_tab t).-1) = t.
+ +
+Lemma is_stdtab_remn t : is_stdtab t -> is_stdtab (remn t).
+ +
+Lemma append_nth_remn t :
+  is_stdtab t -> t != [::] ->
+  append_nth (remn t) (size_tab t).-1 (last_big t (size_tab t).-1) = t.
+ +
+Lemma size_tab_remn t :
+  is_stdtab t -> t != [::] -> size_tab (remn t) = (size_tab t).-1.
+ +
+End StdTabInd.
+ +
+Fixpoint yam_of_stdtab_rec n t :=
+  if n is n'.+1 then
+    (last_big t n') :: yam_of_stdtab_rec n' (remn t)
+  else [::].
+Definition yam_of_stdtab t := yam_of_stdtab_rec (size_tab t) t.
+ +
+Lemma size_yam_of_stdtab_rec n t : size (yam_of_stdtab_rec n t) = n.
+ +
+Theorem yam_of_stdtabK t : is_stdtab t -> stdtab_of_yam (yam_of_stdtab t) = t.
+ +
+Lemma find_append_nth (l : nat) t r :
+  l \notin (to_word t) ->
+  find (fun x : seq nat => l \in x) (append_nth t l r) = r.
+ +
+Lemma size_notin_stdtab_of_yam y : (size y) \notin (to_word (stdtab_of_yam y)).
+ +
+Lemma incr_nth_injl u v i :
+  0 \notin u -> 0 \notin v -> incr_nth u i = incr_nth v i -> u = v.
+ +
+Lemma shape0 d (T : orderType d) (u : seq (seq T)) :
+  [::] \notin u -> 0 \notin (shape u).
+ +
+Lemma append_nth_injl d (T : inhOrderType d) (u v : seq (seq T)) (l : T) p :
+  [::] \notin u -> [::] \notin v ->
+  append_nth u l p = append_nth v l p -> u = v.
+ +
+Lemma stdtab_of_yam_nil y : is_yam y -> [::] \notin (stdtab_of_yam y).
+ +
+Lemma stdtab_of_yam_inj x y :
+  is_yam x -> is_yam y -> stdtab_of_yam x = stdtab_of_yam y -> x = y.
+ +
+Lemma shape_yam_of_stdtab t :
+  is_stdtab t -> evalseq (yam_of_stdtab t) = shape t.
+ +
+Lemma size_yam_of_stdtab t :
+  is_stdtab t -> size (yam_of_stdtab t) = size_tab t.
+ +
+Lemma part_yam_of_stdtab t :
+  is_stdtab t -> is_part (evalseq (yam_of_stdtab t)).
+ +
+Lemma yam_of_stdtabP t : is_stdtab t -> is_yam (yam_of_stdtab t).
+ +
+Theorem stdtab_of_yamK y : is_yam y -> yam_of_stdtab (stdtab_of_yam y) = y.
+ +
+End Bijection.
+ +
+Lemma eq_inv_is_row d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+      (u1 : seq T1) (u2 : seq T2) :
+  eq_inv u1 u2 -> is_row u1 -> is_row u2.
+ +
+Lemma is_row_stdE d (T : inhOrderType d) (w : seq T) :
+  is_row (std w) = is_row w.
+ +
+
+ +
+

Sigma type for standard tableaux

+ +
+
+Section StdtabOfShape.
+ +
+Definition is_stdtab_of_shape sh := [pred t | (is_stdtab t) && (shape t == sh) ].
+Definition enum_stdtabsh sh : seq (seq (seq nat)) :=
+  map stdtab_of_yam (enum_yameval sh).
+ +
+Variable sh : intpart.
+ +
+Structure stdtabsh : Set :=
+  StdtabSh {
+      stdtabshval :> seq (seq nat);
+      _ : is_stdtab_of_shape sh stdtabshval
+  }.
+ +
+Let stdtabsh_enum : seq stdtabsh := pmap insub (enum_stdtabsh sh).
+ +
+Fact finite_stdtabsh : Finite.axiom stdtabsh_enum.
+ +
+ +
+Lemma stdtabshP (t : stdtabsh) : is_stdtab t.
+ +
+Lemma shape_stdtabsh (t : stdtabsh) : shape t = sh.
+ +
+Lemma enum_stdtabshE : map val (enum {: stdtabsh}) = enum_stdtabsh sh.
+ +
+Fact hyper_stdtabsh_subproof :
+  is_stdtab_of_shape sh (stdtab_of_yam (locked (hyper_yameval sh))).
+Definition hyper_stdtabsh := StdtabSh hyper_stdtabsh_subproof.
+ +
+End StdtabOfShape.
+#[export] Hint Resolve stdtabshP : core.
+ +
+Section StdtabCombClass.
+ +
+Variable n : nat.
+ +
+Definition is_stdtab_of_n := [pred t | (is_stdtab t) && (size_tab t == n) ].
+ +
+Structure stdtabn : Set :=
+  StdtabN {stdtabnval :> seq (seq nat); _ : is_stdtab_of_n stdtabnval}.
+ +
+Definition enum_stdtabn : seq (seq (seq nat)) :=
+  map (stdtab_of_yam \o val) (enum ({: yamn n})).
+Let stdtabn_enum : seq stdtabn := pmap insub enum_stdtabn.
+ +
+Fact finite_stdtabn : Finite.axiom stdtabn_enum.
+ +
+Lemma stdtabnP (s : stdtabn) : is_stdtab s.
+ +
+Lemma size_tab_stdtabn (s : stdtabn) : size_tab s = n.
+ +
+Lemma sumn_shape_stdtabnE (Q : stdtabn) : (sumn (shape Q)) = n.
+ +
+Lemma is_part_shape_deg (Q : stdtabn) : is_part_of_n n (shape Q).
+Definition shape_deg (Q : stdtabn) := IntPartN (is_part_shape_deg Q).
+ +
+End StdtabCombClass.
+ +
+Section StdtabnOfStdtabsh.
+ +
+Variables (n : nat) (sh : intpartn n).
+ +
+Fact stdtabn_of_sh_subproof (t : stdtabsh sh) : is_stdtab_of_n n t.
+Definition stdtabn_of_sh t := StdtabN (stdtabn_of_sh_subproof t).
+ +
+Lemma shape_deg_stdtabn_of_sh t :
+  shape_deg (stdtabn_of_sh t) = sh.
+ +
+End StdtabnOfStdtabsh.
+ +
+
+ +
+

Conjugate of a standard tableau

+ +
+
+Section ConjTab.
+ +
+Variables (disp: _) (T : inhOrderType disp).
+ +
+Definition conj_tab (t : seq (seq T)) : seq (seq T) :=
+  let c := conj_part (shape t) in
+  mkseq (fun i => mkseq (fun j => get_tab t (j, i)) (nth 0 c i)) (size c).
+ +
+Lemma size_conj_tab t : size (conj_tab t) = size (conj_part (shape t)).
+ +
+Lemma shape_conj_tab t : shape (conj_tab t) = conj_part (shape t).
+ +
+Lemma get_conj_tab t :
+  is_part (shape t) ->
+  forall i j, get_tab (conj_tab t) (i, j) = get_tab t (j, i).
+ +
+Lemma eq_from_shape_get_tab (t u : seq (seq T)) :
+  shape t = shape u -> get_tab t =1 get_tab u -> t = u.
+ +
+Lemma conj_tab_shapeK t : is_part (shape t) -> conj_tab (conj_tab t) = t.
+ +
+Lemma conj_tabK t : is_tableau t -> conj_tab (conj_tab t) = t.
+ +
+Lemma append_nth_conj_tab (t : seq (seq T)) l i :
+  is_part (shape t) ->
+  is_add_corner (shape t) i ->
+  conj_tab (append_nth t l i) = append_nth (conj_tab t) l (nth 0 (shape t) i).
+ +
+End ConjTab.
+ +
+Example conj_tab_expl1 :
+  conj_tab [:: [:: 0; 1; 4]; [:: 2; 3]] ==
+           [:: [:: 0; 2]; [:: 1; 3]; [:: 4]].
+ Example conj_tab_expl2 :
+  conj_tab [:: [:: 0; 1; 3; 4]; [:: 2; 5]; [:: 6]] ==
+           [:: [:: 0; 2; 6]; [:: 1; 5]; [:: 3]; [:: 4]].
+ +
+Lemma stdtab_get_tabNE t :
+  is_stdtab t ->
+  forall rc1 rc2,
+    in_shape (shape t) rc1 ->
+    in_shape (shape t) rc2 ->
+    rc1 != rc2 -> get_tab t rc1 != get_tab t rc2.
+ +
+Lemma is_stdtab_conj t : is_stdtab t -> is_stdtab (conj_tab t).
+ +
+Lemma conj_stdtabnP n (t : stdtabn n):
+  is_stdtab_of_n n (conj_tab t).
+Canonical conj_stdtabn n (t : stdtabn n) := StdtabN (conj_stdtabnP t).
+ +
+Lemma conj_stdtabshP (sh : intpart) (t : stdtabsh sh) :
+  is_stdtab_of_shape (conj_part sh) (conj_tab t).
+Canonical conj_stdtabsh {sh : intpart} (t : stdtabsh sh) :=
+  StdtabSh (conj_stdtabshP t).
+ +
+Definition stdtabshcast m n (eq_mn : m = n) t :=
+  let: erefl in _ = n := eq_mn return stdtabsh n in t.
+ +
+Lemma val_stdtabshcast m n (eq_mn : m = n) t :
+  val (stdtabshcast eq_mn t) = val t.
+ +
+Lemma conj_stdtabsh_bij sh : bijective (@conj_stdtabsh sh).
+ +
+Lemma card_stdtabsh_conj_part (sh : intpart) :
+  #|{: stdtabsh (conj_part sh)}| = #|{: stdtabsh sh}|.
+ +
+#[export] Hint Resolve stdtabnP stdtabshP : core.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.subseq.html b/combi/1.1.0/Combi.Combi.subseq.html new file mode 100644 index 00000000..336f6f26 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.subseq.html @@ -0,0 +1,277 @@ + + + + + +Combi.Combi.subseq: Subsequence of a sequence as a fintype + + + + +
+ + + +
+ +

Library Combi.Combi.subseq: Subsequence of a sequence as a fintype

+ +
+
+ +
+ +
+
+
+ +
+

Subsequence of a sequence as a fintype

+ +We define a sigma-type subseqs w for subsequence of a given sequence w +We show that subseqs w is canonically a finType. We define the three +following constructor + +
+ +
    +
  • Subseqs Pf == construct a subseqs w from a proof subseq x w. + +
    • +
  • sub_nil w == the empty sequence [::] as a subseqs w. + +
    • +
  • sub_full w == w as as a subseqs w. + +
    + + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+Require Import tools combclass sorted.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+TODO: these probably should be contributed to path.v +
+ +

A few lemmas about subseq and rcons

+ +
+
+Section RCons.
+ +
+Variable (T : eqType).
+Implicit Type s w : seq T.
+Implicit Type a b l : T.
+ +
+Lemma subseq_rcons_eq s w l : subseq s w = subseq (rcons s l) (rcons w l).
+ +
+Lemma subseq_rcons_neq s si w wn :
+  si != wn -> subseq (rcons s si) (rcons w wn) = subseq (rcons s si) w.
+ +
+End RCons.
+ +
+Section SubseqSorted.
+ +
+Variable (T : eqType) (leT : rel T).
+Implicit Type s : seq T.
+ +
+Lemma sorted_subseqP s1 s2 :
+  transitive leT -> irreflexive leT -> sorted leT s1 -> sorted leT s2 ->
+  reflect {subset s1 <= s2} (subseq s1 s2).
+ +
+End SubseqSorted.
+ +
+Section SubseqSortedIn.
+ +
+Variable (T : eqType) (leT : rel T).
+Implicit Type s : seq T.
+ +
+Lemma sorted_subseq_inP s1 s2 :
+  {in s2 & &, transitive leT} -> {in s2, irreflexive leT} ->
+  sorted leT s1 -> sorted leT s2 ->
+  reflect {subset s1 <= s2} (subseq s1 s2).
+ +
+End SubseqSortedIn.
+ +
+
+ +
+

Subsequence of a sequence as a fintype

+ +
+ +We define a dependent type SubSeq w and provide it with a Canonical +finType structure + +
+
+ +
+Section FinType.
+ +
+Variables (T : choiceType) (w : seq T).
+ +
+Structure subseqs : predArgType :=
+  Subseqs {subseqsval :> seq T; _ : subseq subseqsval w}.
+ +
+Implicit Type (s : subseqs).
+ +
+Lemma to_mask_spec s : {m : (size w).-tuple bool | mask m w == s}.
+Definition to_mask s := let: exist m _ := to_mask_spec s in m.
+ +
+Lemma to_maskK : cancel to_mask (fun m => Subseqs (mask_subseq (val m) w)).
+ +
+ +
+Lemma subseqsP s : subseq s w.
+ +
+Definition sub_nil : subseqs := Subseqs (sub0seq w).
+Definition sub_full : subseqs := Subseqs (subseq_refl w).
+ +
+Lemma mask1E x0 i :
+  i < size w -> mask [tuple val x == i | x < size w] w = [:: nth x0 w i].
+ +
+Lemma mask_injP :
+  reflect (injective (fun m : (size w).-tuple bool => mask (val m) w))
+          (uniq w).
+ +
+Lemma Subseqs_maskK :
+  uniq w -> cancel (fun m => Subseqs (mask_subseq (val m) w)) to_mask.
+ +
+Lemma enum_subseqsE :
+  perm_eq
+    (enum {: subseqs})
+    (undup [seq Subseqs (mask_subseq (val m) w) | m : (size w).-tuple bool]).
+ +
+Lemma val_enum_subseqs :
+  perm_eq
+    (map val (enum {: subseqs}))
+    (undup [seq mask (val m) w | m : (size w).-tuple bool]).
+ +
+Lemma seq_masks_uniq :
+  uniq w -> uniq [seq mask (val m) w | m : (size w).-tuple bool].
+ +
+Lemma subseqs_masks_uniq :
+  uniq w ->
+  uniq [seq Subseqs (mask_subseq (val m) w) | m : (size w).-tuple bool].
+ +
+End FinType.
+ +
+Section Bigop.
+ +
+Context {R : Type} {idx : R} {op : Monoid.com_law idx}.
+#[local] Notation "1" := idx.
+#[local] Notation "'*%M'" := op (at level 0).
+#[local] Notation "x * y" := (op x y).
+Context {T : choiceType}.
+ +
+Lemma big_subseqs (F : seq T -> R) (s : seq T) :
+  uniq s ->
+  \big[*%M/1]_(i : subseqs s) F i =
+  \big[*%M/1]_(m : (size s).-tuple bool) F (mask m s).
+ +
+Lemma big_subseqs_cond (P : pred (seq T)) (F : seq T -> R) (s : seq T) :
+  uniq s ->
+  \big[*%M/1]_(i : subseqs s | P i) F i =
+  \big[*%M/1]_(m : (size s).-tuple bool | P (mask m s)) F (mask m s).
+ +
+Lemma big_subseqs0 (F : seq T -> R) :
+  \big[*%M/1]_(i : subseqs [::]) F i = F [::].
+ +
+Lemma big_subseqs_cons (F : seq T -> R) (a : T) (s : seq T) :
+  uniq (a :: s) ->
+  \big[*%M/1]_(i : subseqs (a :: s)) F i =
+  \big[*%M/1]_(i : subseqs s) F (a :: i) * \big[*%M/1]_(i : subseqs s) F (i).
+ +
+Lemma big_subseqs_cons_cond
+      (F : seq T -> R) (P : pred (seq T)) (a : T) (s : seq T) :
+  uniq (a :: s) ->
+  \big[*%M/1]_(i : subseqs (a :: s) | P i) F i =
+  \big[*%M/1]_(i : subseqs s | P (a :: i)) F (a :: i) *
+  \big[*%M/1]_(i : subseqs s | P i) F (i).
+ +
+Lemma big_subseqs_undup (F : seq T -> R) (s : seq T) :
+  idempotent_op op ->
+  \big[*%M/1]_(i : subseqs s) F i =
+  \big[*%M/1]_(m : (size s).-tuple bool) F (mask m s).
+ +
+Lemma big_subseqs_undup_cond (F : seq T -> R) (P : pred (seq T)) (s : seq T) :
+  idempotent_op op ->
+  \big[*%M/1]_(i : subseqs s | P i) F i =
+  \big[*%M/1]_(m : (size s).-tuple bool | P (mask m s)) F (mask m s).
+ +
+End Bigop.
+ +
+
+ +
+

Relating sub sequences of iota and being sorted

+ +
+
+Lemma sorted_subseq_iota_rcons s n :
+  subseq s (iota 0 n) = sorted ltn (rcons s n).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.tableau.html b/combi/1.1.0/Combi.Combi.tableau.html new file mode 100644 index 00000000..05c9e2d8 --- /dev/null +++ b/combi/1.1.0/Combi.Combi.tableau.html @@ -0,0 +1,563 @@ + + + + + +Combi.Combi.tableau: Young Tableaux + + + + +
+ + + +
+ +

Library Combi.Combi.tableau: Young Tableaux

+ +
+
+ +
+ +
+
+
+ +
+

Young Tableaux over an ordtype

+ + +
+ +We define the notion of (semistandard) Young tableau over an ordType +denoted T. + +
+ +
    +
  • is_row r == r is sorted + +
    • +
  • dominate u v == u dominate v, that is v is longer than u and + the i-th letter of u is strictly larger than the i-th letter of v. + this is an order relation on seq T. + +
    • +
  • is_tableau t == t of type (seq (seq T)) is a tableau that is a sequence + of non empty rows which is sorted for the dominate order. + +
    • +
  • get_tab t (r, c) == the element of t of coordinate (r, c), + or inhabitant T if (r, c) is not is the tableau + +
    • +
  • to_word t == the row reading of the tableau t + +
    • +
  • size_tab t == the size (number of boxes) of t + +
    • +
  • filter_gtnX_tab n t == the sub-tableau of t formed by the element smaller + than n. + +
    + + +
    • +
  • tabsh_reading sh w == w is the row reading of a tableau of shape sh + +
    • +
+ +
+ +In the following tableaux are considered on 'I_n.+1 for a given n. + +
+ +
    +
  • is_tab_of_shape sh t == t is a tableau of shape sh. + +
    • +
  • tabsh sh == a sigma-type for the predicate is_tab_of_shape sh where + sh is a partition of d (type 'P_d). This is canonically + a subFinType. + +
    • +
  • tabrowconst pf == if pf is a proof that size sh <= n.+1 then + construct the tableau (of type tabsh sh) whose i-th row + contains only i's as elements of 'I_n.+1. + +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+Require Import tools partition ordtype sorted.
+ +
+Set Implicit Arguments.
+ +
+Open Scope N.
+ +
+Import Order.Theory.
+ +
+
+ +
+

Specialization of sorted Lemmas

+ +
+
+Section Rows.
+ +
+Context disp (T : inhOrderType disp).
+ +
+Implicit Type l : T.
+Implicit Type r : seq T.
+ +
+Notation is_row := (sorted <=%O).
+ +
+Definition is_row1P Z r := sortedP Z (e := <=%O) (s := r).
+Definition is_rowP Z r := sorted2P Z (@le_trans _ T) (@le_refl _ T) r.
+Definition is_row_cons := sorted_cons (@le_refl _ T).
+Definition is_row_consK := sorted_consK (T := T) (R := <=%O).
+Definition is_row_rcons := sorted_rcons (T := T) (R := <=%O).
+Definition is_row_rconsK := sorted_rconsK (T := T) (R := <=%O).
+Definition is_row_last := sorted_last (@le_refl _ T).
+Definition is_row_take := take_sorted (T := T) (leT := <=%O).
+Definition is_row_drop := drop_sorted (T := T) (leT := <=%O).
+Definition is_row_cat2 := cat_sorted2 (T := T) (leT := <=%O).
+Definition head_leq_last_row :=
+  head_leq_last_sorted (@le_trans _ T) (@le_refl _ T).
+Lemma row_lt_by_pos Z r p q:
+  is_row r -> p < size r -> q < size r ->
+  (nth Z r p < nth Z r q)%O -> p < q.
+ +
+End Rows.
+ +
+Notation is_row := (sorted <=%O).
+ +
+
+ +
+

Dominance order for rows

+ +
+
+Section Dominate.
+ +
+Context {disp} {T : inhOrderType disp}.
+ +
+Implicit Type l : T.
+Implicit Type r u v : seq T.
+Implicit Type t : seq (seq T).
+ +
+Lemma in_shape_tab_size t i j :
+  in_shape (shape t) (i, j) -> i < size t.
+ Lemma in_shape_tab i j t :
+  in_shape (shape t) (i, j) -> j < size (nth [::] t i).
+ +
+Lemma is_row_set_nth l r pos :
+  is_row r -> (l < nth l r pos)%O ->
+  (forall n : nat, (l < nth l r n)%O -> pos <= n) ->
+  is_row (set_nth l r pos l).
+ +
+Fixpoint dominate_rec u v :=
+  if u is u0 :: u' then
+    if v is v0 :: v' then (u0 > v0)%O && (dominate_rec u' v')
+    else false
+  else true.
+ +
+Definition dominate u v :=
+  (size u <= size v) &&
+   (all (fun i => nth inh u i > nth inh v i)%O (iota 0 (size u))).
+ +
+Lemma dominate_recE : dominate =2 dominate_rec.
+ +
+Lemma dominateP u v :
+  reflect (size u <= size v /\
+           forall i, i < size u -> (nth inh u i > nth inh v i)%O)
+          (dominate u v).
+ +
+Lemma dominate_trans : transitive dominate.
+ +
+Definition dominate_rev_trans := rev_trans dominate_trans.
+ +
+Lemma dominate_rcons v u l : dominate u v -> dominate u (rcons v l).
+ +
+Lemma dominate_take v u n : dominate u (take n v) -> dominate u v.
+ +
+Lemma dominate_cut u v w:
+  size u <= size v -> dominate u (v ++ w) -> dominate u v.
+ +
+Lemma dominate_head u v :
+  u != [::] -> dominate u v -> (head inh v < head inh u)%O.
+ +
+Lemma dominate_tl a u b v :
+  dominate (a :: u) (b :: v) -> dominate u v.
+ +
+End Dominate.
+Arguments dominate_trans {disp T}.
+Arguments dominate_rev_trans {disp T}.
+ +
+
+ +
+

Tableaux : definition and basic properties

+ +
+
+Section Tableau.
+ +
+Context disp (T : inhOrderType disp).
+ +
+Implicit Type l : T.
+Implicit Type r w : seq T.
+Implicit Type t : seq (seq T).
+ +
+Fixpoint is_tableau t :=
+  if t is t0 :: t'
+  then [&& (t0 != [::]), is_row t0, dominate (head [::] t') t0 & is_tableau t']
+  else true.
+ +
+Definition get_tab t (rc : nat * nat) := nth inh (nth [::] t rc.1) rc.2.
+ +
+Definition to_word t := flatten (rev t).
+ +
+ +
+Lemma is_tableauP t :
+  reflect
+    [/\ forall i, i < size t -> (nth [::] t i) != [::],
+        forall i, is_row (nth [::] t i) &
+        forall i j, i < j -> dominate (nth [::] t j) (nth [::] t i)]
+    (is_tableau t).
+ +
+Lemma get_tab_default t rc :
+  ~~ in_shape (shape t) rc -> get_tab t rc = inh.
+ +
+Lemma to_word_cons r t : to_word (r :: t) = to_word t ++ r.
+Lemma to_word_rcons r t : to_word (rcons t r) = r ++ to_word t.
+ +
+Lemma mem_to_word t rc :
+  in_shape (shape t) rc -> get_tab t rc \in (to_word t).
+ +
+Lemma to_wordK t : rev (reshape (rev (shape t)) (to_word t)) = t.
+ +
+Lemma tableau_is_row r t : is_tableau (r :: t) -> is_row r.
+ +
+Lemma is_tableau_rconsK t (tn : seq T) :
+  is_tableau (rcons t tn) -> is_tableau t.
+ +
+Lemma is_tableau_catl t1 t2 :
+  is_tableau (t1 ++ t2) -> is_tableau t1.
+ +
+Lemma is_tableau_catr t1 t2 :
+  is_tableau (t1 ++ t2) -> is_tableau t2.
+ +
+Lemma is_part_sht t : is_tableau t -> is_part (shape t).
+ +
+Lemma tab_eqP p q :
+  is_tableau p -> is_tableau q ->
+  reflect (forall i, nth [::] p i = nth [::] q i) (p == q).
+ +
+Lemma is_tableau_sorted_dominate t :
+  is_tableau t =
+  [&& is_part (shape t),
+   all (sorted <=%O) t &
+   sorted (fun (r s : seq T) => dominate s r) t].
+ +
+Lemma is_tableau_getP t :
+  reflect
+    [/\ is_part (shape t),
+     (forall (r c : nat), in_shape (shape t) (r, c.+1) ->
+                          (get_tab t (r, c) <= get_tab t (r, c.+1))%O) &
+     (forall (r c : nat), in_shape (shape t) (r.+1, c) ->
+                          (get_tab t (r, c) < get_tab t (r.+1, c))%O)]
+    (is_tableau t).
+ +
+
+ +
+

Cuting rows and tableaux

+ +
+
+Lemma row_dominate (u v : seq T) :
+  is_row (u ++ v) -> dominate u v -> u = [::].
+ +
+Lemma filter_gt_row r n :
+  is_row r -> filter (>%O n) r = take (count (>%O n) r) r.
+ +
+Lemma filter_le_row n r :
+  is_row r -> filter (<=%O n) r = drop (count (>%O n) r) r.
+ +
+Lemma count_gt_dominate r1 r0 n :
+  dominate r1 r0 -> (count (>%O n) r1) <= (count (>%O n) r0).
+ +
+Lemma filter_gt_dominate r1 r0 n :
+  is_row r0 -> is_row r1 -> dominate r1 r0 ->
+  dominate (filter (>%O n) r1) (filter (>%O n) r0).
+ +
+Definition filter_gt_tab n :=
+  [fun t : (seq (seq T)) => filter (fun r => r != [::])
+                                   [seq [seq x <- i | (n > x)%O] | i <- t]].
+ +
+Lemma to_word_filter_nnil t : to_word (filter (fun r => r != [::]) t) = to_word t.
+ +
+Lemma filter_to_word t p : filter p (to_word t) = to_word (map (filter p) t).
+ +
+Lemma head_filter_gt_tab n t :
+  is_tableau t ->
+  head [::] (filter_gt_tab n t) = [seq x <- head [::] t | (x < n)%O].
+ +
+Lemma is_tableau_filter_gt t n :
+  is_tableau t -> is_tableau (filter_gt_tab n t).
+ +
+
+ +
+

The size of a tableau

+ +
+
+Definition size_tab t := sumn (shape t).
+ +
+Lemma tab0 t : is_tableau t -> size_tab t = 0 -> t = [::].
+ +
+Lemma size_to_word t : size (to_word t) = size_tab t.
+ +
+End Tableau.
+ +
+ +
+
+ +
+

Tableaux from their row reading

+ +
+
+Section TableauReading.
+ +
+Context disp (A : inhOrderType disp).
+ +
+Definition tabsh_reading (sh : seq nat) (w : seq A) :=
+  (size w == sumn sh) && (is_tableau (rev (reshape (rev sh) w))).
+ +
+Lemma tabsh_readingP (sh : seq nat) (w : seq A) :
+  reflect
+    (exists tab, [/\ is_tableau tab, shape tab = sh & to_word tab = w])
+    (tabsh_reading sh w).
+ +
+End TableauReading.
+ +
+
+ +
+

Sigma type for tableaux

+ +
+
+Section FinType.
+ +
+Context {disp} {T : inhFinOrderType disp}.
+Variables (d : nat) (sh : 'P_d).
+ +
+Definition is_tab_of_shape (sh : seq nat) :=
+  [pred t : seq (seq T) | (is_tableau t) && (shape t == sh) ].
+ +
+Structure tabsh : predArgType :=
+  TabSh {tabshval :> seq (seq T); _ : is_tab_of_shape sh tabshval}.
+ +
+Implicit Type (t : tabsh).
+ +
+Lemma tabshP t : is_tableau t.
+ +
+Lemma shape_tabsh t : shape t = sh.
+ +
+Lemma tabsh_to_wordK t : rev (reshape (rev sh) (to_word (val t))) = t.
+ +
+Let tabsh_enum :
+  seq tabsh := pmap insub
+              [seq rev (reshape (rev sh) (val w)) | w in {: d.-tuple T}].
+ +
+Lemma finite_tabsh : Finite.axiom tabsh_enum.
+ +
+ +
+Lemma to_word_enum_tabsh :
+  perm_eq
+    [seq to_word (tabshval t) | t : tabsh]
+    [seq x <- [seq (i : seq _) | i : d.-tuple T] | tabsh_reading sh x].
+ +
+End FinType.
+ +
+Notation "'tabsh[' T ']' mu" :=
+  (tabsh (T := T : inhFinOrderType _) mu) (at level 10).
+ +
+#[export] Hint Resolve tabshP : core.
+ +
+
+ +
+TODO : Generalise to any finite order using Order.enum_val when released +
+
+Section OrdTableau.
+ +
+Variable n : nat.
+Variables (d : nat) (sh : 'P_d).
+ +
+Implicit Type (t : tabsh['I_n.+1] sh).
+ +
+Lemma all_ltn_nth_tabsh t (i : nat) :
+  all (fun x : 'I_n.+1 => (i <= x)%O) (nth [::] t i).
+ +
+Lemma size_tabsh t : size t <= n.+1.
+ +
+Hypothesis Hszs : size sh <= n.+1.
+Lemma tabrowconst_subproof :
+  is_tab_of_shape
+    sh (take (size sh) [tuple nseq (nth 0 sh (i : 'I_n.+1)) i | i < n.+1]).
+Definition tabrowconst := TabSh (tabrowconst_subproof).
+ +
+End OrdTableau.
+ +
+
+ +
+

Tableaux and increasing maps

+ +
+
+Section IncrMap.
+ +
+Context disp1 disp2 (T1 : inhOrderType disp1) (T2 : inhOrderType disp2).
+Variable F : T1 -> T2.
+ +
+Lemma shape_map_tab (t : seq (seq T1)) :
+  shape [seq map F r | r <- t] = shape t.
+ +
+Lemma get_map_tab (t : seq (seq T1)) rc :
+  in_shape (shape t) rc ->
+  get_tab [seq [seq F i | i <- r0] | r0 <- t] rc = F (get_tab t rc).
+ +
+Lemma to_word_map_tab (t : seq (seq T1)) :
+  to_word [seq map F r | r <- t] = map F (to_word t).
+ +
+Lemma incr_tab (t : seq (seq T1)) :
+  {in (to_word t) &, {homo F : x y / (x < y)%O}} ->
+  (is_tableau t) = (is_tableau [seq map F r | r <- t]).
+ +
+End IncrMap.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Combi.vectNK.html b/combi/1.1.0/Combi.Combi.vectNK.html new file mode 100644 index 00000000..56ff07ec --- /dev/null +++ b/combi/1.1.0/Combi.Combi.vectNK.html @@ -0,0 +1,167 @@ + + + + + +Combi.Combi.vectNK: Integer Vector of Given Sums and Sizes + + + + +
+ + + +
+ +

Library Combi.Combi.vectNK: Integer Vector of Given Sums and Sizes

+ +
+
+ +
+ +
+
+
+ +
+

Integer vectors of sum n and size k

+ + +
+ +
    +
  • vect_n_k n k == the list of integer vectors of sum n and size k + +
    • +
  • cut_k k s == the list of the cutting of s in k slices (the result is of + type seq (seq T)) + +
    • +
  • cut3 s == the list of the cutting of s in 3 slices (the result is of type + seq (seq T) * (seq T) * (seq T)) + +
    • +
+ +
+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq.
+Require Import tools.
+ +
+Set Implicit Arguments.
+ +
+Section VectNK.
+ +
+Fixpoint vect_n_k n k :=
+  if k is k'.+1
+  then flatten (mkseq (fun i => map (cons i) (vect_n_k (n-i) k') ) n.+1)
+  else if n is 0 then [:: [::]] else [::].
+ +
+Lemma vect_n_k_in n k s : sumn s == n -> size s == k -> s \in vect_n_k n k.
+ +
+Lemma in_vect_n_k n k s : s \in vect_n_k n k -> (sumn s == n) && (size s == k).
+ +
+Lemma vect_n_kP n k s : s \in vect_n_k n k = (sumn s == n) && (size s == k).
+ +
+Lemma vect_0_k k : vect_n_k 0 k = [:: nseq k 0].
+ +
+Lemma count_mem_vect_n_k_eq_1 n k s :
+  s \in vect_n_k n k -> count_mem s (vect_n_k n k) = 1.
+ +
+Lemma uniq_vect_n_k n k : uniq (vect_n_k n k).
+ +
+End VectNK.
+ +
+
+ +
+Cutting a seq in k slices : the result is of type seq (seq T) +
+
+ +
+Section CutK.
+ +
+Variable T : eqType.
+Implicit Type (s : seq T) (ss : seq (seq T)).
+ +
+Lemma mem_shape_vect_n_k ss :
+  (shape ss) \in vect_n_k (size (flatten ss)) (size ss).
+ +
+Definition cut_k k s := [seq reshape sh s | sh <- vect_n_k (size s) k].
+ +
+Lemma cut_k_flatten ss : ss \in cut_k (size ss) (flatten ss).
+ +
+Lemma flatten_equiv_cut_k s ss : s == flatten ss <-> ss \in cut_k (size ss) s.
+ +
+Lemma size_cut_k k s ss : ss \in (cut_k k s) -> size ss = k.
+ +
+End CutK.
+ +
+
+ +
+Cutting a seq in 3 slices : the result is of type + seq (seq T) * (seq T) * (seq T) +
+
+ +
+Section Cut3.
+ +
+Variable T : eqType.
+Implicit Type (s : seq T) (ss : seq (seq T)).
+ +
+Let match3 :=
+  fun ss => match ss return (seq T) * (seq T) * (seq T) with
+              | [:: a; b; c] => (a, b, c) | _ => ([::], [::], [::]) end.
+ +
+Definition cut3 s : seq ((seq T) * (seq T) * (seq T)) :=
+  [seq match3 ss | ss <- cut_k 3 s].
+ +
+Lemma cat3_equiv_cut3 s a b c : s == a ++ b ++ c <-> (a, b, c) \in cut3 s.
+ +
+End Cut3.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.Erdos_Szekeres.Erdos_Szekeres.html b/combi/1.1.0/Combi.Erdos_Szekeres.Erdos_Szekeres.html new file mode 100644 index 00000000..3544f58a --- /dev/null +++ b/combi/1.1.0/Combi.Erdos_Szekeres.Erdos_Szekeres.html @@ -0,0 +1,88 @@ + + + + + +Combi.Erdos_Szekeres.Erdos_Szekeres: The Erdös-Szekeres theorem + + + + +
+ + + +
+ +

Library Combi.Erdos_Szekeres.Erdos_Szekeres: The Erdös-Szekeres theorem

+ +
+
+ +
+ +
+
+
+ +
+

The Erdös-Szekeres theorem on monotonic subsequences.

+ + +
+ +A proof of the Erdös Szekeres theorem about longest increasing and +decreasing subsequences. The theorem is Erdos_Szekeres and +says that any sequence s of length at least n*m+1 over a totally ordered +type contains +
    +
  • either a nondecreasing subsequence of length n+1; + +
    • +
  • or a strictly decreasing subsequence of length m+1. + +
    • +
+We prove it as a corollary of Greene's theorem on the Robinson-Schensted +correspondance. Note that there are other proofs which require less theory. + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq fintype.
+From mathcomp Require Import tuple finfun finset bigop path order.
+ +
+Require Import partition tableau Schensted ordtype Greene Greene_inv.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+Open Scope N.
+ +
+Lemma Greene_rel_one (T : eqType) (s : seq T) (R : rel T) :
+  exists t : seq T, subseq t s /\ sorted R t /\ size t = (Greene_rel R s) 1.
+ +
+Theorem Erdos_Szekeres disp (T : inhOrderType disp) (m n : nat) (s : seq T) :
+  size s > m * n ->
+  (exists t, subseq t s /\ sorted <=%O t /\ size t > m) \/
+  (exists t, subseq t s /\ sorted >%O t /\ size t > n).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.HookFormula.Frobenius_ident.html b/combi/1.1.0/Combi.HookFormula.Frobenius_ident.html new file mode 100644 index 00000000..da23f995 --- /dev/null +++ b/combi/1.1.0/Combi.HookFormula.Frobenius_ident.html @@ -0,0 +1,108 @@ + + + + + +Combi.HookFormula.Frobenius_ident + + + + +
+ + + +
+ +

Library Combi.HookFormula.Frobenius_ident

+ +
+
+ +
+

Combi.hook.Frobenius_ident : Frobenius identity

+ +
+
+
+ +
+

A proof of Frobenius identity:

+ + +
+ +The goal of this file is to prove the following identities: +
+Frobenius_ident n :
+    n`! = \sum_(p : 'P_n) (n`! %/ (hook_length_prod p))^2. +
+ +and +
+Theorem Frobenius_ident_rat n :
+    1 / (n`!)%:Q = \sum_(p : 'P_n) 1 / (hook_length_prod p)%:Q ^+ 2. +
+ + +
+
+ +
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq
+        ssrint div rat fintype finset bigop path ssralg ssrnum order.
+ +
+Set Implicit Arguments.
+ +
+Require Import ordtype partition tableau Schensted std stdtab hook.
+ +
+Section Identity.
+ +
+Variable n : nat.
+ +
+#[local] Notation stpn := (stdtabn n * stdtabn n)%type.
+Lemma card_stpn_shape :
+  #|[set p : stpn | shape p.1 == shape p.2]| =
+    \sum_(sh : 'P_n) #|{: stdtabsh sh}|^2.
+ +
+Lemma card_stpn_shape_hook :
+  #|[set p : stpn | shape p.1 == shape p.2]| =
+    \sum_(sh : 'P_n) (n`! %/ (hook_length_prod sh))^2.
+ +
+Theorem Frobenius_ident :
+  n`! = \sum_(p : 'P_n) (n`! %/ (hook_length_prod p))^2.
+ +
+#[local] Open Scope ring_scope.
+ +
+Import GRing.Theory.
+Import Num.Theory.
+ +
+Theorem Frobenius_ident_rat :
+  1 / (n`!)%:Q = \sum_(p : 'P_n) 1 / (hook_length_prod p)%:Q ^+ 2.
+ +
+End Identity.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.HookFormula.hook.html b/combi/1.1.0/Combi.HookFormula.hook.html new file mode 100644 index 00000000..0825617b --- /dev/null +++ b/combi/1.1.0/Combi.HookFormula.hook.html @@ -0,0 +1,818 @@ + + + + + +Combi.HookFormula.hook: A proof of the Hook-Length formula + + + + +
+ + + +
+ +

Library Combi.HookFormula.hook: A proof of the Hook-Length formula

+ +
+
+ +
+ +
+
+
+ +
+

A proof of the Hook-Length formula

+ + +
+ +This file contains a proof of the Frame-Robinson-Thrall (see [FRT]) hook-Length +formula for the number of standard Young tableau. It follows +closely the probabilistic proof of [GNW]. + +
+ +
    +
  • [FRT] -- J. S. Frame, G. de B. Robinson and R. M. Thrall, + The hook graphs of the symmetric group, Canad. J. Math. 6 (1954), 316-324. + +
    + + +
    • +
  • [GNW] -- Greene, C., Nijenhuis, A. and Wilf, H. S., A probabilistic + proof of a formula for the number of Young tableaux of a given + shape, Adv. in Math. 31 (1979), 104–109. + +
    • +
+ +
+ +Here are the contents of the file: + +
+ +Basic notions: boxes, hook, corners... + +
+ +
    +
  • corner_box sh (r, c) == (r, c) are the coordinate of a corner of sh + +
    • +
  • arm_length sh (r, c) == the arm length of sh of the box (r, c) + +
    • +
  • leg_length sh (r, c) == the leg length of sh of the box (r, c) + +
    • +
  • hook_length sh (r, c) == the hook length of sh of the box (r, c) + +
    + + +
    • +
  • in_hook (r, c) (k, l) == the box (k, l) is in the hook of the box (r, c) + +
    • +
  • hook_box_indices (r, c) == a sequence of indexes for boxes in the hook of + (r, c): leg boxes are inl k, arm boxes are inr + +
    • +
  • hook_box (r, c) n == the coordinate of the box of index n + +
    • +
  • hook_boxes (r, c) == sequence of the boxes in the hool of (r, c) + +
    • +
+ +
+ +The probabilistic algorithm: + +
+ +
    +
  • is_trace p A B == the trace of a trial path in the partition p, that is + the pair of the vertical and horizontal projection (see GNW). + Specifically A and B are two non-empty strictly increasing + sequences of naturals ending at a corner of p. + +
    + + +
    • +
  • trace_seq last == all stricly increasing sequences ending by last + +
    • +
  • enum_trace Alpha Beta == all trace ending at (Alpha, Beta) + +
    + + +
    • +
  • walk_to_corner m (i, j) == the probabilistic distribution of traces of a + hook path of length at most m starting at (i, j) + +
    • +
  • choose_corner p == the probalistic distribution of a trace starting at + a uniformly chosen box + +
    + + +
    • +
  • starts_at (r, c) t == the trace t starts as (r, c) + +
    • +
  • starts_at (r, c) t == the trace t ends as (r, c) + +
    • +
+ +
+ +Formulas: + +
+ +
    +
  • RHSL3 p a b A B == the right hand side of Lemma 3 for a path starting + at (a, b) with projections A and B + +
    • +
  • RHSL3_trace p t == the right hand side of Lemma 3 for trace t in p + +
    + + +
    • +
  • hook_length_prod sh == the product of the hook length of sh + +
    • +
+ +
+ +Finally the main Theorem is stated as: + +
+ +
+Theorem HookLengthFormula (p : intpart) :
+  #|{: stdtabsh p}| = (sumn p)`! %/ (hook_length_prod p). +
+ + +
+ + +
+
+ +
+Require Import Misc Ccpo.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import ssrint div rat ssralg ssrnum.
+ +
+Require Import tools subseq partition Yamanouchi stdtab Qmeasure.
+ +
+Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Import Num.Theory.
+ +
+#[local] Open Scope nat_scope.
+ +
+
+ +
+

Recursion for the number of Yamanouchi words and standard tableaux

+ +
+
+Lemma card_yama_rec (p : intpart) :
+  p != empty_intpart ->
+  #|{: yameval p}| =
+      \sum_(i <- rem_corners p) #|{: yameval (decr_nth_intpart p i)}|.
+ +
+Lemma card_yama0 : #|{: yameval empty_intpart}| = 1.
+ +
+Lemma card_yam_stdtabE (p : intpart) :
+  #|{: yameval p}| = #|{: stdtabsh p}|.
+ +
+#[local] Open Scope ring_scope.
+ +
+Lemma card_stdtabsh_rat_rec (F : intpart -> rat) :
+  F empty_intpart = 1 ->
+  ( forall p : intpart,
+      p != empty_intpart ->
+      F p = \sum_(i <- rem_corners p) F (decr_nth_intpart p i) ) ->
+  forall p : intpart, F p = #|{: stdtabsh p}|%:Q.
+ +
+Close Scope ring_scope.
+ +
+
+ +
+

Boxes, Hooks and corners

+ +
+ + Corner Boxes +
+
+ +
+Definition corner_box sh rc :=
+  is_rem_corner sh rc.1 && (rc.2 == (nth 0 sh rc.1).-1).
+ +
+Lemma corner_box_in_part sh rc :
+  corner_box sh rc -> in_shape sh rc.
+ +
+Lemma corner_box_conj_part sh u v :
+  is_part sh -> corner_box sh (u, v) -> corner_box (conj_part sh) (v, u).
+ +
+
+ +
+Arm Leg and Hook lengths +
+
+ +
+Definition arm_length sh rc := ((nth 0 sh rc.1) - rc.2).-1.
+Definition leg_length sh rc := (arm_length (conj_part sh) (rc.2, rc.1)).
+Definition hook_length sh rc := (arm_length sh rc + leg_length sh rc).+1.
+ +
+
+ +
+The hook length product +
+
+Definition hook_length_prod sh := (\prod_(b : box_in sh) hook_length sh b)%N.
+#[local] Notation HLF sh := (((sumn sh)`!)%:Q / (hook_length_prod sh)%:Q)%R.
+ +
+Lemma hook_length_geq1 sh rc : hook_length sh rc >= 1.
+ #[local] Hint Resolve hook_length_geq1 : core.
+ +
+Lemma hook_length_conj_part sh r c :
+  is_part sh -> hook_length (conj_part sh) (r, c) = hook_length sh (c, r).
+ +
+Lemma arm_length_ler sh r c j :
+  is_part sh -> r < j -> in_shape sh (j, c) ->
+  arm_length sh (j, c) <= arm_length sh (r, c).
+ +
+Lemma arm_length_ltl sh r c j :
+  is_part sh -> c < j -> in_shape sh (r, j) ->
+  arm_length sh (r, j) < arm_length sh (r, c).
+ +
+Lemma leg_length_ltr sh r c j :
+  is_part sh -> r < j -> in_shape sh (j, c) ->
+  leg_length sh (j, c) < leg_length sh (r, c).
+ +
+Lemma leg_length_lel sh r c j :
+  is_part sh -> c < j -> in_shape sh (r, j) ->
+  leg_length sh (r, j) <= leg_length sh (r, c).
+ +
+Lemma hook_length_ltl sh r c j :
+  is_part sh -> c < j -> in_shape sh (r, j) ->
+  hook_length sh (r, j) < hook_length sh (r, c).
+ +
+Lemma hook_length_ltr sh r c j :
+  is_part sh -> r < j -> in_shape sh (j, c) ->
+  hook_length sh (j, c) < hook_length sh (r, c).
+ +
+Lemma hook_length1_corner sh rc :
+  is_part sh -> in_shape sh rc ->
+  hook_length sh rc = 1 -> corner_box sh rc.
+ +
+Lemma corner_arm_length0 sh rc :
+  is_part sh -> corner_box sh rc -> arm_length sh rc = 0.
+ +
+Lemma corner_leg_length0 sh rc :
+  is_part sh -> corner_box sh rc -> leg_length sh rc = 0.
+ +
+Lemma corner_hook_length1 sh rc :
+  is_part sh -> corner_box sh rc -> hook_length sh rc = 1.
+ +
+Lemma arm_length_corner_box sh r c u v :
+  is_part sh ->
+  r <= u -> c <= v -> corner_box sh (u, v) ->
+  arm_length sh (r, c) = arm_length sh (u, c) + arm_length sh (r, v).
+ +
+Lemma leg_length_corner_box sh r c u v :
+  is_part sh ->
+  r <= u -> c <= v -> corner_box sh (u, v) ->
+  leg_length sh (r, c) = leg_length sh (u, c) + leg_length sh (r, v).
+ +
+Lemma hook_length_corner_box sh r c u v :
+  is_part sh -> r <= u -> c <= v -> corner_box sh (u, v) ->
+  hook_length sh (r, c) = hook_length sh (u, c) + hook_length sh (r, v) - 1.
+ +
+Lemma arm_length_incr_nth_row sh r c :
+  in_shape sh (r, c) ->
+  arm_length (incr_nth sh r) (r, c) = (arm_length sh (r, c)).+1.
+ +
+Lemma arm_length_incr_nth_nrow sh r c i :
+  i != r -> arm_length (incr_nth sh i) (r, c) = arm_length sh (r, c).
+ +
+Lemma hook_length_incr_nth_row sh r c :
+  is_part sh -> is_add_corner sh r -> in_shape sh (r, c) ->
+  hook_length (incr_nth sh r) (r, c) = (hook_length sh (r, c)).+1.
+ +
+Lemma hook_length_incr_nth_col sh r i :
+  is_part sh -> is_add_corner sh r ->
+  in_shape sh (i, (nth 0 sh r)) ->
+  hook_length (incr_nth sh r) (i, nth 0 sh r)
+  = (hook_length sh (i, nth 0 sh r)).+1.
+ +
+Lemma hook_length_incr_nth sh i r c :
+  is_part sh -> is_add_corner sh i ->
+  in_shape sh (r, c) ->
+  i != r -> nth 0 sh i != c ->
+  hook_length (incr_nth sh i) (r, c) = hook_length sh (r, c).
+ +
+Open Scope ring_scope.
+ +
+Lemma hook_length_pred sh rc :
+  (hook_length sh rc)%:~R - 1 = ((hook_length sh rc).-1)%:~R :> rat.
+ +
+Lemma prod_hook_length_quot_row p Alpha Beta :
+  is_part p -> corner_box p (Alpha, Beta) ->
+  \prod_(i <- enum_box_in (decr_nth p Alpha) | i.1 == Alpha)
+     ( (hook_length p i)%:Q /
+        (hook_length (decr_nth p Alpha) i)%:Q ) =
+  \prod_(0 <= j < Beta) (1 + ((hook_length p (Alpha, j))%:Q - 1)^-1).
+ +
+Close Scope ring_scope.
+ +
+Section FindCorner.
+ +
+Variable p : intpart.
+Implicit Types (r c k l : nat) (rc kl : nat * nat).
+ +
+
+ +
+

Hook boxes

+ +
+
+ +
+#[local] Notation conj := (conj_part p).
+ +
+Definition in_hook rc kl :=
+  let: (r, c) := rc in let: (k, l) := kl in
+  ((r == k) && (c < l < nth 0 p r)) ||
+  ((c == l) && (r < k < nth 0 conj c)).
+ +
+Lemma in_hook_shape rc kl :
+   in_shape p rc -> in_hook rc kl -> in_shape p kl.
+ +
+Definition hook_box_indices rc : seq (nat+nat) :=
+  [seq inl k | k <- iota rc.1.+1 ((nth 0 conj rc.2).-1 - rc.1)] ++
+  [seq inr k | k <- iota rc.2.+1 ((nth 0 p rc.1).-1 - rc.2)].
+Definition hook_box rc n : nat * nat :=
+  match n with inl k => (k,rc.2) | inr k => (rc.1,k) end.
+Definition hook_boxes rc := [seq hook_box rc n | n <- hook_box_indices rc].
+ +
+Lemma size_hook_box_indices rc :
+  size (hook_box_indices rc) = (hook_length p rc).-1.
+ +
+#[local] Lemma ltnPred a b : a < b -> (a <= b.-1).
+ +
+#[local] Lemma iota_hookE a b c : a < b -> b < a.+1 + (c.-1 - a) = (b < c).
+ +
+Lemma in_hook_boxesP rc kl : (kl \in hook_boxes rc) = (in_hook rc kl).
+ +
+Lemma hook_boxes_empty rc :
+  in_shape p rc -> hook_boxes rc = [::] -> corner_box p rc.
+ +
+
+ +
+

Traces

+ +
+
+Definition is_trace (A B : seq nat) :=
+  [&& (A != [::]), (B != [::]),
+    sorted ltn A, sorted ltn B &
+    corner_box p (last 0 A, last 0 B) ].
+ +
+Lemma is_trace_tll a A B : A != [::] -> is_trace (a :: A) B -> is_trace A B.
+ +
+Lemma is_trace_tlr b A B : B != [::] -> is_trace A (b :: B) -> is_trace A B.
+ +
+Lemma is_trace_lastr (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> is_trace (a :: A) [:: last b B].
+ +
+Lemma is_trace_lastl (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> is_trace [:: last a A] (b :: B).
+ +
+Lemma sorted_in_leq_last A a : sorted ltn A -> a \in A -> a <= last 0 A.
+ +
+Lemma sorted_leq_last A a : sorted ltn (a :: A) -> a <= last a A.
+ +
+Lemma is_trace_in_in_shape (A B : seq nat) : is_trace A B ->
+  forall a b, a \in A -> b \in B -> in_shape p (a, b).
+ +
+Lemma is_trace_in_shape (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> in_shape p (a, b).
+ +
+Lemma trace_size_arm_length (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> size B <= arm_length p (a, b).
+ +
+Lemma trace_size_leg_length (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> size A <= leg_length p (a, b).
+ +
+Lemma size_tracer (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> size B < hook_length p (a, b).
+ +
+Lemma size_tracel (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) -> size A < hook_length p (a, b).
+ +
+Lemma is_trace_corner_nil (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) ->
+  (size (hook_box_indices (a, b)) == 0) = (A == [::]) && (B == [::]).
+ +
+Lemma hook_length_last_rectangle (a b : nat) (A B : seq nat) :
+  is_trace (a :: A) (b :: B) ->
+  hook_length p (a, b)
+  = hook_length p (last a A, b) + hook_length p (a, (last b B)) - 1.
+ +
+Definition trace_seq (last : nat) : seq (seq nat) :=
+  [seq rcons tr last | tr : subseqs (iota 0 last)].
+ +
+Definition enum_trace (Alpha Beta : nat) : seq ((seq nat) * (seq nat)) :=
+  [seq (A, B) | A <- trace_seq Alpha, B <- trace_seq Beta].
+ +
+Lemma trace_seq_uniq l : uniq (trace_seq l).
+ +
+Lemma enum_trace_uniq (Alpha Beta : nat) : uniq (enum_trace Alpha Beta).
+ +
+Lemma trace_corner_box (Alpha Beta : nat) :
+  corner_box p (Alpha, Beta) ->
+  forall A B, A \in trace_seq Alpha -> B \in trace_seq Beta -> is_trace A B.
+ +
+Lemma trace_seqlP (A B : seq nat) :
+  is_trace A B -> A \in trace_seq (last 0 A).
+ +
+Lemma trace_seqrP (A B : seq nat) :
+  is_trace A B -> B \in trace_seq (last 0 B).
+ +
+Lemma enum_traceP (Alpha Beta : nat) :
+  corner_box p (Alpha, Beta) ->
+  forall A B,
+    (A, B) \in enum_trace Alpha Beta =
+    [&& (is_trace A B), (last 0 A == Alpha) & (last 0 B == Beta)].
+ +
+
+ +
+

The probabilistic algorithm

+ +
+
+ +
+Fixpoint walk_to_corner_rec (m : nat) (i j : nat) : distr (seq nat * seq nat) :=
+  if m is m'.+1 then
+    let s := hook_box_indices (i, j) in
+    (if size s is _.+1
+     then Mlet (Uniform (unif_def (inl 0%N) s))
+       (fun n => match n with
+        | inl k => Mlet (walk_to_corner_rec m' k j) (fun X => Munit (i::X.1, X.2))
+        | inr k => Mlet (walk_to_corner_rec m' i k) (fun X => Munit (X.1, j::X.2))
+        end)
+      else Munit ([::i],[::j]))
+   else Munit ([::i],[::j]).
+Definition walk_to_corner m ij := walk_to_corner_rec m ij.1 ij.2.
+ +
+Lemma walk_to_corner0_simpl r c :
+  walk_to_corner 0 (r, c) = Munit ([:: r], [:: c]).
+ +
+Lemma walk_to_corner_end_simpl m r c :
+  size (hook_box_indices (r, c)) = 0 ->
+  walk_to_corner m (r, c) = Munit ([:: r], [:: c]).
+ +
+Lemma walk_to_corner_simpl m r c :
+  forall (Hs : (size (hook_box_indices (r, c)) != 0)),
+    walk_to_corner m.+1 (r, c) =
+    Mlet (Uniform (mkunif (hook_box_indices (r, c)) Hs))
+      (fun n => match n with
+       | inl k => Mlet (walk_to_corner m (k, c)) (fun X => Munit (r::X.1, X.2))
+       | inr k => Mlet (walk_to_corner m (r, k)) (fun X => Munit (X.1, c::X.2))
+       end).
+ +
+Open Scope ring_scope.
+ +
+Lemma walk_to_corner_inv m r c :
+  mu (walk_to_corner m (r, c))
+     (fun HS => [&& size HS.1 != 0, size HS.2 != 0,
+                    head 0 HS.1 == r& head 0 HS.2 == c]%N%:Q)
+      = 1.
+ +
+#[local] Definition charfun A B := fun x : seq nat * seq nat => (x == (A, B))%:Q.
+ +
+Lemma walk_to_corner_emptyl m rc (A B : seq nat) :
+  (A == [::])%B -> mu (walk_to_corner m rc) (charfun A B) = 0.
+ +
+Lemma walk_to_corner_emptyr m rc (A B : seq nat) :
+  (B == [::])%B -> mu (walk_to_corner m rc) (charfun A B) = 0.
+ +
+Lemma charfun_simpll a A B :
+  (fun x => charfun (a :: A) B (a :: x.1, x.2)) == charfun A B.
+ +
+Lemma charfun_simplr b A B :
+  (fun x => charfun A (b :: B) (x.1, b :: x.2)) == charfun A B.
+ +
+Lemma walk_to_corner_decomp m a b (A B : seq nat) :
+  (size (hook_box_indices (a, b)) != 0)%N ->
+  is_trace (a::A) (b::B) ->
+  mu (walk_to_corner m.+1 (a, b)) (charfun (a :: A) (b :: B))
+  =
+  ( mu (walk_to_corner m (a, head O B)) (charfun (a :: A) B) +
+    mu (walk_to_corner m (head O A, b)) (charfun A (b :: B))
+  ) / (size (hook_box_indices (a, b)))%:Q.
+ +
+Lemma mu_walk_to_corner_is_trace rc m :
+  in_shape p rc ->
+  ((hook_length p rc) <= m.+1)%N ->
+  mu (walk_to_corner m rc) (fun X => (is_trace X.1 X.2)%:Q) = 1.
+ +
+
+ +
+Right hand size formula of Lemma 3. +
+
+Definition RHSL3 (a b : nat) (A B : seq nat) : rat :=
+  \prod_(i <- belast a A) (1 / ((hook_length p (i, last b (b :: B)))%:Q - 1)) *
+  \prod_(j <- belast b B) (1 / ((hook_length p (last a (a :: A), j))%:Q - 1)).
+ +
+Lemma Lemma3 m a b (A B : seq nat) :
+  (size A + size B <= m)%N -> is_trace (a :: A) (b :: B) ->
+  mu (walk_to_corner m (a, b)) (charfun (a :: A) (b :: B)) = RHSL3 a b A B.
+ +
+
+ +
+Choose a box in p +
+
+ +
+Definition choose_corner : distr ((seq nat) * (seq nat)) :=
+  Mlet (Random (sumn p).-1)
+       (fun n => let (r, c) := (reshape_index p n, reshape_offset p n) in
+                 walk_to_corner (hook_length p (r, c)).-1 (r, c)).
+ +
+Section EndsAt.
+ +
+Definition starts_at rc t := (head O t.1 == rc.1) && (head O t.2 == rc.2).
+Definition ends_at rc t := (last O t.1 == rc.1) && (last O t.2 == rc.2).
+Definition RHSL3_trace t :=
+  (RHSL3 (head O t.1) (head O t.2) (behead t.1) (behead t.2)).
+ +
+Variable (Alpha Beta : nat).
+Hypothesis Hcorn : corner_box p (Alpha, Beta).
+ +
+Lemma sumnpSPE : (sumn p).-1.+1 = sumn p.
+ +
+Lemma reshape_index_walk_to i (r := reshape_index p i) (c := reshape_offset p i) :
+  (i < sumn p)%N ->
+  mu (walk_to_corner (hook_length p (r, c)).-1 (r, c))
+     (fun pair => (ends_at (Alpha, Beta) pair)%:Q) =
+  \sum_(X <- enum_trace Alpha Beta | starts_at (r, c) X) RHSL3_trace X.
+ +
+Lemma prob_choose_corner_ends_at :
+  mu choose_corner (fun pair => (ends_at (Alpha, Beta) pair)%:Q) =
+  1 / (sumn p)%:Q * \sum_(X <- enum_trace Alpha Beta) RHSL3_trace X.
+ +
+End EndsAt.
+ +
+
+ +
+

The proof

+ +
+
+Section Formula.
+ +
+Variable T : countType.
+Variable R : comPzRingType.
+Variable alpha : T -> R.
+ +
+Lemma expand_prod_add1_seq (S : seq T) :
+  uniq S ->
+  \prod_(i <- S) (1 + alpha i) = \sum_(s : subseqs S) \prod_(i <- s) alpha i.
+ +
+End Formula.
+ +
+Section Theorem2.
+ +
+Variable (Alpha Beta : nat).
+Hypothesis Hcorn : corner_box p (Alpha, Beta).
+ +
+Let p' := decr_nth p Alpha.
+ +
+Fact Hcrn : is_rem_corner p Alpha.
+Hint Resolve Hcrn : core.
+ +
+Fact Hp : incr_nth p' Alpha = p.
+Fact Hpart' : is_part p'.
+Let Hpartc' : is_part (conj_part p') := is_part_conj Hpart'.
+Hint Resolve Hpart' Hpartc' : core.
+ +
+Fact HBeta : Beta = (nth O p Alpha).-1.
+Fact HBeta' : Beta = (nth O p' Alpha).
+ +
+Lemma Formula1 :
+  (hook_length_prod p)%:Q / (hook_length_prod p')%:Q =
+  ( \prod_(0 <= i < Alpha) (1 + ((hook_length p (i, Beta) )%:Q - 1)^-1) ) *
+  ( \prod_(0 <= j < Beta) (1 + ((hook_length p (Alpha, j))%:Q - 1)^-1) ).
+ +
+Lemma SimpleCalculation :
+  \sum_(X <- enum_trace Alpha Beta) RHSL3_trace X =
+  (hook_length_prod p)%:Q / (hook_length_prod p')%:Q.
+ +
+Theorem Theorem2 :
+  mu choose_corner (fun pair => (ends_at (Alpha, Beta) pair)%:Q) =
+  (HLF p') / (HLF p).
+ +
+End Theorem2.
+ +
+Open Scope ring_scope.
+ +
+Lemma ends_at_rem_cornerE :
+  (fun X : seq nat * seq nat =>
+     \sum_(i0 <- rem_corners p) (ends_at (i0, (nth O p i0).-1) X)%:Q)
+    == (fun X => (corner_box p (last O X.1, last O X.2))%:Q).
+ +
+Corollary Corollary4 :
+  p != empty_intpart ->
+  \sum_(i <- rem_corners p) (HLF (decr_nth p i)) / (HLF p) = 1.
+ +
+Corollary Corollary4_eq :
+  p != empty_intpart ->
+  \sum_(i <- rem_corners p) (HLF (decr_nth_intpart p i)) = HLF p.
+ +
+End FindCorner.
+ +
+Theorem HookLengthFormula_rat (p : intpart) :
+  ( (#|{: stdtabsh p}|)%:Q = HLF p )%R.
+ +
+Lemma hook_length_prod_non0 (p : intpart) : (hook_length_prod p) != 0.
+ +
+Lemma hook_length_prod_nat (p : intpart) :
+  #|{: stdtabsh p}| * (hook_length_prod p) = (sumn p)`!.
+ +
+Lemma hook_length_prod_div (p : intpart) : (hook_length_prod p) %| (sumn p)`!.
+ +
+Theorem HookLengthFormula (p : intpart) :
+  #|{: stdtabsh p}| = (sumn p)`! %/ (hook_length_prod p).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.Greene.html b/combi/1.1.0/Combi.LRrule.Greene.html new file mode 100644 index 00000000..99d76622 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.Greene.html @@ -0,0 +1,887 @@ + + + + + +Combi.LRrule.Greene: Greene monotone subsequence numbers + + + + +
+ + + +
+ +

Library Combi.LRrule.Greene: Greene monotone subsequence numbers

+ +
+
+ +
+ +
+
+
+ +
+

Greene monotone subsequences numbers

+ + +
+ +We define the following notions and notations: + +
+ +
    +
  • trivIseq U == u : seq {set T} contains pairwise disjoint sets. + +
    • +
+ +
+ +In the following, we denote: + +
+ +
    +
  • wt : an n.-tuple Alph for an alphabet Alph. + +
    • +
  • R : a relation on Alph + +
    • +
+ +
+ +Then we define: + +
+ +
    +
  • extractpred wt P == the sequence of the entries of wt whose index verify + the predicate P. + +
    • +
  • extract wt S == the sequence of the entries of wt supported by S, + that is, whose index are in S. + +
    • +
  • P \is a k.-supp[R, wt] == P : {set {set 'I_N}} is a k-support of wt + w.r.t R. This means that P is of cardinality at most k, contains + pairwise disjoint sets and finally that for all S in P, the + subsequence of wt of support S is sorted for the relation R. + +
    • +
  • Greene_rel_t R wt k == the maximal cardinality of the union of a k-support + of wt w.r.t R + +
    • +
  • Greene_rel R s k == Greene_rel_t R (in_tuple s) k + +
    • +
+ +
+ +We now suppose that Alph is canonically ordered. Moreover t is a tableau +on Alph. Then + +
+ +
    +
  • tabrows t == the sequence of the supports in the row-reading of t of + the rows of t + +
    • +
  • tabcols t == the sequence of the supports in the row-reading of t of + the columns of t + +
    • +
  • tabrowsk t k == the sequence of the supports in the row-reading of t of + the k first rows of t + +
    • +
  • tabcolsk t k == the sequence of the supports in the row-reading of t of + the k first columns of t + +
    + + +
    • +
  • Greene_row s k == Greene number for nondecreasing subsequence + +
    • +
  • Greene_col s k == Greene number for strictly decreasing subsequence + +
    + + +
    • +
  • ksupp_inj R1 R2 k u1 u2 == for each k-support for R1 of u1 there + exists a k-support for R2 of u2 of the same size. + +
    • +
+ +
+ +Among the main results is Theorem Greene_row_tab which assert that the +k-th row Greene number of the row reading of a tableau is the sum of the +length of the k first rows of its shape. Similarly Theorem Greene_col_tab +link column green number with the length of the column of the shape (with is +the same as the length of the row of the conjugate. + +
+ +As a consequence two tableaux whose row readings have the same row (or column) +Greene numbers have the same shape (this is Theorem Greene_row_tab_eq_shape and +Greene_col_tab_eq_shape). + +
+
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm.
+ +
+Require Import sorted tools subseq partition tableau.
+Require Import Schensted congr plactic ordcast.
+ +
+Set Implicit Arguments.
+ +
+Import Order.Theory.
+ +
+Open Scope bool.
+ +
+Lemma unlift_seqE n (l : seq 'I_n.+1) :
+  sorted (fun i j : 'I_n.+1 => i < j) (ord0 :: l) ->
+  exists l1 : seq 'I_n,
+    sorted (fun i j : 'I_n => i < j) l1 /\ l = map (lift ord0) l1.
+ +
+Lemma ord0_in_map_liftF n (l : seq 'I_n) :
+  ord0 \in [seq lift ord0 i | i <- l] = false.
+ +
+Lemma mem_enum_seqE n (l : seq 'I_n) :
+  sorted (fun i j : 'I_n => val i < val j) l ->
+  [seq i <- enum 'I_n | i \in l] = l.
+ +
+
+ +
+

Trivial intersection sequences

+ +
+
+Section TrivISeq.
+ +
+Variable T : finType.
+ +
+Lemma bigcup_seq_cover (u : seq {set T}) :
+  \bigcup_(s <- u) s = cover [set s in u].
+ +
+Lemma card_seq (s : seq T) : #|[set i in s]| <= size s.
+ +
+Definition trivIseq (u : seq {set T}) : Prop :=
+  forall i j, i < j < size u -> [disjoint (nth set0 u i) & (nth set0 u j)].
+ +
+Lemma trivIseq_consK u0 u : trivIseq (u0 :: u) -> trivIseq u.
+ +
+Lemma trivIsubseq u v :
+  subseq u v -> trivIseq v -> trivIseq u.
+ +
+Lemma trivIs u : trivIseq u -> trivIset [set i | i \in u].
+ +
+Lemma trivIseq_cover S :
+  trivIseq S -> \sum_(i <- S) #|i| = #|\bigcup_(i <- S) i|.
+ +
+Lemma size_coverI (S : {set {set T}}) B :
+  trivIset S -> \sum_(i in S) #|i :&: B| = #|cover S :&: B|.
+ +
+Lemma trivIset_I (S : {set {set T}}) B :
+  trivIset S -> \sum_(i in S) #|i :&: B| <= #|B|.
+ +
+End TrivISeq.
+ +
+Lemma trivIseq_map (T1 T2 : finType) (f : T1 -> T2) (S : seq {set T1}) :
+  injective f -> trivIseq S -> trivIseq (map (fun s : {set T1} => f @: s) S).
+ +
+Lemma set_nil (T : finType) : [set s : T in [::]] = set0.
+ +
+Lemma cover_nil (T : finType) : #|cover [set s : {set T} in [::]]| = 0.
+ +
+Lemma subseq_take (T : eqType) k (l : seq T) : subseq (take k l) l.
+ +
+Section BigTrivISeq.
+ +
+Variable T : finType.
+Variables (R : Type) (idx : R) (op : Monoid.com_law idx).
+ +
+Lemma bigop_trivIseq (S : seq {set T}) F :
+  all (fun s => s != set0) S -> trivIseq S ->
+  \big[op/idx]_(i in [set x in S]) F i = \big[op/idx]_(i <- S) F i.
+ +
+End BigTrivISeq.
+ +
+Require Import ordtype.
+ +
+
+ +
+

Greene subsequence numbers

+

Definition and basic properties

+ +
+
+Section GreeneDef.
+ +
+Context {Alph : eqType}.
+ +
+Definition extractpred n (wt : n.-tuple Alph) (P : pred 'I_n) :=
+  [seq tnth wt i | i <- enum P].
+ +
+Lemma extractIE n (wt : n.-tuple Alph) P :
+  extractpred wt P = [seq tnth wt i | i <- enum 'I_n & P i].
+ +
+Lemma extractmaskE n (wt : n.-tuple Alph) P :
+  extractpred wt P = mask [seq P i | i <- enum 'I_n] wt.
+ +
+Lemma extsubsIm n wt (P1 P2 : pred 'I_n) :
+  subpred P1 P2 -> subseq (extractpred wt P1) (extractpred wt P2).
+ +
+Lemma extsubsm n (w : n.-tuple Alph) (P : pred 'I_n) :
+  subseq (extractpred w P) w.
+ +
+Variable R : rel Alph.
+Hypothesis HR : transitive R.
+ +
+Variable N : nat.
+Variable wt : N.-tuple Alph.
+ +
+Definition extract := [fun s : {set 'I_N} => extractpred wt (mem s)].
+ +
+Lemma size_extract s : size (extract s) = #|s|.
+ +
+Lemma extsubsI (s1 s2 : {set 'I_N}) :
+  s1 \subset s2 -> subseq (extract s1) (extract s2).
+ +
+Lemma sorted_extract_cond s P :
+  sorted R (extract s) -> sorted R (extract (s :&: P)).
+ +
+Definition ksupp (k : nat) :=
+  [qualify a P : {set {set 'I_N}} |
+   [&& (#|P| <= k)%N, trivIset P & [forall (s | s \in P), sorted R (extract s)]]].
+ +
+#[local] Notation "k .-supp" := (ksupp k)
+  (at level 2, format "k .-supp") : form_scope.
+ +
+Lemma ksupp0 k : set0 \is a k.-supp.
+ +
+Definition Greene_rel_t k := \max_(P | P \is a k.-supp) #|cover P|.
+ +
+Notation Ik k := [set i : 'I_N | i < k].
+ +
+Lemma iotagtnk k : [seq x <- iota 0 N | gtn k x] = iota 0 (minn N k).
+ +
+Lemma sizeIk k : #|Ik k| = minn N k.
+ +
+Lemma extract0 : extract set0 = [::].
+ +
+Lemma extract1 i : extract [set i] = [:: tnth wt i].
+ +
+Lemma extractS (l : seq 'I_N) :
+  sorted (fun i j : 'I_N => val i < val j) l ->
+  extract [set i in l] = [seq tnth wt i | i <- l].
+ +
+Lemma extract2 i j :
+  val i < val j -> extract [set i; j] = [:: tnth wt i; tnth wt j].
+ +
+Lemma Greene_rel_t_inf k : Greene_rel_t k >= minn N k.
+ +
+Lemma Greene_rel_t_sup k : Greene_rel_t k <= N.
+ +
+Lemma Greene_rel_t_0 : Greene_rel_t 0 = 0.
+ +
+End GreeneDef.
+ +
+Notation "k '.-supp' [ R , wt ]" := (@ksupp _ R _ wt k)
+  (at level 2, format "k '.-supp' [ R , wt ]") : form_scope.
+ +
+Section Cast.
+ +
+Context {T : eqType}.
+ +
+Lemma ksupp_cast R (w1 w2 : seq T) (H : w1 = w2) k Q :
+  Q \is a k.-supp[R, in_tuple w1] ->
+  (cast_set (congr1 size H)) @: Q \is a k.-supp[R, in_tuple w2].
+ +
+Lemma eq_Greene_rel_t (R1 R2 : rel T) N (u : N.-tuple T) :
+  R1 =2 R2 -> Greene_rel_t R1 u =1 Greene_rel_t R2 u.
+ +
+Lemma Greene_rel_t_cast R M N (Heq : M = N) k (V : M.-tuple T) :
+  Greene_rel_t R (tcast Heq V) k = Greene_rel_t R V k.
+ +
+Lemma Greene_rel_t_uniq (leT : rel T) N (u : N.-tuple T) :
+  uniq u ->
+  Greene_rel_t leT u =1 Greene_rel_t (fun x y => (x != y) && (leT x y)) u.
+ +
+End Cast.
+ +
+
+ +
+

Greene numbers of a concatenation

+ +
+
+Section GreeneCat.
+ +
+Context {Alph : eqType}.
+Variable R : rel Alph.
+Hypothesis HR : transitive R.
+ +
+Variable M N : nat.
+Variable V : M.-tuple Alph.
+Variable W : N.-tuple Alph.
+ +
+#[local] Notation lsh := (@lshift M N).
+#[local] Notation rsh := (@rshift M N).
+ +
+Lemma enumIMN : enum 'I_(M + N) = map lsh (enum 'I_M) ++ map rsh (enum 'I_N).
+ +
+Let lsplit := [fun s : {set 'I_(M+N)} => lsh @^-1: s].
+Let rsplit := [fun s : {set 'I_(M+N)} => rsh @^-1: s].
+ +
+Lemma disjoint_inj (sM : {set 'I_M}) (sN : {set 'I_N}) :
+  [disjoint lsh @: sM & rsh @: sN].
+ +
+Lemma splitsetK (s : {set 'I_(M+N)}) :
+  s = (lsh @: lsplit s) :|: (rsh @: rsplit s).
+ +
+Lemma cutcover (p : {set {set 'I_(M + N)}}) :
+  cover p = lsh @: (cover (lsplit @: p)) :|: rsh @: (cover (rsplit @: p)).
+ +
+Lemma extlsplit s :
+  extract V (lsplit s) =
+  extract [tuple of V ++ W] (s :&: [set i : 'I_(M+N)| i < M]).
+ +
+Lemma extrsplit s :
+  extract W (rsplit s) =
+  extract [tuple of V ++ W] (s :&: [set i : 'I_(M+N)| i >= M]).
+ +
+#[local] Notation scover := (fun x => #|cover x|).
+ +
+Lemma Greene_rel_t_cat k :
+  Greene_rel_t R [tuple of V ++ W] k <= Greene_rel_t R V k + Greene_rel_t R W k.
+ +
+End GreeneCat.
+ +
+Section GreeneSeq.
+ +
+Context {Alph : eqType}.
+Variable R : rel Alph.
+Hypothesis HR : transitive R.
+ +
+Definition Greene_rel u := [fun k => Greene_rel_t R (in_tuple u) k].
+ +
+Lemma Greene_rel_cat k v w :
+  Greene_rel (v ++ w) k <= Greene_rel v k + Greene_rel w k.
+ +
+Let negR := (fun a b => ~~(R a b)).
+Hypothesis HnegR : transitive negR.
+ +
+Lemma Greene_rel_seq r k : sorted negR r -> Greene_rel r k = minn (size r) k.
+ +
+End GreeneSeq.
+ +
+Lemma eq_Greene_rel (T : eqType) (R1 R2 : rel T) u :
+  R1 =2 R2 -> Greene_rel R1 u =1 Greene_rel R2 u.
+ +
+Lemma Greene_rel_uniq (T : eqType) (leT : rel T) u :
+  uniq u ->
+  Greene_rel leT u =1 Greene_rel (fun x y => (x != y) && (leT x y)) u.
+ +
+Lemma size_cover_inj (T1 T2 : finType) (F : T1 -> T2) (P : {set {set T1}}):
+  injective F -> #|cover P| = #|cover [set F @: S | S : {set T1} in P]|.
+ +
+
+ +
+

Injection of k-supports

+ +
+
+Section GreeneInj.
+ +
+Context {T1 T2 : eqType}.
+Variable R1 : rel T1.
+Variable R2 : rel T2.
+ +
+Definition ksupp_inj k (u1 : seq T1) (u2 : seq T2) :=
+  forall s1, s1 \is a k.-supp[R1, in_tuple u1] ->
+             exists s2, (#|cover s1| == #|cover s2|)
+                          && (s2 \is a k.-supp[R2, in_tuple u2]).
+ +
+Lemma leq_Greene k (u1 : seq T1) (u2 : seq T2) :
+  ksupp_inj k u1 u2 -> Greene_rel R1 u1 k <= Greene_rel R2 u2 k.
+ +
+End GreeneInj.
+ +
+
+ +
+

Reverting a word on the dual alphabet

+ +
+
+Section Rev.
+ +
+Context {Alph : eqType}.
+Implicit Type u v w : seq Alph.
+Implicit Type p : seq nat.
+Implicit Type t : seq (seq Alph).
+ +
+#[local] Definition revset n (s : {set 'I_n}) : {set 'I_n} :=
+  [set rev_ord x | x in s].
+ +
+Lemma revsetK {n} : involutive (@revset n).
+ +
+Lemma ksupp_inj_rev (leT : rel Alph) u k :
+  ksupp_inj leT (fun y x => leT x y) k u (rev u).
+ +
+Lemma Greene_rel_rev (leT : rel Alph) u :
+  Greene_rel leT u =1 Greene_rel (fun y x => leT x y) (rev u).
+ +
+End Rev.
+ +
+
+ +
+

Greene number for tableaux

+ +
+
+Section GreeneRec.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Implicit Type u v w : seq Alph.
+Implicit Type t : seq (seq Alph).
+ +
+Let sym_cast m n (i : 'I_(m + n)) : 'I_(n + m) := cast_ord (addnC m n) i.
+ +
+ +
+#[local] Definition shiftset s0 sh :=
+  [fun s : {set 'I_(sumn sh)} => (sym_cast \o (@lshift (sumn sh) s0)) @: s].
+ +
+#[local] Fixpoint shrows sh : seq {set 'I_(sumn sh)} :=
+  if sh is s0 :: sh then
+    [set (sym_cast \o (@rshift (sumn sh) s0)) i | i in 'I_s0] ::
+    map (@shiftset s0 sh) (shrows sh)
+  else [::].
+#[local] Fixpoint shcols sh : seq {set 'I_(sumn sh)} :=
+  if sh is s0 :: sh then
+    [seq (sym_cast (@rshift (sumn sh) s0 i)) |:
+         (@shiftset s0 sh (nth set0 (shcols sh) i))
+    | i <- enum 'I_s0]
+  else [::].
+ +
+Let cast_set_tab t :=
+  [fun s : {set 'I_(sumn (shape t))} => (cast_ord (esym (size_to_word t))) @: s].
+ +
+Definition tabrows t := map (cast_set_tab t) (shrows (shape t)).
+Definition tabcols t := map (cast_set_tab t) (shcols (shape t)).
+Definition tabrowsk t := [fun k => take k (tabrows t)].
+Definition tabcolsk t := [fun k => take k (tabcols t)].
+ +
+Lemma size_shcols_cons s0 sh : size (shcols (s0 :: sh)) = s0.
+ +
+Lemma size_shcols sh : size (shcols sh) = head 0 sh.
+ +
+Lemma size_tabcols t : size (tabcols t) = size (head [::] t).
+ +
+Lemma size_shrows sh : size (shrows sh) = size sh.
+ +
+Lemma size_tabrows t : size (tabrows t) = size t.
+ +
+Lemma shcol_cards sh :
+  is_part sh ->
+  map (fun S : {set 'I_(sumn sh)} => #|S|) (shcols sh) = conj_part sh.
+ +
+Lemma shrows_cards sh :
+  map (fun S : {set 'I_(sumn sh)} => #|S|) (shrows sh) = sh.
+ +
+Lemma tabrows_non0 t :
+  is_part (shape t) -> forall s, s \in tabrows t -> s != set0.
+ +
+Lemma trivIseq_shrows sh : trivIseq (shrows sh).
+ +
+Lemma trivIset_tabrowsk k t : trivIset [set s | s \in (tabrowsk t k)].
+ +
+Lemma trivIseq_shcols sh : trivIseq (shcols sh).
+ +
+Lemma trivIseq_tabcols (t : seq (seq Alph)) : trivIseq (tabcols t).
+ +
+Lemma trivIset_tabcolsk k t : trivIset [set s | s \in (tabcolsk t k)].
+ +
+
+ +
+

Induction step: adding a row to a tableau

+ +
+
+Section Induction.
+ +
+Variable (t0 : seq Alph) (t : seq (seq Alph)).
+ +
+Lemma size_to_word_cons : size (to_word t) + size t0 = size (to_word (t0 :: t)).
+ +
+#[local] Definition cast_cons := cast_ord size_to_word_cons.
+#[local] Definition rsh_rec := (cast_cons \o (@rshift (size (to_word t)) (size t0))).
+#[local] Definition lsh_rec := (cast_cons \o (@lshift (size (to_word t)) (size t0))).
+ +
+Lemma lshift_recP : injective lsh_rec.
+Lemma rshift_recP : injective rsh_rec.
+ +
+Lemma lrshift_recF i j : (lsh_rec i == rsh_rec j) = false.
+ +
+Lemma rshift_in_lshift_recF i (s : {set 'I_(size (to_word t))}) :
+  rsh_rec i \in [set lsh_rec x | x in s] = false.
+ +
+Lemma lshift_in_rshift_recF i (s : {set 'I_(size t0)}) :
+  lsh_rec i \in [set rsh_rec x | x in s] = false.
+ +
+Lemma disjoint_inj_rec (st : {set 'I_(size (to_word t))}) (s0 : {set 'I_(size t0)}) :
+  [disjoint lsh_rec @: st & rsh_rec @: s0].
+ +
+#[local] Definition lsplit_rec :=
+  [fun s : {set 'I_(size (to_word (t0 :: t)))} => lsh_rec @^-1: s].
+#[local] Definition rsplit_rec :=
+  [fun s : {set 'I_(size (to_word (t0 :: t)))} => rsh_rec @^-1: s].
+ +
+Lemma split_recabK (s : {set 'I_(size (to_word (t0 :: t)))}) :
+  s = (lsh_rec @: lsplit_rec s) :|: (rsh_rec @: rsplit_rec s).
+ +
+Lemma split_rec_cover (p : {set {set 'I_(size (to_word (t0 :: t)))}}) :
+  cover p =
+  lsh_rec @: (cover (lsplit_rec @: p)) :|: rsh_rec @: (cover (rsplit_rec @: p)).
+ +
+Lemma lcast_com :
+  (cast_ord (esym (size_to_word (t0 :: t))))
+    \o sym_cast \o (@lshift (sumn (shape t)) (size t0))
+  =1 lsh_rec \o (cast_ord (esym (size_to_word t))).
+ +
+Lemma rcast_com :
+ (cast_ord (esym (size_to_word (t0 :: t))))
+   \o sym_cast \o (@rshift (sumn (shape t)) (size t0)) =1 rsh_rec.
+ +
+Lemma enumIsize_to_word :
+  enum 'I_(size (to_word (t0 :: t))) =
+  map lsh_rec (enum 'I_(size (to_word t))) ++ map rsh_rec (enum 'I_(size t0)).
+ +
+Lemma extract_tabrows_0 :
+  extract (in_tuple (to_word (t0 :: t))) (nth set0 (tabrows (t0 :: t)) 0) = t0.
+ +
+Lemma extract_tabrows_rec i :
+  i < size t ->
+  extract (in_tuple (to_word (t0 :: t))) (nth set0 (tabrows (t0 :: t)) i.+1) =
+  extract (in_tuple (to_word t)) (nth set0 (tabrows t) i).
+ +
+Lemma tabcols_cons :
+  tabcols (t0 :: t) =
+    [seq rsh_rec i |: (lsh_rec @: (nth set0 (tabcols t) i))
+      | i <- enum 'I_(size t0)].
+ +
+Lemma size_tabcols_cons : size (tabcols (t0 :: t)) = size t0.
+ +
+Lemma extract_tabcols_rec i :
+  i < size t0 ->
+  extract (in_tuple (to_word (t0 :: t))) (nth set0 (tabcols (t0 :: t)) i) =
+  rcons (extract (in_tuple (to_word t)) (nth set0 (tabcols t) i))
+        (nth inh t0 i).
+ +
+Lemma lsplit_rec_tab k :
+  head 0 (shape t) <= size t0 ->
+  cover [set lsplit_rec x | x in [set s in tabcolsk (t0 :: t) k]] =
+  cover [set s in tabcolsk t k].
+ +
+Lemma rsplit_rec_tab k :
+  cover [set rsplit_rec x | x in [set s in (tabcolsk (t0 :: t)) k]] =
+  [set i : 'I_(size t0) | i < k].
+ +
+Lemma cover_tabcols_rec k :
+  head 0 (shape t) <= size t0 ->
+  cover [set s in (tabcolsk (t0 :: t) k)] =
+  rsh_rec @: [set s : 'I_(size t0) | s < k]
+           :|: lsh_rec @: cover [set s in (tabcolsk t k)].
+ +
+End Induction.
+ +
+Lemma size_cover_tabrows k t :
+  is_part (shape t) ->
+  #|cover [set s | s \in (tabrowsk t k)]| = sumn (take k (shape t)).
+ +
+Lemma sorted_leqX_tabrows t i :
+  is_tableau t -> i < size (tabrows t) ->
+  sorted <=%O (extract (in_tuple (to_word t)) (nth set0 (tabrows t) i)).
+ +
+Lemma ksupp_leqX_tabrowsk k t :
+  is_tableau t ->
+  [set s | s \in (tabrowsk t k)] \is a k.-supp[<=%O, in_tuple (to_word t)].
+ +
+Lemma size_cover_tabcolsk k t :
+  is_part (shape t) ->
+  #|cover [set s | s \in (tabcolsk t k)]| = \sum_(l <- (shape t)) minn l k.
+ +
+Lemma cover_tabcols t :
+  is_part (shape t) -> cover [set s | s \in tabcols t] = setT.
+ +
+Lemma sorted_gt_tabcols t i :
+  is_tableau t -> i < size (tabcols t) ->
+  sorted >%O (extract (in_tuple (to_word t)) (nth set0 (tabcols t) i)).
+ +
+Lemma ksupp_gt_tabcolsk k t :
+  is_tableau t ->
+  [set s | s \in (tabcolsk t k)] \is a k.-supp[>%O, in_tuple (to_word t)].
+ +
+End GreeneRec.
+ +
+
+ +
+

Greene numbers of a tableau

+ +
+
+Section GreeneTab.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+ +
+Implicit Type t : seq (seq Alph).
+ +
+Definition Greene_row := Greene_rel (@Order.le _ Alph).
+Definition Greene_col := Greene_rel (@Order.gt _ Alph).
+ +
+Lemma gt_trans : transitive (@Order.gt _ Alph).
+ +
+Lemma size_row_extract t U V :
+  is_tableau t -> V \in (tabcols t) ->
+  sorted <=%O (extract (in_tuple (to_word t)) U) ->
+  #|U :&: V| <= 1.
+ +
+Lemma tabcol_cut t :
+  is_part (shape t) -> forall B, \sum_(U <- tabcols t) #|B :&: U| = #|B|.
+ +
+Lemma shape_tabcols t:
+  is_tableau t ->
+  conj_part (shape t) = [seq #|s : {set 'I_(size (to_word t))}| | s <- tabcols t].
+ +
+Theorem Greene_row_tab k t :
+  is_tableau t -> Greene_row (to_word t) k = sumn (take k (shape t)).
+ +
+Theorem Greene_col_tab k t :
+  is_tableau t -> Greene_col (to_word t) k = sumn (take k (conj_part (shape t))).
+ +
+End GreeneTab.
+ +
+
+ +
+

Recovering a shape from Greene numbers on tableaux

+ +
+
+Theorem Greene_row_tab_eq_shape
+        d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+        (t1 : seq (seq T1)) (t2 : seq (seq T2)) :
+  is_tableau t1 -> is_tableau t2 ->
+  (forall k, Greene_row (to_word t1) k = Greene_row (to_word t2) k) ->
+  (shape t1 = shape t2).
+ +
+Theorem Greene_col_tab_eq_shape
+        d1 d2 (T1 : inhOrderType d1) (T2 : inhOrderType d2)
+        (t1 : seq (seq T1)) (t2 : seq (seq T2)) :
+  is_tableau t1 -> is_tableau t2 ->
+  (forall k, Greene_col (to_word t1) k = Greene_col (to_word t2) k) ->
+  (shape t1 = shape t2).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.Greene_inv.html b/combi/1.1.0/Combi.LRrule.Greene_inv.html new file mode 100644 index 00000000..5d9c6562 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.Greene_inv.html @@ -0,0 +1,1158 @@ + + + + + +Combi.LRrule.Greene_inv: Greene subsequence theorem + + + + +
+ + + +
+ +

Library Combi.LRrule.Greene_inv: Greene subsequence theorem

+ +
+
+ +
+ +
+
+
+ +
+

Greene subsequence theorem

+ + +
+ +The goal of this file is to show that row and column Greene numbers are plactic +invariants. As a consequence, for any word w, the Greene numbers of w are +equal to the Greene numbers of the row reading of its insertion tableau. This +ultimately allows to prove the reciprocal of Sch_plact, that is + +
+ + Theorem plactic_RS u v : u =Pl v <-> RS u == RS v. + +
+ +The main tool of the proof is surgery of k-supports along plactic rewriting. +As a consequence, the whole file is rather technical with no external use except +for the final theorems. To keep notation short while avoiding to pollute the +global namespace we enclosed the different cases into Coq modules. +
+ + +
+ +Here is the content of the file: + +
+ +Greene numbers and duality: + +
+ +The following function allows to transfer k-support through duality: +
    +
  • rev_ord_cast w i == size w - i : 'I_(size (revdual w)) + +
    • +
  • rev_set w S == the image of S by rev_ord_cast w + +
    • +
  • rev_ksupp w P == the image of P by rev_set w + +
    • +
  • rev_ksupp_inv w == the inverse of rev_ksupp w + +
    • +
+ +
+ +The Greene numbers of a word and its reversed dual agrees: Lemmas +Greene_col_dual and Greene_row_dual. +
+ + +
+ +Swapping two letters: + +
+ +In Module Swap, we fix two words u v and two letters l0 l1. +We denote x the word x := u l0 l1 v. Then we define: +
    +
  • pos0 u v l0 l1 == the position of l0 in x as a 'I_(size x) + +
    • +
  • pos1 u v l0 l1 == the position of l1 in x as a 'I_(size x) + +
    • +
  • swap i == exchange l0 and l1 and fixes all the other positions. + +
    • +
  • swap_set S == then image of S by swap + +
    • +
+We prove then various lemmas such as swap_size_cover asserting that swaping +keep the size of the cover. +
+ + +
+ +In Module NoSetContainingBoth, we consider x := u a b v. +We assume that (the position of) a and b are not in the same set of a +given k-support P. We construct a k-support Q for y := u b a v of the +same cover size: +
    +
  • swap_set S == exchange the position of a and b in + S : {set 'I_(size x)} and return the result as a {set 'I_(size y)} + +
    • +
  • Q P == for a k-support P for x, then Q P is a k-support for y + of the same cardinality, assuming that a and b are not in the + same set of P. + +
    • +
+The main results are lemmas ksupp_Q and size_cover_Q. + +
+ +
+ + +
+ +In Module SetContainingBothLeft, we denote x := u b a c v and consider +a given k-support P containing both a and c. We want to deal at once with +the case where a < b <= c and R := <= and c < b <= a and R := >. +So we encaspulate the hypothesis in a record hypRabc which contains among +other that R b c holds but not R b a. Specifically: +
    +
  • hypRabc R a b c == a record encapsulating that a R b R c for a total + strict or non strict order R. + +
    • +
+We construct in lemmas RabcLeqX and RabcGtnX the two records dealing with +the two preceding cases. + +
+ +We are looking for a k-support for y := u b c a v of the same size. To be +able to apply the preceding module, we need to construct another k-support Q +for x with the same cover than P, but such that a and c are not in the +same set. There are two cases: + +
+ +1- if b is not in cover P: We define: + +
+ + Qbnotin P == P with a replaced by b. This is a still a k-support + for x of the same cardinality by lemmas ksupp_bnotin and + size_cover_bnotin. + +
+ +2- if b is in cover P: Due to monotonic condition, b must be in a + different set T in P than the set S which contains a and b. The + idea is that S and T should exchange their part in c :: v. We + therefore define: + +
+ + S1 S T == the elements of S which are on the left of a + + the elements of T which are in v + S1 S T == the elements of T which are on the left of b + + the elements of S which are in c :: v + Qbin P S T == P where we have replaced S and T by S1 and T1. + It is a k-support for x (Lemma ksupp_bin) with the same cover + than p (Lemma cover_bin). + +
+ +Then we prove consecutively two theorems: +
    +
  • SetContainingBothLeft.exists_Q_noboth which allows to to assume that there + is a k-support where no set contains both a and c. + +
    • +
  • SetContainingBothLeft.exists_Qy which find a k-support for + y := u b c a v of the same size of cover. + +
    • +
+ +
+ +This allow to show that each plactic rewriting rule leave the Greene numbers +invariant. +
+ + +
+ +We conclude by the main results: +
    +
  • Greene_row_invar_plactic and Greene_col_invar_plactic asserting than + Greene numbers are plactic invariants. + +
    • +
  • plactic_RS which shows that the plactic classes are the fiber of the + Robinson-Schensted map. + +
    • +
  • shape_RS_revdual asserting that the tableaux associated to a word and its + revdual have the same shapes. + +
    • +
  • RS_rev_uniq asserting that reverting a uniq word conjugate its insertion + tableau. + +
    + + + +
    • +
+
+
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm.
+Require Import tools ordcast ordtype subseq partition tableau Yamanouchi stdtab.
+Require Import Schensted congr plactic Greene.
+ +
+Set Implicit Arguments.
+ +
+Import Order.Theory.
+ +
+Open Scope bool.
+ +
+Section Duality.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Lemma extract_cut (N : nat) (wt : N.-tuple Alph) (i : 'I_N) (S : {set 'I_N}) :
+  i \in S ->
+  extract wt S =
+  extract wt (S :&: [set j : 'I_N | j < i]) ++
+          (tnth wt i) :: extract wt (S :&: [set j : 'I_N | j > i]).
+ +
+
+ +
+

k-Support and order duality

+ +
+
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Variable w : word.
+Variable k : nat.
+ +
+#[local] Definition rev_ord_cast : 'I_(size w) -> 'I_(size (revdual w)) :=
+  (cast_ord (size_revdual w)) \o (@rev_ord _).
+#[local] Definition rev_set (s : {set 'I_(size w)}) : {set 'I_(size (revdual w))} :=
+  [set rev_ord_cast i | i in s].
+ +
+#[local] Definition rev_ksupp (P : {set {set 'I_(size w)}}) :
+  {set {set 'I_(size (revdual w))}} :=
+  [set rev_set u | u in P].
+#[local] Definition rev_ksupp_inv (S : {set {set 'I_(size (revdual w))}}) :
+  {set {set 'I_(size w)}} :=
+  [set rev_ord_cast @^-1: s | s : {set 'I_(_)} in S].
+ +
+Lemma rev_ord_cast_inj : injective rev_ord_cast.
+ +
+Lemma rev_set_inj : injective rev_set.
+ +
+Lemma rev_ksuppK : cancel rev_ksupp rev_ksupp_inv.
+ +
+Lemma rev_ksuppKV : cancel rev_ksupp_inv rev_ksupp.
+ +
+#[local] Lemma irev_w i : i < size w -> size w - i.+1 < size w.
+ +
+Lemma rev_enum :
+  enum 'I_(size (revdual w)) = rev [seq rev_ord_cast i | i : 'I_(size w)].
+ +
+Lemma extract_rev_set S :
+  extract (in_tuple (revdual w)) (rev_set S) = revdual (extract (in_tuple w) S).
+ +
+Lemma is_row_dual T :
+  sorted <=%O (extract (in_tuple w) T) =
+  sorted <=%O (extract (in_tuple (revdual w)) (rev_set T)).
+ +
+Lemma is_col_dual T :
+  sorted >%O (extract (in_tuple w) T) =
+  sorted >%O (extract (in_tuple (revdual w)) (rev_set T)).
+ +
+Lemma size_rev_ksupp P : #|rev_ksupp P| = #|P|.
+ +
+Lemma trivIset_setrev P : trivIset P = trivIset (rev_ksupp P).
+ +
+Lemma rev_is_ksupp_row P :
+  (P \is a k.-supp[<=%O, in_tuple w]) =
+  (rev_ksupp P \is a k.-supp[<=%O, in_tuple (revdual w)]).
+ +
+Lemma rev_is_ksupp_col P :
+  (P \is a k.-supp[>%O, in_tuple w]) =
+  (rev_ksupp P \is a k.-supp[>%O, in_tuple (revdual w)]).
+ +
+Lemma size_cover_rev P : #|cover (rev_ksupp P)| = #|cover P|.
+ +
+Lemma Greene_row_dual : Greene_row w k = Greene_row (revdual w) k.
+ +
+Lemma Greene_col_dual : Greene_col w k = Greene_col (revdual w) k.
+ +
+End Duality.
+ +
+
+ +
+

Swaping two letters in a word and its k-supports

+ +
+
+Module Swap.
+Section Swap.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Variable R : rel Alph.
+ +
+Variable u v : word.
+Variable l0 l1 : Alph.
+Let x := u ++ [:: l0; l1] ++ v.
+ +
+Lemma pos0_subproof : size u < size x.
+Definition pos0 := Ordinal pos0_subproof.
+Lemma pos1_subproof : (size u).+1 < size x.
+Definition pos1 := Ordinal pos1_subproof.
+ +
+Lemma tnth_pos0 : tnth (in_tuple x) pos0 = l0.
+ +
+Lemma tnth_pos1 : tnth (in_tuple x) pos1 = l1.
+ +
+Lemma pos01F : (pos0 == pos1) = false.
+ +
+Definition swap (i : 'I_(size x)) : 'I_(size x) :=
+  if i == pos0 then pos1 else if i == pos1 then pos0 else i.
+ +
+Lemma swap_invol : involutive swap.
+Lemma swap_inj : injective swap.
+ +
+Lemma swap0 : swap pos0 = pos1.
+Lemma swap1 : swap pos1 = pos0.
+Lemma swapL (i : 'I_(size x)) : i < size u -> swap i = i.
+Lemma swapR (i : 'I_(size x)) : i > (size u).+1 -> swap i = i.
+ +
+Definition swap_set :=
+  [fun s : {set 'I_(size x)} => swap @: s : {set 'I_(size x)}].
+Lemma swap_set_invol : involutive swap_set.
+Lemma swap_set_inj : injective swap_set.
+ +
+Lemma swap_cover (P : {set {set 'I_(size x)}}) :
+  cover (swap_set @: P) = swap_set (cover P).
+ +
+Lemma swap_size_cover (P : {set {set 'I_(size x)}}) :
+  #|cover (swap_set @: P)| = #|cover P|.
+ +
+Lemma enum_cut : enum 'I_(size x) =
+                 [seq i <- enum 'I_(size x) | val i < size u]
+                   ++ [:: pos0; pos1]
+                   ++ [seq i <- enum 'I_(size x) | val i >= (size u) + 2].
+ +
+Lemma size_cut_sizeu :
+  size [seq i <- enum 'I_(size x) | val i < size u] = size u.
+ +
+End Swap.
+End Swap.
+ +
+
+ +
+

Case where no sets contains both

+ + +
+ +The goal of this module is the following: given a k-support P for the word +u ++ [:: a; b] ++ v which doesn't have as subsequence containing both +a and b to construct a k-support Q for u ++ [:: b; a] ++ v with +the same cover size. These statements are Lemmas ksupp_Q and size_cover_Q + +
+
+Module NoSetContainingBoth.
+ +
+Section Case.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Variable R : rel Alph.
+ +
+Variable u v : word.
+Variable a b : Alph.
+Variable k : nat.
+Let x := u ++ [:: a; b] ++ v.
+Let y := u ++ [:: b; a] ++ v.
+ +
+Variable P : {set {set 'I_(size x)}}.
+Hypothesis Px : P \is a k.-supp[R, in_tuple x].
+ +
+Notation posa := (Swap.pos0 u v a b).
+Notation posb := (Swap.pos1 u v a b).
+Notation swapX := (@Swap.swap _ u v a b).
+Notation swap_setX := (Swap.swap_set u v a b).
+ +
+Hypothesis HnoBoth :
+  forall S : {set 'I_(size x)}, S \in P -> ~ ((posa \in S) && (posb \in S)).
+ +
+Lemma Hcast : size x = size y.
+ +
+
+ +
+This is essentially : set cast_ord Hcast x | x in swap_setX S. +
+
+Definition swap_set : {set 'I_(size x)} -> {set 'I_(size y)} :=
+  (fun S : {set 'I_(size x)} => [set cast_ord Hcast x | x in S]) \o swap_setX.
+
+ +
+This is essentially : set swap_set x | x in P +
+
+Definition Q : {set {set 'I_(size y)}} := imset swap_set (mem P).
+ +
+Lemma swap_set_inj : injective swap_set.
+ +
+Lemma extract_swap_set S :
+  S \in P -> extract (in_tuple y) (swap_set S) = extract (in_tuple x) S.
+ +
+Lemma ksupp_Q : Q \is a k.-supp[R, in_tuple y].
+ +
+Lemma size_cover_Q : #|cover P| == #|cover Q|.
+ +
+End Case.
+ +
+End NoSetContainingBoth.
+ +
+
+ +
+

Cover surgery

+ +
+
+Section CoverSurgery.
+ +
+Variable N : nat.
+Variable S : {set 'I_N}.
+Variable P Q : {set {set 'I_N}}.
+ +
+Lemma trivIset_coverU1 :
+  trivIset P -> [disjoint S & cover P] -> trivIset (S |: P).
+ +
+Lemma disjoint_cover (A B : {set 'I_N}) :
+  [disjoint cover P & cover Q] -> A \in P -> B \in Q -> [disjoint A & B].
+ +
+Lemma trivIset_coverU :
+  trivIset P -> trivIset Q -> [disjoint cover P & cover Q] -> trivIset (P :|: Q).
+ +
+Lemma trivIset_coverD1 : trivIset P -> S \in P -> [disjoint S & cover (P :\ S)].
+ +
+End CoverSurgery.
+ +
+
+ +
+

Case where a set in P contains both a and c

+ + +
+ +In Module SetContainingBothLeft, we denote x := u b a c v and consider +a given k-support P containing both a and c. We construct another +k-support Q for x with the same cover than P, but such that a and c +are not in the same set. +
+
+Module SetContainingBothLeft.
+ +
+
+ +
+

Generic order hypothesis

+ +
+
+Section RelHypothesis.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Implicit Type a b c : Alph.
+ +
+Record hypRabc R a b c := HypRabc {
+                      Rtrans : transitive R;
+                      Hbc : R b c;
+                      Hba : ~~ R b a;
+                      Hxba : forall l, R l a -> R l b;
+                      Hbax : forall l, R b l -> R a l
+                  }.
+ +
+Lemma RabcLeqX a b c :
+  (a < b <= c)%O -> hypRabc <=%O a b c.
+ +
+Lemma RabcGtnX a b c :
+  (a < b <= c)%O -> hypRabc >%O c b a.
+ +
+End RelHypothesis.
+ +
+Section Case.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Implicit Type u v w r : word.
+ +
+Variable R : rel Alph.
+Variable u v : word.
+Variable a b c : Alph.
+ +
+Hypothesis HRabc : hypRabc R a b c.
+ +
+Let x := u ++ [:: b; a; c] ++ v.
+ +
+Variable k : nat.
+Variable P : {set {set 'I_(size x)}}.
+Hypothesis Px : P \is a k.-supp[R, in_tuple x].
+ +
+Notation posb := (Swap.pos0 u (c :: v) b a).
+Notation posa := (Swap.pos1 u (c :: v) b a).
+Notation swap := (@Swap.swap _ u (c :: v) b a).
+Notation swap_set := (Swap.swap_set u (c :: v) b a).
+ +
+Lemma posc_subproof : (size u).+2 < size x.
+Definition posc := Ordinal posc_subproof.
+ +
+Lemma tnth_posc : tnth (in_tuple x) posc = c.
+ +
+Variable S : {set 'I_(size x)}.
+Hypothesis HS : S \in P.
+Hypothesis Hposa : (posa \in S).
+Hypothesis Hposc : (posc \in S).
+ +
+
+ +
+

Case where b is not in cover P

+ + +
+ +Replacing a by b gives another k-support. + +
+
+Section BNotIn.
+ +
+Hypothesis HbNin : posb \notin (cover P).
+ +
+Lemma posbinSF : posb \in S = false.
+ +
+Definition Qbnotin : {set {set 'I_(size x)}} := imset swap_set (mem P).
+ +
+Lemma size_cover_bnotin : #|cover P| == #|cover Qbnotin|.
+ +
+Lemma inPQE T : T != swap_set S -> T \in Qbnotin -> T \in P.
+ +
+Lemma sorted_extract_swap_set T :
+  T != swap_set S -> T \in Qbnotin -> sorted R (extract (in_tuple x) T).
+ +
+Lemma extract_SE :
+  extract (in_tuple x) S =
+  (extract (in_tuple x) (S :&: [set j : 'I_(size x) | j < size u])) ++
+    a :: c ::
+    extract (in_tuple x) (S :&: [set j : 'I_(size x) | (size u).+2 < j]).
+ +
+Lemma extract_swap_setSE :
+  extract (in_tuple x) (swap_set S) =
+  (extract (in_tuple x) (S :&: [set j : 'I_(size x) | j < size u])) ++
+    b :: c ::
+    extract (in_tuple x) (S :&: [set j : 'I_(size x) | (size u).+2 < j]).
+ +
+Lemma sorted_extract_swap_set_S : sorted R (extract (in_tuple x) (swap_set S)).
+ +
+Lemma ksupp_bnotin : Qbnotin \is a k.-supp[R, in_tuple x].
+ +
+Lemma Qbnotin_noboth T : T \in Qbnotin -> ~ ((posa \in T) && (posc \in T)).
+ +
+End BNotIn.
+ +
+
+ +
+

Case where b is in cover P

+ + +
+ +We assume that it is in a set T and switch the right part of S and T + +
+
+Section BIn.
+ +
+Variable T : {set 'I_(size x)}.
+Hypothesis HT : T \in P.
+Hypothesis Hposb : posb \in T.
+ +
+Lemma TSneq : T != S.
+ +
+Lemma TS_disjoint : [disjoint S & T].
+ +
+Lemma posb_inSF : (posb \in S) = false.
+ +
+Lemma posa_inTF : (posa \in T) = false.
+ +
+Lemma posc_inTF : (posc \in T) = false.
+ +
+Definition S1 := (S :&: [set j : 'I_(size x) | j <= posa])
+                   :|: (T :&: [set j : 'I_(size x) | j > posc]).
+Definition T1 := (T :&: [set j : 'I_(size x) | j <= posb])
+                   :|: (S :&: [set j : 'I_(size x) | j >= posc]).
+ +
+Lemma S1_subsST : S1 \subset (S :|: T).
+ +
+Lemma T1_subsST : T1 \subset (S :|: T).
+ +
+Lemma coverS1T1 : cover [set S1; T1] = (S :|: T).
+ +
+Lemma disjointS1T1 : [disjoint S1 & T1].
+ +
+Lemma ST_cover_disjoint : [disjoint S :|: T & cover (P :\: [set S; T])].
+ +
+Definition Qbin : {set {set 'I_(size x)}} :=
+  [set S1; T1] :|: (P :\: [set S; T]).
+ +
+Lemma trivIset_Qbin : trivIset Qbin.
+ +
+Lemma cover_bin : cover Qbin = cover P.
+ +
+Lemma enumUltV (U V : {set 'I_(size x)}) d :
+  (forall l, l \in U -> l <= d) ->
+  (forall l, l \in V -> l > d) ->
+  enum (mem (U :|: V)) = enum U ++ enum V.
+ +
+Lemma extract_S1E :
+  extract (in_tuple x) S1 =
+  extract (in_tuple x) (S :&: [set j : 'I_(size x) | j <= posa]) ++
+  extract (in_tuple x) (T :&: [set j : 'I_(size x) | j > posc]).
+ +
+Lemma extract_T1E :
+  extract (in_tuple x) T1 =
+  extract (in_tuple x) (T :&: [set j : 'I_(size x) | j <= posb]) ++
+  extract (in_tuple x) (S :&: [set j : 'I_(size x) | j >= posc]).
+ +
+Lemma extract_Sa :
+  extract (in_tuple x) (S :&: [set j : 'I_(size x) | j <= posa]) =
+  rcons (extract (in_tuple x) (S :&: [set j : 'I_(size x) | j < posa])) a.
+ +
+Lemma extract_cS :
+  extract (in_tuple x) (S :&: [set j : 'I_(size x) | j >= posc]) =
+  c :: extract (in_tuple x) (S :&: [set j : 'I_(size x) | j > posc]).
+ +
+Lemma extract_Tb :
+  extract (in_tuple x) (T :&: [set j : 'I_(size x) | j <= posb]) =
+  rcons (extract (in_tuple x) (T :&: [set j : 'I_(size x) | j < posb])) b.
+ +
+Lemma extract_bT :
+  extract (in_tuple x) (T :&: [set j : 'I_(size x) | j >= posb]) =
+  b :: extract (in_tuple x) (T :&: [set j : 'I_(size x) | j > posc]).
+ +
+Lemma sorted_extract_S1 : sorted R (extract (in_tuple x) S1).
+ +
+Lemma sorted_extract_T1 : sorted R (extract (in_tuple x) T1).
+ +
+Lemma ksupp_bin : Qbin \is a k.-supp[R, in_tuple x].
+ +
+Lemma posaS1_bin : posa \in S1.
+ +
+Lemma posaT1_bin : posb \in T1.
+ +
+Lemma Qbin_noboth U : U \in Qbin -> ~ ((posa \in U) && ((posc \in U))).
+ +
+End BIn.
+ +
+
+ +
+

Existence theorem for k-supports

+ +
+
+Theorem exists_Q_noboth :
+  exists Q : {set {set 'I_(size x)}},
+    [/\ Q \is a k.-supp[R, in_tuple x], #|cover Q| = #|cover P| &
+      forall S, S \in Q -> ~ ((posa \in S) && ((posc \in S)))].
+ +
+Let x' := (u ++ [:: b]) ++ [:: a; c] ++ v.
+Let y := (u ++ [:: b]) ++ [:: c; a] ++ v.
+ +
+Theorem exists_Qy :
+  exists Q : {set {set 'I_(size y)}},
+    #|cover Q| = #|cover P| /\ Q \is a k.-supp[R, in_tuple y].
+ +
+End Case.
+End SetContainingBothLeft.
+ +
+
+ +
+

Greene numbers are invariant by each plactic rules

+ +
+
+Section GreeneInvariantsRule.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Variable u v1 w v2 : word.
+Variable k : nat.
+ +
+
+ +
+Rule: [:: c; a; b] => if (a <= b < c)%Ord then [:: [:: a; c; b]] else [::] +
+
+Lemma ksuppRow_inj_plact1i :
+  v2 \in plact1i v1 -> ksupp_inj <=%O <=%O k (u ++ v1 ++ w) (u ++ v2 ++ w).
+ +
+Corollary Greene_row_leq_plact1i :
+  v2 \in plact1i v1 ->
+  Greene_row (u ++ v1 ++ w) k <= Greene_row (u ++ v2 ++ w) k.
+ +
+
+ +
+Rule: [:: b; a; c] => + if (a < b <= c)%Ord then [:: [:: b; c; a]] else [::] +
+
+Lemma ksuppRow_inj_plact2 :
+  v2 \in plact2 v1 -> ksupp_inj <=%O <=%O k (u ++ v1 ++ w) (u ++ v2 ++ w).
+ +
+Corollary Greene_row_leq_plact2 :
+  v2 \in plact2 v1 ->
+  Greene_row (u ++ v1 ++ w) k <= Greene_row (u ++ v2 ++ w) k.
+ +
+
+ +
+Rule: [:: a; c; b] => + if (a <= b < c)%Ord then [:: [:: c; a; b]] else [::] +
+
+Lemma ksuppCol_inj_plact1 :
+  v2 \in plact1 v1 -> ksupp_inj >%O >%O k (u ++ v1 ++ w) (u ++ v2 ++ w).
+ +
+Corollary Greene_col_leq_plact1 :
+  v2 \in plact1 v1 ->
+  Greene_col (u ++ v1 ++ w) k <= Greene_col (u ++ v2 ++ w) k.
+ +
+
+ +
+Rule: [:: b; c; a] => + if (a < b <= c)%Ord then [:: [:: b; a; c]] else [::] +
+
+Lemma ksuppCol_inj_plact2i :
+  v2 \in plact2i v1 -> ksupp_inj >%O >%O k (u ++ v1 ++ w) (u ++ v2 ++ w).
+ +
+Corollary Greene_col_leq_plact2i :
+  v2 \in plact2i v1 ->
+  Greene_col (u ++ v1 ++ w) k <= Greene_col (u ++ v2 ++ w) k.
+ +
+End GreeneInvariantsRule.
+ +
+
+ +
+

Deducing the other comparisons by duality

+ +
+
+Section GreeneInvariantsDual.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+Implicit Type u v w : word.
+ +
+Lemma Greene_row_leq_plact2i u v1 w v2 k :
+  v2 \in plact2i v1 ->
+  Greene_row (u ++ v1 ++ w) k <= Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_row_leq_plact1 u v1 w v2 k :
+  v2 \in plact1 v1 ->
+  Greene_row (u ++ v1 ++ w) k <= Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_leq_plact1i u v1 w v2 k :
+  v2 \in plact1i v1 ->
+  Greene_col (u ++ v1 ++ w) k <= Greene_col (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_leq_plact2 u v1 w v2 k :
+  v2 \in plact2 v1 ->
+  Greene_col (u ++ v1 ++ w) k <= Greene_col (u ++ v2 ++ w) k.
+ +
+
+ +
+

Invariance by the rules

+ +
+
+Lemma Greene_row_invar_plact1 u v1 w v2 k :
+  v2 \in plact1 v1 ->
+  Greene_row (u ++ v1 ++ w) k = Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_row_invar_plact1i u v1 w v2 k :
+  v2 \in plact1i v1 ->
+  Greene_row (u ++ v1 ++ w) k = Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_row_invar_plact2 u v1 w v2 k :
+  v2 \in plact2 v1 ->
+  Greene_row (u ++ v1 ++ w) k = Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_row_invar_plact2i u v1 w v2 k :
+  v2 \in plact2i v1 ->
+  Greene_row (u ++ v1 ++ w) k = Greene_row (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_invar_plact1 u v1 w v2 k :
+  v2 \in plact1 v1 ->
+  Greene_col (u ++ v1 ++ w) k = Greene_col (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_invar_plact1i u v1 w v2 k :
+  v2 \in plact1i v1 ->
+  Greene_col (u ++ v1 ++ w) k = Greene_col (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_invar_plact2 u v1 w v2 k :
+  v2 \in plact2 v1 ->
+  Greene_col (u ++ v1 ++ w) k = Greene_col (u ++ v2 ++ w) k.
+ +
+Lemma Greene_col_invar_plact2i u v1 w v2 k :
+  v2 \in plact2i v1 ->
+  Greene_col (u ++ v1 ++ w) k = Greene_col (u ++ v2 ++ w) k.
+ +
+End GreeneInvariantsDual.
+ +
+
+ +
+

Main theorem

+ +
+
+Section GreeneInvariants.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+
+ +
+

Row Greene number

+ +
+
+Theorem Greene_row_invar_plactic u v :
+  u =Pl v -> forall k, Greene_row u k = Greene_row v k.
+ +
+Corollary Greene_row_RS k w : Greene_row w k = sumn (take k (shape (RS w))).
+ +
+Corollary plactic_shapeRS_row_proof u v :
+  u =Pl v -> shape (RS u) = shape (RS v).
+ +
+
+ +
+

Column Greene number

+ +
+
+Theorem Greene_col_invar_plactic u v :
+  u =Pl v -> forall k, Greene_col u k = Greene_col v k.
+ +
+Corollary Greene_col_RS k w :
+  Greene_col w k = sumn (take k (conj_part (shape (RS w)))).
+ +
+Corollary plactic_shapeRS u v : u =Pl v -> shape (RS u) = shape (RS v).
+ +
+
+ +
+

Robinson-Schensted and the plactic monoid

+ +
+
+Theorem plactic_RS u v : u =Pl v <-> RS u == RS v.
+ +
+Corollary RS_tabE (t : seq (seq Alph)) : is_tableau t -> RS (to_word t) = t.
+ +
+End GreeneInvariants.
+ +
+Section GreenEqShape.
+ +
+Context d1 d2 (S : inhOrderType d1) (T : inhOrderType d2).
+ +
+Corollary Greene_row_eq_shape_RS (s : seq S) (t : seq T) :
+  (forall k, Greene_row s k = Greene_row t k) -> shape (RS s) = shape (RS t).
+ +
+Corollary Greene_col_eq_shape_RS (s : seq S) (t : seq T) :
+  (forall k, Greene_col s k = Greene_col t k) -> shape (RS s) = shape (RS t).
+ +
+End GreenEqShape.
+ +
+
+ +
+

Reverting words

+ +
+
+Section RevConj.
+ +
+Context d {Alph : inhOrderType d}.
+Implicit Type s : seq Alph.
+ +
+Corollary shape_RS_revdual s : shape (RS (revdual s)) = shape (RS s).
+ +
+Theorem RS_rev_uniq s : uniq s -> RS (rev s) = conj_tab (RS s).
+ +
+End RevConj.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.Schensted.html b/combi/1.1.0/Combi.LRrule.Schensted.html new file mode 100644 index 00000000..31a2cd90 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.Schensted.html @@ -0,0 +1,1123 @@ + + + + + +Combi.LRrule.Schensted: The Robinson-Schensted correspondence + + + + +
+ + + +
+ +

Library Combi.LRrule.Schensted: The Robinson-Schensted correspondence

+ +
+
+ +
+ +
+
+
+ +
+

The Robinson-Schensted correspondence

+ + +
+ +This file is a formalization of Schensted's algorithm and the Robinson-Schensted +correspondence. In the latter, it is easier to first store record the insertion +as a Yamanouchi word, that is the reverted sequence of the index of the rows +where the elements were inserted. In a second step, we translate the Yamanouchi +word in a standard tableau. + +
+ +Note: There is some duplication is this file, essentially for pedagogical +purpose. And also because it was my first serious Coq/Mathcomp development ;-) + +
+ +Here are the contents: + +
+ +Insertion in a row: + +
+ +
    +
  • inspred r l i == l bump the i letter of r + +
    • +
  • bump r l == l bump when inserted in r + +
    + + +
    • +
  • mininspred r l == the position of the letter bumped by l + +
    • +
  • inspos r l == the position of the letter bumped by l defined recursively + +
    + + +
    • +
  • insmin r l == the insertion of l in r defined using mininspred + +
    • +
  • insrow r l == the insertion of l in r defined resursively + +
    • +
  • ins r l == the insertion of l in r defined using inspos + +
    • +
+ +
+ + +
+ +Schensted algorithm: + +
+ +
    +
  • Sch w == Schented algorithm applied to w + +
    • +
  • subseqrow s w == s is a nondeceasing subsequence of w + +
    • +
  • subseqrow_n s w n == s is a nondeceasing subsequence of w of size n + +
    • +
+The two main results are +
    +
  • Corollary size_ndec_Sch : subseqrow s w -> (size s) <= size (Sch w). + +
    • +
  • Corollary exist_size_Sch w : exists s : seq T, subseqrow_n s w (size (Sch w)). + +
    • +
+ +
+ + +
+ +Robinson-Schensted bumping: + +
+ +
    +
  • bumped r l == the letter bumped by l when inserted in r, l itself + if there is no bumping + +
    • +
  • bumprow r l == bump l in r of type (option T) * (seq T) + +
    + + +
    • +
  • instab t l == insert the letter l in the tableau t + +
    • +
  • RS w == the insertion tableau (P-symbol) of w. It is a tableau as stated + in Theorem is_tableau_RS. + +
    • +
+ +
+ + +
+ +Inverting the Robinson-Schensted map: + +
+ +
    +
  • invbump b s == b has been bumped by s that is head b s < b + +
    • +
  • invbumprow b s == the inverse bumping of b in the row s. The result is + a pair (r, l) such that bumprow r l give back (some b, s). + This is lemma bumprowinvK. + +
    • +
  • invins b s == the r in (r, l) := invbumprow b s + +
    • +
  • invbumped b s the l in (r, l) := invbumprow b s. + +
    + + +
    • +
  • instabnrow t l == a pair (s, n) where s is the result of the insertion + of l in t and n is the index of the row where the insertion + stopped. As stated in Lemma instabnrowE s = instab t l. + +
    • +
  • invinstabnrow s n == the inverse insertion, that the pair (t, l) such that + instabnrow t l = (s, n). This is Theorem invinstabnrowK. See also + Theorem instabnrowinvK. + +
    + + +
    • +
  • RSmap w == the Robinson-Schensted map where the recording tableau is + returned as a Yamanouchi word. + +
    • +
  • is_RSpair (P, Q) == P is a tableau, Q is a Yammanouchi word and the shape + of P is equal to the evaluation of Q. + +
    • +
+ +
+ +The main result is of course Theorem RSmap_spec w : is_RSpair (RSmap w). + +
+ +
    +
  • RSmapinv tab yam == the Robinson-Schensted inverse map of the pair + (tab, yam) + +
    • +
  • RSmapinv2 p == the uncurrying of RSmapinv. + +
    • +
+ +
+ +The bijectivity of RSmap and RSmapinv2 are stated in +
    +
  • Theorem RSmapK stated as RSmapinv2 (RSmap w) = w. + +
    • +
  • and Theorem RSmapinv2K as RSmap (RSmapinv2 pair) = pair. + +
    • +
+Moreover RS preserve the content: +
    +
  • Theorem perm_RS w : perm_eq w (to_word (RS w)). + +
    • +
+ +
+ + +
+ +A sigma type for Robinson-Schensted pairs: + +
+ +
    +
  • rspair T == a sigma type for RS pair with tableau in type T. + +
    • +
  • RSbij w == the rspair T associated to w : seq T + +
    • +
  • RSbijinv p == the seq T associated to p : rspair T + +
    • +
+ +
+ +On has Lemma bijRS : bijective RSbij. + +
+ +Robinson-Schensted classes: + +
+ +
    +
  • RSclass t == the list of word w having t as RS tableau. This is stated + in Lemma RSclassE which says w \in RSclass tab = (RS w == tab). + +
    • +
+ +
+ +Robinson-Schensted with standard recording tableau: + +
+ +
    +
  • is_RStabpair (P, Q) == P and Q are two tableau of the same shape, + and Q is standard. + +
    • +
  • rstabpair T == a sigma type for is_RStabpair. + +
    + + +
    • +
  • RStabmap w == the RS map applied to w is is a RS pair as stated in + Theorem RStabmap_spec. + +
    + + +
    • +
  • RStab w == the RS map applied to w as a rstabpair T. + +
    • +
  • RStabinv pair == the word w associated to a rstabpair + +
    • +
+ +
+ +Again, one has Lemma bijRStab : bijective RStab. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm fingroup.
+Require Import tools partition Yamanouchi ordtype subseq tableau std stdtab.
+ +
+Set Implicit Arguments.
+ +
+Open Scope N.
+ +
+Import Order.Theory.
+ +
+Section NonEmpty.
+ +
+Context disp (T : inhOrderType disp).
+
+ +
+

Schensted's algorithm

+ +
+ +

Row insertion

+ +
+
+Section Insert.
+ +
+Variable Row : seq T.
+Hypothesis HRow : is_row Row.
+Variable l : T.
+ +
+Definition inspred i := (l < nth l Row i)%O.
+Definition bump := (l < (last l Row))%O.
+ +
+Lemma notbump : ~~bump = (l >= (last l Row))%O.
+ +
+Lemma transf : bump -> (l < (nth l Row (size Row).-1))%O.
+ +
+Lemma inspred_any_bump i : inspred i -> bump.
+ +
+Definition mininspred : nat :=
+  if ltP l (last l Row) is Order.LtlNotGe Hlast
+  then ex_minn (ex_intro inspred (size Row).-1 (transf Hlast))
+  else size Row.
+Definition insmin := set_nth l Row mininspred l.
+ +
+Lemma bump_mininspredE (Hbump : bump) :
+  mininspred = ex_minn (ex_intro inspred (size Row).-1 (transf Hbump)).
+ +
+Lemma nbump_mininspredE : ~~bump -> mininspred = size Row.
+ +
+Fixpoint insrow r l : seq T :=
+  if r is l0 :: r then
+    if (l < l0)%O then l :: r
+    else l0 :: (insrow r l)
+  else [:: l].
+ +
+Fixpoint inspos r (l : T) : nat :=
+  if r is l0 :: r' then
+    if (l < l0)%O then 0
+    else (inspos r' l).+1
+  else 0.
+ +
+Notation pos := (inspos Row l).
+Definition ins := set_nth l Row pos l.
+ +
+Lemma inspos_leq_size : pos <= size Row.
+ +
+Lemma inspos_lt_size_ins : pos < size ins.
+ +
+Lemma nth_inspos_ins : nth l ins pos = l.
+ +
+Lemma nbump_insposE : ~~bump -> mininspred = pos.
+ +
+Lemma inspred_inspos : bump -> inspred pos.
+ +
+Lemma inspred_mininspred : bump -> inspred mininspred.
+ +
+Lemma nth_lt_inspos i : i < pos -> (nth l Row i <= l)%O.
+ +
+Lemma inspredN_lt_inspos i : i < pos -> ~~ (inspred i).
+ +
+Lemma bump_insposE : bump -> mininspred = pos.
+ +
+Lemma insposE : mininspred = pos.
+ +
+Lemma inspos_leq_exP i : inspred i -> pos <= i.
+ +
+Lemma insE : insmin = ins.
+ +
+Lemma insrowE : insmin = insrow Row l.
+ +
+Lemma bump_inspos_lt_size : bump -> pos < size Row.
+ +
+Lemma nbump_inspos_eq_size : ~~bump -> pos = size Row.
+ +
+Lemma lt_inspos_nth i : i < size Row -> (nth l Row i <= l)%O -> i < pos.
+ +
+Lemma insrow_head_lt : (head l (insrow Row l) <= l)%O.
+ +
+Lemma ins_head_lt : (head l ins <= l)%O.
+ +
+Lemma is_row_ins : is_row ins.
+ +
+Lemma bump_size_ins : bump -> size ins = size Row.
+ +
+Lemma nbump_size_ins : ~~bump -> size ins = (size Row).+1.
+ +
+Lemma nbump_ins_rconsE : ~~bump -> ins = rcons Row l.
+ +
+Lemma size_ins_inf : (size Row) <= size ins.
+ +
+Lemma size_ins_sup : size ins <= (size Row).+1.
+ +
+Lemma ins_leq i : i < size Row -> (nth l ins i <= nth l Row i)%O.
+ +
+Lemma ins_non_nil : ins != [::].
+ +
+Lemma size_ins_non_0 : 0 < size ins.
+ +
+End Insert.
+ +
+Lemma bump_nil l : bump [::] l = false.
+ +
+Lemma bump_tail l0 r l : bump (l0 :: r) l -> (l0 <= l -> bump r l)%O.
+ +
+
+ +
+

The algorithm

+ +
+
+Section Schensted.
+ +
+Implicit Type l : T.
+Implicit Type r w s : seq T.
+ +
+Fixpoint Sch_rev w := if w is l0 :: w' then ins (Sch_rev w') l0 else [::].
+Definition Sch w := Sch_rev (rev w).
+ +
+Lemma Sch_rcons l w : Sch (rcons w l) = ins (Sch w) l.
+ +
+Lemma is_row_Sch w : is_row (Sch w).
+ +
+Lemma Sch_size w : size (Sch w) <= size w.
+ +
+
+ +
+

Schensted's algorithem specifications

+ +
+
+Definition subseqrow s w := subseq s w && is_row s.
+Definition subseqrow_n s w n := [&& subseq s w , (size s == n) & is_row s].
+ +
+Theorem Sch_exists w i :
+  i < size (Sch w) ->
+  exists s : seq T, (last inh s == nth inh (Sch w) i) && subseqrow_n s w i.+1.
+ +
+Theorem Sch_leq_last w s si:
+  subseqrow (rcons s si) w ->
+  size s < size (Sch w) /\ (nth inh (Sch w) (size s) <= si)%O.
+ +
+Corollary size_ndec_Sch w s : subseqrow s w -> (size s) <= size (Sch w).
+ +
+Corollary exist_size_Sch w : exists s : seq T, subseqrow_n s w (size (Sch w)).
+ +
+End Schensted.
+ +
+Theorem Sch_max_size (w : seq T) :
+  size (Sch w) = \max_(s : subseqs w | is_row s) size s.
+ +
+
+ +
+

Robinson-Schensted bumping

+ +
+ +

bumping a letter

+ +
+
+Section Bump.
+ +
+Variable Row : seq T.
+Hypothesis HRow : is_row Row.
+Variable l : T.
+ +
+Definition bumped := nth l Row (inspos Row l).
+Notation ins := (ins Row l).
+Notation inspos := (inspos Row l).
+Notation insRow := (insrow Row l).
+Notation bump := (bump Row l).
+ +
+Lemma lt_bumped : bump -> (l < bumped)%O.
+ +
+Fixpoint bumprow r l : (option T) * (seq T) :=
+  if r is l0 :: r then
+    if (l < l0)%O then (Some l0, l :: r)
+    else let: (lr, rr) := bumprow r l in (lr, l0 :: rr)
+  else (None, [:: l]).
+ +
+Notation bumpRow := (bumprow Row l).
+ +
+Lemma ins_bumprowE : insRow = bumpRow.2.
+ +
+Lemma bump_bumprowE : bump -> bumpRow = (Some bumped, ins).
+ +
+Lemma nbump_bumprowE : ~~bump -> bumpRow = (None, ins).
+ +
+Lemma head_ins_lt_bumped i : bump -> (head i ins < bumped)%O.
+ +
+Lemma bumprow_size :
+  let: (lr, tr) := bumpRow in
+  (size Row).+1 == (size tr) + if lr is Some _ then 1 else 0.
+ +
+Lemma bumprow_count p :
+  let: (lr, tr) := bumpRow in
+  count p Row + (p l) == count p tr + if lr is Some ll then (p ll) else 0.
+ +
+End Bump.
+ +
+Lemma bumprow_rcons r l : is_row (rcons r l) -> bumprow r l = (None, rcons r l).
+ +
+Section Dominate.
+ +
+Implicit Type l : T.
+Implicit Type r u v : seq T.
+ +
+Lemma dominate_inspos r1 r0 l :
+  is_row r0 -> is_row r1 -> dominate r1 r0 ->
+  bump r0 l -> inspos r0 l >= inspos r1 (bumped r0 l).
+ +
+Lemma bump_dominate r1 r0 l :
+  is_row r0 -> is_row r1 -> bump r0 l ->
+  dominate r1 r0 -> dominate (ins r1 (bumped r0 l)) (ins r0 l).
+ +
+Lemma dominateK_inspos r1 r0 l0 :
+  is_row r0 -> is_row r1 -> dominate (ins r1 (bumped r0 l0)) (ins r0 l0) ->
+  bump r0 l0 -> inspos r0 l0 >= inspos r1 (bumped r0 l0).
+ +
+Lemma bump_dominateK r1 r0 l0 :
+  is_row r0 -> is_row r1 -> bump r0 l0 ->
+  dominate (ins r1 (bumped r0 l0)) (ins r0 l0) -> dominate r1 r0.
+ +
+End Dominate.
+ +
+
+ +
+

The insertion tableau

+ +
+
+Section Tableaux.
+ +
+Implicit Type l : T.
+Implicit Type r w : seq T.
+Implicit Type t : seq (seq T).
+ +
+Fixpoint instab t l : seq (seq T) :=
+  if t is t0 :: t' then
+    let: (lr, rr) := bumprow t0 l in
+    if lr is Some ll then rr :: (instab t' ll) else rr :: t'
+  else [:: [:: l]].
+ +
+Lemma head_instab (t0 : seq T) t l :
+  is_row t0 -> head [::] (instab (t0 :: t) l) = ins t0 l.
+ +
+Theorem is_tableau_instab t l : is_tableau t -> is_tableau (instab t l).
+ +
+Lemma instab_non_nil t l : instab t l != [::].
+ +
+Fixpoint RS_rev w : seq (seq T) :=
+  if w is w0 :: wr then instab (RS_rev wr) w0 else [::].
+Definition RS w := RS_rev (rev w).
+ +
+Theorem is_tableau_RS w : is_tableau (RS w).
+ +
+End Tableaux.
+ +
+
+ +
+

Inverting a bump

+ +
+
+Section InverseBump.
+ +
+Implicit Type a b l : T.
+Implicit Type r s w : seq T.
+Implicit Type t : seq (seq T).
+ +
+Definition invbump b s := ((head b s) < b)%O.
+ +
+Fixpoint invbumprow b s : (seq T) * T :=
+  if s is l0 :: s then
+    if (b <= head b s)%O
+    then (b :: s, l0)
+    else let: (rr, lr) := invbumprow b s in (l0 :: rr, lr)
+  else ([::], b).
+ +
+Definition invins b s := (invbumprow b s).1.
+Definition invbumped b s := (invbumprow b s).2.
+ +
+Lemma head_lt_invins b s i :
+  s != [::] -> invbump b s -> (head i s <= head i (invins b s))%O.
+ +
+Lemma is_row_invins b s : is_row s -> is_row (invins b s).
+ +
+Lemma head_leq_invbumped b s :
+  s != [::] -> is_row s -> (head inh s <= (invbumped b s))%O.
+ +
+Lemma invbumprowK r a :
+  is_row r -> bump r a ->
+  (invbumprow (bumped r a) (ins r a)) = (r, a).
+ +
+Lemma bumprowinvK b s :
+  s != [::] -> is_row s -> invbump b s ->
+  (bumprow (invins b s) (invbumped b s)) = (Some b, s).
+ +
+
+ +
+

The Robinson-Schensted insertion with recording

+ +
+
+Fixpoint instabnrow t l : seq (seq T) * nat :=
+  if t is t0 :: t then
+    let: (lr, rr) := bumprow t0 l
+    in if lr is Some ll then
+         let: (tres, nres) := instabnrow t ll
+         in (rr :: tres, nres.+1)
+       else (rr :: t, 0)
+  else ([:: [:: l]], 0).
+ +
+Lemma instabnrowE t l : (instabnrow t l).1 = instab t l.
+ +
+Lemma shape_instabnrow t l :
+  is_tableau t ->
+  let: (tr, nrow) := instabnrow t l in shape tr = incr_nth (shape t) nrow.
+ +
+End InverseBump.
+ +
+Section Inverse.
+ +
+Implicit Type a b l : T.
+Implicit Type r s w : seq T.
+Implicit Type t u : seq (seq T).
+ +
+Lemma is_rem_corner_instabnrow t l : is_tableau t ->
+    let: (res, nrow) := instabnrow t l in is_rem_corner (shape res) nrow.
+ +
+
+ +
+

Invertion a Robinson-Schensted step

+ +
+
+Fixpoint invinstabnrow t nrow : seq (seq T) * T :=
+  if t is t0 :: t
+  then if nrow is nrow.+1
+       then let: (tr, lr) := invinstabnrow t nrow in
+            let: (t0r, l0r) := invbumprow lr t0 in
+            (t0r :: tr, l0r)
+       else if t0 is l0 :: t0
+            then if t0 == [::]
+                 then (t, l0)
+                 else ((belast l0 t0) :: t, last l0 t0)
+            else ([::], inh)
+  else ([::], inh).
+ +
+Theorem invinstabnrowK t l :
+  is_tableau t -> invinstabnrow (instab t l) (instabnrow t l).2 = (t, l).
+ +
+Lemma invbump_geq_head t tin l nrow :
+  t != [::] -> is_tableau t -> invinstabnrow t nrow = (tin, l) ->
+  (l >= head l (head [::] t))%O.
+ +
+Lemma invbump_dom r0 t tin l nrow :
+  t != [::] -> is_tableau t -> invinstabnrow t nrow = (tin, l) ->
+  r0 != [::] -> dominate (head [::] t) r0 -> invbump l r0.
+ +
+Theorem instabnrowinvK t nrow :
+  is_tableau t -> t != [::] -> is_rem_corner (shape t) nrow ->
+  let: (tin, l) := invinstabnrow t nrow in (instabnrow tin l) = (t, nrow).
+ +
+
+ +
+

Robinson-Schensted correspondence

+ +
+ +

The Robinson-Schensted map

+ +
+
+Fixpoint RSmap_rev w : (seq (seq T)) * (seq nat) :=
+  if w is w0 :: wtl
+  then let: (t, rows) := RSmap_rev wtl in
+       let: (tr, nrow) := instabnrow t w0 in
+       (tr, nrow :: rows)
+  else ([::], [::]).
+Definition RSmap w := RSmap_rev (rev w).
+ +
+Lemma RSmapE w : (RSmap w).1 = RS w.
+ +
+Lemma size_RSmap2 w : size ((RSmap w).2) = size w.
+ +
+Lemma is_tableau_RSmap1 w : is_tableau (RSmap w).1.
+ +
+Lemma shape_RSmap_eq w : shape (RSmap w).1 = evalseq (RSmap w).2.
+ +
+Lemma is_yam_RSmap2 w : is_yam (RSmap w).2.
+ +
+Definition is_RSpair pair :=
+  let: (P, Q) := pair in
+  [&& is_tableau (T:=T) P, is_yam Q & (shape P == evalseq Q)].
+ +
+Theorem RSmap_spec w : is_RSpair (RSmap w).
+ +
+
+ +
+

The inverse Robinson-Schensted inverse map

+ +
+
+Fixpoint RSmapinv tab yam :=
+  if yam is nrow :: yam'
+  then let: (tr, lr) := invinstabnrow tab nrow in
+       rcons (RSmapinv tr yam') lr
+  else [::].
+Definition RSmapinv2 pair := RSmapinv (pair.1) (pair.2).
+ +
+Theorem RSmapK w : RSmapinv2 (RSmap w) = w.
+ +
+Lemma behead_incr_nth (s : seq nat) nrow :
+  behead (incr_nth s nrow.+1) = incr_nth (behead s) nrow.
+ +
+Lemma size_invins b s : size (invins b s) = (size s).
+ +
+Lemma yam_tail_non_nil (l : nat) (s : seq nat) :
+  is_yam (l.+1 :: s) -> s != [::].
+ +
+Lemma shape_instabnrowinv1 t nrow yam :
+  is_yam (nrow :: yam) -> shape t == evalseq (nrow :: yam) ->
+  shape (invinstabnrow t nrow).1 == evalseq yam.
+ +
+Lemma head_tableau_non_nil h t : is_tableau (h :: t) -> h != [::].
+ +
+Lemma is_tableau_instabnrowinv1 (s : seq (seq T)) nrow :
+  is_tableau s -> is_rem_corner (shape s) nrow ->
+  is_tableau (invinstabnrow s nrow).1.
+ +
+Theorem RSmapinv2K pair : is_RSpair pair -> RSmap (RSmapinv2 pair) = pair.
+ +
+End Inverse.
+ +
+
+ +
+

Statistics preserved by the Robinson-Schensted map

+ +
+
+Section Statistics.
+ +
+Implicit Type t : seq (seq T).
+ +
+Lemma size_instab t l : is_tableau t -> size_tab (instab t l) = (size_tab t).+1.
+ +
+Theorem size_RS w : size_tab (RS w) = size w.
+ +
+Lemma count_instab t l p :
+  is_tableau t -> count p (to_word (instab t l)) = (p l) + count p (to_word t).
+ +
+Theorem count_RS w p : count p w = count p (to_word (RS w)).
+ +
+Theorem perm_RS w : perm_eq w (to_word (RS w)).
+ +
+End Statistics.
+ +
+
+ +
+

Sigma types and bijections

+ +
+
+Section Bijection.
+ +
+Notation Pair := (seq (seq T) * seq nat : Type).
+ +
+Structure rspair : predArgType :=
+  RSpair { pyampair :> Pair; _ : is_RSpair pyampair }.
+ +
+Lemma pyampair_inj : injective pyampair.
+ +
+Definition RSbij w := RSpair (RSmap_spec w).
+Definition RSbijinv (ps : rspair) := RSmapinv2 ps.
+ +
+Lemma bijRS : bijective RSbij.
+ +
+End Bijection.
+ +
+
+ +
+

Robinson-Schensted classes

+ +
+
+Section Classes.
+ +
+Definition RSclass :=
+  [fun tab => [seq RSmapinv2 (tab, y) | y <- enum_yameval (shape tab)] ].
+ +
+Lemma RSclassP tab :
+  is_tableau tab -> all (fun w => RS w == tab) (RSclass tab).
+ +
+Lemma RSclass_countE w : count_mem w (RSclass (RS w)) = 1.
+ +
+Lemma mem_RSclass w : w \in (RSclass (RS w)).
+ +
+Lemma RSclassE tab w :
+  is_tableau tab -> w \in RSclass tab = (RS w == tab).
+ +
+End Classes.
+ +
+End NonEmpty.
+ +
+
+ +
+

The Robinson-Schensted map with standard recording tableau

+ +
+
+Lemma RSperm n (p : 'S_n) : is_stdtab (RS (wordperm p)).
+ +
+Lemma RSstdE (p : seq nat) : is_stdtab (RS p) = is_std p.
+ +
+Section QTableau.
+Context disp (T : inhOrderType disp).
+ +
+Notation TabPair := (seq (seq T) * seq (seq nat) : Type).
+ +
+Definition is_RStabpair (pair : TabPair) :=
+  let: (P, Q) := pair in [&& is_tableau P, is_stdtab Q & (shape P == shape Q)].
+ +
+Structure rstabpair : predArgType :=
+  RSTabPair { pqpair :> TabPair; _ : is_RStabpair pqpair }.
+ +
+ +
+Lemma pqpair_inj : injective pqpair.
+ +
+Definition RStabmap (w : seq T) :=
+  let (p, q) := (RSmap w) in (p, stdtab_of_yam q).
+ +
+Lemma RStabmapE (w : seq T) : (RStabmap w).1 = RS w.
+ +
+Theorem RStabmap_spec w : is_RStabpair (RStabmap w).
+ +
+Lemma shape_RStabmapE (w : seq T) : shape (RStabmap w).1 = shape (RStabmap w).2.
+ +
+Lemma is_stdtab_RStabmap2 (w : seq T) : is_stdtab (RStabmap w).2.
+ +
+Definition RStab w := RSTabPair (RStabmap_spec w).
+Definition RStabinv (pair : rstabpair) :=
+  let: (P, Q) := pqpair pair in RSmapinv2 (P, yam_of_stdtab Q).
+ +
+Lemma RStabK : cancel RStab RStabinv.
+Lemma RStabinvK : cancel RStabinv RStab.
+Lemma bijRStab : bijective RStab.
+ +
+End QTableau.
+ +
+Section Tests.
+ +
+Goal (insrow [:: 1; 1; 2; 3; 5] 2) = [:: 1; 1; 2; 2; 5].
+ +
+Goal (insrow [:: 1; 1; 2; 3; 5] 2) = [:: 1; 1; 2; 2; 5].
+ +
+Goal (ins [:: 1; 1; 2; 3; 5] 2) = [:: 1; 1; 2; 2; 5].
+ +
+Goal (Sch [:: 2; 5; 1; 6; 4; 3]) = [:: 1; 3; 6].
+ +
+Goal (RS [:: 2; 5; 1; 6; 4; 3]) = [:: [:: 1; 3; 6]; [:: 2; 4]; [:: 5]].
+ +
+Goal (to_word (RS [:: 2; 5; 1; 6; 4; 3])) = [:: 5; 2; 4; 1; 3; 6].
+ +
+Goal is_tableau (RS [:: 2; 5; 1; 6; 4; 3]).
+ +
+Goal (invbumprow 3 [:: 1; 1; 2; 2; 5]) = ([:: 1; 1; 2; 3; 5], 2).
+ +
+Goal (invbumprow 3 [:: 1; 1; 2; 2; 3]) = ([:: 1; 1; 2; 3; 3], 2).
+ +
+Goal instabnrow [:: [:: 1; 3; 6]; [:: 2; 4]; [:: 5]] 3 =
+               ([:: [:: 1; 3; 3]; [:: 2; 4; 6]; [:: 5]], 1).
+ +
+Goal invinstabnrow [:: [:: 1; 3; 3]; [:: 2; 4; 6]; [:: 5]] 1 =
+                  ([:: [:: 1; 3; 6]; [:: 2; 4]; [:: 5]], 3).
+ +
+Goal is_part [:: 0] = false.
+ +
+Goal evalseq [::] = [::].
+ +
+Goal evalseq [:: 0; 1; 2; 0; 1; 3] = [:: 2; 2; 1; 1].
+ +
+Goal (RSmapinv2 (RSmap [:: 4; 1; 2; 1; 3; 2])) = [:: 4; 1; 2; 1; 3; 2].
+ +
+End Tests.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.Yam_plact.html b/combi/1.1.0/Combi.LRrule.Yam_plact.html new file mode 100644 index 00000000..e28080fc --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.Yam_plact.html @@ -0,0 +1,187 @@ + + + + + +Combi.LRrule.Yam_plact: Plactic classes and Yamanouchi words + + + + +
+ + + +
+ +

Library Combi.LRrule.Yam_plact: Plactic classes and Yamanouchi words

+ +
+
+ +
+ +
+
+
+ +
+

Plactic classes and Yamanouchi words

+ + +
+ +The goal of this file is to show that the Yamanouchi words of a given +evaluation form a plactic class. + +
+ +
    +
  • yamtab sh == the uniq Yamanouchi tableau of shape sh: the i-th line + contains only i's as in: 33 2222 11111 0000000. + +
    • +
+ +
+ +The main result is Corollary yam_plactic_shape + +
+ + + y =Pl z <-> (is_yam z /\ evalseq y = evalseq z). + + +
+ +We also show that for all partition sh the standardization defines a bijection +from Yamanouchi words of evaluation sh and a plactic class. In particular, it +is surjective. This is Theorem plact_from_yam: + +
+ + + w =Pl std (hyper_yam sh) -> { y | is_yam_of_eval sh y & std y = w }. + + +
+
+ +
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq fintype.
+From mathcomp Require Import tuple finfun finset path bigop order.
+ +
+Require Import tools partition Yamanouchi ordtype std tableau stdtab.
+Require Import Schensted congr plactic Greene_inv stdplact.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+Open Scope N.
+ +
+Lemma is_part_incr_nthE sh i j :
+  i.+1 < j -> is_part sh -> is_part (incr_nth (incr_nth sh j) i) ->
+  is_part (incr_nth sh i) = is_part (incr_nth sh j).
+ +
+Lemma is_part_incr_nth1E sh i :
+  is_part sh ->
+  is_part (incr_nth (incr_nth sh i.+1) i) = is_part (incr_nth sh i).
+ +
+Lemma is_yam_plactic y : is_yam y -> forall w, y =Pl w -> is_yam w.
+ +
+
+ +
+

The Yamanouchi tableau

+ +
+
+Fixpoint yamtab_rec i sh :=
+  if sh is s0 :: s then
+    nseq s0 i :: yamtab_rec (i.+1) s
+  else [::].
+Definition yamtab := yamtab_rec 0.
+ +
+Lemma shape_yamtab sh : shape (yamtab sh) = sh.
+ +
+Lemma to_word_yamtab sh : to_word (yamtab sh) = hyper_yam sh.
+ +
+Lemma yamtabP sh : is_part sh -> is_tableau (yamtab sh).
+ +
+Lemma yamtab_rcons sh sn :
+  yamtab (rcons sh sn) = rcons (yamtab sh) (nseq sn (size sh)).
+ +
+Lemma yamtab_unique t :
+  is_tableau t -> is_yam (to_word t) -> t = yamtab (shape t).
+ +
+Corollary RS_yam_RS y : is_yam y -> RS y = yamtab (shape (RS y)).
+ +
+Lemma shape_RS_yam y : is_yam y -> shape (RS y) = evalseq y.
+ +
+Lemma RS_yam y : is_yam y -> RS y = yamtab (evalseq y).
+ +
+Theorem yam_plactic_hyper y : is_yam y -> y =Pl hyper_yam (evalseq y).
+ +
+Corollary yam_plactic_shape y z :
+  is_yam y -> ( y =Pl z <-> (is_yam z /\ evalseq y = evalseq z)).
+ +
+
+ +
+

Yamanouchi words, standardization and plactic classes

+ +
+
+Lemma yam_std_inj : {in is_yam &, injective std}.
+ +
+Lemma auxbijP p (y : yameval p) : is_yam_of_eval p ((RSmap y).2).
+ +
+#[local] Definition auxbij (p : intpart) (y : yameval p) : yameval p :=
+  YamEval (auxbijP y).
+ +
+Lemma auxbij_inj (p : intpart) : injective (@auxbij p).
+ +
+#[local] Definition auxbij_inv (p : intpart) := invF (@auxbij_inj p).
+ +
+Theorem plact_from_yam sh w :
+  is_part sh ->
+  w =Pl std (hyper_yam sh) ->
+  { y | is_yam_of_eval sh y & std y = w }.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.extract.html b/combi/1.1.0/Combi.LRrule.extract.html new file mode 100644 index 00000000..d51dd8a1 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.extract.html @@ -0,0 +1,96 @@ + + + + + +Combi.LRrule.extract: Extracting the implementation to OCaml + + + + +
+ + + +
+ +

Library Combi.LRrule.extract: Extracting the implementation to OCaml

+ +
+
+ +
+ +
+
+
+ +
+

A certified OCaml implementation

+ + +
+ +We extract to OCaml the implementation of the Robinson-Schensted correspondance +and The Littlewood-Richardson Rule. + +
+
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg.
+From mathcomp Require Import mpoly.
+Require Import subseq partition ordtype Schensted congr plactic Greene Greene_inv
+        std stdtab skewtab therule implem.
+ +
+ +
+From Stdlib Require Import Wf_nat.
+Extraction Inline Wf_nat.lt_wf_rec1 Wf_nat.lt_wf_rec
+  Wf_nat.lt_wf_ind Wf_nat.gt_wf_rec Wf_nat.gt_wf_ind.
+ +
+Extract Inductive bool => "bool" [ "true" "false" ].
+ +
+Extract Inductive list => "list" [ "[]" "(::)" ].
+ +
+Extract Inductive prod => "(*)" [ "(,)" ].
+ +
+ +
+Definition disp := Order.NatOrder.nat_display.
+Definition RS := (@RS disp nat).
+Definition RSbijnat := (@RSbij disp nat).
+Definition RSbijinvnat := (@RSbijinv disp nat).
+Definition RStabnat := (@RStab disp nat).
+Definition RStabinvnat := (@RStabinv disp nat).
+ +
+ +
+
+ +
+

The Littlewood-Richardson Rule

+ +
+
+Extraction "src/LRrule/lrcoeff.ml"
+           LRcoeff LRyamtab_list
+.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.freeSchur.html b/combi/1.1.0/Combi.LRrule.freeSchur.html new file mode 100644 index 00000000..61805df1 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.freeSchur.html @@ -0,0 +1,581 @@ + + + + + +Combi.LRrule.freeSchur: Free Schur functions + + + + +
+ + + +
+ +

Library Combi.LRrule.freeSchur: Free Schur functions

+ +
+
+ +
+ +
+
+
+ +
+

Free Schur functions

+ + +
+ +This file is the second step of the proof of the Littewood-Richardson rule. +We translate theorem LRtriple_cat_equiv in a algebraic setting. Specifically, +the main goal of this file is to lift the multiplication of Schur multivariate +polynomials to the non commutative setting. + +
+ +
    +
  • commword n R w == the commutative image of the word w as a multivariate + polynomial (of type {mpoly R[n]}). + +
    • +
  • homlang n d == the type of homogenous langage over 'I_n.+1 of degre d. + that is {set d.-tuple 'I_n}. + +
    • +
  • polylang n R s == the commutative image of the langage s where s is of + type homlang n d. + +
    • +
  • catlang l1 l2 == the concatenation of homogeneous language: given + s1 of degree d1 and s2 of degree d2 return an + homogeneous language of degree d1 + d2. + +
    • +
  • tabwordshape n sh == the set of reading of tableaux over 'I_n.+1 of + shape sh, where sh is of type 'P_d + +
    • +
  • freeSchur n t == the set of words whose recording tableau over 'I_n.+1 + is t, where t is of type stdtabn + +
    • +
  • tabword_of_tuple w == the bijection freeSchur -> tabwordshape as stated + in Theorem tabword_of_tuple_freeSchur: + +
    + + + forall Q : stdtabn d, + [set tabword_of_tuple x | x in freeSchur n0 Q] = tabwordshape n0 (shape_deg Q) + + +
    • +
+ +
+ +The free Littlewood-Richardson rule: + +
+ +
    +
  • LRsupport Q1 Q2 == the set of standard Littlewood-Richardson Q-tableau in + the product of the free Schur function indexed by Q1 + and Q2, that is the set of Q which forms a LRtriple + with Q1 and Q2. + +
    • +
+ +
+ +The main result here is the free LR rule free_LR_rule: + +
+ + + catlang (freeSchur Q1) (freeSchur Q2) = \bigcup_(Q in LRsupport) freeSchur Q. + + +
+ +We then go back to commutative Schur polynomials: + +
+ +
    +
  • hyper_stdtab sh == the hyper standard tableau of shape sh as a seq (seq nat). + +
    • +
  • hyper_stdtabn sh == the hyper standard tableau of shape sh as a + stdtabn d where sh is a 'P_d. + +
    • +
  • LRtab_set Q1 Q2 Q == the set of standard Littlewood-Richardson Q-tableau in + the product of the free Schur function indexed by Q1 + and Q2 of shape Q. + +
    • +
  • LRtab_coeff Q1 Q2 == the Littlewood-Richardson coefficient defined + as the cardinality of LRtab_set Q1 Q2 Q. + +
    • +
+ +
+ +Invariance with the choice of Q1 and Q2: + +
+ +
    +
  • bij_LRsupport Q1 Q2 == a bijection from LRsupport T1 T2 to LRsupport Q1 Q2 + as long as T1 and Q1 have the same shape as well as + T2 and Q2. It is used to show Theorem + LRtab_coeff_shapeE: + +
    + + + shape T1 = P1 -> shape T2 = P2 -> + LRtab_coeff P = #|[set Q in (LRsupport T1 T2) | (shape Q == P)]|. + + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg.
+From mathcomp Require Import ssrcomplements freeg mpoly.
+ +
+Require Import tools ordtype partition Yamanouchi std tableau stdtab.
+Require Import Schensted congr plactic stdplact Yam_plact Greene_inv shuffle.
+Require Import Schur_mpoly.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+
+ +
+

Commutative image of an homogeneous langage

+ +
+
+Section CommutativeImage.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+ +
+Definition commword (w : seq 'I_n) : {mpoly R[n]} := \prod_(i <- w) 'X_i.
+ +
+Lemma perm_commword (u v : seq 'I_n) :
+  perm_eq u v -> commword u = commword v.
+ +
+Lemma commword_morph (u v : seq 'I_n) :
+  commword (u ++ v) = (commword u) * (commword v).
+ +
+Lemma commtuple_morph d1 d2 (u : d1.-tuple 'I_n) (v : d2.-tuple 'I_n) :
+  commword (cat_tuple u v) = (commword u) * (commword v).
+ +
+Definition homlang d := {set d.-tuple 'I_n}.
+Definition polylang d (s : homlang d) := \sum_(w in s) commword w.
+Definition catlang d1 d2 (s1 : homlang d1) (s2 : homlang d2) :
+  homlang (d1 + d2) :=
+ [set cat_tuple w1 w2 | w1 in s1, w2 in s2].
+ +
+Lemma cat_tuple_inj d1 d2 (u x : d1.-tuple 'I_n) (v y : d2.-tuple 'I_n) :
+  cat_tuple u v = cat_tuple x y -> (u, v) = (x, y).
+ +
+Lemma catlangM d1 d2 (s1 : homlang d1) (s2 : homlang d2) :
+  polylang s1 * polylang s2 = polylang (catlang s1 s2).
+ +
+End CommutativeImage.
+ +
+
+ +
+

Row reading of tableau

+ +
+
+Section TableauReading.
+ +
+Context {disp} {A : inhOrderType disp}.
+ +
+Definition tabsh_reading_RS (sh : seq nat) (w : seq A) :=
+  (to_word (RS w) == w) && (shape (RS (w)) == sh).
+ +
+Lemma tabsh_reading_RSP (sh : seq nat) (w : seq A) :
+  reflect
+    (exists tab, [/\ is_tableau tab, shape tab = sh & to_word tab = w])
+    (tabsh_reading_RS sh w).
+ +
+Lemma tabsh_reading_RSE sh :
+  tabsh_reading sh =1 tabsh_reading_RS sh.
+ +
+End TableauReading.
+ +
+
+ +
+

Free Schur functions : lifting Schur functions in the free algebra

+ +
+
+Section FreeSchur.
+ +
+Variable R : comNzRingType.
+ +
+Variable n0 : nat.
+#[local] Notation n := (n0.+1).
+#[local] Notation Schur sh := (Schur n0 R sh).
+#[local] Notation homlang d := (homlang n d).
+ +
+Section Degree.
+ +
+Variable d : nat.
+ +
+Definition tabwordshape (sh : 'P_d) : homlang d :=
+  [set t : d.-tuple 'I_n | tabsh_reading sh t ].
+Definition freeSchur (Q : stdtabn d) : homlang d :=
+  [set t : d.-tuple 'I_n | (RStabmap t).2 == Q].
+ +
+Lemma freeSchurP Q (t : d.-tuple 'I_n) :
+  t \in freeSchur Q = (val t \in langQ Q).
+ +
+Lemma size_RS_tuple (t : d.-tuple 'I_n) : size (to_word (RS t)) == d.
+ +
+
+ +
+ Bijection freeSchur -> tabwordshape +
+
+Definition tabword_of_tuple (t : d.-tuple 'I_n) : d.-tuple 'I_n :=
+  Tuple (size_RS_tuple t).
+ +
+Lemma perm_tabword_of_tuple (t : d.-tuple 'I_n) :
+  perm_eq t (tabword_of_tuple t).
+ +
+Lemma tabword_of_tuple_freeSchur_inj (Q : stdtabn d) :
+  {in (freeSchur Q) &, injective tabword_of_tuple}.
+ +
+Lemma tabword_of_tuple_freeSchur (Q : stdtabn d) :
+  [set tabword_of_tuple x | x in freeSchur Q] = tabwordshape (shape_deg Q).
+ +
+End Degree.
+ +
+
+ +
+

Noncommutative lifting of Schur polynomials

+ +
+
+Lemma SchurE d (Q : stdtabn d) :
+  Schur (shape_deg Q) = polylang R (tabwordshape (shape_deg Q)).
+ +
+
+ +
+

Commutative image of freeSchur language

+ +
+
+Lemma Schur_freeSchurE d (Q : stdtabn d) :
+  Schur (shape_deg Q) = polylang R (freeSchur Q).
+ +
+
+ +
+

The free Littlewood-Richardson rule

+ +
+
+Section FreeLRrule.
+ +
+Variables (d1 d2 : nat).
+Variables (Q1 : stdtabn d1) (Q2 : stdtabn d2).
+ +
+Definition LRsupport :=
+  [set Q : stdtabn (d1 + d2) | pred_LRtriple_fast Q1 Q2 Q ].
+ +
+Lemma free_LR_rule :
+  catlang (freeSchur Q1) (freeSchur Q2) = \bigcup_(Q in LRsupport) freeSchur Q.
+ +
+
+ +
+Alternative proof from LRrule_langQ +
+
+Lemma free_LR_rule_alternate :
+  catlang (freeSchur Q1) (freeSchur Q2) = \bigcup_(Q in LRsupport) freeSchur Q.
+ +
+
+ +
+Passing to commutative image in the free LR rule +
+
+Theorem LR_rule_tab :
+  Schur (shape_deg Q1) * Schur (shape_deg Q2) =
+    \sum_(Q in LRsupport) (Schur (shape_deg Q)).
+ +
+End FreeLRrule.
+ +
+Definition hyper_stdtab sh := RS (std (hyper_yam sh)).
+Lemma hyper_stdtabP sh : is_stdtab (hyper_stdtab sh).
+ +
+Lemma hyper_stdtabnP d (P : 'P_d) : is_stdtab_of_n d (hyper_stdtab P).
+Canonical hyper_stdtabn d (P : 'P_d) := StdtabN (hyper_stdtabnP P).
+ +
+Lemma shape_hyper_stdtabnP d (P : 'P_d) : shape (hyper_stdtabn P) = P.
+Lemma shaped_hyper_stdtabnP d (P : 'P_d) : shape_deg (hyper_stdtabn P) = P.
+ +
+Section Coeffs.
+ +
+Variables d1 d2 : nat.
+Variables (P1 : 'P_d1) (P2 : 'P_d2).
+ +
+Definition LRtab_set (P : 'P_(d1 + d2)) :=
+  [set Q in (LRsupport (hyper_stdtabn P1) (hyper_stdtabn P2)) | (shape Q == P)].
+Definition LRtab_coeff (P : 'P_(d1 + d2)) := #|LRtab_set P|.
+ +
+Theorem LRtab_coeffP :
+  Schur P1 * Schur P2 = \sum_P (Schur P) *+ LRtab_coeff P.
+ +
+Lemma size_RSmapinv2_yam d disp (Typ : inhOrderType disp)
+      (tab : seq (seq Typ)) (T : stdtabn d) :
+  size (RSmapinv2 (tab, yam_of_stdtab T)) = d.
+ +
+
+ +
+

Invariance with respect to the choice of the Q-Tableau

+ +
+
+Section Bij_LRsupport.
+ +
+Section ChangeUT.
+ +
+Variable (U1 T1 : stdtabn d1) (U2 T2 : stdtabn d2).
+Hypothesis Hsh1 : shape U1 = shape T1.
+Hypothesis Hsh2 : shape U2 = shape T2.
+ +
+Section TakeDrop.
+ +
+Context {disp} {T : inhOrderType disp}.
+ +
+Lemma RStabE (w : seq T) : (RStab w).1 = (RS w).
+ +
+Definition changeUT T1 T2 (w : seq T) : seq T :=
+  (RSmapinv2 (RS (take d1 w), yam_of_stdtab T1)) ++
+  (RSmapinv2 (RS (drop d1 w), yam_of_stdtab T2)).
+ +
+Variable w : seq T.
+Hypothesis Htake : shape (RS (take d1 w)) = shape U1.
+Hypothesis Hdrop : shape (RS (drop d1 w)) = shape U2.
+ +
+Lemma changeUtakeP : is_RStabpair (RS (take d1 w), val U1).
+Lemma changeUdropP : is_RStabpair (RS (drop d1 w), val U2).
+Lemma changeTtakeP : is_RStabpair (RS (take d1 w), val T1).
+Lemma changeTdropP : is_RStabpair (RS (drop d1 w), val T2).
+Lemma toDepRSPair (u : seq T) d (t : stdtabn d) :
+  forall H : is_RStabpair (RS u, val t),
+  RSmapinv2 (RS u, yam_of_stdtab t) = RStabinv (RSTabPair H).
+ +
+Lemma plact_changeUT_take : take d1 (changeUT T1 T2 w) =Pl take d1 w.
+ +
+Lemma plact_changeUT_drop : drop d1 (changeUT T1 T2 w) =Pl drop d1 w.
+ +
+Lemma plact_changeUT : changeUT T1 T2 w =Pl w.
+ +
+End TakeDrop.
+ +
+Lemma changeUTK disp (T : inhOrderType disp) (w : seq T) :
+  (take d1 w) \in langQ U1 ->
+  (drop d1 w) \in langQ U2 ->
+  changeUT U1 U2 (changeUT T1 T2 w) = w.
+ +
+Section DefBij.
+ +
+Variable Q : stdtabn (d1 + d2).
+Hypothesis HTriple : pred_LRtriple U1 U2 Q.
+Let w := RSmapinv2 (yamtab (shape Q), yam_of_stdtab Q).
+ +
+Lemma RSpairyamQ : is_RSpair (yamtab (shape Q), yam_of_stdtab Q).
+ +
+Lemma bij_LRsupportP :
+  is_stdtab_of_n (d1 + d2)
+    (RStab (changeUT T1 T2 (RSmapinv2 (yamtab (shape Q), yam_of_stdtab Q)))).2.
+Definition bij_LRsupport := StdtabN bij_LRsupportP.
+ +
+Lemma take_drop_langQ :
+  ((take d1 w) \in langQ U1 /\ (drop d1 w) \in langQ U2).
+ +
+Lemma shape_bij_LRsupport : shape bij_LRsupport = shape Q.
+ +
+Lemma shape_takeRS : shape (RS (take d1 w)) = shape U1.
+ +
+Lemma shape_dropRS : shape (RS (drop d1 w)) = shape U2.
+ +
+Lemma predLR_bij_LRsupport : pred_LRtriple T1 T2 bij_LRsupport.
+ +
+End DefBij.
+ +
+Lemma card_LRtab_set_leq (P : seq nat) :
+  #|[set Q in (LRsupport U1 U2) | (shape Q == P)]| <=
+  #|[set Q in (LRsupport T1 T2) | (shape Q == P)]|.
+ +
+End ChangeUT.
+ +
+Lemma card_LRtab_set_shapeE P (U1 T1 : stdtabn d1) (U2 T2 : stdtabn d2) :
+  shape T1 = shape U1 -> shape T2 = shape U2 ->
+  #|[set Q in (LRsupport U1 U2) | (shape Q == P)]| =
+  #|[set Q in (LRsupport T1 T2) | (shape Q == P)]|.
+ +
+Theorem LRtab_coeff_shapeE (T1 : stdtabn d1) (T2 : stdtabn d2) P :
+  shape T1 = P1 -> shape T2 = P2 ->
+  LRtab_coeff P = #|[set Q in (LRsupport T1 T2) | (shape Q == P)]|.
+ +
+End Bij_LRsupport.
+ +
+End Coeffs.
+ +
+End FreeSchur.
+ +
+
+ +
+

Conjugating tableaux in the free LR rule

+ +
+
+Section Conj.
+ +
+Variables d1 d2 : nat.
+ +
+Lemma LRsupport_conj (T1 : stdtabn d1) (T2 : stdtabn d2):
+  LRsupport (conj_stdtabn T1) (conj_stdtabn T2) =
+            (@conj_stdtabn _) @: (LRsupport T1 T2).
+ +
+Theorem LRtab_coeff_conj (P1 : 'P_d1) (P2 : 'P_d2) (P : 'P_(d1 + d2)) :
+  LRtab_coeff P1 P2 P =
+  LRtab_coeff (conj_intpartn P1) (conj_intpartn P2) (conj_intpartn P).
+ +
+End Conj.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.implem.html b/combi/1.1.0/Combi.LRrule.implem.html new file mode 100644 index 00000000..8a82cbc2 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.implem.html @@ -0,0 +1,673 @@ + + + + + +Combi.LRrule.implem: A Coq implementation of the Littlewood-Richarson rule + + + + +
+ + + +
+ +

Library Combi.LRrule.implem: A Coq implementation of the Littlewood-Richarson rule

+ +
+
+ +
+ +
+
+
+ +
+

A Coq implementation of the Littlewood-Richarson rule

+ + +
+ +This file contains a Coq implementation of the Littlewood-Richardson rule. We +define the following main functions: +
    +
  • LRcoeff inner eval outer == the LR coefficient + +
    • +
  • LRyamtab_list inner eval outer == the list of LR tableaux of shape + outer/inner and evaluation eval + +
    • +
+ +
+ +The following lemma assert that LRcoeff agrees with LRyamtab_list + +
+ + Lemma LRcoeffP inner eval outer : + size (LRyamtab_list inner eval outer) = LRcoeff inner eval outer. + +
+ + +
+
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg.
+From mathcomp Require Import mpoly.
+ +
+Require Import tools combclass partition Yamanouchi ordtype tableau.
+Require Import skewtab Schur_mpoly freeSchur therule.
+ +
+Set Implicit Arguments.
+ +
+Open Scope N.
+ +
+Lemma sorted_is_part p :
+  is_part p -> sorted geq p.
+ +
+Lemma is_part_pad0 n p :
+  is_part p -> sorted geq (pad 0 n p).
+ +
+Lemma head_row_skew_yam innev shape l r :
+  is_part innev -> sorted <=%O (l :: r) ->
+  is_skew_yam innev shape (l :: r) ->
+  l <= head (size innev) r.
+ +
+
+ +
+

Recursive enumeration and counting function

+ +
+
+Section OutEval.
+ +
+#[local] Fixpoint add m n := if m is m'.+1 then add m' n.+1 else n.
+Lemma addE : add =2 addn.
+ +
+Let tsumn := foldl add 0.
+Lemma tsumnE : tsumn =1 sumn.
+ +
+Variable outev : seq nat.
+ +
+
+ +
+Possible choice of a new letter: + +
+ +The new letter l must satisty the following conditions: +
    +
  • mini < l < maxi + +
    • +
  • l is an addable corner of innev + +
    • +
  • adding l to innev still is inside outev + + +
    • +
+
+
+Definition one_letter_choices innev mini maxi :=
+  filter
+    (fun i => is_add_corner innev i && (nth 0 innev i < nth 0 outev i))
+    (iota mini ((minn (size innev) maxi).+1 - mini) ).
+ +
+
+ +
+The possible new rows which dominates the given row + +
+ +Returns a seqence of pairs (r, ev) where +
    +
  • r is the new row + +
    • +
  • ev is the new evaluation + + +
    • +
+
+
+Fixpoint yamtab_rows innev row :=
+  if row is r0 :: tlr then
+    flatten [seq
+               [seq (i :: res.1, incr_nth res.2 i) |
+                i <- one_letter_choices res.2 r0.+1 (head (size innev) res.1)
+               ] |
+             res <- yamtab_rows innev tlr ]
+  else [:: ([::], innev) ].
+ +
+
+ +
+The possible new rows which are at the bottom of the shape + +
+ +
    +
  • maxi is the maximum letter (typically the head of sol below) + +
    • +
  • sh is the shape of the expected rows + +
    • +
  • sol is the already known end of the row + +
    • +
+Returns a seqence of pairs (r, ev) where +
    +
  • r is the new row + +
    • +
  • ev is the new evaluation + + +
    • +
+
+
+Fixpoint yamtab_shift innev maxi sh sol :=
+  if sh is s.+1 then
+    flatten [seq
+               [seq (i :: res.1, incr_nth res.2 i) |
+                i <- one_letter_choices res.2 0 (head maxi res.1)
+               ] |
+             res <- yamtab_shift innev maxi s sol ]
+  else [:: (sol, innev) ].
+ +
+
+ +
+Recursive step of the enumeration functions + +
+ +Compute the list of skew tableaux +
    +
  • of evaluation outev-innev + +
    • +
  • whose row reading is a innev-skew Yamanouchi word + +
    • +
  • of shape outer/inner + +
    • +
  • which are legitimate skew tableaux if lied over row0 shifted by sh0 + +
    • +
+ +
+
+Fixpoint LRyamtab_list_rec innev inner outer sh0 row0 :=
+  if outer is out0 :: out then
+    let inn0 := head 0 inner in let inn := behead inner in
+    
+    let call_rec row := LRyamtab_list_rec row.2 inn out inn0 row.1 in
+    let rowres := yamtab_rows innev (take (out0 - sh0) row0) in
+    let rows :=
+        flatten [seq yamtab_shift res.2 (head (size innev) res.1)
+                     ((minn sh0 out0) - inn0) res.1
+                | res <- rowres ] in
+    flatten [seq [seq row.1 :: tab | tab <- call_rec row ] | row <- rows ]
+  else [:: [::]].
+ +
+
+ +
+Recursive step of the counting functions + +
+ +Compute the number of skew tableaux +
    +
  • of evaluation outev-innev + +
    • +
  • whose row reading is a innev-skew Yamanouchi word + +
    • +
  • of shape outer/inner + +
    • +
  • which are legitimate skew tableaux if lied over row0 shifted by sh0 + +
    • +
+ +
+
+Fixpoint LRyamtab_count_rec innev inner outer sh0 row0 :=
+  if outer is out0 :: out then
+    let inn0 := head 0 inner in let inn := behead inner in
+    
+    let call_rec row := LRyamtab_count_rec row.2 inn out inn0 row.1 in
+    let rowres := yamtab_rows innev (take (out0 - sh0) row0) in
+    tsumn [seq tsumn [seq call_rec row |
+                      row <- yamtab_shift res.2 (head (size innev) res.1)
+                          ((minn sh0 out0) - inn0) res.1 ]
+          | res <- rowres ]
+  else 1.
+ +
+Lemma size_LRyamtab_listE innev inner outer sh0 row0 :
+  size (LRyamtab_list_rec innev inner outer sh0 row0) =
+  LRyamtab_count_rec innev inner outer sh0 row0.
+ +
+
+ +
+Basic lemmas +
+
+Lemma yamtab_shift_drop innev maxi sh y :
+  forall res shape, (res, shape) \in yamtab_shift innev maxi sh y ->
+  drop sh res = y.
+ +
+
+ +
+

LRyamtab_list_rec returns only Yamanouchi words

+ +
+
+Lemma one_letter_choicesP innev mini maxi :
+  forall i, i \in one_letter_choices innev mini maxi ->
+  is_skew_yam innev (incr_nth innev i) [:: i].
+ +
+Lemma yamtab_rowsP innev row :
+  forall res shape, (res, shape) \in yamtab_rows innev row ->
+  is_skew_yam innev shape res.
+ +
+Lemma yamtab_shiftP innev maxi sh row shrow :
+  is_skew_yam innev shrow row ->
+  forall res shape, (res, shape) \in yamtab_shift shrow maxi sh row ->
+  is_skew_yam innev shape res.
+ +
+Lemma LRyamtab_list_recP innev inner outer sh0 row0 y :
+  is_yam_of_eval innev y ->
+  forall res, res \in LRyamtab_list_rec innev inner outer sh0 row0 ->
+  is_yam (to_word res ++ y).
+ +
+
+ +
+

LRyamtab_list_rec returns words whose evaluation in included in outev

+ +
+
+Lemma one_letter_included innev mini maxi :
+  included innev outev ->
+  forall i, i \in one_letter_choices innev mini maxi ->
+                  included (incr_nth innev i) outev.
+ +
+Lemma yamtab_rows_included innev row :
+  included innev outev ->
+  forall res shape, (res, shape) \in yamtab_rows innev row ->
+  included shape outev.
+ +
+Lemma yamtab_shift_included innev maxi sh y :
+  included innev outev ->
+  forall res shape, (res, shape) \in yamtab_shift innev maxi sh y ->
+  included shape outev.
+ +
+Lemma LRyamtab_list_included innev inner outer sh0 row0 y :
+  included innev outev ->
+  is_yam_of_eval innev y ->
+  forall res, res \in LRyamtab_list_rec innev inner outer sh0 row0 ->
+  included (evalseq (to_word res ++ y)) outev.
+ +
+
+ +
+

LRyamtab_list_rec returns fillings of skew shape inner/outer

+ +
+
+Lemma yamtab_rows_size innev row :
+  forall res shape, (res, shape) \in yamtab_rows innev row ->
+  size res = size row.
+ +
+Lemma yamtab_shift_size innev maxi sh y :
+  forall res shape, (res, shape) \in yamtab_shift maxi innev sh y ->
+  size res = sh + size y.
+ +
+Lemma LRyamtab_list_pad0 innev inner outer sh0 row0 :
+      LRyamtab_list_rec innev inner outer sh0 row0 =
+      LRyamtab_list_rec innev (pad 0 (size outer) inner) outer sh0 row0.
+ +
+Lemma LRyamtab_list_size innev inner outer sh0 row0 :
+  forall res, res \in LRyamtab_list_rec innev inner outer sh0 row0 ->
+  size res = size outer.
+ +
+
+ +
+inner is padded with 0 +
+
+Lemma LRyamtab_list_shape0 innev inner outer sh0 row0 :
+  sorted geq (sh0 :: inner) -> is_part (sh0 + size row0 :: outer) ->
+  included inner outer -> size inner = size outer ->
+  forall res, res \in LRyamtab_list_rec innev inner outer sh0 row0 ->
+  shape res = outer / inner.
+ +
+
+ +
+

LRyamtab_list_rec returns legitimate skew_tableaux

+ +
+
+Lemma yamtab_rows_dominate innev row :
+  forall res shape, (res, shape) \in yamtab_rows innev row ->
+  dominate res row.
+ +
+Lemma yamtab_shift_dominate innev maxi sh row y :
+  dominate y row ->
+  forall res shape, (res, shape) \in yamtab_shift innev maxi sh y ->
+  skew_dominate sh res row.
+ +
+Lemma yamtab_rows_is_row innev row :
+  forall res shape, (res, shape) \in yamtab_rows innev row ->
+  is_row res.
+ +
+Lemma yamtab_shift_is_row innev maxi sh y :
+  is_row y ->
+  forall res shape, (res, shape) \in yamtab_shift innev maxi sh y ->
+  is_row res.
+ +
+
+ +
+inner is padded with 0 +
+
+Lemma LRyamtab_list_skew_tableau0 innev inner outer sh0 row0 :
+  sorted geq (sh0 :: inner) -> is_part (sh0 + size row0 :: outer) ->
+  included inner outer -> size inner = size outer ->
+  is_row row0 ->
+  forall res, res \in LRyamtab_list_rec innev inner outer sh0 row0 ->
+  is_skew_tableau (sh0 :: inner) (row0 :: res).
+ +
+
+ +
+

Mutiplicities are all one

+ +
+
+Lemma choose_one_countE shr innev shape mini maxi row l :
+  is_part innev ->
+  is_skew_yam innev shr row ->
+  is_skew_yam innev shape (l :: row) ->
+  included shape outev ->
+  mini <= l <= maxi ->
+  is_row (l :: row) ->
+  (count_mem l) (one_letter_choices shr mini maxi) = 1.
+ +
+Lemma yamtab_rows_countE innev shape row base :
+  is_part innev ->
+  size row = size base ->
+  dominate row base ->
+  is_row row ->
+  is_skew_yam innev shape row ->
+  included shape outev ->
+  count (preim (fst (B:=seq nat)) (pred1 row))
+        (yamtab_rows innev base) = 1.
+ +
+Lemma yamtab_shift_countE inn0 innev shape sh row sol :
+  is_part inn0 ->
+  is_row (row ++ sol) ->
+  size row = sh ->
+  is_skew_yam inn0 innev sol ->
+  is_skew_yam innev shape row ->
+  included shape outev ->
+  count (preim (fst (B:=seq nat)) (pred1 (row ++ sol)))
+        (yamtab_shift innev (head (size inn0) sol) sh sol) = 1.
+ +
+Lemma LRyamtab_list_countE innev inner sh0 row0 yamtab :
+  is_part innev ->
+  sorted geq (sh0 :: inner) ->
+  is_part (sh0 + size row0 :: (outer_shape inner (shape yamtab))) ->
+  size inner = size yamtab ->
+  is_skew_tableau (sh0 :: inner) (row0 :: yamtab) ->
+  is_skew_yam innev outev (to_word yamtab) ->
+  count_mem yamtab
+    (LRyamtab_list_rec innev inner (outer_shape inner (shape yamtab)) sh0 row0) = 1.
+ +
+End OutEval.
+ +
+
+ +
+

The main functions

+ +
+
+Definition LRyamtab_list inner eval outer :=
+  LRyamtab_list_rec eval [::] inner outer (head 1 outer) [::].
+Definition LRcoeff inner eval outer :=
+  LRyamtab_count_rec eval [::] inner outer (head 1 outer) [::].
+ +
+Lemma LRcoeffP inner eval outer :
+  size (LRyamtab_list inner eval outer) = LRcoeff inner eval outer.
+ +
+Lemma LRyamtab_yam inner eval outer tab:
+  tab \in (LRyamtab_list inner eval outer) -> is_yam (to_word tab).
+ +
+Lemma LRyamtab_included inner eval outer tab:
+  tab \in (LRyamtab_list inner eval outer) -> included (evalseq (to_word tab)) eval.
+ +
+Lemma LRyamtab_shape inner eval outer tab :
+  is_part inner -> is_part outer -> included inner outer ->
+  tab \in (LRyamtab_list inner eval outer) -> shape tab = outer / inner.
+ +
+Lemma LRyamtab_skew_tableau inner eval outer tab :
+  is_part inner -> is_part outer -> included inner outer ->
+  tab \in (LRyamtab_list inner eval outer) -> is_skew_tableau inner tab.
+ +
+Lemma LRyamtab_eval inner eval outer tab:
+  is_part inner -> is_part outer -> included inner outer ->
+  is_part eval -> sumn eval = sumn (outer / inner) ->
+  tab \in (LRyamtab_list inner eval outer) -> evalseq (to_word tab) = eval.
+ +
+Lemma count_mem_LRyamtab_list inner eval outer yamtab :
+  is_part inner -> is_part outer -> included inner outer ->
+  is_skew_tableau inner yamtab -> shape yamtab = outer / inner ->
+  is_yam_of_eval eval (to_word yamtab) ->
+  count_mem yamtab (LRyamtab_list inner eval outer) = 1.
+ +
+
+ +
+

The specification of the enumeration function

+ +
+
+Section PackedSpec.
+ +
+Variable inner eval outer : seq nat.
+ +
+Record inputSpec :=
+  InputSpec {
+      inner_part : is_part inner;
+      outer_part : is_part outer;
+      incl : included inner outer;
+      eval_part : is_part eval;
+      sumn_eq : sumn eval = sumn (outer / inner)
+    }.
+ +
+Record outputSpec (tab : seq (seq nat)) :=
+  OutputSpec {
+      skew : is_skew_tableau inner tab;
+      shaps_eq : shape tab = outer / inner;
+      yam_to_word : is_yam (to_word tab);
+      eval_eq : evalseq (to_word tab) = eval
+    }.
+ +
+Lemma outputSpecP tab :
+  inputSpec -> tab \in (LRyamtab_list inner eval outer) -> outputSpec tab.
+ +
+Lemma outputSpec_count_mem tab :
+  inputSpec -> outputSpec tab -> count_mem tab (LRyamtab_list inner eval outer) = 1.
+ +
+End PackedSpec.
+ +
+
+ +
+

The specification on sigma types

+ +
+
+Section Spec.
+ +
+Variables d1 d2 : nat.
+Variables (P1 : 'P_d1) (P2 : 'P_d2).
+Variable P : 'P_(d1 + d2).
+Hypothesis Hincl : included P1 P.
+ +
+Lemma LRyamtabP tab :
+  tab \in (LRyamtab_list P1 P2 P) -> is_yam_of_eval P2 (to_word tab).
+ +
+Lemma LRyamtab_all :
+  all (is_yam_of_eval P2) (map to_word (LRyamtab_list P1 P2 P)).
+Definition LRyam_list :=
+  subType_seq (yameval P2) (map to_word (LRyamtab_list P1 P2 P)).
+ +
+Lemma LRyamtab_spec_recip yam :
+  yam \in LRyam_set P1 P2 P ->
+  count_mem (val yam) (map to_word (LRyamtab_list P1 P2 P)) = 1.
+ +
+Lemma LRyam_spec_recip yam :
+  yam \in LRyam_set P1 P2 P -> count_mem yam LRyam_list = 1.
+ +
+Theorem LRcoeffE : LRyam_coeff P1 P2 P = LRcoeff P1 P2 P.
+ +
+End Spec.
+ +
+From mathcomp Require Import ssralg.
+ +
+
+ +
+

Back to the rule

+ +
+
+Section LR.
+ +
+Variables d1 d2 : nat.
+Variables (P1 : 'P_d1) (P2 : 'P_d2).
+ +
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+Variable (n0 : nat) (R : comNzRingType).
+#[local] Notation n := (n0.+1).
+Notation Schur p := (Schur n0 R p).
+ +
+Theorem LRtab_coeffP :
+  Schur P1 * Schur P2 =
+  \sum_(P : 'P_(d1 + d2) | included P1 P) Schur P *+ LRcoeff P1 P2 P.
+ +
+End LR.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.plactic.html b/combi/1.1.0/Combi.LRrule.plactic.html new file mode 100644 index 00000000..72b2530d --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.plactic.html @@ -0,0 +1,639 @@ + + + + + +Combi.LRrule.plactic: The plactic monoid + + + + +
+ + + +
+ +

Library Combi.LRrule.plactic: The plactic monoid

+ +
+
+ +
+ +
+
+
+ +
+

The plactic monoid

+ + +
+ +Knuth rewriting rules: + +
+ +
    +
  • plact1 == rewriting rule acb -> cab if a <= b < c + +
    • +
  • plact1i == rewriting rule cab -> acb if a <= b < c + +
    • +
  • plact2 == rewriting rule bac -> bca if a < b <= c + +
    • +
  • plact2i == rewriting rule bca -> bac if a < b <= c + +
    • +
  • plactrule == the union of the four preceding rewriting rules + +
    • +
  • plactcongr == the plactic congruence + +
    • +
  • a =Pl b == a and b are plactic equivalent + +
    • +
+ +
+ +Reverse and dual alphabet: + +
+ +
    +
  • revdual s == rev s : seq Dual considered on the dual alphabet + +
    • +
  • from_revdual ds == rev ds where ds : seq Dual + +
    • +
+ +
+ +The two main result in this section are plact_dualE and plact_from_dualE +which assert that plactic equivalence is conserved by revdual and from_revdual. + +
+ +Plactic equivalence and Robinson-Schensted: + +
+ +Main results: +
    +
  • Theorem congr_RS w : w =Pl (to_word (RS w)). + +
    • +
  • Corollary Sch_plact u v : RS u == RS v -> u =Pl v. + +
    • +
+ +
+ + +
+ +Restriction to interval: + +
+ +Thanks to bisimulation we show (Theorem rembig_RS) that the last big letter +goes on the border of the tableau by Robinson-Schensted. As a consequence +filtering large or small letters preserve plactic congruence. These are Lemmas +plactic_filter_geqX, plactic_filter_gtnX, plactic_filter_leqX and +plactic_filter_ltnX + +
+ +Plactic congruence and increasing maps: + +
+ +We finally shows that plactic congruence is preserved by increasing maps: +Lemma plact_map_in_incr. + +
+ + +
+
+ +
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm.
+Require Import tools partition ordtype tableau stdtab Schensted congr.
+ +
+Set Implicit Arguments.
+ +
+Open Scope bool.
+ +
+Import Order.Theory.
+ +
+
+ +
+

Definition of the Plactic monoid

+ +
+
+Section Defs.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w : word.
+ +
+Definition plact1 :=
+  fun s => match s return seq word with
+             | [:: a; c; b] => if (a <= b < c)%O then [:: [:: c; a; b]] else [::]
+             | _ => [::]
+           end.
+ +
+Definition plact1i :=
+  fun s => match s return seq word with
+             | [:: c; a; b] => if (a <= b < c)%O then [:: [:: a; c; b]] else [::]
+             | _ => [::]
+           end.
+ +
+Definition plact2 :=
+  fun s => match s return seq word with
+             | [:: b; a; c] => if (a < b <= c)%O then [:: [:: b; c; a]] else [::]
+             | _ => [::]
+           end.
+ +
+Definition plact2i :=
+  fun s => match s return seq word with
+             | [:: b; c; a] => if (a < b <= c)%O then [:: [:: b; a; c]] else [::]
+             | _ => [::]
+           end.
+ +
+Lemma plact1P u v :
+  reflect (exists a b c,
+             [/\ (a <= b < c)%O, u = [:: a; c; b] & v = [:: c; a; b] ] )
+          (v \in plact1 u).
+ +
+Lemma plact1iP u v :
+  reflect (exists a b c,
+             [/\ (a <= b < c)%O, v = [:: a; c; b] & u = [:: c; a; b] ] )
+          (v \in plact1i u).
+ +
+Lemma plact2P u v :
+  reflect (exists a b c,
+             [/\ (a < b <= c)%O, u = [:: b; a; c] & v = [:: b; c; a] ] )
+          (v \in plact2 u).
+ +
+Lemma plact2iP u v :
+  reflect (exists a b c,
+             [/\ (a < b <= c)%O, v = [:: b; a; c] & u = [:: b; c; a] ] )
+          (v \in plact2i u).
+ +
+Lemma plact1I u v : u \in plact1 v <-> v \in plact1i u.
+ +
+Lemma plact2I u v : u \in plact2 v <-> v \in plact2i u.
+ +
+Definition plactrule := [fun s => plact1 s ++ plact1i s ++ plact2 s ++ plact2i s].
+ +
+Lemma plactruleP u v :
+  reflect ([\/ v \in plact1 u, v \in plact1i u, v \in plact2 u | v \in plact2i u ])
+          (v \in plactrule u).
+ +
+Lemma plactrule_sym u v : v \in (plactrule u) -> u \in (plactrule v).
+ +
+Lemma plact1_homog : forall u : word, all (perm_eq u) (plact1 u).
+ +
+Lemma plact1i_homog : forall u : word, all (perm_eq u) (plact1i u).
+ +
+Lemma plact2_homog : forall u : word, all (perm_eq u) (plact2 u).
+ +
+Lemma plact2i_homog : forall u : word, all (perm_eq u) (plact2i u).
+ +
+Lemma plactrule_homog : forall u : word, all (perm_eq u) (plactrule u).
+ +
+Definition plactcongr := gencongr_multhom plactrule_homog.
+ +
+Lemma plact_equiv : equivalence_rel plactcongr.
+ +
+Lemma plact_refl : reflexive plactcongr.
+ +
+Lemma plact_sym : symmetric plactcongr.
+ +
+Lemma plact_ltrans : left_transitive plactcongr.
+ +
+Lemma plact_trans : transitive plactcongr.
+ +
+Lemma plact_is_congr : congruence_rel plactcongr.
+ +
+Definition plact_cons := congr_cons plact_is_congr.
+Definition plact_rcons := congr_rcons plact_is_congr.
+Definition plact_catl := congr_catl plact_is_congr.
+Definition plact_catr := congr_catr plact_is_congr.
+Definition plact_cat := congr_cat plact_is_congr plact_equiv.
+ +
+Lemma plact_homog u v : plactcongr u v -> perm_eq u v.
+ +
+Lemma size_plact u v : plactcongr u v -> size u = size v.
+ +
+End Defs.
+ +
+Notation "a =Pl b" := (plactcongr a b) (at level 70).
+#[export] Hint Resolve plact_refl : core.
+ +
+Section RowsAndCols.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+Implicit Type u v w : word.
+ +
+Lemma plact_row u v : is_row u -> u =Pl v -> u = v.
+ +
+Lemma sorted_center u v w :
+  sorted >%O (u ++ v ++ w) -> sorted >%O v.
+ +
+Lemma plact_col u v : sorted >%O u -> u =Pl v -> u = v.
+ +
+End RowsAndCols.
+ +
+
+ +
+

Plactic equivalence and reversal

+ +
+
+Section Rev.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+Implicit Type u v w : word.
+ +
+Lemma plact_uniq_rev u v : uniq u -> u =Pl v -> rev u =Pl rev v.
+ +
+Lemma plact_uniq_revE u v : uniq u -> (u =Pl v) = (rev u =Pl rev v).
+ +
+End Rev.
+ +
+
+ +
+

Plactic equivalence and order duality

+ +
+
+Section DualRule.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+Implicit Type u v w : word.
+ +
+Definition revdual := [fun s : seq Alph => rev s : seq Alph^d].
+Definition from_revdual := [fun s : seq Alph^d => (rev s) : seq Alph].
+ +
+Lemma revdualK : cancel revdual from_revdual.
+ +
+Lemma from_revdualK : cancel from_revdual revdual.
+ +
+Lemma size_revdual u : size u = size (revdual u).
+ +
+Lemma plact2dual u v :
+  u \in plact2 v = (revdual u \in @plact1 _ Alph^d (revdual v)).
+ +
+Lemma plact1dual u v :
+  u \in plact1 v = (revdual u \in @plact2 _ Alph^d (revdual v)).
+ +
+Lemma plact1idual u v : u \in plact1i v = (revdual u \in plact2i (revdual v)).
+ +
+Lemma plact2idual u v : u \in plact2i v = (revdual u \in plact1i (revdual v)).
+ +
+End DualRule.
+ +
+Arguments revdual {disp Alph}.
+Arguments from_revdual {disp Alph}.
+ +
+Section PlactDual.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+Implicit Type u v w : word.
+ +
+Theorem plact_revdual u v : u =Pl v -> revdual u =Pl revdual v.
+ +
+Theorem plact_from_revdual (u v : seq Alph^d) :
+  u =Pl v -> from_revdual u =Pl from_revdual v.
+ +
+Theorem plact_dualE u v : u =Pl v <-> revdual u =Pl revdual v.
+ +
+Theorem plact_from_dualE (u v : seq Alph^d) :
+  u =Pl v <-> from_revdual u =Pl from_revdual v.
+ +
+End PlactDual.
+ +
+
+ +
+

Plactic monoid and Robinson-Schensted map

+ +
+
+Section RSToPlactic.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Lemma rcons_rcons w a b : rcons (rcons w a) b = w ++ [:: a; b].
+ +
+Lemma congr_row_1 r b l :
+  is_row (rcons r l) -> (l < b)%O -> rcons (rcons r b) l =Pl b :: rcons r l.
+ +
+Lemma congr_row_2 a r l :
+  is_row (a :: r) -> (l < a)%O -> a :: rcons r l =Pl a :: l :: r.
+ +
+Lemma set_nth_LxR L c R l pos :
+  (size L) = pos -> set_nth l (L ++ c :: R) pos l = L ++ l :: R.
+ +
+Lemma congr_bump r l :
+  r != [::] -> is_row r -> bump r l -> r ++ [:: l] =Pl [:: bumped r l] ++ ins r l.
+ +
+Theorem congr_RS w : w =Pl (to_word (RS w)).
+ +
+Corollary Sch_plact u v : RS u == RS v -> u =Pl v .
+ +
+End RSToPlactic.
+ +
+
+ +
+

Removing the last big letter and plactic congruence

+ +
+
+Section RemoveBig.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Lemma rembig_plact1 u v : u \in (plact1 v) -> rembig u = rembig v.
+ +
+Lemma rembig_plact1i u v : u \in (plact1i v) -> rembig u = rembig v.
+ +
+Lemma rembig_plact2 u v : u \in (plact2 v) -> rembig u = rembig v.
+ +
+Lemma rembig_plact2i u v : u \in (plact2i v) -> rembig u = rembig v.
+ +
+Lemma rembig_plact u v : u \in (plactrule _ v) -> rembig u = rembig v.
+ +
+Theorem rembig_plactcongr u v : u =Pl v -> (rembig u) =Pl (rembig v).
+ +
+Lemma inspos_rcons l r b : (l < b)%O -> inspos r l = inspos (rcons r b) l.
+ +
+Lemma bump_bumprow_rconsE l r b :
+  is_row (rcons r b) -> (l < b)%O -> bump r l ->
+  let: (lres, rres) := bumprow r l in
+  bumprow (rcons r b) l = (lres, rcons rres b).
+ +
+Lemma nbump_bumprow_rconsE l r b :
+  is_row (rcons r b) -> (l < b)%O -> ~~bump r l ->
+  let: (lres, rres) := bumprow r l in bumprow (rcons r b) l = (Some b, rres).
+ +
+Lemma allLeq_is_row_rcons w b :
+  allLeq w b -> forall row, row \in RS w -> is_row (rcons row b).
+ +
+Lemma last_ins_lt r l b : (l < b -> last b r < b -> last b (ins r l) < b)%O.
+ +
+Lemma bumped_lt r b l : is_row r -> (l < b -> last b r < b -> bumped r l < b)%O.
+ +
+Lemma bisimul_instab t l b lb :
+  is_tableau t -> (l < b)%O ->
+  (forall row, row \in t -> is_row (rcons row b)) ->
+  (forall j : nat, j < lb -> (last b (nth [::] t j) < b)%O) ->
+  let tres := (append_nth t b lb) in
+  instab tres l = append_nth (instab t l) b (last_big (instab tres l) b).
+ +
+Theorem rembig_RS_last_big a v :
+  RS (a :: v) = append_nth (RS (rembig (a :: v)))
+                           (maxL a v)
+                           (last_big (RS (a :: v)) (maxL a v)).
+ +
+Theorem rembig_RS a v :
+  {i | RS (a :: v) = append_nth (RS (rembig (a :: v))) (maxL a v) i}.
+ +
+End RemoveBig.
+ +
+Section RestrIntervSmall.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Lemma plact1_ge L u v1 w v2 :
+   v2 \in plact1 v1 ->
+   [seq x <- u ++ v1 ++ w | (L >= x)%O] =Pl [seq x <- u ++ v2 ++ w | (L >= x)%O].
+ +
+Lemma plact2_ge L u v1 w v2 :
+   v2 \in plact2 v1 ->
+   [seq x <- u ++ v1 ++ w | (L >= x)%O] =Pl [seq x <- u ++ v2 ++ w | (L >= x)%O].
+ +
+Lemma plactic_filter_ge L u v :
+  u =Pl v -> [seq x <- u | (L >= x)%O] =Pl [seq x <- v | (L >= x)%O].
+ +
+Lemma plactic_filter_gt L u v :
+  u =Pl v -> [seq x <- u | (L > x)%O] =Pl [seq x <- v | (L > x)%O].
+ +
+End RestrIntervSmall.
+ +
+Section RestrIntervBig.
+ +
+Context disp (Alph : inhOrderType disp).
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w r : word.
+ +
+Variable L : Alph.
+Notation leL := (@Order.le _ Alph L).
+Notation geL := (@Order.ge _ Alph^d L).
+Notation ltL := (@Order.lt _ Alph L).
+Notation gtL := (@Order.gt _ Alph^d L).
+ +
+Lemma leL_geLdualE u :
+  filter leL u = from_revdual (filter geL (revdual u)).
+ +
+Lemma ltL_gtLdualE u :
+  filter ltL u = from_revdual (filter gtL (revdual u)).
+ +
+Lemma plactic_filter_le u v : u =Pl v -> filter leL u =Pl filter leL v.
+ +
+Lemma plactic_filter_lt u v : u =Pl v -> filter ltL u =Pl filter ltL v.
+ +
+End RestrIntervBig.
+ +
+
+ +
+

Plactic congruence and increasing maps

+ +
+
+Section IncrMap.
+ +
+Context disp1 disp2 (T1 : inhOrderType disp1) (T2 : inhOrderType disp2).
+Variable F : T1 -> T2.
+Variable u v : seq T1.
+ +
+Hypothesis Hincr : {in u &, {homo F : x y / (x < y)%O}}.
+ +
+Lemma subset_abc l a b c r :
+  {subset l ++ [:: a; b; c] ++ r <= u} -> [/\ a \in u, b \in u & c \in u].
+ +
+Lemma plact_map_in_incr : u =Pl v -> (map F u) =Pl (map F v).
+ +
+End IncrMap.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.shuffle.html b/combi/1.1.0/Combi.LRrule.shuffle.html new file mode 100644 index 00000000..84367c4b --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.shuffle.html @@ -0,0 +1,558 @@ + + + + + +Combi.LRrule.shuffle: Shuffle and shifted shuffle + + + + +
+ + + +
+ +

Library Combi.LRrule.shuffle: Shuffle and shifted shuffle

+ +
+
+ +
+ +
+
+
+ +
+

Shuffle, shifted shuffle and plactic classes

+ + +
+ +This file contains the core of Schützenberger proof of the Littlewood-Richardson +rule. Namely, the theorem saying that the shifted shuffle of two plactic classes +is a union of plactic classes. The set of those class is described by the +pred_LRtriple predicate. + +
+ +However, to prove the rule, instead of taking the road of plactic Schur +function, we follow Duchamp-Hivert-Thibon using free Schur functions. We +nevertheless prove Schützenberger theorem Schutzenberger_shuffle_plact but +we don't use it. + +
+ +Here are the new notions: + +
+ +
    +
  • shuffle u v == the sequence of shuffling of u and v + +
    • +
  • shiftn n u == add n to all the entries of the sequence u + +
    • +
  • sfiltergtn n v == filter all the entries smaller than n in v + +
    • +
  • sfilterleq n v == filter and substract n to the entries of v larger + than n in v + +
    + + +
    • +
  • shsh u v == the shifted shuffle of u and v + +
    + + +
    • +
  • langQ t == the set of words whose Q-symbol is t + +
    • +
+ +
+ +The Littlewood-Richardson triple: + +
+ +
    +
  • LRtriple t1 t2 t == there exists three words p1 p2 p of respective + P-symbol t1 t2 t such that p appears in the shifted shuffle + of p1 and p2. Otherwise said, the row reading of t appears + in the shifted shuffle of the plactic classes of t1 and t2. + This is an inductive proposition. + +
    • +
  • pred_LRtriple t1 t2 t == a predicate deciding LRtriple t1 t2 t + +
    • +
  • pred_LRtriple_fast t1 t2 t == a faster predicate deciding LRtriple t1 t2 t + only computing the shifted shuffle of (the row reading of) t1 with + the plactic class of t2 + +
    • +
+ +
+ +The main statement is LRtriple_cat_equiv which stats the equivalence, +for two standard tableau t1 and t2 and two words u1 u2 of the right length, +between the two following statements: +
    +
  • u1 \in langQ t1 /\ u2 \in langQ t2 + +
    • +
  • exists t, LRtriple t1 t2 t /\ u1 ++ u2 \in langQ t + +
    • +
+ +
+ +Then to get a first version of the LR rule, we will need to do the necessary +abgebraic translation of the rule and recode the triple by some standard +skew tableaux. + +
+
+ +
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype fintype choice.
+From mathcomp Require Import order seq tuple finfun finset perm binomial bigop.
+Require Import tools vectNK subseq partition Yamanouchi ordtype std tableau stdtab.
+Require Import Schensted plactic Greene_inv stdplact.
+ +
+Set Implicit Arguments.
+ +
+Open Scope bool.
+ +
+Import Order.TTheory.
+ +
+
+ +
+

The shuffle of two sequences

+ +
+
+Section Defs.
+ +
+Variable Alph : eqType.
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w : word.
+ +
+
+ +
+A well founded recursive definition of shuffle (a :: u) v +
+
+Fixpoint shuffle_from_rec a u shuffu' v {struct v} :=
+  if v is b :: v' then
+    [seq a :: w | w <- shuffu' v] ++
+    [seq b :: w | w <- shuffle_from_rec a u shuffu' v']
+  else [:: (a :: u)].
+ +
+Fixpoint shuffle u v {struct u} :=
+  if u is a :: u' then
+    shuffle_from_rec a u' (shuffle u') v
+  else [:: v].
+ +
+Lemma shuffle_nil u : shuffle u [::] = [:: u].
+ +
+Lemma size_shuffle u v :
+  size (shuffle u v) = binomial ((size u) + (size v)) (size u).
+ +
+Lemma shuffleC u v : perm_eq (shuffle u v) (shuffle v u).
+ +
+Lemma perm_shuffle u v : all (perm_eq (u ++ v)) (shuffle u v).
+ +
+Lemma all_size_shuffle u v :
+  all (fun s => size s == size u + size v) (shuffle u v).
+ +
+
+ +
+

Shuffling two disjoint sequences

+ +
+
+Lemma all_in_shufflel u v :
+  predI (mem u) (mem v) =i pred0 ->
+  all (fun s => filter (mem u) s == u) (shuffle u v).
+ +
+Lemma all_in_shuffler u v :
+  predI (mem u) (mem v) =i pred0 ->
+  all (fun s => filter (mem v) s == v) (shuffle u v).
+ +
+Lemma mem_shuffle_predU u v s Pu Pv:
+  predI Pu Pv =i pred0 ->
+  filter Pu s = u -> filter Pv s = v -> size s = size u + size v ->
+  filter (predU Pu Pv) s = s.
+ +
+Lemma mem_shuffle_pred u v s Pu Pv:
+  predI Pu Pv =i pred0 ->
+  filter Pu s = u -> filter Pv s = v -> size s = size u + size v ->
+  s \in shuffle u v.
+ +
+Lemma mem_shuffle u v s :
+  predI (mem u) (mem v) =i pred0 ->
+  [&& filter (mem u) s == u,
+      filter (mem v) s == v &
+      (size s == size u + size v)] = (s \in shuffle u v).
+ +
+End Defs.
+ +
+
+ +
+

The shifted shuffle of two standard words

+ +
+
+Section ShiftedShuffle.
+ +
+Implicit Type u v w : seq nat.
+ +
+Definition shiftn n := map (addn n).
+Definition sfiltergtn n := [fun v => filter (gtn n) v].
+Definition sfilterleq n := [fun v => map (subn^~ n) (filter (leq n) v)].
+ +
+Lemma shiftuK n : cancel (shiftn n) (map (subn^~ n)).
+ +
+Lemma sfilterleqK n v : shiftn n (sfilterleq n v) = [seq x <- v | n <= x].
+ +
+Lemma sfilterleqE n u v : u = sfilterleq n v <-> shiftn n u = filter (leq n) v.
+ +
+Lemma mem_sfilterleqK n v i : i \in (sfilterleq n v) = (i + n \in v).
+ +
+Lemma perm_shiftn_std n s :
+  is_std s -> perm_eq (shiftn n s) (iota n (size s)).
+ +
+Definition shsh u v := shuffle u (shiftn (size u) v).
+ +
+Lemma std_shsh u v : is_std u -> is_std v -> all is_std (shsh u v).
+ +
+Lemma pred0_std u v : is_std u -> [predI u & shiftn (size u) v] =i pred0.
+ +
+Lemma shsh_sfiltergtn p u v :
+  is_std u -> p \in shsh u v -> u = sfiltergtn (size u) p.
+ +
+Lemma shsh_sfilterleq p u v :
+  is_std u -> p \in shsh u v -> v = sfilterleq (size u) p.
+ +
+Lemma mem_shsh p u v :
+  is_std u ->
+  (p \in shsh u v) =
+  ((sfiltergtn (size u) p == u) && (sfilterleq (size u) p == v)).
+ +
+Lemma shift_plactcongr n u v : (u =Pl v) = (shiftn n u =Pl shiftn n v).
+ +
+
+ +
+

Schützenberger theorem

+ +
+
+Theorem Schutzenberger_shuffle_plact u1 v1 w1 w2 :
+  is_std u1 -> is_std v1 -> w1 \in shsh u1 v1 ->
+  (w1 =Pl w2) ->
+  exists u2 v2, [/\ u1 =Pl u2, v1 =Pl v2 & w2 \in shsh u2 v2].
+ +
+End ShiftedShuffle.
+ +
+
+ +
+

Shifted shuffle and inverse standardized

+ +
+
+Section LRTriple.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Let word := seq Alph.
+ +
+Implicit Type a b c : Alph.
+Implicit Type u v w : word.
+Implicit Type t : seq (seq nat).
+ +
+Lemma perm_sfiltergtn p n :
+  is_std p -> perm_eq (sfiltergtn n p) (iota 0 (minn n (size p))).
+ +
+Lemma perm_sfilterleq p n :
+  is_std p -> perm_eq (sfilterleq n p) (iota 0 ((size p) - n)).
+ +
+Lemma size_sfiltergtn p n :
+  is_std p -> size (sfiltergtn n p) = minn n (size p).
+Lemma size_sfilterleq p n :
+  is_std p -> size (sfilterleq n p) = (size p) - n.
+ +
+Lemma sfiltergtn_is_std p n : is_std p -> is_std (sfiltergtn n p).
+Lemma sfilterleq_is_std p n : is_std p -> is_std (sfilterleq n p).
+ +
+Lemma size_sfiltergtn_cat u v :
+  size (sfiltergtn (size u) (invstd (std (u ++ v)))) = size u.
+Lemma size_sfilterleq_cat u v :
+  size (sfilterleq (size u) (invstd (std (u ++ v)))) = size v.
+ +
+Lemma index_leq_filter s (P : pred nat) i j :
+  P i -> P j ->
+  (index i (filter P s)) <= (index j (filter P s)) = (index i s <= index j s).
+ +
+Lemma index_sfilterleq n s i :
+  index i (sfilterleq n s) = index (i + n) (filter (leq n) s).
+ +
+Lemma index_invstd l i :
+  is_std l -> i < size l -> index i (invstd l) = nth inh l i.
+ +
+Lemma invstd_catgtn u v :
+  invstd (std u) = sfiltergtn (size u) (invstd (std (u ++ v))).
+ +
+Lemma invstd_catleq u v :
+  invstd (std v) = sfilterleq (size u) (invstd (std (u ++ v))).
+ +
+
+ +
+This is essentially the product rule of FQSym +
+
+Theorem invstd_cat_in_shsh u v :
+  invstd (std (u ++ v)) \in shsh (invstd (std u)) (invstd (std v)).
+ +
+
+ +
+Free Schur functions as predicates +
+
+Definition langQ t := [pred w : word | (RStabmap w).2 == t].
+ +
+Lemma langQE u t : (u \in langQ t) = (RS (invstd (std u)) == t).
+ +
+Lemma size_langQ t u : u \in langQ t -> size u = size_tab t.
+ +
+
+ +
+

Littlewood-Richardson-Schützenberger triple

+ +
+
+Inductive LRtriple t1 t2 t : Prop :=
+  LRTriple :
+    forall p1 p2 p, RS p1 = t1 -> RS p2 = t2 -> RS p = t ->
+                    p \in shsh p1 p2 -> LRtriple t1 t2 t.
+Definition pred_LRtriple (t1 t2 : seq (seq nat)) :=
+  [pred t : (seq (seq nat)) |
+   has (fun p => RS p == t)
+       (flatten [seq shsh p1 p2 | p1 <- RSclass _ t1, p2 <- RSclass _ t2])].
+Definition pred_LRtriple_fast (t1 t2 : seq (seq nat)) :=
+  [pred t : (seq (seq nat)) |
+   has (fun p2 => to_word t \in shsh (to_word t1) p2) (RSclass _ t2)].
+ +
+Lemma LRtripleP t1 t2 t :
+  is_stdtab t1 -> is_stdtab t2 ->
+  reflect (LRtriple t1 t2 t) (pred_LRtriple t1 t2 t).
+ +
+Lemma filter_gt_RS d (T : inhOrderType d) (w : seq T) n :
+  RS (filter (>%O n) w) = filter_gt_tab n (RS w).
+ +
+Lemma pred_LRtriple_fast_filter_gt t1 t2 t :
+  is_stdtab t1 -> is_stdtab t ->
+  pred_LRtriple_fast t1 t2 t -> t1 = filter_gt_tab (size_tab t1) t.
+ +
+Lemma LRtriple_fastE t1 t2 t :
+  is_stdtab t1 -> is_stdtab t2 -> is_stdtab t ->
+  pred_LRtriple t1 t2 t = pred_LRtriple_fast t1 t2 t.
+ +
+Lemma is_stdtab_of_n_LRtriple t1 t2 t :
+  is_stdtab t1 -> is_stdtab t2 -> LRtriple t1 t2 t ->
+  is_stdtab_of_n ((size_tab t1) + (size_tab t2)) t.
+ +
+Theorem LRtriple_cat_langQ t1 t2 u1 u2:
+  is_stdtab t1 -> is_stdtab t2 -> u1 \in langQ t1 -> u2 \in langQ t2 ->
+  LRtriple t1 t2 (RStabmap (u1 ++ u2)).2.
+ +
+
+ +
+This is essentially the free LR rule: + +
+ +The concatenation of langQ t1 and langQ t2 is the union of langQ t for +t such that LRtriple t1 t2 t. + +
+
+Theorem LRtriple_cat_equiv t1 t2 :
+  is_stdtab t1 -> is_stdtab t2 ->
+  forall u1 u2 : word,
+  ( (u1 \in langQ t1 /\ u2 \in langQ t2) <->
+    [/\ size u1 = size_tab t1, size u2 = size_tab t2 &
+     exists t, LRtriple t1 t2 t /\ u1 ++ u2 \in langQ t] ).
+ +
+
+ +
+

A longer alternative road

+ In the following we goes along the proof of Schützenberger theorem + but we end up duplicating the whole proof: compare the statement and proofs + of +
    +
  • sfiltergtn_invstd below with invstd_catgtn + +
    • +
  • sfilterleq_invstd below with invstd_catleq + +
    • +
+ +
+
+ +
+Lemma sfiltergtn_invstd w n :
+  n <= size w -> sfiltergtn n (invstd (std w)) = invstd (std (take n w)).
+ +
+Lemma sfilterleq_invstd w n :
+  n <= size w -> sfilterleq n (invstd (std w)) = invstd (std (drop n w)).
+ +
+
+ +
+Yet another form of Schützenberger theorem +
+
+Theorem LRrule_langQ t1 t2 w :
+  is_stdtab t1 ->
+  (exists u v, [/\ w = u ++ v, u \in langQ t1 & v \in langQ t2]) <->
+  (exists t, LRtriple t1 t2 t /\ w \in langQ t).
+ +
+
+ +
+Alternative proof from LRtriple_cat_equiv +
+
+Theorem LRrule_langQ_alternate t1 t2 w :
+  is_stdtab t1 -> is_stdtab t2 ->
+  (exists u v, [/\ w = u ++ v, u \in langQ t1 & v \in langQ t2]) <->
+  (exists t, LRtriple t1 t2 t /\ w \in langQ t).
+ +
+
+ +
+

Conjugating LRtriple

+ +
+
+Theorem LRtriple_conj t1 t2 t :
+  is_stdtab t1 -> is_stdtab t2 -> is_stdtab t ->
+  LRtriple t1 t2 t -> LRtriple (conj_tab t1) (conj_tab t2) (conj_tab t).
+ +
+Theorem pred_LRtriple_conj t1 t2 t :
+  is_stdtab t1 -> is_stdtab t2 -> is_stdtab t ->
+  pred_LRtriple t1 t2 t = pred_LRtriple (conj_tab t1) (conj_tab t2) (conj_tab t).
+ +
+End LRTriple.
+ +
+Arguments langQ {disp} [Alph].
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.stdplact.html b/combi/1.1.0/Combi.LRrule.stdplact.html new file mode 100644 index 00000000..070c53f3 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.stdplact.html @@ -0,0 +1,239 @@ + + + + + +Combi.LRrule.stdplact: Plactic congruences and standardization + + + + +
+ + + +
+ +

Library Combi.LRrule.stdplact: Plactic congruences and standardization

+ +
+
+ +
+ +
+
+
+ +
+

Plactic congruences and standardization

+ + +
+ +The goal of this file is to show a few important properties of plactic +congruence related to standardisation. + +
+ +The first one is easy: it says that standardization preserve plactic congruence. +This is Theorem std_plact and is stated as u =Pl v -> std u =Pl std v. + +
+ +The next series of results relates the result of the RS map applied on +u and std u: +
    +
  • the insertion tableau have the same shape: Theorem shape_RS_std; + +
    • +
  • the recording tableau are equal: Theorem RSmap_std and RStabmap_std. + +
    • +
+ +
+ +And finally, inverting a standard word amount to exchange the insertion and +the recording tableaux: Theorem invseqRSPQE, and Corollaries invseqRSE, +invstdRSE, RSTabmapstdE and RSinvstdE. + +
+
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import perm fingroup.
+ +
+Require Import tools combclass ordcast partition Yamanouchi ordtype std tableau.
+Require Import stdtab sorted Schensted congr plactic Greene Greene_inv.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+ +
+
+ +
+

Plactic congruence and standardization

+ +
+
+Section StdRS.
+ +
+Context {disp} {Alph : inhOrderType disp}.
+Implicit Type s u v w : seq Alph.
+Implicit Type p : seq nat.
+Implicit Type t : seq (seq Alph).
+ +
+Lemma std_plact1 (u v1 w v2 : seq Alph) :
+  v2 \in plact1 v1 -> std (u ++ v1 ++ w) =Pl std (u ++ v2 ++ w).
+ +
+Lemma std_plact2 (u v1 w v2 : seq Alph) :
+  v2 \in plact2 v1 -> std (u ++ v1 ++ w) =Pl std (u ++ v2 ++ w).
+ +
+Theorem std_plact u v : u =Pl v -> std u =Pl std v.
+ +
+
+ +
+

Robinson-Schensted correspondence and standardization

+ +
+
+Lemma cast_enum u (S : {set 'I_(size u)}) :
+  enum (mem (cast_set (esym (size_std u)) S)) =
+  map (cast_ord (esym (size_std u))) (enum (mem S)).
+ +
+Lemma sorted_std_extract u (S : {set 'I_(size u)}) :
+  sorted <=%O (extractpred (in_tuple u) (mem S)) =
+  sorted <=%O (extractpred (in_tuple (std u))
+                           (mem (cast_set (esym (size_std u)) S))).
+ +
+Lemma ksupp_inj_std u k : ksupp_inj <=%O <=%O k u (std u).
+ +
+Lemma ksupp_inj_stdI u k : ksupp_inj <=%O <=%O k (std u) u.
+ +
+Lemma Greene_std u k : Greene_row (std u) k = Greene_row u k.
+ +
+Theorem shape_RS_std u : shape (RS (std u)) = shape (RS u).
+ +
+End StdRS.
+ +
+Theorem RSmap_std disp (T : inhOrderType disp) (w : seq T) :
+  (RSmap (std w)).2 = (RSmap w).2.
+ +
+Corollary RStabmap_std disp (T : inhOrderType disp) (w : seq T) :
+  (RStabmap (std w)).2 = (RStabmap w).2.
+ +
+
+ +
+

Robinson-Schensted correspondence and inversion

+ +
+
+Section KsuppInj.
+ +
+Variable s t : seq nat.
+Hypothesis Hinv : invseq s t.
+ +
+#[local] Lemma Hinvst : linvseq s t.
+#[local] Lemma Hinvts : linvseq t s.
+ +
+#[local] Definition val2pos :=
+  fun (i : 'I_(size s)) => Ordinal (linvseq_ltn_szt Hinvst (ltn_ord i)).
+ +
+Lemma val2posE : val \o val2pos =1 nth (size t) s.
+Lemma val2pos_inj : injective val2pos.
+ +
+Lemma val2pos_enum (p : {set 'I_(size s)}) :
+  
+  sorted <=%O [seq tnth (in_tuple s) i | i <- enum (mem p)] ->
+  enum (mem [set val2pos x | x in p]) = [seq val2pos x | x <- enum p].
+ +
+Lemma ksupp_inj_invseq k : ksupp_inj <=%O <=%O k s t.
+ +
+End KsuppInj.
+ +
+Lemma Greene_invseq s t k : invseq s t -> Greene_row s k = Greene_row t k.
+ +
+Lemma shape_invseq s t : invseq s t -> shape (RS s) = shape (RS t).
+ +
+Lemma std_rcons_shiftinv t tn :
+  is_std (rcons t tn) -> std t = map (shiftinv_pos tn) t.
+ +
+Lemma posbig_invseq s0 s t tn :
+  invseq (s0 :: s) (rcons t tn) -> posbig (s0 :: s) = tn.
+ +
+Lemma nth_std_pos s i x :
+  is_std s -> i < size s -> i != posbig s -> nth x s i < (size s).-1.
+ +
+Lemma linvseqK s0 s t tn :
+  invseq (s0 :: s) (rcons t tn) -> linvseq (rembig (s0 :: s)) (std t).
+ +
+Lemma invseqK s0 s t tn :
+  invseq (s0 :: s) (rcons t tn) -> invseq (rembig (s0 :: s)) (std t).
+ +
+Theorem invseqRSPQE s t :
+  invseq s t -> (RStabmap s).1 = (RStabmap t).2.
+ +
+Corollary invseqRSE s t :
+  invseq s t -> RStabmap s = ((RStabmap t).2, (RStabmap t).1).
+ +
+Corollary invstdRSE s :
+  is_std s -> let (P, Q) := RStabmap (invstd s) in RStabmap s = (Q, P).
+ +
+Corollary RSTabmapstdE disp (T : inhOrderType disp) (w : seq T) :
+  (RStabmap (invstd (std w))).1 = (RStabmap (std w)).2.
+ +
+Corollary RSinvstdE disp (T : inhOrderType disp) (w : seq T) :
+  RS (invstd (std w)) = (RStabmap w).2.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.LRrule.therule.html b/combi/1.1.0/Combi.LRrule.therule.html new file mode 100644 index 00000000..d2c86c70 --- /dev/null +++ b/combi/1.1.0/Combi.LRrule.therule.html @@ -0,0 +1,439 @@ + + + + + +Combi.LRrule.therule: The Littlewood-Richardson rule + + + + +
+ + + +
+ +

Library Combi.LRrule.therule: The Littlewood-Richardson rule

+ +
+
+ +
+ +
+
+
+ +
+

The Littlewood-Richardson rule

+ + +
+ +The goal of this file is to formalize the final step ot the proof: a bijection +beetween LR-standard tableau as defined in LRsupport Q1 Q2 and LR-Yamanouchi +tableaux. + +
+ +Below we use the following notation: +
    +
  • d1 and d2 are nat, + +
    • +
  • P1 : a partition of d1 (of type: inpartn d1). + +
    • +
  • P2 : a partition of d2 (of type: inpartn d2). + +
    • +
  • P : a partition of d1 + d2 (of type: inpartn d1 + d2). + +
    • +
+ +
+ + +
+ +We define the following: +
    +
  • is_skew_reshape_tableau P P1 w == w is the row reading of a skew tableau of + shape P/P1. Equivalently, the P/P1-reshape of w is a skew tableau. + +
    • +
  • bijLRyam P P1 == a map from seq nat to seq (seq nat) which defines a + bijection between LR yamanouchi tableaux and LR-standard tableaux + +
    • +
  • bijLR P P1 == the sigma-type version of bijLRyam: + (yam : yameval P2) -> stdtabn (d1 + d2) + +
    + + +
    • +
  • LRyam_set P1 P2 P == the set of LR Yamanouchi words. + +
    • +
  • LRyam_coeff P1 P2 P == the LR coefficient defined as the cardinality of + the set of LR Yamanouchi words. + +
    + + +
    • +
  • LRyam_enum P1 P2 P == a list for the LR Yamanouchi words. + +
    • +
  • LRyam_compute P1 P2 P == the length LRyam_enum, allows to compute LR-coeffs + and LR-tableaux from the definition inside coq. As an + implementation, its a very slow way to compute LR-coeff. Indeed, + the set of Yamanouchi words where we are looking for LR words is + much too large. The module implem will provide a much efficient + way using backtracking instead of filtering to enumerate those. + +
    + + +
    • +
  • yamrow n == the trivial Yamanouchi word of size n whis is constant to 0 + +
    • +
+ +
+ + +
+ +The main theorem is Theorem LRyam_coeffP: + +
+ + + Schur P1 * Schur P2 = + \sum_(P : 'P_(d1 + d2) | included P1 P) Schur P *+ LRyam_coeff P. + + +
+ +As a corollary we provide the two Pieri rules Pieri_rowpartn and +Pieri_colpartn. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg.
+From mathcomp Require Import mpoly.
+ +
+Require Import tools ordcast combclass partition skewpart Yamanouchi ordtype.
+Require Import std tableau stdtab Schensted congr plactic Greene_inv.
+Require Import stdplact Yam_plact skewtab shuffle Schur_mpoly freeSchur.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+Open Scope N.
+#[local] Open Scope Combi.
+ +
+
+ +
+

Gluing a standard tableaux with a skew tableau

+ +
+
+Section LR.
+ +
+Lemma to_word_map_shiftn sh t :
+  to_word (map (shiftn sh) t) = shiftn sh (to_word t).
+ +
+Lemma filter_le_shiftn sh t :
+  [seq x - sh | x <- [seq sh + i | i <- t] & sh <= x] = t.
+ +
+Lemma filter_gt_shiftn sh t :
+  [seq x <- [seq sh + i | i <- t] | gtn sh x] = [::].
+ +
+Lemma shiftn_skew_dominate n sh u v :
+  skew_dominate sh u v -> skew_dominate sh (shiftn n u) (shiftn n v).
+ +
+Lemma is_skew_tableau_map_shiftn sh inn t :
+  is_skew_tableau inn t -> is_skew_tableau inn (map (shiftn sh) t).
+ +
+Lemma join_stdtab s t :
+  is_stdtab s -> is_skew_tableau (shape s) t ->
+  is_tableau (join_tab s (map (shiftn (size_tab s)) t)).
+ +
+Lemma join_stdtab_in_shuffle s t :
+  is_stdtab s ->
+  size s <= size t ->
+  to_word (join_tab s (map (shiftn (size_tab s)) t)) \in
+       shsh (to_word s) (to_word t).
+ +
+Variables d1 d2 : nat.
+ +
+Section TheRule.
+ +
+Variables (P1 : 'P_d1) (P2 : 'P_d2).
+ +
+Lemma sfilterleq_LRsupportP Q :
+  Q \in LRsupport (hyper_stdtabn P1) (hyper_stdtabn P2) ->
+  exists y : yameval P2, std y = (sfilterleq d1 (to_word Q)).
+ +
+Lemma filter_gt_to_word disp (T : inhOrderType disp) n (t : seq (seq T)) :
+  filter (>%O n) (to_word t) = to_word (filter_gt_tab n t).
+ +
+Lemma filter_le_to_word disp (T : inhOrderType disp) n (t : seq (seq T)) :
+  filter (<=%O n) (to_word t) = to_word (filter_le_tab n t).
+ +
+
+ +
+

Littlewood-Richardson Yamanouchi tableaux

+ +
+
+Section OneCoeff.
+ +
+Variable P : 'P_(d1 + d2).
+Hypothesis Hincl : included P1 P.
+ +
+Lemma sumn_diff_shape_intpartE : sumn (P / P1) = sumn P2.
+ +
+Definition is_skew_reshape_tableau (P P1 : seq nat) (w : seq nat) :=
+  is_skew_tableau P1 (skew_reshape P1 P w).
+Definition LRyam_set :=
+  [set y : yameval P2 | is_skew_reshape_tableau P P1 y].
+Definition LRyam_coeff := #|LRyam_set|.
+ +
+Lemma is_skew_reshape_tableauP (w : seq nat) :
+  size w = sumn (P / P1) ->
+  reflect
+    (exists tab, [/\ is_skew_tableau P1 tab,
+                  shape tab = P / P1 & to_word tab = w])
+    (is_skew_reshape_tableau P P1 w).
+ +
+Lemma size_leq_skew_reshape (y : seq nat) :
+  size (RS (std (hyper_yam P1))) <= size (skew_reshape P1 P y).
+ +
+
+ +
+

The final bijection

+ +
+
+Definition bijLRyam :=
+  [fun y : seq nat =>
+     join_tab (hyper_stdtabn P1) (map (shiftn d1) (skew_reshape P1 P (std y)))].
+ +
+Lemma pred_LRtriple_fast_bijLRyam (yam : yameval P2) :
+  is_skew_reshape_tableau P P1 yam ->
+  pred_LRtriple_fast (hyper_stdtabn P1) (hyper_stdtabn P2) (bijLRyam yam).
+ +
+Lemma bijLRyamP (yam : yameval P2) :
+  is_skew_reshape_tableau P P1 yam -> is_stdtab_of_n (d1 + d2) (bijLRyam yam).
+ +
+Definition bijLR (yam : yameval P2) : stdtabn (d1 + d2) :=
+  if (boolP (is_skew_reshape_tableau P P1 yam)) is AltTrue pf then
+    StdtabN (bijLRyamP pf)
+  else
+    hyper_stdtabn P.
+ +
+Lemma bijLR_LRsupport yam :
+  yam \in LRyam_set ->
+  bijLR yam \in LRsupport (hyper_stdtabn P1) (hyper_stdtabn P2).
+ +
+Lemma filtergtn_LRsupport Q :
+  Q \in LRsupport (hyper_stdtabn P1) (hyper_stdtabn P2) ->
+  filter_gt_tab d1 Q = hyper_stdtabn P1.
+ +
+Lemma size_zip2 (T : Type) (s t : seq (seq T)) :
+  [seq size p.1 + size p.2 | p <- zip s t] =
+  [seq p.1 + p.2 | p <- zip (map size s) (map size t)].
+ +
+Lemma shape_bijLR yam : yam \in LRyam_set -> shape (bijLR yam) = P.
+ +
+Lemma filterleq_LRsupport Q :
+  Q \in LRtab_set P1 P2 P ->
+  (skew_reshape P1 P [seq x <- to_word Q | d1 <= x]) = filter_le_tab d1 Q.
+ +
+Lemma sfilterleq_LRsupport_skew Q :
+  Q \in LRtab_set P1 P2 P ->
+        is_skew_reshape_tableau P P1 (sfilterleq d1 (to_word Q)).
+ +
+Lemma bijLR_surj Q :
+  Q \in LRtab_set P1 P2 P -> exists2 yam, yam \in LRyam_set & bijLR yam = Q.
+ +
+Lemma bijLR_inj : {in LRyam_set &, injective bijLR}.
+ +
+Lemma bijLR_image : LRtab_set P1 P2 P = [set bijLR x | x in LRyam_set].
+ +
+Theorem LRyam_coeffE : LRtab_coeff P1 P2 P = LRyam_coeff.
+ +
+
+ +
+

A slow way to compute LR coefficients in coq:

+ + +
+ +We enumerate Yamanouchi words and filter those who are row reading of the skew +tableaux of shape P/P1. This is very innefficient. A better way is to use +backtracking as in implem. + +
+
+ +
+Definition LRyam_enum (P1 P2 P : seq nat) :=
+  [seq x <- enum_yameval P2 | is_skew_reshape_tableau P P1 x].
+Definition LRyam_compute (P1 P2 P : seq nat) := size (LRyam_enum P1 P2 P).
+ +
+Lemma LRcoeff_computeP : LRyam_compute P1 P2 P = LRyam_coeff.
+ +
+End OneCoeff.
+ +
+Lemma included_shape_filter_gt_tab disp (T : inhOrderType disp) (n : T) t :
+  is_tableau t -> included (shape (filter_gt_tab n t)) (shape t).
+ +
+Lemma LRtab_set_included (P : 'P_(d1 + d2)) Q :
+  Q \in LRtab_set P1 P2 P -> included P1 P.
+ +
+
+ +
+

The statement of the Littlewood-Richardson rule

+ +
+
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+Variable (n0 : nat) (R : comNzRingType).
+#[local] Notation n := (n0.+1).
+Notation Schur p := (Schur n0 R p).
+ +
+Theorem LRyam_coeffP :
+  Schur P1 * Schur P2 =
+  \sum_(P : 'P_(d1 + d2) | included P1 P) Schur P *+ LRyam_coeff P.
+ +
+End TheRule.
+ +
+
+ +
+

Pieri's rules

+ +
+
+Section Pieri.
+ +
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+Variable (n0 : nat) (R : comNzRingType).
+#[local] Notation n := (n0.+1).
+Notation Schur p := (Schur n0 R p).
+ +
+Lemma yamrowP :
+  is_yam_of_eval (intpart_of_intpartn (rowpartn d2)) (ncons d2 0%N [::]).
+Definition yamrow : yameval (rowpartn d2) := YamEval yamrowP.
+ +
+Lemma is_row_yamrow : is_row (ncons d2 0%N [::]).
+ +
+Lemma yam_of_rowpart d y : is_yam_of_eval (rowpartn d) y -> y = ncons d 0%N [::].
+ +
+Theorem LRyam_coeff_rowpart (P1 : 'P_d1) (P : 'P_(d1 + d2)) :
+included P1 P -> LRyam_coeff P1 (rowpartn d2) P = hb_strip P1 P.
+ +
+Theorem Pieri_rowpartn (P1 : 'P_d1) :
+  Schur P1 * Schur (rowpartn d2) =
+  \sum_(P : 'P_(d1 + d2) | hb_strip P1 P) Schur P.
+ +
+Theorem LRyam_coeff_colpartn (P1 : 'P_d1) (P : 'P_(d1 + d2)) :
+  included P1 P -> LRyam_coeff P1 (colpartn d2) P = vb_strip P1 P.
+ +
+Theorem Pieri_colpartn (P1 : 'P_d1) :
+  Schur P1 * Schur (colpartn d2) =
+  \sum_(P : 'P_(d1 + d2) | vb_strip P1 P) Schur P.
+ +
+End Pieri.
+ +
+End LR.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.Cauchy.html b/combi/1.1.0/Combi.MPoly.Cauchy.html new file mode 100644 index 00000000..a320f5be --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.Cauchy.html @@ -0,0 +1,507 @@ + + + + + +Combi.MPoly.Cauchy: Cauchy formula for symmetric polynomials + + + + +
+ + + +
+ +

Library Combi.MPoly.Cauchy: Cauchy formula for symmetric polynomials

+ +
+
+ +
+ +
+
+
+ +
+

Cauchy formula for symmetric polynomials

+ + +
+ +In this file we fix two non zero natural m and n and consider the two sets +of variables X := (x_i)_{i < m} and Y := (y_j)_{j < n}. We also consider +the variables z_{i,j} := x_i * y_j. + +
+ +We encode polynomial in X \cup Y as polynomials in X whose coefficient are +polynomials in Y. We denote by mz a monomial in the Z. + +
+ +
    +
  • monX mz == the monomial in X obtained by setting all the y_i to 1. + +
    • +
  • monsY mz == the m.-tuple of whose i-th element is the monomial in Y + obtained by putting x_i to 1 and all the others to 0. + +
    • +
  • Ymon ms == given ms : m.-tuple 'X_{1.. n} assemble them to get a + monomial in the Z. + +
    • +
  • polXY m n R == polynomial in m variable whose coefficients are polynomials + in n over the commutative ring R. This is canonically a + algType over R. + +
    • +
  • polXY_scale c p == base ring multiplication for elements of polXY m n R + +
    • +
  • p(X) == the image of p by the canonical inclusion algebra morphism + polX -> polXY + +
    • +
  • p(Y) == the image of p by the canonical inclusion algebra morphism + polY -> polXY + +
    • +
  • p(XY) == the polynomial in polXY from a polynomials in the Z_{i,j}. + +
    • +
  • Cauchy_kernel d == the Cauchy reproducing kernel in degree d, that is + the sum of all monomial in x_i*y_i of degree d + which is defined as 'h_d(XY) + +
    • +
  • co_hp la p == if p is symmetric in X, returns the coefficient of p on + 'hp[la] + +
    • +
  • co_hpXY la mu p == if p is symmetric both in X and Y, returns the + coefficient of p on 'hp[la](X) 'hp[mu](Y). + +
    • +
+ +
+ +The main result is Theorem homsymdotss which asserts that Schur function are +orthonormal for the scalar product. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import ssrint rat ssralg ssrnum algC matrix vector.
+From mathcomp Require Import sesquilinear.
+From mathcomp Require Import mpoly.
+ +
+Require Import tools partition ordtype.
+Require Import antisym Schur_mpoly Schur_altdef sympoly homogsym permcent.
+ +
+Require ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import GRing.Theory Num.Theory.
+#[local] Open Scope ring_scope.
+ +
+#[local] Lemma pchar0_rat : [pchar rat] =i pred0.
+ #[local] Lemma pchar0_algC : [pchar algC] =i pred0.
+ #[local] Hint Resolve pchar0_algC pchar0_rat : core.
+ +
+Reserved Notation "p '(Y)'" (at level 20, format "p '(Y)'").
+Reserved Notation "p '(X)'" (at level 20, format "p '(X)'").
+Reserved Notation "p '(XY)'" (at level 20, format "p '(XY)'").
+ +
+
+ +
+

Polynomials in two sets of variables

+ +
+
+Section CauchyKernel.
+ +
+Variables (m0 n0 : nat).
+Notation m := m0.+1.
+Notation n := n0.+1.
+Notation mxvec_index := (@mxvec_index m n).
+ +
+Let vecmx_index : 'I_(m * n) -> 'I_m * 'I_n :=
+  (enum_val \o cast_ord (esym (mxvec_cast m n))).
+ +
+Lemma vecmx_indexK i : mxvec_index (vecmx_index i).1 (vecmx_index i).2 = i.
+Lemma mxvec_indexK i j : vecmx_index (mxvec_index i j) = (i, j).
+ +
+Section Big.
+ +
+Variables (R : Type) (idx : R).
+Variable op : Monoid.com_law idx.
+ +
+Lemma big_mxvec_index P F :
+  \big[op/idx]_(i : 'I_(m * n) | P i) F i =
+   \big[op/idx]_(i < m)
+    \big[op/idx]_(j < n | P (mxvec_index i j)) F (mxvec_index i j).
+ +
+End Big.
+ +
+Definition monX (mon : 'X_{1.. m*n}) : 'X_{1.. m} :=
+  [multinom (\sum_(j < n) mon (mxvec_index i j))%N | i < m].
+ +
+Lemma mdeg_monX mon : mdeg (monX mon) = mdeg mon.
+ +
+Definition monsY (mz : 'X_{1.. m*n}) :=
+  [tuple [multinom mz (mxvec_index i j) | j < n] | i < m].
+ +
+Definition Ymon (ms : m.-tuple 'X_{1.. n}) :=
+  [multinom (tnth ms (vecmx_index i).1) (vecmx_index i).2 | i < m*n].
+ +
+Lemma monsYK : cancel monsY Ymon.
+Lemma YmonK : cancel Ymon monsY.
+Lemma monsY_bij : bijective monsY.
+ Lemma Ymon_bij : bijective Ymon.
+ +
+Lemma mdeg_tnth_monsY mz i :
+  mdeg (tnth (monsY mz) i) = tnth (monX mz) i.
+ +
+Variable (R : comNzRingType).
+ +
+#[local] Notation polZ := {mpoly R[m * n]}.
+#[local] Notation polX := {mpoly R[m]}.
+#[local] Notation polY := {mpoly R[n]}.
+Definition polXY := {mpoly polY[m]}.
+Definition polXY_scale (c : R) (p : polXY) : polXY := c%:MP *: p.
+#[local] Notation "c *:M p" := (polXY_scale c p)
+  (at level 40, left associativity).
+ +
+ +
+Fact scale_polXYA a b p : a *:M (b *:M p) = (a * b) *:M p.
+ Fact scale_polXY1m p : 1 *:M p = p.
+ Fact scale_polXYDr c p1 p2 :
+  c *:M (p1 + p2) = c *:M p1 + c *:M p2.
+ Fact scale_polXYDl p c1 c2 :
+  (c1 + c2) *:M p = c1 *:M p + c2 *:M p.
+ +
+Lemma scale_polXYE (c : R) (p : polXY) : c *: p = c *:M p.
+ +
+Fact polXY_scaleAl (c : R) (p q : polXY) : c *: (p * q : polXY) = (c *: p) * q.
+ +
+Fact polXY_scaleAr (c : R) (p q : polXY) : c *: (p * q : polXY) = p * (c *: q).
+ +
+Definition polX_XY : polX -> polXY := map_mpoly (mpolyC n (R := R)).
+ +
+Fact polX_XY_is_zmod_morphism : zmod_morphism polX_XY.
+ +
+Fact polX_XY_is_monoid_morphism : monoid_morphism polX_XY.
+ +
+Fact polX_XY_is_linear : linear polX_XY.
+ +
+Definition polY_XY : polY -> polXY := mpolyC m (R := {mpoly R[n]}).
+ +
+Fact polY_XY_is_zmod_morphism : zmod_morphism polY_XY.
+ +
+Fact polY_XY_is_monoid_morphism : monoid_morphism polY_XY.
+ +
+Fact polY_XY_is_linear : linear polY_XY.
+ +
+Notation "p '(Y)'" := (polY_XY p).
+Notation "p '(X)'" := (polX_XY p).
+ +
+Lemma polyXY_scale p q : p(X) * q(Y) = q *: p(X).
+ +
+Lemma symmX d (la : 'P_d) : 'hm[la](X) = 'hm[la].
+ +
+Definition evalXY : polZ -> polXY :=
+  mmap ((@mpolyC m _) \o (@mpolyC n R))
+       (fun i => 'X_((vecmx_index i).1) (X) * 'X_((vecmx_index i).2) (Y)).
+Notation "p '(XY)'" := (evalXY p).
+ +
+Fact evalXY_is_zmod_morphism : zmod_morphism evalXY.
+ +
+Fact evalXY_is_monoid_morphism : monoid_morphism evalXY.
+ +
+Fact evalXY_is_linear : linear evalXY.
+ +
+Lemma evalXY_XE mz :
+  'X_[mz](XY) = 'X_[monX mz](X) * \prod_(i < m) 'X_[tnth (monsY mz) i](Y).
+ +
+Lemma evalXY_homog d p : p \is d.-homog -> p(XY) \is d.-homog.
+ +
+Lemma sympXY k : 'p_k(XY) = 'p_k(X) * 'p_k(Y).
+ +
+Lemma prod_sympXY d (la : 'P_d) : 'hp[la](XY) = 'hp[la](X) * 'hp[la](Y).
+ +
+
+ +
+

The Cauchy reproducing kernel

+ +
+
+Definition Cauchy_kernel d := 'h_d(XY).
+ +
+Lemma Cauchy_kernel_dhomog d : Cauchy_kernel d \is d.-homog.
+ +
+Section BijectionFam.
+ +
+Variable d : nat.
+ +
+Fact famY_subproof (mz : 'X_{1.. (m * n) < d.+1}) i :
+    (mdeg (tnth (monsY (val mz)) i) < d.+1)%N.
+Definition famY mz : {ffun 'I_m -> 'X_{1.. n < d.+1}} :=
+  [ffun i : 'I_m => BMultinom (famY_subproof mz i)].
+ +
+Let famYinv_fun (ff : {ffun 'I_m -> 'X_{1.. n < d.+1}}) :=
+  let mz := [multinom (ff (vecmx_index i).1 (vecmx_index i).2) | i < m * n]
+  in if (mdeg mz < d.+1)%N then mz else 0%MM.
+Fact famYinv_subproof ff : (mdeg (famYinv_fun ff) < d.+1)%N.
+Definition famYinv ff := BMultinom (famYinv_subproof ff).
+ +
+Lemma famY_bij (mon : 'X_{1.. m}) :
+  mdeg mon = d ->
+  {on [pred i in
+       family (fun i0 (j : 'X_{1.. n < d.+1}) => mdeg j == tnth mon i0)],
+    bijective famY}.
+ +
+End BijectionFam.
+ +
+Variable d : nat.
+ +
+
+ +
+

Cauchy formula for complete and monomial symmetric polynomials

+ +
+
+Lemma Cauchy_symm_symh :
+  Cauchy_kernel d = \sum_(la : 'P_d) ('h[la] : polY) *: ('m[la] : polXY).
+ +
+Lemma Cauchy_homsymm_homsymh :
+  Cauchy_kernel d = \sum_(la : 'P_d) 'hm[la](X) * 'hh[la](Y).
+ +
+
+ +
+

Cauchy formula for Schur symmetric polynomials

+ +
+
+Lemma Cauchy_homsyms_homsyms :
+  Cauchy_kernel d = \sum_(la : 'P_d) 'hs[la](X) * 'hs[la](Y).
+ +
+
+ +
+Unused lemma +
+
+Lemma Cauchy_kernel_symmetric : Cauchy_kernel d \is symmetric.
+ +
+
+ +
+Unused lemma +
+
+Lemma Cauchy_kernel_coeff_symmetric mon :
+  (Cauchy_kernel d)@_mon \is symmetric.
+ +
+
+ +
+Unused lemma +
+
+Lemma Cauchy_kernel_coeff_homog mon :
+  (Cauchy_kernel d)@_mon \is d.-homog.
+ +
+End CauchyKernel.
+ +
+Notation "p '(Y)'" := (@polY_XY _ _ _ p).
+Notation "p '(X)'" := (@polX_XY _ _ _ p).
+Notation "p '(XY)'" := (@evalXY _ _ _ p).
+ +
+Section CauchyKernelField.
+Variable R : fieldType.
+ +
+
+ +
+

Cauchy formula for power sum symmetric polynomials

+ +
+
+Lemma Cauchy_homsymp_zhomsymp m n d :
+  [pchar R] =i pred0 ->
+  Cauchy_kernel m n R d =
+  \sum_(la : 'P_d) 'hp[la](X) * ((zcard la)%:R^-1 *: 'hp[la](Y)).
+ +
+End CauchyKernelField.
+ +
+
+ +
+

Cauchy kernel and scalar product of symmetric functions

+ +
+
+Section Scalar.
+ +
+Variable n0 d : nat.
+#[local] Notation n := n0.+1.
+Hypothesis Hd : (d <= n)%N.
+ +
+#[local] Notation HSC := {homsym algC[n, d]}.
+#[local] Notation polXY := (polXY n0 n0 algC).
+#[local] Notation pol := {mpoly algC[n]}.
+#[local] Notation "p '(Y)'" := (@polY_XY n0 n0 _ p).
+#[local] Notation "p '(X)'" := (@polX_XY n0 n0 _ p).
+ +
+#[local] Notation "''hsC[' la ]" := ('hs[la] : HSC).
+#[local] Notation "''hpC[' la ]" := ('hp[la] : HSC).
+ +
+Definition co_hp (la : 'P_d) : pol -> algC :=
+  homsymdot^~ 'hp[la] \o in_homsym d (R := algC).
+Definition co_hpXY (la mu : 'P_d) : polXY -> algC :=
+  locked (co_hp la \o map_mpoly (co_hp mu)).
+ +
+Fact co_hp_is_zmod_morphism la : zmod_morphism (co_hp la).
+ +
+Fact co_hp_is_scalar la : scalar (co_hp la).
+ +
+Lemma co_hp_hp la mu : co_hp la 'hp[mu] = (zcard mu)%:R * (mu == la)%:R.
+ +
+Fact co_hpXY_is_zmod_morphism la mu : zmod_morphism (co_hpXY la mu).
+ +
+Lemma co_hpYE la (p q : pol) :
+  map_mpoly (co_hp la) (p(X) * q(Y)) = (co_hp la q) *: p.
+ +
+Lemma co_hprXYE la mu (p q : pol) :
+  co_hpXY la mu (p(X) * q(Y)) = (co_hp la p) * (co_hp mu q).
+ +
+Lemma coord_zsympsps (la mu : 'P_d) :
+  (\sum_nu
+    (coord 'hp (enum_rank la) 'hsC[nu]) *
+    ((zcard mu)%:R * coord 'hp (enum_rank mu) 'hsC[nu]))
+  = (la == mu)%:R.
+ +
+Lemma coord_zsymspsp (la mu : 'P_d) :
+  (\sum_nu
+    (coord 'hp (enum_rank nu) 'hsC[la]) *
+    ((zcard nu)%:R * coord 'hp (enum_rank nu) 'hsC[mu]))
+  = (la == mu)%:R.
+ +
+
+ +
+

Schur function are orthonormal

+ +
+
+Theorem homsymdotss la mu : '[ 'hsC[la] | 'hsC[mu] ] = (la == mu)%:R.
+ +
+Theorem homsyms_orthonormal : orthonormal homsymdot ('hs : seq HSC).
+ +
+End Scalar.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.MurnaghanNakayama.html b/combi/1.1.0/Combi.MPoly.MurnaghanNakayama.html new file mode 100644 index 00000000..b2cc422f --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.MurnaghanNakayama.html @@ -0,0 +1,510 @@ + + + + + +Combi.MPoly.MurnaghanNakayama: Murnaghan-Nakayama rule + + + + +
+ + + +
+ +

Library Combi.MPoly.MurnaghanNakayama: Murnaghan-Nakayama rule

+ +
+
+ +
+ +
+
+
+ +
+

The Murnaghan-Nakayama rule

+ + +
+ +See the page +Murnaghan-Nakayama on Wikipedia for a statement. The fixpoint MN_coeff la mu +implement the recursive version, as stated in +
+Theorem MN_coeff_consE la m0 mu :
+  MN_coeff la (m0 :: mu) =
+  \sum_(sh : 'P_(sumn mu) | ribbon sh la)
+   MN_coeff sh mu * (-1) ^+ (ribbon_height sh la).-1. +
+ +and the base case +
+Lemma MN_coeff0 : MN_coeff [::] [::] = 1. +
+ +The Murnaghan-Nakayama rule stated in terms of symmetric polynomials is then +
+Theorem MN_coeffP d (la : 'P_d) :
+  'p[la] = \sum_(sh : 'P_d) (MN_coeff sh la)%:~R *: 's[sh] :> {sympoly R[n]}. +
+ +There is a second implementation which goes bottom up, adding ribbons instead +of removing them. It allows to compute skew Murnaghan-Nakayama coefficients. + +
+ +We provide the following definitions: + +
+ +
    +
  • MN_coeff la mu == then Murnaghan-Nakayama coefficients. That is the + alternating number of ribbon filling of the shape la + with content mu, defined recursively. + +
    • +
  • MN_coeff_fast la mu == fast version of MN_coeff la mu + +
    • +
  • MN_coeff_rec la nu mu == The alternating number of ribbon filling of + the skew shape la / nu with content mu, defined recursively. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import ssralg ssrint perm fingroup tuple vector rat.
+From mathcomp Require Import ssrcomplements mpoly.
+ +
+Require Import sorted tools ordtype permuted partition skewpart.
+Require Import antisym Schur_mpoly Schur_altdef sympoly homogsym.
+ +
+Set Implicit Arguments.
+ +
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+#[local] Reserved Notation "''a_' k"
+      (at level 8, k at level 2, format "''a_' k").
+#[local] Reserved Notation "m # s"
+      (at level 40, left associativity, format "m # s").
+ +
+
+ +
+

Product of an alternating polynomial and a power sum

+ +
+
+Section MultAlternSymp.
+ +
+Variable n0 : nat.
+Variable R : comNzRingType.
+ +
+#[local] Notation n := n0.+1.
+#[local] Notation rho := (rho n).
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+ +
+Lemma mult_altern_symp_pol p d :
+  'a_(mpart p + rho) * (symp_pol n R d.+1) =
+   \sum_(i < n) 'a_(mpart p + rho + U_(i) *+ d.+1).
+ +
+Lemma mult_altern_oapp p d :
+  is_part p -> size p <= n ->
+  'a_(mpart p + rho) * (symp_pol n R d.+1) =
+  \sum_(i < n) oapp (fun ph => (-1) ^+ ph.2.-1 *: 'a_(mpart ph.1 + rho)) 0
+   (add_ribbon p d i).
+ +
+Lemma mult_altern_pmap p d :
+  is_part p -> size p <= n ->
+  'a_(mpart p + rho) * (symp_pol n R d.+1) =
+  \sum_(psh <- pmap (add_ribbon p d) (iota 0 n))
+   (-1) ^+ (psh.2).-1 *: 'a_(mpart psh.1 + rho).
+ +
+End MultAlternSymp.
+ +
+
+ +
+

Product of a Schur polynomial and a power sum

+ +
+
+Section MultSymsSympIDomain.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+#[local] Notation SF := {sympoly int[n]}.
+ +
+Lemma syms_sympM_oapp_int d (la : 'P_d) m :
+  m != 0%N -> size la <= n ->
+  's[la] * 'p_m =
+  \sum_(i < n) oapp (fun ph => (-1) ^+ ph.2.-1 *: 's[ph.1]) 0
+   (add_ribbon_intpartn la m.-1 i) :> SF.
+ +
+End MultSymsSympIDomain.
+ +
+Section MultSymsSymp.
+ +
+Variable n0 : nat.
+Variable R : comNzRingType.
+#[local] Notation n := n0.+1.
+#[local] Notation SF := {sympoly R[n]}.
+ +
+Lemma syms_sympM_oapp d (la : 'P_d) m :
+  m != 0%N ->
+  's[la] * 'p_m =
+  \sum_(i < n) oapp (fun ph => (-1) ^+ ph.2.-1 *: 's[ph.1]) 0
+   (add_ribbon_intpartn la m.-1 i) :> SF.
+ +
+Lemma syms_sympM_pmap d (la : 'P_d) m :
+  m != 0%N ->
+  's[la] * 'p_m =
+  \sum_(ph <- pmap (add_ribbon_intpartn la m.-1) (iota 0 n))
+   (-1) ^+ ph.2.-1 *: 's[ph.1] :> SF.
+ +
+
+ +
+The following theorem is a step of the Murnaghan-Nakayama rule +
+
+Theorem syms_sympM d (la : 'P_d) m :
+  m != 0%N ->
+  's[la] * 'p_m =
+  \sum_(sh : 'P_(m + d) | ribbon la sh)
+   (-1) ^+ (ribbon_height la sh).-1 *: 's[sh] :> SF.
+ +
+End MultSymsSymp.
+ +
+
+ +
+

Murnaghan-Nakayama coefficients

+ +
+ + We define those for any sequence of nat, but MN_coeff should only be used when sumn la == sumn mu. +
+
+Fixpoint MN_coeff (la mu : seq nat) : int :=
+  if mu is m0 :: m then
+    foldr (fun sh acc =>
+             if ribbon sh la then
+               MN_coeff sh m * (-1) ^+ (ribbon_height sh la).-1 + acc
+             else acc)
+          0 (enum_partn (sumn m))
+  else 1.
+ +
+
+ +
+Base case +
+
+Lemma MN_coeff0 : MN_coeff [::] [::] = 1.
+ +
+
+ +
+Recursive step +
+
+Theorem MN_coeff_consE la m0 mu :
+  MN_coeff la (m0 :: mu) =
+  \sum_(sh : 'P_(sumn mu) | ribbon sh la)
+   MN_coeff sh mu * (-1) ^+ (ribbon_height sh la).-1.
+ +
+Section Tests.
+
+ +
+Tests : +
+sage: s(p[2,1,1])
+-s[1, 1, 1, 1] - s[2, 1, 1] + s[3, 1] + s[4]
+
+ +
+
+Goal ([seq x | x <- [seq (p, MN_coeff p [:: 2; 1; 1]) | p <- enum_partn 4]
+               & x.2 != 0%R] =
+      [:: ([:: 4], Posz 1);
+      ([:: 3; 1], Posz 1);
+      ([:: 2; 1; 1], Negz 0);
+      ([:: 1; 1; 1; 1], Negz 0)])%N.
+ +
+
+ +
+Tests : +
+sage: s(p[4,2,1,1])
+s[1, 1, 1, 1, 1, 1, 1, 1] + s[2, 1, 1, 1, 1, 1, 1] - s[3, 1, 1, 1, 1, 1]
+ - 2*s[3, 3, 2] - s[4, 1, 1, 1, 1] + 2*s[4, 2, 1, 1] - s[5, 1, 1, 1]
+ - s[6, 1, 1] + s[7, 1] + s[8]
+
+ +
+
+Goal ([seq x | x <- [seq (p, MN_coeff p [:: 4; 2; 1; 1]) | p <- enum_partn 8]
+               & x.2 != 0%R] =
+      [:: ([:: 8], Posz 1);
+      ([:: 7; 1], Posz 1);
+      ([:: 3; 3; 2], Negz 1);
+      ([:: 6; 1; 1], Negz 0);
+      ([:: 4; 2; 1; 1], Posz 2);
+      ([:: 5; 1; 1; 1], Negz 0);
+      ([:: 4; 1; 1; 1; 1], Negz 0);
+      ([:: 3; 1; 1; 1; 1; 1], Negz 0);
+      ([:: 2; 1; 1; 1; 1; 1; 1], Posz 1);
+      ([:: 1; 1; 1; 1; 1; 1; 1; 1], Posz 1)])%N.
+ +
+End Tests.
+ +
+
+ +
+

Murnaghan-Nakayama rule

+ +
+
+Section MNRule.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+ +
+Theorem MN_coeffP_int d (la : 'P_d) :
+  'p[la] = \sum_(sh : 'P_d) MN_coeff sh la *: 's[sh] :> {sympoly int[n]}.
+ +
+Variable R : comNzRingType.
+#[local] Notation SF := {sympoly R[n]}.
+#[local] Notation HSF := {homsym R[n, _]}.
+ +
+Theorem MN_coeffP d (la : 'P_d) :
+  'p[la] = \sum_(sh : 'P_d) (MN_coeff sh la)%:~R *: 's[sh] :> SF.
+ +
+Corollary MN_coeff_homogP d (la : 'P_d) :
+  'hp[la] = \sum_(sh : 'P_d) (MN_coeff sh la)%:~R *: 'hs[sh] :> HSF.
+ +
+End MNRule.
+ +
+
+ +
+MN_coeff_rec should only be used when sumn la == sum nu + sumn mu. +
+
+Fixpoint MN_coeff_rec (la nu mu : seq nat) : int :=
+  if mu is m0 :: m then
+    foldr (fun psh acc =>
+             MN_coeff_rec la psh.1 m * (-1) ^+ psh.2.-1 + acc)
+          0
+          [seq psh | psh <- pmap (add_ribbon nu m0.-1) (iota 0 (size la))
+                   & included psh.1 la]
+  else (la == nu).
+Definition MN_coeff_fast la mu := MN_coeff_rec la [::] mu.
+ +
+Lemma MN_coeff_rec_szE la nu m0 mu :
+  MN_coeff_rec la nu (m0 :: mu) =
+  \sum_(psh <- pmap (add_ribbon nu m0.-1) (iota 0 (size la))
+       | included psh.1 la)
+   MN_coeff_rec la psh.1 mu * (-1) ^+ psh.2.-1.
+ +
+Lemma MN_coeff_rec_notincl la nu mu :
+  0%N \notin mu -> is_part nu -> ~~ included nu la ->
+  MN_coeff_rec la nu mu = 0.
+ +
+Lemma MN_coeff_rec_consE n la nu m0 mu :
+  m0 != 0%N -> size la <= n ->
+  MN_coeff_rec la nu (m0 :: mu) =
+  \sum_(psh <- pmap (add_ribbon nu m0.-1) (iota 0 n) | included psh.1 la)
+   MN_coeff_rec la psh.1 mu * (-1) ^+ psh.2.-1.
+ +
+Section Tests.
+ +
+
+ +
+Tests : +
+sage: s(p[3, 3, 1, 1]).coefficient([5, 2, 1])
+-2
+
+ +
+
+Goal MN_coeff_fast [:: 5; 2; 1]%N [:: 3; 3; 1; 1]%N = - 2%:R.
+
+ +
+Tests : +
+sage: s(p[5, 2, 1]).coefficient([3, 3, 1, 1])
+1
+
+ +
+
+Goal MN_coeff_fast [:: 3; 3; 1; 1]%N [:: 5; 2; 1]%N = 1%:R.
+ +
+
+ +
+Tests : +
+sage: s(p[6, 5, 5, 4, 2, 1]).coefficient([12, 5, 2, 2, 1, 1])
+4
+
+ +
+
+Goal MN_coeff_fast [:: 12; 5; 2; 2; 1; 1]%N [:: 6; 5; 5; 4; 2; 1]%N = 4%:R.
+
+ +
+Tests : +
+sage: s(p[6, 5, 5, 4, 2, 1]).coefficient([12, 5, 3, 1, 1, 1])
+-2
+
+ +
+
+Goal MN_coeff_fast [:: 12; 5; 3; 1; 1; 1]%N [:: 6; 5; 5; 4; 2; 1]%N = - 2%:R.
+ Goal MN_coeff_fast [:: 12; 5; 3; 2; 1]%N [:: 6; 5; 5; 4; 2; 1]%N = - 3%:R.
+ Goal MN_coeff_fast [:: 12; 5; 4; 1; 1]%N [:: 6; 5; 5; 4; 2; 1]%N = 2%:R.
+ Goal MN_coeff_fast [:: 12; 5; 4; 2]%N [:: 6; 5; 5; 4; 2; 1]%N = 4%:R.
+ +
+Local Open Scope int_scope.
+ +
+ +
+Let partn := rev (enum_partn 5).
+Goal [seq [seq MN_coeff_fast la mu | mu <- partn] | la <- partn]
+ = [:: [:: 1; -1; 1; 1; -1; -1; 1];
+       [:: 4; -2; 1; 0; 0; 1; -1];
+       [:: 6; 0; 0; -2; 0; 0; 1];
+       [:: 5; -1; -1; 1; 1; -1; 0];
+       [:: 4; 2; 1; 0; 0; -1; -1];
+       [:: 5; 1; -1; 1; -1; 1; 0];
+       [:: 1; 1; 1; 1; 1; 1; 1]].
+ +
+End Tests.
+ +
+Section FastImplem.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+ +
+Lemma syms_prod_sympM_int dn (nu : 'P_dn) dm (mu : 'P_dm) :
+  's[nu] * 'p[mu] =
+  \sum_(la : 'P_(dn + dm)) MN_coeff_rec la nu mu *: 's[la] :> {sympoly int[n]}.
+ +
+Section ComRing.
+ +
+Variable R : comNzRingType.
+#[local] Notation SF := {sympoly R[n]}.
+#[local] Notation HSF := {homsym R[n, _]}.
+ +
+Theorem syms_prod_sympM dn (nu : 'P_dn) dm (mu : 'P_dm) :
+  's[nu] * 'p[mu] =
+  \sum_(la : 'P_(dn + dm)) (MN_coeff_rec la nu mu)%:~R *: 's[la] :> SF.
+ +
+Corollary homsyms_homsympM dn (nu : 'P_dn) dm (mu : 'P_dm) :
+  'hs[nu] *h 'hp[mu] =
+  \sum_(la : 'P_(dn + dm)) (MN_coeff_rec la nu mu)%:~R *: 'hs[la] :> HSF.
+ +
+Corollary MN_coeff_recP d (la : 'P_d) :
+  'p[la] = \sum_(sh : 'P_d) (MN_coeff_fast sh la)%:~R *: 's[sh] :> SF.
+ +
+Corollary MN_coeff_rec_homogP d (la : 'P_d) :
+  'hp[la] = \sum_(sh : 'P_d) (MN_coeff_fast sh la)%:~R *: 'hs[sh] :> HSF.
+ +
+End ComRing.
+ +
+End FastImplem.
+ +
+
+ +
+The two implementations coincide +
+
+Corollary MN_coeffE d (la mu : 'P_d) : MN_coeff_fast la mu = MN_coeff la mu.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.Schur_altdef.html b/combi/1.1.0/Combi.MPoly.Schur_altdef.html new file mode 100644 index 00000000..3d64e1fc --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.Schur_altdef.html @@ -0,0 +1,833 @@ + + + + + +Combi.MPoly.Schur_altdef: Alternants definition of Schur polynomials + + + + +
+ + + +
+ +

Library Combi.MPoly.Schur_altdef: Alternants definition of Schur polynomials

+ +
+
+ +
+ +
+
+
+ +
+

Definition of Schur polynomials as quotient of alternant and Kostka numbers

+ + +
+ +In the combinatorial definitions below, we use the following notations: +
    +
  • n : an integer denoting the number of variable. + +
    • +
  • la mu : partitions of an integer d + +
    • +
  • m : a monomial for the ring of polynomials {mpoly R[n]} + +
    • +
  • w : a sequence of 'I_n + +
    • +
  • t : a tableau + +
    • +
+ +
+ +We define the following notions: + +
+ +
    +
  • add_mesym la S == for a set S of 'I_n the partition obtained by + incrementing the parts in S if its a partition. If not + the value is unspecified. + +
    • +
  • setdiff la mu == the set of elements i of 'I_n for which the + i-th part of la is smaller than the one of mu. + +
    • +
+ +
+ +Kostka numbers: + +
+ +
    +
  • eval w == the evaluation of w as a n.-tuple nat. + +
    • +
  • KostkaTab la m == the set of tableaux of shape la and evaluation m. + +
    • +
  • KostkaMon la m == the number of Kostka tableau : #|KostkaTab la m|. + +
    • +
  • Kostka la mu == + +
    • +
  • 'K(la, mu) == the Kostka number associated to la and mu + as a nat in nat_scope and as an element of R (inferred from + the context) in ring_scope. + +
    • +
  • Kostak_rec la mu == a Coq implementation of the computation of the Kostka + number. It suppose that sumn la = sumn mu. + +
    • +
  • 'K^-1(la, mu) == the inverse Kostka number, that is the coefficient of the + inverse Kostka matrix computed using matrix inversion. + +
    • +
  • eqeval t la == the evaluation of t is la. More precisely, the evaluation + of the row reading of the tableau t is equal to the monomial + associated to la + +
    • +
+ +
+ +Kostka numbers and standard tableaux: + +
+ +In this section, we fix two nat n and d with hypothesis Hd : d <= n.+1. + +
+ +
    +
  • tabnat_of_ord Hd t == the standard tableau of shape la corresponding to + the tableau t : tabsh n la + +
    • +
  • tabord_of_nat Hd t == the tableau over 'I_n.+1 of shape la + corresponding to t : stdtabsh la + +
    • +
+ +
+ +Extension of tableaux: + +
+ +In this section, we fix +
    +
  • a non empty sequence of integers rcons s m whose size is less than n. + +
    • +
  • la : a partition of sumn s whose size is less than n. + +
    • +
  • mu : a partition of (sumn s) + m whose size is less than n. + +
    • +
+We are given some proofs Hs Hla Hmu that all these partition are of size +less than n. We suppose moreover that mu/la is an horizontal border +strip. The proof is denoted Hstrip. + +
+ +We denote by +
    +
  • T : any tableau of shape la and evaluation s. + +
    • +
  • Tm : any tableau of shape mu and evaluation rcons s m. + +
    • +
+Then we define: + +
+ +
    +
  • shape_res_tab Hsz Tm == the shape obtained by removing from Tm the boxes + containing size s + +
    • +
  • res_tab Hsz Hmu Hstrip Tm == the tableau obtained by removing from Tm + the boxes containing size s if its of shape la. Otherwise the result + is unspecified. + +
    • +
  • ext_tab Hsz Hmu Hstrip T == the tableau obtained by filling the boxes of mu/la + with size s. + +
    • +
+ +
+ +Then ext_tab and res_tab are two inverse bijections. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg ssrint fingroup perm.
+From mathcomp Require Import mpoly.
+ +
+Require Import tools combclass ordtype sorted partition tableau.
+Require Import skewpart skewtab antisym Schur_mpoly freeSchur therule.
+Require Import std stdtab unitriginv presentSn.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+Import GRing.Theory.
+ +
+#[local] Open Scope ring_scope.
+#[local] Open Scope Combi_scope.
+ +
+#[local] Reserved Notation "''a_' k"
+      (at level 8, k at level 2, format "''a_' k").
+#[local] Reserved Notation "''e_' k"
+      (at level 8, k at level 2, format "''e_' k").
+#[local] Reserved Notation "m # s"
+      (at level 40, left associativity, format "m # s").
+ +
+
+ +
+Alternating and symmetric polynomial +
+
+Section Alternant.
+ +
+Variables (n : nat) (R : comNzRingType).
+ +
+#[local] Notation rho := (rho n).
+#[local] Notation "''e_' k" := (mesym n R k).
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+ +
+Lemma alt_syme (m : 'X_{1..n}) k :
+  'a_(m + rho) * 'e_k =
+  \sum_(h : {set 'I_n} | #|h| == k) 'a_(m + mesym1 h + rho).
+ +
+#[local] Open Scope nat_scope.
+ +
+Section HasIncr.
+ +
+Variables (d k : nat) (la : 'P_d) (h : {set 'I_n}).
+ +
+#[local] Definition hasincr :=
+  has (fun i => (nth 0 (mpart la + mesym1 h)%MM i).+1 ==
+                 nth 0 (mpart la + mesym1 h)%MM i.+1) (iota 0 n.-1).
+ +
+Lemma hasincr0 : hasincr -> 'a_(mpart la + mesym1 h + rho) = 0%R :> {mpoly R[n]}.
+ +
+Lemma not_hasincr_part :
+  size la <= n -> ~~ hasincr ->
+  is_part_of_n (d + #|h|) (rem_trail0 (mpart la + mesym1 h)%MM).
+ +
+Let add_mpart_mesym :=
+  if [&& size la <= n, #|h| == k & ~~ hasincr]
+  then (rem_trail0 (mpart la + mesym1 h)%MM)
+  else rowpartn (d + k) .
+Lemma add_mpart_mesymP : is_part_of_n (d + k) add_mpart_mesym.
+Definition add_mesym : 'P_(d + k) := IntPartN add_mpart_mesymP.
+ +
+Lemma add_mesymE :
+  size la <= n -> #|h| == k -> ~~ hasincr ->
+  mpart add_mesym = (mpart la + mesym1 h)%MM.
+ +
+End HasIncr.
+ +
+Definition setdiff (la mu : seq nat) : {set 'I_n} :=
+  [set i : 'I_n | nth 0 la i < nth 0 mu i].
+ +
+Lemma card_setdiff d k (la : 'P_d) (mu : 'P_(d + k)) :
+  size mu <= n -> size la <= n -> vb_strip la mu -> #|setdiff la mu| = k.
+ +
+Lemma nth_add_setdiff d k (la : 'P_d) (mu : 'P_(d + k)) :
+  size mu <= n -> size la <= n -> vb_strip la mu ->
+  forall i,
+  nth 0 [seq (mpart la) i0 + (mesym1 (setdiff la mu)) i0 | i0 : 'I_n] i =
+  nth 0 mu i.
+ +
+Lemma nohasincr_setdiff d k (la : 'P_d) (mu : 'P_(d + k)) :
+  size mu <= n -> size la <= n ->
+  vb_strip la mu -> ~~ hasincr la (setdiff la mu).
+ +
+Lemma add_mesymK d k (la : 'P_d) :
+  size la <= n ->
+  {in [pred mu : 'P_(d + k) | vb_strip la mu && (size mu <= n)],
+  cancel (fun mu : 'P_(d + k) => setdiff la (val mu)) (add_mesym k la)}.
+ +
+
+ +
+

Piery's rule multplying an alternant by an elementary polynomial

+ +
+
+Theorem alt_mpart_syme d (la : 'P_d) k :
+  size la <= n ->
+  ('a_(mpart la + rho) * 'e_k =
+  \sum_(mu : 'P_(d + k) | vb_strip la mu && (size mu <= n))
+     'a_(mpart mu + rho))%R.
+ +
+Lemma vb_strip_rem_col0 d (la : 'P_d) :
+  vb_strip (conj_part (behead (conj_part la))) la.
+ +
+Lemma vb_strip_lexi (d1 k : nat) (la : 'P_(d1 + k)) (mu : seq nat) :
+  vb_strip mu la ->
+  sumn mu = d1 ->
+  is_part mu -> (val la <= incr_first_n mu k :> seqlexi nat)%O.
+ +
+End Alternant.
+ +
+
+ +
+

Cauchy-Jacobi definition of Schur function

+ +
+
+Section SchurAlternantDef.
+ +
+Variable (n0 : nat) (R : comNzRingType).
+#[local] Notation n := (n0.+1).
+#[local] Notation rho := (rho n).
+#[local] Notation "''s_[' k ']'" := (Schur n0 R k).
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+ +
+Lemma Schur_cast d d' (la : 'P_d) (Heq : d = d') :
+  Schur n0 R (cast_intpartn Heq la) = 's_[la].
+ +
+Theorem alt_SchurE d (la : 'P_d) :
+  size la <= n -> 'a_rho * 's_[la] = 'a_(mpart la + rho).
+ +
+Import LeqGeqOrder.
+ +
+Lemma mcoeff_alt_SchurE d (la mu : 'P_d) :
+  size la <= n -> size mu <= n ->
+  ('a_rho * Schur n0 R la)@_(mpart mu + rho) = (la == mu)%:R.
+ +
+End SchurAlternantDef.
+ +
+
+ +
+

Schur polynomials are symmetric at last

+ +
+
+Section IdomainSchurSym.
+ +
+Variable (n0 : nat) (R : idomainType).
+#[local] Notation n := (n0.+1).
+#[local] Notation rho := (rho n).
+#[local] Notation "''s_' k" := (Schur n0 R k).
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+ +
+Theorem alt_uniq d (la : 'P_d) (s : {mpoly R[n]}) :
+  size la <= n -> 'a_rho * s = 'a_(mpart la + rho) -> s = 's_la.
+ +
+Theorem Schur_sym_idomain d (la : 'P_d) : 's_la \is symmetric.
+ +
+End IdomainSchurSym.
+ +
+Section RingSchurSym.
+ +
+Variable (n0 : nat) (R : nzRingType).
+#[local] Notation n := (n0.+1).
+#[local] Notation "''s_' k" := (Schur n0 R k).
+ +
+Theorem Schur_sym d (la : 'P_d) : 's_la \is symmetric.
+ +
+Lemma Schur_homog (d : nat) (la : 'P_d) : 's_la \is d.-homog.
+ +
+End RingSchurSym.
+ +
+
+ +
+

Kostka numbers

+ +
+
+#[local] Open Scope nat_scope.
+ +
+Section DefsKostkaMon.
+ +
+Variables (d : nat) (la : 'P_d) (n : nat).
+Implicit Type m : 'X_{1..n.+1}.
+Definition eval (w : seq 'I_n.+1) := [tuple count_mem i w | i < n.+1].
+Definition KostkaTab m := [set t : tabsh la | eval (to_word t) == m].
+Definition KostkaMon m := #|KostkaTab m|.
+ +
+Lemma sumn_eval w : sumn (eval w) = size w.
+ +
+Lemma KostkaMon_sumeval m :
+  mdeg m != sumn la -> KostkaMon m = 0.
+ +
+Lemma evalE (R : nzRingType) m w :
+  (\prod_(v <- w) 'X_v)@_m = (eval w == m)%:R :> R.
+ +
+Lemma Kostka_Coeff (R : nzRingType) m : (Schur n R la)@_m = (KostkaMon m)%:R.
+ +
+Lemma perm_KostkaMon m1 m2 :
+  perm_eq m1 m2 -> KostkaMon m1 = KostkaMon m2.
+ +
+Lemma tab_eval_partdom (t : tabsh la) : partdom (eval (to_word t)) la.
+ +
+Lemma KostkaMon_partdom m : KostkaMon m != 0 -> partdom m la.
+ +
+End DefsKostkaMon.
+ +
+Section KostkaEq.
+ +
+Variables (d : nat) (la : 'P_d).
+ +
+Lemma Kostka_mnmwiden n (m : 'X_{1..n.+1}) :
+  KostkaMon la m = KostkaMon la (mnmwiden m).
+ +
+End KostkaEq.
+ +
+Section Kostka.
+ +
+Variable d : nat.
+Implicit Type la : 'P_d.
+ +
+Definition Kostka la mu :=
+  KostkaMon la (mpart (n := (size mu).-1.+1) mu).
+#[local] Notation "''K' ( la , mu )" := (Kostka la mu)
+  (at level 8, format "''K' ( la , mu )") : nat_scope.
+#[local] Notation "''K' ( la , mu )" := ((Kostka la mu)%:R : int)
+  (at level 8, format "''K' ( la , mu )") : ring_scope.
+ +
+#[local] Arguments mpart n s : clear implicits.
+ +
+Lemma mpartS n mu :
+  size mu <= n -> mnmwiden (mpart n mu) = mpart n.+1 mu.
+ +
+Lemma Kostka_any la mu n :
+  size mu <= n.+1 -> 'K(la, mu) = KostkaMon la (mpart n.+1 mu).
+ +
+Lemma Kostka_sumnE la mu : d != sumn mu -> Kostka la mu = 0.
+ +
+Lemma Kostka_size0 la mu :
+  size la > size mu -> 'K(la, mu) = 0.
+ +
+Lemma Kostka_partdom (la mu : 'PDom_d) : 'K(la, mu) != 0 -> (mu <= la)%O.
+ +
+Lemma Kostka0 (la mu : 'PDom_d) : ~~ (mu <= la)%O -> 'K(la, mu) = 0.
+ +
+Lemma Kostka_diag la : 'K(la, la) = 1.
+ +
+End Kostka.
+ +
+
+ +
+

Converting between standard tableau and tableau over 'I_n.

+ + +
+ +The type stdtabsh sh is a subtype of seq nat whereas tabsh sh is a subtype +of seq 'I_n.+1. We provide two conversion functions tabnat_of_ord and +tabord_of_nat +
+
+Section StdKostka.
+ +
+Variables (d : nat) (la : 'P_d).
+ +
+Section Nvar.
+ +
+Variable n : nat.
+Hypothesis Hd : d <= n.+1.
+ +
+Let td := [tuple ((i < d) : nat) | i < n.+1].
+ +
+Lemma stdtabsh_eval_to_word (t : stdtabsh la) :
+  eval [seq inord i | i <- to_word t] = td.
+ +
+Lemma tabsh_is_std (t : tabsh la) :
+  eval (to_word t) = td -> is_std [seq (i : nat) | i : 'I_n.+1 <- to_word t].
+ +
+Definition tabnat_of_ord_fun (t : tabsh la) :=
+  if (eval (to_word t) == td)
+  then [seq [seq (i : nat) | i : 'I_n.+1 <- r] | r <- t]
+  else hyper_stdtabsh la.
+ +
+Definition tabord_of_nat_fun (t : stdtabsh la) : seq (seq 'I_n.+1) :=
+  [seq map inord r | r <- t].
+ +
+Lemma tabnat_of_ord_subproof (t : tabsh la) :
+  is_stdtab_of_shape la (tabnat_of_ord_fun t).
+Definition tabnat_of_ord (t : tabsh la) :=
+  StdtabSh (sh := la) (tabnat_of_ord_subproof t).
+ +
+Lemma tabord_of_nat_subproof (t : stdtabsh la) :
+  is_tab_of_shape la (tabord_of_nat_fun t).
+Definition tabord_of_nat (t : stdtabsh la) :=
+  TabSh (sh := la) (tabord_of_nat_subproof t).
+ +
+Lemma tabord_of_natK : cancel tabord_of_nat tabnat_of_ord.
+ +
+Lemma tabnat_of_ordK :
+  {in [pred t : tabsh la | eval (to_word t) == td],
+   cancel tabnat_of_ord tabord_of_nat }.
+ +
+End Nvar.
+ +
+Lemma KostkaStd : Kostka la (colpartn d) = #|{: stdtabsh la}|.
+ +
+End StdKostka.
+ +
+Definition eqeval n (t : (seq (seq 'I_n.+1))) (la : seq nat) :=
+  eval (to_word t) == mpart la.
+ +
+Lemma eqevalP n (t : (seq (seq 'I_n.+1))) (la : seq nat) :
+  size la <= n.+1 ->
+  reflect (forall i : 'I_n.+1, count_mem i (to_word t) = nth 0 la i)
+          (eqeval t la).
+ +
+
+ +
+

Restricting and extending tableaux

+ +
+
+Section BijectionExtTab.
+ +
+Variable n : nat.
+ +
+#[local] Open Scope nat_scope.
+ +
+Variables (s : seq nat) (m : nat).
+Hypothesis (Hsz : size s < n.+1).
+#[local] Notation sz := (Ordinal Hsz).
+#[local] Lemma Hszrcons : size (rcons s m) <= n.+1.
+ +
+#[local] Notation P := ('P_(sumn s)).
+#[local] Notation Pm := ('P_((sumn s) + m)).
+Variable (mu : Pm).
+#[local] Notation Tm := (tabsh mu).
+Hypothesis Hmu : size mu <= n.+1.
+ +
+Lemma shape_res_tab_subproof (t : Tm) :
+  is_part_of_n (sumn s) (
+                 if eqeval t (rcons s m) then shape (filter_gt_tab sz t)
+                 else locked rowpartn (sumn s)).
+Definition shape_res_tab (t : Tm) := IntPartN (shape_res_tab_subproof t).
+ +
+Lemma hb_strip_shape_res_tab (t : Tm) :
+  eqeval t (rcons s m) -> hb_strip (shape_res_tab t) mu.
+ +
+Variable (la : P).
+Hypothesis Hstrip : hb_strip la mu.
+#[local] Notation T := (tabsh la).
+ +
+#[local] Definition Hlamu := size_included (hb_strip_included Hstrip).
+#[local] Definition Hla : size la <= n.+1 := leq_trans Hlamu Hmu.
+ +
+Definition res_tab (t : Tm) : T :=
+  if insub (filter_gt_tab sz t) is Some tr then tr
+  else locked (tabrowconst Hla).
+#[local] Definition ext_tab_fun (t : T) :=
+  if eqeval t s then join_tab t (skew_reshape la mu (nseq m sz))
+  else locked (tabrowconst Hmu).
+ +
+#[local] Lemma sumndiff : sumn (mu / la) = m.
+ +
+Lemma ext_tab_subproof t : is_tab_of_shape mu (ext_tab_fun t).
+Definition ext_tab t := TabSh (ext_tab_subproof t).
+ +
+Lemma res_tabP (t : Tm) :
+  shape (filter_gt_tab sz t) == la ->
+  is_tab_of_shape la (filter_gt_tab sz t).
+ +
+Lemma eval_res_tab (t : Tm) :
+  shape (filter_gt_tab sz t) == la ->
+  eqeval t (rcons s m) -> eqeval (res_tab t) s.
+ +
+Lemma eval_ext_tab (t : T) :
+  eqeval t s -> eqeval (ext_tab t) (rcons s m).
+ +
+Lemma res_tabK (t : Tm) :
+  shape (filter_gt_tab sz t) == la ->
+  eqeval t (rcons s m) -> ext_tab (res_tab t) = t.
+ +
+Corollary res_tab_inj :
+  {in [set x : Tm | eqeval x (rcons s m) & shape_res_tab x == la] &,
+      injective res_tab}.
+ +
+Lemma filter_ext_tab (t : T) :
+  eqeval t s -> filter_gt_tab sz (ext_tab t) = t.
+ +
+Lemma ext_tabK (t : T) : eqeval t s -> res_tab (ext_tab t) = t.
+ +
+Corollary ext_tab_inj : {in [pred t : T | eqeval t s] &, injective ext_tab }.
+ +
+Lemma card_eq_eval :
+  #|[set t : tabsh['I_n.+1] mu |
+     (eqeval t (rcons s m)) && (shape (filter_gt_tab sz t) == la)]|
+  = #|[set t : tabsh['I_n.+1] la | eqeval t s]|.
+ +
+End BijectionExtTab.
+ +
+
+ +
+

Recursive computation of Kostka numbers

+ +
+
+Section KostkaRec.
+ +
+#[local] Open Scope nat_scope.
+ +
+Lemma Kostka_ind d (la : 'P_d) m mu :
+  d = m + sumn mu ->
+  Kostka la (m :: mu) =
+  \sum_(nu : 'P_(sumn mu) | hb_strip nu la) Kostka nu mu.
+ +
+Fixpoint Kostka_rec (la mu : seq nat) : nat :=
+  if mu is m :: mu then
+    sumn [seq Kostka_rec nu mu | nu <- enum_partn (sumn mu) & hb_strip nu la]
+  else la == [::].
+ +
+Example Kostka_expl1 :
+  Kostka_rec [:: 4; 3] [:: 3; 2; 2] == 2.
+ Example Kostka_expl2 :
+  Kostka_rec [:: 4; 3; 1] [:: 3; 2; 2; 1] == 5.
+ Example Kostka_expl3 :
+  Kostka_rec [:: 4; 4; 3; 1] [:: 3; 3; 2; 2; 1; 1] == 25.
+ +
+Lemma Kostka_rec_size0 la mu :
+  size la > size mu -> Kostka_rec la mu = 0.
+ +
+Lemma Kostka_recE d (la : 'P_d) mu :
+  sumn mu = d -> Kostka_rec la mu = Kostka la mu.
+ +
+End KostkaRec.
+ +
+Notation "''K' ( la , mu )" := (Kostka la mu)
+  (at level 8, format "''K' ( la , mu )") : nat_scope.
+Notation "''K' ( la , mu )" := (Kostka la mu)%:R
+  (at level 8, format "''K' ( la , mu )") : ring_scope.
+ +
+Lemma Kostka_unitrig (R : comUnitRingType) d :
+  unitrig (fun la mu : 'PDom_d => 'K(la, mu)%:R : R).
+ +
+
+ +
+

Inverse Kostka numbers

+ +
+
+Definition KostkaInv d : 'P_d -> 'P_d -> int :=
+  Minv (fun la mu : 'PDom_d => 'K(la, mu)%:R : int).
+ +
+Lemma KostkaInv_unitrig d :
+  unitrig (fun la mu : 'PDom_d => KostkaInv la mu).
+ +
+Notation "''K^-1' ( la , mu )" := ((KostkaInv la mu)%:~R)
+  (at level 8, format "''K^-1' ( la , mu )") : ring_scope.
+ +
+
+ +
+

Straightening of alternant polynomials

+ +
+
+Section AlternStraighten.
+ +
+Variable n0 : nat.
+Variable R : comNzRingType.
+ +
+#[local] Notation n := n0.+1.
+#[local] Notation rho := (rho n).
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+#[local] Notation "m # s" := [multinom m (s i) | i < n].
+ +
+Lemma alt_straight_step (m : 'X_{1..n}) (i : nat) :
+  i < n0 -> m (inord i.+1) != 0%N ->
+  'a_(m + rho) = - 'a_(m # 's_i - U_(inord i) + U_(inord i.+1) + rho).
+ +
+Lemma alt_straight_add_ribbon0 (p : seq nat) (i : 'I_n) (d : nat) :
+  is_part p -> size p <= n ->
+  add_ribbon p d i == None -> 'a_(mpart p + rho + U_(i) *+ d.+1) = 0%R.
+ +
+Lemma alt_straight_add_ribbon (p : seq nat) (i : 'I_n) (d : nat) :
+  is_part p -> size p <= n ->
+  forall res h, add_ribbon p d i = Some (res, h) ->
+    'a_(mpart p + rho + U_(i) *+ d.+1) = (-1) ^+ h.-1 *: 'a_(mpart res + rho).
+ +
+End AlternStraighten.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.Schur_mpoly.html b/combi/1.1.0/Combi.MPoly.Schur_mpoly.html new file mode 100644 index 00000000..1b67f28b --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.Schur_mpoly.html @@ -0,0 +1,169 @@ + + + + + +Combi.MPoly.Schur_mpoly: Schur symmetric polynomials + + + + +
+ + + +
+ +

Library Combi.MPoly.Schur_mpoly: Schur symmetric polynomials

+ +
+
+ +
+ +
+
+
+ +
+

Combinatorial definition of Schur symmetric polynomials

+ + +
+ +
    +
  • Schur n0 R la == The Schur polynomial associated to the partition la in + {mpoly R[n0.+1]} as the sum of all tableau of shape + la over the alphabet 'I_n0.+1. + +
    • +
+ +
+ +We give some values for particular partition such as small one, rows and columns. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg.
+From mathcomp Require Import ssrcomplements freeg mpoly.
+ +
+Require Import tools ordtype partition tableau.
+ +
+Set Implicit Arguments.
+ +
+Import Order.TTheory.
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+Section Schur.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+Variable R : nzRingType.
+ +
+Lemma mons2mE s : 'X_[s2m s] = \prod_(i <- s) 'X_i :> {mpoly R[n]}.
+ +
+Definition Schur d (sh : 'P_d) : {mpoly R[n]} :=
+  \sum_(t : tabsh sh) \prod_(i <- to_word t) 'X_i.
+ +
+Lemma Schur_tabsh_readingE d (sh : 'P_d) :
+  Schur sh =
+  \sum_(t : d.-tuple 'I_n | tabsh_reading sh t) \prod_(i <- t) 'X_i.
+ +
+
+ +
+

Some particular Schur functions

+ +
+
+ +
+Lemma Schur0 (sh : 'P_0) : Schur sh = 1.
+ +
+Lemma Schur_oversize d (sh : 'P_d) : (size sh > n)%N -> Schur sh = 0.
+ +
+
+ +
+Equivalent definition of symh symmetric function +
+
+ +
+Lemma tabwordshape_row d (w : d.-tuple 'I_n) :
+  tabsh_reading (rowpartn d) w = sorted leq [seq val i | i <- w].
+ +
+Lemma symh_basisE d :
+  \sum_(s in basis n d) 'X_[s2m s] = Schur (rowpartn d).
+ +
+End Schur.
+ +
+Section SchurComRingType.
+ +
+Variable n0 : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+ +
+Lemma perm_enum_basis d :
+  perm_eq [seq s2m (tval s) | s in basis n d]
+          [seq val m | m in [set m : 'X_{1..n < d.+1} | mdeg m == d]].
+ +
+Lemma Schur_rowpartn d :
+  \sum_(m : 'X_{1..n < d.+1} | mdeg m == d) 'X_[m] = Schur n0 R (rowpartn d).
+ +
+
+ +
+The definition of column Schur symmetric polynomials agrees with mesym + from mpoly +
+
+ +
+Lemma tabwordshape_col d (w : d.-tuple 'I_n) :
+  tabsh_reading (colpartn d) w = sorted >%O w.
+ +
+Lemma mesym_SchurE d :
+  mesym n R d = Schur n0 R (colpartn d).
+ +
+Lemma Schur1 (sh : 'P_1) : Schur n0 R sh = \sum_(i < n) 'X_i.
+ +
+End SchurComRingType.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.antisym.html b/combi/1.1.0/Combi.MPoly.antisym.html new file mode 100644 index 00000000..47cf76d9 --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.antisym.html @@ -0,0 +1,636 @@ + + + + + +Combi.MPoly.antisym: Antisymmetric polynomials and Vandermonde product + + + + +
+ + + +
+ +

Library Combi.MPoly.antisym: Antisymmetric polynomials and Vandermonde product

+ +
+
+ +
+ +
+
+
+ +
+

Antisymmetric polynomials

+ + +
+ +Monomials and partitions: + +
+ +
    +
  • mpart s == the multi-monomial whose exponent are s if size s is smaller + than the number of variables. + +
    • +
  • partm m == the partition obtained by sorting the exponent of m. + +
    • +
  • m \is dominant == the exponent of m are sorted in reverse order. + +
    • +
+ +
+ +Antisymmetric polynomials: + +
+ +
    +
  • p \is antisym == p is an antisymmetric polynomial. This is a keyed predicate + closed by submodule operations submodPred. + +
    • +
+ +
+ +Vandermonde products and determinants: + +
+ +
    +
  • alternpol f == the alternating sunm of the permuted of f. + +
    • +
  • rho == the multi-monomial [n-1, n-2, ..., 1, 0] + +
    • +
  • Vanprod n R == the Vandermonde product in {mpoly R[n]}, that is the product + \prod_(i < j) ('X_i - 'X_j) . + +
    + + +
    • +
  • antim s == the n x n - matrix whose (i, j) coefficient is + 'X_i^(s j - rho j) + +
    • +
  • Vanmx == the Vandermonde matrix 'X_i^(n - 1 - j) = 'X_i^(rho j) . + +
    • +
  • Vandet == the Vandermonde determinant + +
    • +
+ +
+ +The main results are the Vandermonde determinant expansion: + +
+ +
    +
  • Vanprod_alt : Vanprod = alternpol 'X_[(rho n)] + +
    • +
  • Vandet_VanprodE : Vandet = Vanprod + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg ssrint fingroup perm zmodp binomial.
+From mathcomp Require Import ssrcomplements freeg mpoly.
+ +
+Require Import tools permcomp presentSn sorted partition.
+ +
+Set Implicit Arguments.
+ +
+Import LeqGeqOrder.
+ +
+#[local] Reserved Notation "''a_' k"
+      (at level 8, k at level 2, format "''a_' k").
+#[local] Reserved Notation "m # s"
+      (at level 40, left associativity, format "m # s").
+ +
+#[local] Notation "''II_' n" := ('I_n * 'I_n)%type (at level 8, n at level 2).
+ +
+Open Scope group_scope.
+Open Scope nat_scope.
+ +
+
+ +
+

monomials and partitions

+ +
+
+Section MonomPart.
+ +
+Variable n : nat.
+Implicit Type m : 'X_{1.. n}.
+ +
+Definition dominant : qualifier 0 'X_{1.. n} :=
+  [qualify m : 'X_{1.. n} | sorted geq m].
+Definition mpart (s : seq nat) :=
+  if size s <= n then [multinom (nth 0 s i)%N | i < n] else mnm0.
+ +
+Lemma dominant_eq m1 m2 :
+  m1 \is dominant -> m2 \is dominant -> perm_eq m1 m2 -> m1 = m2.
+ +
+Fact partmP m : is_part (sort geq [seq d <- m | d != 0]).
+Definition partm m := locked (IntPart (partmP m)).
+Lemma partmE m : partm m = sort geq [seq d <- m | d != 0] :> seq nat.
+ +
+Lemma size_partm m : size (partm m) <= n.
+ +
+Lemma mpart_is_dominant sh : is_part sh -> mpart sh \is dominant.
+ +
+Lemma is_dominant_partm m :
+  m \is dominant -> partm m = [seq d <- m | d != 0] :> seq nat.
+ +
+Lemma is_dominant_nth_partm m (i : 'I_n) :
+  m \is dominant -> nth 0 (partm m) i = m i.
+ +
+Lemma partmK m : m \is dominant -> mpart (partm m) = m.
+ +
+Lemma mpartK sh :
+  is_part sh -> size sh <= n -> partm (mpart sh) = sh :> seq nat.
+ +
+Lemma mpartE s i : size s <= n -> mpart s i = nth 0 s i.
+ +
+Lemma mpart0 : @mpart [::] = 0%MM.
+ +
+Lemma perm_mpart s1 s2 : perm_eq s1 s2 -> perm_eq (mpart s1) (mpart s2).
+ +
+Lemma perm_partm m1 m2 : perm_eq m1 m2 -> partm m1 = partm m2.
+ +
+Lemma partm_permK m : perm_eq m (mpart (partm m)).
+ +
+Lemma sumn_mpart sh : size sh <= n -> sumn (mpart sh) = sumn sh.
+ +
+Lemma mdeg_mpart sh : size sh <= n -> mdeg (mpart sh) = sumn sh.
+ +
+Lemma sumn_partm m : sumn (partm m) = mdeg m.
+ +
+#[local] Notation "m # s" := [multinom m (s i) | i < n].
+ +
+Lemma mnm_perm m1 m2 : perm_eq m1 m2 -> {s : 'S_n | m1 == m2 # s}.
+ +
+Lemma perm_mpart_partm m : {s : 'S_n | (mpart (partm m)) # s == m}.
+ +
+Lemma mpart_partm_perm m : {s : 'S_n | (mpart (partm m)) == m # s}.
+ +
+End MonomPart.
+ +
+Arguments mpart [n] s.
+Arguments dominant {n}.
+ +
+Import GRing.Theory.
+#[local] Open Scope ring_scope.
+#[local] Definition simplexp := (expr0, expr1, scale1r, scaleN1r,
+                              mulrN, mulNr, mulrNN).
+ +
+
+ +
+

Change of scalar in multivariate polynomials

+ +
+
+Section ScalarChange.
+ +
+Variables R S : nzRingType.
+Variable mor : {rmorphism R -> S}.
+Variable n : nat.
+ +
+Lemma map_mpolyX (m : 'X_{1..n}) : map_mpoly mor 'X_[m] = 'X_[m].
+ +
+Lemma msym_map_mpoly s (p : {mpoly R[n]}) :
+  msym s (map_mpoly mor p) = map_mpoly mor (msym s p).
+ +
+End ScalarChange.
+ +
+
+ +
+

Characteristic of multivariate polynomials

+ +
+
+Lemma char_mpoly n (R : nzRingType) : [pchar R] =i [pchar {mpoly R[n]}].
+ +
+
+ +
+

Symmetric and Antisymmetric multivariate polynomials

+ +
+
+Section MPolySym.
+ +
+Variable n : nat.
+Variable R : nzRingType.
+ +
+Implicit Types p q r : {mpoly R[n]}.
+ +
+Lemma issym_tpermP p :
+  reflect (forall i j, msym (tperm i j) p = p) (p \is symmetric).
+ +
+Definition antisym : qualifier 0 {mpoly R[n]} :=
+  [qualify p | [forall s, msym s p == (-1) ^+ s *: p]].
+ +
+Fact antisym_key : pred_key antisym.
+Canonical antisym_keyed := KeyedQualifier antisym_key.
+ +
+Lemma isantisymP p :
+  reflect (forall s, msym s p = (-1) ^+ s *: p) (p \is antisym).
+ +
+Lemma isantisym_tpermP p :
+  reflect (forall i j, msym (tperm i j) p = if (i != j) then - p else p)
+          (p \is antisym).
+ +
+Lemma antisym_pchar2 : (2 \in [pchar R]) -> symmetric =i antisym.
+ +
+Lemma perm_smalln : n <= 1 -> forall s : 'S_n, s = 1%g.
+ +
+Lemma sym_smalln : n <= 1 -> (@symmetric n R) =i predT.
+ +
+Lemma antisym_smalln : n <= 1 -> antisym =i predT.
+ +
+Lemma antisym_zmod : zmod_closed antisym.
+ +
+ +
+Lemma antisym_submod_closed : submod_closed antisym.
+ +
+ +
+Lemma sym_anti p q :
+  p \is antisym -> q \is symmetric -> p * q \is antisym.
+ +
+Lemma anti_anti p q :
+  p \is antisym -> q \is antisym -> p * q \is symmetric.
+ +
+#[local] Notation "m # s" := [multinom m (s i) | i < n].
+ +
+Lemma isantisym_msupp p (s : 'S_n) (m : 'X_{1..n}) : p \is antisym ->
+  (m#s \in msupp p) = (m \in msupp p).
+ +
+Import Order Order.Syntax Order.TotalTheory.
+ +
+Lemma mlead_antisym_sorted (p : {mpoly R[n]}) : p \is antisym ->
+  forall (i j : 'I_n), i <= j -> (mlead p) j <= (mlead p) i.
+ +
+End MPolySym.
+ +
+Arguments antisym {n R}.
+ +
+Lemma issym_eltrP n (R : nzRingType) (p : {mpoly R[n.+1]}) :
+  reflect (forall i, i < n -> msym 's_i p = p) (p \is symmetric).
+ +
+Lemma isantisym_eltrP n (R : nzRingType) (p : {mpoly R[n.+1]}) :
+  reflect (forall i, i < n -> msym 's_i p = - p) (p \is antisym).
+ +
+
+ +
+

Alternating polynomials

+ +
+
+Definition alternpol n (R : nzRingType) (f : {mpoly R[n]}) : {mpoly R[n]} :=
+  \sum_(s : 'S_n) (-1) ^+ s *: msym s f.
+ +
+Section AlternIDomain.
+ +
+Variable n : nat.
+Variable R : idomainType.
+Hypothesis Hchar : ~~ (2 \in [pchar R]).
+ +
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+#[local] Notation "m # s" := [multinom m (s i) | i < n].
+ +
+Lemma sym_antisym_char_not2 :
+  n >= 2 -> forall p : {mpoly R[n]}, p \is symmetric -> p \is antisym -> p = 0.
+ +
+Definition rho := [multinom (n - 1 - i)%N | i < n].
+ +
+Lemma rho_iota : rho = rev (iota 0 n) :> seq nat.
+ +
+Lemma rho_uniq : uniq rho.
+ +
+Lemma mdeg_rho : mdeg rho = 'C(n, 2).
+ +
+Lemma alt_homog : 'a_(rho) \is 'C(n, 2).-homog.
+ +
+Lemma alt_anti m : 'a_m \is antisym.
+ +
+
+ +
+

The leading monomial of an antisymmetric polynomial

+ +
+
+Section LeadingMonomial.
+ +
+Variable p : {mpoly R[n]}.
+ +
+Implicit Types q r : {mpoly R[n]}.
+ +
+Hypothesis Hpn0 : p != 0.
+Hypothesis Hpanti : p \is antisym.
+ +
+Lemma sym_antiE q : (q \is symmetric) = (p * q \is antisym).
+ +
+Lemma isantisym_msupp_uniq (m : 'X_{1..n}) : m \in msupp p -> uniq m.
+ +
+Hypothesis Hphomog : p \is 'C(n , 2).-homog.
+ +
+Lemma isantisym_mlead_iota : mlead p = rev (iota 0 n) :> seq nat.
+ +
+Lemma isantisym_mlead_rho : mlead p = rho.
+ +
+End LeadingMonomial.
+ +
+Lemma isantisym_alt (p : {mpoly R[n]}) :
+  p != 0 -> p \is antisym -> p \is ('C(n, 2)).-homog -> p = p@_(rho) *: 'a_rho.
+ +
+End AlternIDomain.
+ +
+
+ +
+

Vandermonde product

+ +
+
+Definition Vanprod {n} {R : nzRingType} : {mpoly R[n]} :=
+  \prod_(p : 'II_n | p.1 < p.2) ('X_p.1 - 'X_p.2).
+ +
+Section EltrP.
+ +
+Variable n i : nat.
+Implicit Type (p : 'II_n.+1).
+ +
+#[local] Definition eltrp p := ('s_i p.1, 's_i p.2).
+#[local] Definition predi p := (p.1 < p.2) && (p != (inord i, inord i.+1)).
+ +
+Lemma eltrpK : involutive eltrp.
+ +
+Lemma predi_eltrp p : i < n -> predi p -> predi (eltrp p).
+ +
+Lemma predi_eltrpE p : i < n -> predi p = predi ('s_i p.1, 's_i p.2).
+ +
+End EltrP.
+ +
+Lemma Vanprod_anti n (R : comNzRingType) : @Vanprod n R \is antisym.
+ +
+Lemma sym_VanprodM n (R : comNzRingType) (p : {mpoly R[n]}) :
+  p \is symmetric -> Vanprod * p \is antisym.
+ +
+Section Vanprod.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+ +
+#[local] Notation Delta := (@Vanprod n R).
+#[local] Notation "'X_ i" := (@mpolyX n R U_(i)). #[local] Notation rho := (rho n).
+#[local] Notation "''a_' k" := (alternpol 'X_[k]).
+ +
+Lemma polyX_inj (i j : 'I_n) : 'X_i = 'X_j -> i = j.
+ +
+Lemma diffX_neq0 (i j : 'I_n) : i != j -> 'X_i - 'X_j != 0.
+ +
+Lemma msuppX1 i : msupp 'X_i = [:: U_(i)%MM].
+ +
+Let abound b : {mpoly R[n]} :=
+  \prod_(p : 'II_n | p.1 < p.2 <= b) ('X_p.1 - 'X_p.2).
+Let rbound b := [multinom (b - i)%N | i < n].
+ +
+Lemma mesymlm_rbound b : (mesymlm n b <= rbound b)%MM.
+ +
+Lemma coeffXdiff (b : 'I_n) (k : 'X_{1..n}) (i : 'I_n) :
+  (k <= rbound b)%MM -> ('X_i - 'X_b)@_k = (k == U_(i)%MM)%:R.
+ +
+Lemma coeff_prodXdiff (b : 'I_n) (k : 'X_{1..n}) :
+  (k <= rbound b)%MM ->
+  (\prod_(i < n | i < b) ('X_i - 'X_b))@_k = (k == mesymlm n b)%:R.
+ +
+Lemma mcoeff_arbound b : b < n -> (abound b)@_(rbound b) = 1.
+ +
+Lemma Vanprod_coeff_rho : Delta@_rho = 1.
+ +
+Corollary Vanprod_neq0 : Delta != 0.
+ +
+Lemma Vanprod_dhomog : Delta \is 'C(n, 2).-homog.
+ +
+End Vanprod.
+ +
+Theorem Vanprod_alt_int n :
+  Vanprod = alternpol 'X_[rho n] :> {mpoly int[n]}.
+ +
+Corollary Vanprod_alt n (R : nzRingType) :
+  Vanprod = alternpol 'X_[rho n] :> {mpoly R[n]}.
+ +
+From mathcomp Require Import matrix.
+ +
+
+ +
+

Vandermonde matrix and determinant

+ +
+
+Section VandermondeDet.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+ +
+#[local] Notation "''a_' k" := (@alternpol n R 'X_[k]).
+#[local] Notation rho := (rho n).
+ +
+Definition antim (s : seq nat) : 'M[ {mpoly R[n]} ]_n :=
+  \matrix_(i, j < n) 'X_i ^+ (nth 0 s j + (n - 1) - j)%N.
+Definition Vanmx : 'M[ {mpoly R[n]} ]_n :=
+  \matrix_(i, j < n) 'X_i ^+ (n - 1 - j).
+Definition Vandet := \det Vanmx.
+ +
+#[local] Open Scope ring_scope.
+ +
+Lemma Vanmx_antimE : Vanmx = antim [::].
+ +
+Lemma alt_detE s : 'a_(s + rho) = \det (antim s).
+ +
+Corollary Vandet_VanprodE : Vandet = Vanprod.
+ +
+Lemma mcoeff_alt (m : 'X_{1..n}) : uniq m -> ('a_m)@_m = 1.
+ +
+Lemma alt_uniq_non0 (m : 'X_{1..n}) : uniq m -> 'a_m != 0.
+ +
+Lemma alt_rho_non0 : 'a_rho != 0.
+ +
+Lemma alt_alternate (m : 'X_{1..n}) (i j : 'I_n) :
+  i != j -> m i = m j -> 'a_m = 0.
+ +
+Lemma alt_add1_0 (m : 'X_{1..n}) i :
+  (nth 0%N m i).+1 = nth 0%N m i.+1 -> 'a_(m + rho) = 0.
+ +
+End VandermondeDet.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.homogsym.html b/combi/1.1.0/Combi.MPoly.homogsym.html new file mode 100644 index 00000000..1a9289fb --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.homogsym.html @@ -0,0 +1,870 @@ + + + + + +Combi.MPoly.homogsym: Homogenous Symmetric Polynomials + + + + +
+ + + +
+ +

Library Combi.MPoly.homogsym: Homogenous Symmetric Polynomials

+ +
+
+ +
+ +
+
+
+ +
+

The Space of Homogeneous Symmetric Polynomials

+ + +
+ +In this file we study the vector space of homogeneous symmetric polynomials. +The main goal is to construct its classical basis and Cauchy's scalar product. + +
+ +
    +
  • f \is d.-homsym == f is a homogenerous of degree d symmetric polynomial. + +
    • +
  • f \is [in R[n], d.-homsym] == idem specifying the ring and number of + variables. + +
    • +
  • {homsym R[n, d]} == the space of homogenerous of degree d symmetric + polynomials in n variables over R. + +
    • +
  • p *h q == the product of two homogeneous symmetric polynomial as a + homogeneous symmetric polynomials. + +
    • +
+ +
+ +The classical bases: + +
+ +
    +
  • 'he[la] == the elementary hom. sym. poly. associated to la + +
    • +
  • 'hh[la] == the complete hom. sym. poly. associated to la + +
    • +
  • 'hp[la] == the power sum hom. sym. poly. associated to la + +
    • +
  • 'hm[la] == the monomial hom. sym. poly. associated to la + +
    • +
  • 'hs[la] == the Schur hom. sym. poly. associated to la + +
    + + +
    • +
  • in_homsym d p == if p is a polynomial {mpoly R[n]} which is both + symmetric and homogeneous of degree d, return it as a + {homsym R[n, d]}. It is canonically linear. + +
    + + +
    • +
  • 'he == the elementary hom. sym. basis + +
    • +
  • 'hh == the complete hom. sym. basis + +
    • +
  • 'hp == the power sum hom. sym. basis + +
    • +
  • 'hm == the monomial hom. sym. basis + +
    • +
  • 'hs == the Schur hom. sym. basis + +
    • +
+ +
+ +The omega involution + +
+ +
    +
  • omegahomsym f == the image of f under the omega involution. + +
    • +
+ +
+ +Changing the base ring and the number of variables: + +
+ +
    +
  • intpart_of_mon m == if m is the monomial x_1^{i_1}x_2^{i_2}...x_n^{i_n} + returns the integer partition n^{i_n}...2^{i_2}1^{i_1} + +
    • +
  • intpartn_of_mon H == the same as an intpart_of_mon d where H is a proof of + mnmwgt m = d + +
    + + +
    • +
  • map_homsym mor f == change the base ring of the hom. sym. poly f using + the ring morphism mor. + This is canonically a zmodule morphism. + +
    • +
  • cnvarhomsym n f == change the number of variables of the hom. sym. poly + f by sending elementary to elementary. This is + canonically linear. + +
    • +
+ +
+ +The scalar product: + +
+ +
    +
  • '[ u | v ] == the scalar product of hom. sym. poly., only defined over + the field algC. + +
    • +
  • '[ u | v ] _(n, d) == the scalar product of hom. sym. poly. specifying + the number of variable and degree. + +
    • +
+ +
+ +The main results are symbm_basis, symbe_basis, symbs_basis, +symbh_basis, symbp_basis which asserts that they are all bases (if the +characteristic of the base ring is zero for symbp_basis), and the definition +of the scalar product. + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import ssralg matrix vector ssrnum algC archimedean.
+From mathcomp Require Import sesquilinear.
+From mathcomp Require Import fingroup perm.
+From mathcomp Require Import ssrcomplements freeg mpoly.
+ +
+Require Import tools sorted ordtype permuted partition permcent.
+Require Import antisym Schur_mpoly Schur_altdef sympoly.
+ +
+Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Import ssrnum algC GRing.Theory Num.Theory.
+ +
+#[local] Open Scope nat_scope.
+#[local] Open Scope ring_scope.
+ +
+Reserved Notation "d .-homsym" (at level 1, format "d .-homsym").
+Reserved Notation "[ 'in' R [ n ] , d .-homsym ]"
+  (at level 0, R, n at level 2, d at level 0,
+     format "[ 'in' R [ n ] , d .-homsym ]").
+Reserved Notation "{ 'homsym' T [ n , d ] }"
+  (at level 0, T, n, d at level 2, format "{ 'homsym' T [ n , d ] }").
+ +
+Reserved Notation "''he[' k ]" (at level 8, k at level 2, format "''he[' k ]").
+Reserved Notation "''hh[' k ]" (at level 8, k at level 2, format "''hh[' k ]").
+Reserved Notation "''hp[' k ]" (at level 8, k at level 2, format "''hp[' k ]").
+Reserved Notation "''hm[' k ]" (at level 8, k at level 2, format "''hm[' k ]").
+Reserved Notation "''hs[' k ]" (at level 8, k at level 2, format "''hs[' k ]").
+ +
+Reserved Notation "''he'" (at level 8, format "''he'").
+Reserved Notation "''hh'" (at level 8, format "''hh'").
+Reserved Notation "''hp'" (at level 8, format "''hp'").
+Reserved Notation "''hm'" (at level 8, format "''hm'").
+Reserved Notation "''hs'" (at level 8, format "''hs'").
+ +
+Reserved Notation "p *h q" (at level 20, format "p *h q").
+ +
+Reserved Notation "'[ p | q ]"
+  (at level 0, format "'[hv' ''[' p | '/ ' q ] ']'").
+Reserved Notation "'[ p | q ] _( R , n )"
+  (at level 0, format "'[hv' ''[' p | '/ ' q ] ']' '_(' R , n )").
+ +
+ +
+Definition ishomogsym1 {n} {R : nzRingType} (d : nat) :
+  qualifier 0 {sympoly R[n]} := [qualify p | sympol p \is d.-homog].
+ +
+Module SymPolyHomogKey.
+Fact homogsym1_key {n} {R : nzRingType} d : pred_key (@ishomogsym1 n R d).
+ Definition homogsym1_keyed {n R} d := KeyedQualifier (@homogsym1_key n R d).
+End SymPolyHomogKey.
+Canonical SymPolyHomogKey.homogsym1_keyed.
+ +
+Notation "d .-homsym" := (@ishomogsym1 _ _ d) : form_scope.
+Notation "[ 'in' R [ n ] , d .-homsym ]" := (@ishomogsym1 n R d) : form_scope.
+ +
+
+ +
+

Homogeneous symmetric polynomials

+ +
+
+Section DefType.
+ +
+Variable n : nat.
+Variable R : nzRingType.
+Variable d : nat.
+ +
+Implicit Types p q : {sympoly R[n]}.
+ +
+Definition is_homsym p := sympol p \is d.-homog.
+ +
+Lemma homsymE p : (p \is d.-homsym) = (is_homsym p).
+ +
+Hypothesis Hvar : (d <= n.+1)%N.
+ +
+Record homogsym : predArgType :=
+  HomogSym {homsym :> {sympoly R[n]}; _ : is_homsym homsym}.
+ +
+ +
+Lemma homsym_inj : injective homsym.
+ +
+End DefType.
+ +
+Bind Scope ring_scope with homogsym.
+ +
+Notation "{ 'homsym' T [ n , d ] }" := (homogsym n T d).
+ +
+Section HomogSymLModType.
+ +
+Variable n : nat.
+Variable R : nzRingType.
+Variable d : nat.
+ +
+#[local] Notation is_homsym := (@is_homsym n R d).
+ +
+Fact is_homsym_submod_closed : submod_closed is_homsym.
+ +
+Lemma homsym_is_linear :
+  linear (@homsym n R d : {homsym R[n, d]} -> {sympoly R[n]}).
+ +
+Fact homsym_is_dhomog (x : {homsym R[n, d]}) : sympol x \is d.-homog.
+ Definition dhomog_of_homogsym (p : {homsym R[n, d]}) :=
+  DHomog (homsym_is_dhomog p).
+ +
+Fact dhomog_of_sym_is_linear : linear dhomog_of_homogsym.
+ +
+End HomogSymLModType.
+ +
+
+ +
+

Homogeneous symmetric polynomials as a vector space

+ +
+
+Section Vector.
+ +
+Variable n0 : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+ +
+Variable d : nat.
+#[local] Notation SF := {sympoly R[n]}.
+Implicit Type (la : 'P_d).
+ +
+Definition homsymm la : {homsym R[n, d]} := HomogSym (symm_homog n R la).
+Definition homsyme la : {homsym R[n, d]} := HomogSym (prod_syme_homog n R la).
+Definition homsymh la : {homsym R[n, d]} := HomogSym (prod_symh_homog n R la).
+Definition homsymp la : {homsym R[n, d]} := HomogSym (prod_symp_homog n R la).
+Definition homsyms la : {homsym R[n, d]} := HomogSym (syms_homog n0 R la).
+ +
+Lemma homsymmE (f : {homsym R[n, d]}) :
+  f = \sum_(l : 'P_d) f@_(mpart l) *: homsymm l.
+ +
+Fact homogsym_vecaxiom :
+  Vector.axiom #|[set p : 'P_d | (size p <= n)%N]| {homsym R[n, d]}.
+ +
+End Vector.
+ +
+Notation "''he[' k ]" := (homsyme _ _ k).
+Notation "''hh[' k ]" := (homsymh _ _ k).
+Notation "''hp[' k ]" := (homsymp _ _ k).
+Notation "''hm[' k ]" := (homsymm _ _ k).
+Notation "''hs[' k ]" := (homsyms _ _ k).
+ +
+
+ +
+

Products of homogeneous symmetric polynomials

+ +
+
+Section HomogSymProd.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+Variable c d : nat.
+ +
+Fact homsymprod_subproof (p : {homsym R[n, c]}) (q : {homsym R[n, d]}) :
+  homsym p * homsym q \is (c + d).-homsym.
+ Canonical homsymprod p q : {homsym R[n, c + d]} :=
+  HomogSym (homsymprod_subproof p q).
+Fact homsymprod_is_bilinear : bilinear_for *:%R *:%R homsymprod.
+ +
+#[local] Notation "p *h q" := (homsymprod p q).
+ +
+Lemma homsymprod0r p : p *h 0 = 0.
+Lemma homsymprodBr p q1 q2 : p *h (q1 - q2) = p *h q1 - p *h q2.
+ Lemma homsymprodNr p q : p *h (- q) = - p *h q.
+ Lemma homsymprodDr p q1 q2 : p *h (q1 + q2) = p *h q1 + p *h q2.
+ Lemma homsymprodMnr p q m : p *h (q *+ m) = (p *h q) *+ m.
+ Lemma homsymprod_sumr p I r (P : pred I) (q : I -> {homsym R[n, d]}) :
+  p *h (\sum_(i <- r | P i) q i) = \sum_(i <- r | P i) p *h q i.
+ Lemma homsymprodZr a p q : p *h (a *: q) = a *: (p *h q).
+ +
+Lemma homsymprod0l p : 0 *h p = 0.
+ Lemma homsymprodNl p q : (- q) *h p = - q *h p.
+ Lemma homsymprodDl p q1 q2 : (q1 + q2) *h p = q1 *h p + q2 *h p.
+ Lemma homsymprodBl p q1 q2 : (q1 - q2) *h p = q1 *h p - q2 *h p.
+ Lemma homsymprodMnl p q m : (q *+ m) *h p = q *h p *+ m.
+ Lemma homsymprod_suml p I r (P : pred I) (q : I -> {homsym R[n, c]}) :
+  (\sum_(i <- r | P i) q i) *h p = \sum_(i <- r | P i) q i *h p.
+ Lemma homsymprodZl p a q : (a *: q) *h p = a *: q *h p.
+ +
+End HomogSymProd.
+ +
+Notation "p *h q" := (homsymprod p q).
+ +
+Section HomSymProdGen.
+ +
+Variable n0 : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+#[local] Notation HSF := {homsym R[n, _]}.
+ +
+Section Merge.
+Variables (d1 d2 : nat) (la : 'P_d1) (mu : 'P_d2).
+ +
+Lemma homsymprod_hh : 'hh[la] *h 'hh[mu] = 'hh[la +|+ mu] :> HSF.
+ Lemma homsymprod_he : 'he[la] *h 'he[mu] = 'he[la +|+ mu] :> HSF.
+ Lemma homsymprod_hp : 'hp[la] *h 'hp[mu] = 'hp[la +|+ mu] :> HSF.
+ End Merge.
+ +
+Section Cons.
+Variables (d l0 : nat) (la : seq nat).
+Hypotheses (Hla : is_part_of_n d la)
+           (Hlla : is_part_of_n (l0 + d)%N (l0 :: la)).
+Notation Plla := (IntPartN Hlla).
+Notation Pla := (IntPartN Hla).
+ +
+Lemma homsymprod_h1h : 'hh[Plla] = 'hh[rowpartn l0] *h 'hh[Pla] :> HSF.
+ Lemma homsymprod_h1e : 'he[Plla] = 'he[rowpartn l0] *h 'he[Pla] :> HSF.
+ Lemma homsymprod_h1p : 'hp[Plla] = 'hp[rowpartn l0] *h 'hp[Pla] :> HSF.
+ End Cons.
+ +
+End HomSymProdGen.
+ +
+Section InHomSym.
+ +
+Variable n0 d : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+#[local] Notation Pol := {mpoly R[n]}.
+#[local] Notation HSF := {homsym R[n, d]}.
+ +
+#[local] Notation "''pi_' d" :=
+  (pihomog mdeg d) (at level 5, format "''pi_' d").
+ +
+
+ +
+TODO: Contribute to Pierre-Yves's multinomials +
+
+Lemma msym_pihomog nv s (p : {mpoly R[nv]}) :
+  msym s ('pi_d p) = 'pi_d (msym s p).
+ +
+Lemma pihomog_sym nv (p : {mpoly R[nv]}) :
+  p \is symmetric -> 'pi_d p \is symmetric.
+ +
+Definition in_homsym (p : Pol) : HSF :=
+  \sum_(la : 'P_d) p@_(mpart la) *: 'hm[la].
+ +
+Fact in_homsym_is_linear : linear in_homsym.
+ +
+Lemma in_homsymE (p : HSF) : in_homsym p = p.
+ +
+End InHomSym.
+ +
+
+ +
+

The omega involution

+ +
+
+Section OmegaHomSym.
+ +
+Variable n0 d : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+#[local] Notation HSF := {homsym R[n, d]}.
+Implicit Types (p q : HSF) (la : intpartn d).
+ +
+Fact omegahomsym_subproof p : is_homsym d (omegasf p).
+Definition omegahomsym p : HSF := HomogSym (omegahomsym_subproof p).
+Fact omegahomsym_is_linear : linear omegahomsym.
+ +
+Lemma omega_homsymh la :
+  (head 0%N la <= n)%N -> omegahomsym 'hh[la] = 'he[la].
+Lemma omega_homsyme la :
+  (head 0%N la <= n)%N -> omegahomsym 'he[la] = 'hh[la].
+Lemma omega_homsyms la :
+  (d <= n)%N -> omegahomsym 'hs[la] = 'hs[conj_intpartn la].
+Lemma omega_homsymp la :
+  (head 0%N la <= n)%N -> omegahomsym 'hp[la] = (-1) ^+ (d - size la) *: 'hp[la].
+ +
+End OmegaHomSym.
+ +
+Section OmegaProd.
+ +
+Variable n0 : nat.
+#[local] Notation n := (n0.+1).
+Variable R : comNzRingType.
+ +
+Lemma omegahomsym_rmorph c d (p : {homsym R[n, c]}) (q : {homsym R[n, d]}) :
+  omegahomsym (p *h q) = (omegahomsym p) *h (omegahomsym q).
+ +
+End OmegaProd.
+ +
+
+ +
+

The classical bases of homogeneous symmetric polynomials

+ +
+
+Section HomSymField.
+ +
+Variable n0 d : nat.
+#[local] Notation n := (n0.+1).
+Variable R : fieldType.
+#[local] Notation HSF := {homsym R[n, d]}.
+ +
+#[local] Notation Basis := (#|{: 'P_d}|.-tuple HSF).
+Definition symbe : Basis := [tuple of [seq 'he[la] | la : 'P_d]].
+Definition symbh : Basis := [tuple of [seq 'hh[la] | la : 'P_d]].
+Definition symbm : Basis := [tuple of [seq 'hm[la] | la : 'P_d]].
+Definition symbs : Basis := [tuple of [seq 'hs[la] | la : 'P_d]].
+Definition symbp : Basis := [tuple of [seq 'hp[la] | la : 'P_d]].
+ +
+Hypothesis Hd : (d <= n)%N.
+ +
+Lemma basis_homsym : [set p : 'P_d | (size p <= n)%N] =i {: 'P_d}.
+ +
+Lemma dim_homsym :
+  \dim (fullv (vT := HSF)) = #|{: 'P_d}|.
+ +
+Lemma symbm_free : free symbm.
+ +
+Lemma symbm_basis : basis_of fullv symbm.
+ +
+Lemma symbs_basis : basis_of fullv symbs.
+Lemma symbs_free : free symbs.
+ +
+Theorem mcoeff_symbs (la : 'P_d) f :
+  coord symbs (enum_rank la) f =
+  (alternpol 'X_[rho n] * sympol (homsym f))@_(mpart la + rho n).
+ +
+#[local] Notation E := [tuple mesym n R i.+1 | i < n].
+ +
+Definition intpart_of_mon m : seq nat :=
+  rev (flatten [tuple nseq (m i) i.+1 | i < n]).
+ +
+Lemma intpart_of_monP m : mnmwgt m = d -> is_part_of_n d (intpart_of_mon m).
+Canonical intpartn_of_mon m (H : mnmwgt m = d) := IntPartN (intpart_of_monP H).
+ +
+#[local] Lemma Esym : (forall i : 'I_n, E`_i \is symmetric).
+ +
+Lemma comp_symbe m (H : mnmwgt m = d) :
+  'X_[m] \mPo E = 'he[intpartn_of_mon H].
+ +
+Lemma in_homsym_comp_symbe m (H : mnmwgt m = d) :
+  in_homsym d ('X_[m] \mPo E) = 'he[intpartn_of_mon H].
+ +
+Lemma symbe_basis : basis_of fullv symbe.
+Lemma symbe_free : free symbe.
+ +
+Lemma symbh_basis : basis_of fullv symbh.
+Lemma symbh_free : free symbh.
+ +
+Lemma symbp_basis : [pchar R] =i pred0 -> basis_of fullv symbp.
+Lemma symbp_free : [pchar R] =i pred0 -> free symbp.
+ +
+End HomSymField.
+ +
+Notation "''he'" := (symbe _ _ _).
+Notation "''hh'" := (symbh _ _ _).
+Notation "''hp'" := (symbp _ _ _).
+Notation "''hm'" := (symbm _ _ _).
+Notation "''hs'" := (symbs _ _ _).
+ +
+
+ +
+

Changing the base field

+ +
+
+Section ChangeField.
+ +
+Variable R S : fieldType.
+Variable mor : {rmorphism R -> S}.
+ +
+Variable n0 d : nat.
+#[local] Notation n := (n0.+1).
+#[local] Notation HSFR := {homsym R[n, d]}.
+#[local] Notation HSFS := {homsym S[n, d]}.
+ +
+Fact map_sympoly_d_homog (p : HSFR) : map_sympoly mor p \is d.-homsym.
+Definition map_homsym (p : HSFR) : HSFS := HomogSym (map_sympoly_d_homog p).
+ +
+Fact map_homsym_is_zmod_morphism : zmod_morphism map_homsym.
+ +
+Lemma map_homsym_is_scalable : scalable_for (mor \; *:%R) map_homsym.
+ +
+Lemma coord_map_homsym (b : #|{: 'P_d}|.-tuple HSFR) j (f : HSFR) :
+  basis_of fullv b ->
+  basis_of fullv (map_tuple map_homsym b) ->
+  coord (map_tuple map_homsym b) j (map_homsym f) = mor (coord b j f).
+ +
+Lemma map_homsymm la : map_homsym 'hm[la] = 'hm[la].
+ Lemma map_homsyme la : map_homsym 'he[la] = 'he[la].
+ Lemma map_homsymh la : map_homsym 'hh[la] = 'hh[la].
+ Lemma map_homsymp la : map_homsym 'hp[la] = 'hp[la].
+ Lemma map_homsyms la : map_homsym 'hs[la] = 'hs[la].
+ +
+Lemma map_homsymbm : map_tuple map_homsym 'hm = 'hm.
+ Lemma map_homsymbe : map_tuple map_homsym 'he = 'he.
+ Lemma map_homsymbh : map_tuple map_homsym 'hh = 'hh.
+ Lemma map_homsymbp : map_tuple map_homsym 'hp = 'hp.
+ Lemma map_homsymbs : map_tuple map_homsym 'hs = 'hs.
+ +
+End ChangeField.
+ +
+
+ +
+

Extracting coords

+ +
+
+Section Coord.
+ +
+Variable n0 d : nat.
+#[local] Notation n := (n0.+1).
+Variable R : fieldType.
+#[local] Notation HSF := {homsym R[n, d]}.
+Implicit Type (la : 'P_d).
+ +
+Lemma symbmE la : ('hm)`_(enum_rank la) = 'hm[la] :> HSF.
+ Lemma symbeE la : ('he)`_(enum_rank la) = 'he[la] :> HSF.
+ Lemma symbhE la : ('hh)`_(enum_rank la) = 'hh[la] :> HSF.
+ Lemma symbpE la : ('hp)`_(enum_rank la) = 'hp[la] :> HSF.
+ Lemma symbsE la : ('hs)`_(enum_rank la) = 'hs[la] :> HSF.
+ +
+#[local] Lemma er_eqE (la mu : 'P_d) :
+  (enum_rank la == enum_rank mu) = (la == mu).
+ +
+#[local] Notation coord := (coord (vT := HSF)).
+ +
+Hypothesis (Hd : (d <= n)%N).
+Lemma coord_symbm la mu : coord 'hm (enum_rank mu) 'hm[la] = (la == mu)%:R.
+ Lemma coord_symbe la mu : coord 'he (enum_rank mu) 'he[la] = (la == mu)%:R.
+ Lemma coord_symbh la mu : coord 'hh (enum_rank mu) 'hh[la] = (la == mu)%:R.
+ Lemma coord_symbs la mu : coord 'hs (enum_rank mu) 'hs[la] = (la == mu)%:R.
+ +
+Lemma coord_symbp (char0 : [pchar R] =i pred0) la mu :
+  coord 'hp (enum_rank mu) 'hp[la] = (la == mu)%:R.
+ +
+End Coord.
+ +
+
+ +
+

Changing the number of variables

+ +
+
+Section ChangeNVar.
+ +
+Variable R : comNzRingType.
+Variable m0 n0 : nat.
+#[local] Notation m := m0.+1.
+#[local] Notation n := n0.+1.
+Variable d : nat.
+Hypothesis Hd : (d <= m)%N || (n0 < m)%N.
+ +
+Fact cnvarhomsym_subproof (p : {homsym R[m, d]}) :
+  (cnvarsym n0 p) \is d.-homsym.
+Definition cnvarhomsym (p : {homsym R[m, d]}) : {homsym R[n, d]} :=
+  HomogSym (cnvarhomsym_subproof p).
+Fact cnvarhomsym_is_linear : linear cnvarhomsym.
+ +
+Lemma cnvarhomsyme la : cnvarhomsym 'he[la] = 'he[la].
+Lemma cnvarhomsymh la : cnvarhomsym 'hh[la] = 'hh[la].
+Lemma cnvarhomsymp la : cnvarhomsym 'hp[la] = 'hp[la].
+Lemma cnvarhomsymm la : cnvarhomsym 'hm[la] = 'hm[la].
+Lemma cnvarhomsyms la : cnvarhomsym 'hs[la] = 'hs[la].
+ +
+End ChangeNVar.
+ +
+#[local] Lemma pchar0_algC : [pchar algC] =i pred0.
+ #[local] Hint Resolve pchar0_algC : core.
+ +
+
+ +
+

The scalar product

+ +
+
+Section ScalarProduct.
+ +
+Context {n0 d : nat}.
+#[local] Notation n := (n0.+1).
+#[local] Notation HSF := {homsym algC[n, d]}.
+ +
+Implicit Type (p q u v : HSF).
+ +
+Definition homsymdot p q : algC :=
+  \sum_(i < #|{: 'P_d}|)
+    (zcard (enum_val i))%:R * (coord 'hp i p) * (coord 'hp i q)^*.
+Notation "''[' u | v ]" := (homsymdot u v) : ring_scope.
+ +
+Lemma homsymdotE p q :
+  '[ p | q ] =
+  \sum_(la : 'P_d) (zcard la)%:R *
+    (coord 'hp (enum_rank la) p) * (coord 'hp (enum_rank la) q)^*.
+Lemma homsymdotC p q : '[p | q] = ('[q | p])^*.
+Fact homsymdot_is_bilinear : bilinear_for *%R (Num.conj \; *%R) homsymdot.
+Notation "''[' u | v ]" := (homsymdot u v) : ring_scope.
+ +
+Fact homsymdot_is_hermitian p q : '[p | q] = (-1) ^+ false * '[q | p]^*.
+ +
+Lemma homsymdot0l p : '[0 | p] = 0.
+ Lemma homsymdotNl p q : '[- q | p] = - '[q | p].
+ Lemma homsymdotDl p q1 q2 : '[q1 + q2 | p] = '[q1 | p] + '[q2 | p].
+ Lemma homsymdotBl p q1 q2 : '[q1 - q2 | p] = '[q1 | p] - '[q2 | p].
+ Lemma homsymdotMnl p q n : '[q *+ n | p] = '[q | p] *+ n.
+ Lemma homsymdot_suml p I r (P : pred I) (q : I -> HSF) :
+  '[\sum_(i <- r | P i) q i | p] = \sum_(i <- r | P i) '[q i | p].
+ Lemma homsymdotZl p a q : '[a *: q | p] = a * '[q | p].
+ +
+Lemma homsymdot0r p : '[p | 0] = 0.
+ Lemma homsymdotNr p q : '[p | - q] = - '[p | q].
+ Lemma homsymdotDr p q1 q2 : '[p | q1 + q2] = '[p | q1] + '[p | q2].
+ Lemma homsymdotBr p q1 q2 : '[p | q1 - q2] = '[p | q1] - '[p | q2].
+ Lemma homsymdotMnr p q n : '[p | q *+ n] = '[p | q] *+ n.
+ Lemma homsymdot_sumr p I r (P : pred I) (q : I -> HSF) :
+  '[p | \sum_(i <- r | P i) q i] = \sum_(i <- r | P i) '[p | q i].
+ Lemma homsymdotZr a p q : '[p | a *: q] = a^* * '[p | q].
+ +
+Hypothesis (Hd : (d <= n)%N).
+ +
+Fact homsymdot_is_dot f : f != 0 -> 0 < '[f | f].
+ +
+
+ +
+Formulas for other bases will be proved in Cauchy +
+
+Lemma homsymdotpp la mu :
+  '['hp[la] | 'hp[mu]] = (zcard la)%:R * (la == mu)%:R.
+ +
+Lemma homsymdot_omegasf f g :
+  '[omegahomsym f |omegahomsym g ] = '[f | g].
+ +
+Lemma homsymp_orthogonal : pairwise_orthogonal homsymdot 'hp.
+ +
+End ScalarProduct.
+ +
+Notation "''[' u | v ]" := (homsymdot u v) : ring_scope.
+Notation "''[' u | v ] _( n , d )" :=
+  (@homsymdot n d u v) (only parsing) : ring_scope.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.MPoly.sympoly.html b/combi/1.1.0/Combi.MPoly.sympoly.html new file mode 100644 index 00000000..a4dea71d --- /dev/null +++ b/combi/1.1.0/Combi.MPoly.sympoly.html @@ -0,0 +1,1521 @@ + + + + + +Combi.MPoly.sympoly: Symmetric Polynomials + + + + +
+ + + +
+ +

Library Combi.MPoly.sympoly: Symmetric Polynomials

+ +
+
+ +
+ +
+
+
+ +
+

The Ring of Symmetric Polynomials

+ + +
+ +
    +
  • {sympoly R[n]} == the ring of symmetric polynomial in n variable over R. + +
    • +
  • sympol f == the coercion from {sympoly R[n]} to {mpoly R[n]} + +
    • +
+ +
+ +Classical bases + +
+ +
    +
  • 'e_k == the k-th elementary symmetric polynomial + +
    • +
  • 'h_k == the k-th complete homogeneous symmetric polynomial + +
    • +
  • 'p_k == the k-th power sum symmetric polynomial + +
    + + +
    • +
  • prod_gen G la == given a familly of generators G : nat -> {sympoly R[n]} + the product \prod_(i <- la) G i. + +
    + + +
    • +
  • 'e[mu] == the product of elementary symmetric polynomial + +
    • +
  • 'h[mu] == the product of complete homogeneous symmetric polynomial + +
    • +
  • 'p[mu] == the product of power sum symmetric polynomial + +
    • +
  • 'm[mu] == the monomial symmetric polynomial + +
    • +
  • 's[mu] == the Schur symmetric polynomial + +
    + + +
    • +
  • coeff_prodgen Co la mu == the coefficient of the product 'g[la] + on 'g_[mu] assuming that co : forall d : nat, 'P_d -> R gives + the coefficients of 'f_i on 'g_[mu] + +
    • +
+ +
+ +Change of scalars + +
+ +
    +
  • map_sympoly M == the ring morphism {sympoly R[n]} -> {sympoly S[n]} + obtained by the change of scalar M : {rmorphism R -> S} + +
    • +
+ +
+ +Formula for basis changes + +
+ +
    +
  • perm_partn la == the number of composition which are permuted of la + +
    • +
  • prod_partsum la == the product of the sum of all the prefix of la + +
    • +
+ +
+ +We list here a few theorems expressing a basis in another one. See the file +for a more comprehensive list. The rule is that we call syma_to_symb when +we expand a genrator of syma in symb. We call syma_symb the expansion +of a basis element of syma in symb + +
+ +
    +
  • e and h : syme_to_symh symh_to_syme + +
    • +
  • s and m : syms_symm symm_syms + +
    • +
  • s and h : syms_symh symh_syms + +
    • +
  • h and p : symh_to_symp and Newton's formulas Newton_symh Newton_symh1 + +
    • +
  • e and p : Newton's formulas Newton_syme1 + +
    • +
  • h and m : symh_to_symm + +
    • +
  • p and m : symp_to_symm + +
    • +
+ +
+ +The omega involution + +
+ +
    +
  • omegasf == the omega involution, that is the unique ring morphism + exchanging 'e_k and 'h_k wheneven k is smaller than the + number of variables. + +
    • +
+ +
+ +Change of the number of variables + +
+ +
    +
  • sympolyf R m == the algebra morphism expanding any symetric polynomial + (in {sympoly R[m]}) as a polynomial in the 'e_i + (in {mpoly R[m]}) by the fundamental theorem of symmetric + polynomials. + +
    • +
  • cnvarsym R m n == the canonical algebra morphism + {sympoly R[m.+1]} -> {sympoly R[n.+1]} sending 'e_i to 'e_i + computed thanks to the fundamental theorem. + +
    • +
+ +
+ +We show that if d ≤ m or n ≤ m, for any partition in 'P_d the change of +number of variables sends a basis element 'b[la] to the same element. These +are lemmas + +
+ +cnvar_prodsyme, cnvar_prodsymh, cnvar_prodsymp, cnvar_syms +and cnvar_symm. + +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import ssralg ssrint fingroup perm.
+From mathcomp Require Import ssrcomplements freeg mpoly.
+ +
+Require Import sorted tools ordtype permuted partition skewpart composition.
+Require Import Yamanouchi std tableau stdtab skewtab permcent.
+Require Import antisym Schur_mpoly therule Schur_altdef unitriginv.
+ +
+Set Implicit Arguments.
+ +
+#[local] Open Scope ring_scope.
+Import GRing.Theory.
+ +
+Lemma boolRP (R : nzRingType) (b : bool) : reflect (b%:R = 0 :> R) (~~b).
+ +
+Section MultinomCompl.
+ +
+Variables (n : nat) (R : comNzRingType).
+ +
+Lemma mnm_n0E : @all_equal_to 'X_{1..0} 0%MM.
+ +
+Lemma eq_mnm1 (i j : 'I_n) : (U_(i)%MM == U_(j)%MM) = (i == j).
+ +
+Lemma mcoeffXU (i j : 'I_n) : ('X_i @_U_(j) : R) = (i == j)%:R.
+ +
+End MultinomCompl.
+ +
+Reserved Notation "{ 'sympoly' T [ n ] }"
+  (at level 0, T, n at level 2, format "{ 'sympoly' T [ n ] }").
+Reserved Notation "''e_' k" (at level 8, k at level 2, format "''e_' k").
+Reserved Notation "''h_' k" (at level 8, k at level 2, format "''h_' k").
+Reserved Notation "''p_' k" (at level 8, k at level 2, format "''p_' k").
+Reserved Notation "''e[' k ]" (at level 0, format "''e[' k ]").
+Reserved Notation "''h[' k ]" (at level 0, format "''h[' k ]").
+Reserved Notation "''p[' k ]" (at level 0, format "''p[' k ]").
+Reserved Notation "''m[' k ]" (at level 0, format "''m[' k ]").
+Reserved Notation "''s[' k ]" (at level 0, format "''s[' k ]").
+ +
+Section DefType.
+ +
+Variable n : nat.
+Variable R : nzRingType.
+ +
+Record sympoly : predArgType :=
+  SymPoly {sympol :> {mpoly R[n]}; _ : sympol \is symmetric}.
+ +
+ +
+Lemma sympol_inj : injective sympol.
+ +
+End DefType.
+ +
+Bind Scope ring_scope with sympoly.
+ +
+Notation "{ 'sympoly' T [ n ] }" := (sympoly n T).
+ +
+Section SymPolyRingType.
+ +
+Variable n : nat.
+Variable R : nzRingType.
+ +
+ +
+Fact sympol_is_linear : linear (@sympol n R).
+ Fact sympol_is_monoid_morphism : monoid_morphism (@sympol n R).
+ +
+Lemma sympolP (x : {sympoly R[n]}) : sympol x \is symmetric.
+ +
+End SymPolyRingType.
+ +
+#[export] Hint Resolve sympolP : core.
+ +
+Section SymPolyComRingType.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+ +
+ +
+End SymPolyComRingType.
+ +
+Section SymPolyIdomainType.
+ +
+Variable n : nat.
+Variable R : idomainType.
+ +
+ +
+End SymPolyIdomainType.
+ +
+Section Bases.
+ +
+Variable n : nat.
+ +
+Variable R : comNzRingType.
+Implicit Type m : 'X_{1.. n}.
+ +
+#[local] Notation "m # s" := [multinom m (s i) | i < n]
+  (at level 40, left associativity, format "m # s").
+ +
+
+ +
+

Elementary symmetric polynomials

+ +
+
+Fact syme_sym d : mesym n R d \is symmetric.
+Canonical syme d : {sympoly R[n]} := SymPoly (syme_sym d).
+#[local] Notation "''e_' k" := (syme k).
+ +
+Lemma syme_geqnE d : d > n -> 'e_d = 0.
+ Lemma syme_homog d : sympol 'e_d \is d.-homog.
+ +
+
+ +
+

Complete homogeneous symmetric polynomials

+ +
+
+Definition symh_pol_bound b d : {mpoly R[n]} :=
+  \sum_(m : 'X_{1..n < b} | mdeg m == d) 'X_[m].
+Definition symh_pol d : {mpoly R[n]} := symh_pol_bound d.+1 d.
+Lemma mcoeff_symh_pol_bound b d m :
+  b > d -> (symh_pol_bound b d)@_m = (mdeg m == d)%:R.
+Lemma mcoeff_symh_pol d m : (symh_pol d)@_m = (mdeg m == d)%:R.
+ Lemma symh_pol_any b d : d < b -> symh_pol d = symh_pol_bound b d.
+Fact symh_sym d : symh_pol d \is symmetric.
+Canonical symh d : {sympoly R[n]} := SymPoly (symh_sym d).
+#[local] Notation "''h_' k" := (symh k).
+ +
+Lemma mcoeff_symh d m : 'h_d@_m = (mdeg m == d)%:R.
+ Lemma symh_homog d : sympol 'h_d \is d.-homog.
+ +
+
+ +
+

Power sum symmetric polynomials

+ +
+
+Definition symp_pol d : {mpoly R[n]} := \sum_(i < n) 'X_i ^+ d.
+Fact symp_sym d : symp_pol d \is symmetric.
+Canonical symp d : {sympoly R[n]} := SymPoly (symp_sym d).
+#[local] Notation "''p_' k" := (symp k).
+ +
+Lemma symp_homog d : sympol 'p_d \is d.-homog.
+ +
+
+ +
+

Monomial symmetric polynomials

+ +
+
+Definition symm_pol (sh : n.-tuple nat) : {mpoly R[n]} :=
+  \sum_(p : permuted sh) 'X_[Multinom p].
+Lemma mcoeff_symm_pol sh m : (symm_pol sh)@_m = (perm_eq sh m)%:R.
+Fact symm_sym sh : symm_pol sh \is symmetric.
+Definition symm sh : {sympoly R[n]} :=
+  if size sh <= n then SymPoly (symm_sym (mpart sh)) else 0 : {sympoly R[n]}.
+Notation "''m[' k ]" := (symm k).
+ +
+Lemma symm_oversize sh : n < size sh -> 'm[sh] = 0.
+ Lemma mcoeff_symm sh m :
+  size sh <= n -> 'm[sh]@_m = (perm_eq (mpart (n := n) sh) m)%:R.
+ Lemma symm_homog d (sh : 'P_d) : sympol 'm[sh] \is d.-homog.
+ +
+Lemma symE (p q : {sympoly R[n]}) :
+  reflect (forall m, m \is dominant -> p@_m = q@_m) (p == q).
+ +
+
+ +
+

Expansion of symmetric polynomials on monomials

+ +
+
+Lemma sym_symmE (p : {sympoly R[n]}) :
+  p = \sum_(m <- msupp p | m \is dominant) p@_m *: 'm[partm m].
+ +
+Lemma size_mpart_in_supp (f : {mpoly R[n]}) d (p : 'P_d) :
+  f \is d.-homog -> mpart p \in msupp f -> size p <= n.
+ +
+Lemma dominant_mpart d m :
+  m \is dominant -> mdeg m = d -> { p : 'P_d | m = mpart p }.
+ +
+Lemma homog_symmE d (f : {sympoly R[n]}) :
+  sympol f \is d.-homog -> f = \sum_(l : 'P_d) f@_(mpart l) *: 'm[l].
+ +
+Lemma symm_unique d (f : {sympoly R[n]}) c :
+  f = \sum_(l : 'P_d) (c l) *: 'm[l] ->
+  forall l : 'P_d, size l <= n -> c l = f@_(mpart l).
+ +
+Lemma symm_unique0 d c :
+  \sum_(l : 'P_d) (c l) *: 'm[l] = 0 ->
+  forall l : 'P_d, size l <= n -> c l = 0.
+ +
+Lemma sum_symmE d (f : {sympoly R[n]}) :
+  \sum_(l : 'P_d) f@_(mpart l) *: 'm[l] =
+  \sum_(l <- [seq val p | p <- enum {: 'P_d}]) f@_(mpart l) *: 'm[l].
+ +
+
+ +
+

Basis at degree 0

+ +
+
+Lemma syme0 : 'e_0 = 1.
+ +
+Lemma symp0 : 'p_0 = n%:R.
+ +
+Lemma symh0 : 'h_0 = 1.
+ +
+Lemma symm0 : 'm[[::]] = 1.
+ +
+
+ +
+

All basis agrees at degree 1

+ +
+
+Lemma syme1 : val ('e_1) = \sum_(i < n) 'X_i.
+ +
+Lemma sympe1E : 'p_1 = 'e_1.
+ +
+Lemma symhe1E : 'h_1 = 'e_1.
+ +
+End Bases.
+ +
+Notation "''e_' k" := (syme _ _ k).
+Notation "''h_' k" := (symh _ _ k).
+Notation "''p_' k" := (symp _ _ k).
+Notation "''m[' k ]" := (symm _ _ k).
+ +
+Section ChangeBaseMonomial.
+ +
+Variables (n : nat) (R : comNzRingType).
+#[local] Notation SP := {sympoly R[n]}.
+ +
+Lemma expUmpartE nv k :
+  (U_(ord0) *+ k)%MM = mpart (rowpartn k) :> 'X_{1..nv.+1}.
+ +
+Lemma expUmpartNE nv k i (P : intpartn k.+1) :
+  ((U_(i) *+ k.+1)%MM == mpart P :> 'X_{1..nv.+1})
+  = (i == ord0) && (P == rowpartn k.+1).
+ +
+Lemma symp_to_symm k : 'p_k.+1 = 'm[rowpartn k.+1] :> SP.
+ +
+Lemma symh_to_symm k : 'h_k = \sum_(l : 'P_k) 'm[l] :> SP.
+ +
+Lemma syme_to_symm k : 'e_k = 'm[colpartn k] :> SP.
+ +
+End ChangeBaseMonomial.
+ +
+
+ +
+

Schur symmetric polynomials

+ +
+
+Section Schur.
+ +
+Variable n0 : nat.
+Variable R : comNzRingType.
+ +
+#[local] Notation n := n0.+1.
+ +
+Definition syms d (la : 'P_d) : {sympoly R[n]} :=
+  SymPoly (Schur_sym n0 R la).
+ +
+#[local] Notation "''s[' k ]" := (syms k).
+ +
+Lemma syms_homog d (la : 'P_d) : sympol 's[la] \is d.-homog.
+ +
+Lemma syms0 (la : 'P_0) : 's[la] = 1.
+ +
+Lemma syms1 (la : 'P_1) : 's[la] = \sum_(i < n) 'X_i :> {mpoly R[n]}.
+ +
+Lemma syms_rowpartn d : 's[rowpartn d] = 'h_d.
+ +
+Lemma syms_colpartn d : 's[colpartn d] = 'e_d.
+ +
+Lemma syms_oversize d (la : 'P_d) : n < size la -> 's[la] = 0.
+ +
+End Schur.
+ +
+Notation "''s[' k ]" := (syms _ _ k).
+ +
+
+ +
+

Multiplicative bases.

+ + +
+ +Given a family of generators 'g_k, we define 'g[la] as the product of the +generators \prod(i <- la) 'g_i. + +
+
+Section ProdGen.
+ +
+Variable n : nat.
+Variable R : comNzRingType.
+#[local] Notation SF := {sympoly R[n]}.
+ +
+Section Defs.
+ +
+Variable gen : nat -> SF.
+Hypothesis gen_homog : forall d, sympol (gen d) \is d.-homog.
+ +
+Definition prod_gen d (sh : 'P_d) := \prod_(i <- sh) gen i.
+ +
+#[local] Notation "''g_' k" := (gen k) (at level 8, format "''g_' k").
+#[local] Notation "''g[' k ]" := (prod_gen k) (at level 8, format "''g[' k ]").
+ +
+Lemma prod_gen_homog d (sh : 'P_d) : sympol 'g[sh] \is d.-homog.
+ +
+Lemma prod_gen0 (l : 'P_0) : 'g[l] = 1.
+ +
+Lemma prod_genM c d (l : 'P_c) (k : 'P_d) : 'g[l] * 'g[k] = 'g[l +|+ k].
+ +
+Lemma prod_gen_colpartn d : 'g[colpartn d] = 'g_1 ^+ d.
+ +
+Lemma prod_gen_cast d1 d2 (eq_d : d1 = d2) (la : 'P_d1) :
+  'g[cast_intpartn eq_d la] = 'g[la].
+ +
+End Defs.
+ +
+Variable gA gB : nat -> SF.
+Variable co : forall (d : nat), 'P_d -> R.
+ +
+#[local] Notation "''gA_' k" := (gA k) (at level 8, format "''gA_' k").
+#[local] Notation "''gA[' k ]" :=
+  (prod_gen gA k) (at level 8, format "''gA[' k ]").
+#[local] Notation "''gB_' k" := (gB k) (at level 8, format "''gB_' k").
+#[local] Notation "''gB[' k ]" :=
+  (prod_gen gB k) (at level 8, format "''gB[' k ]").
+ +
+Fixpoint coeff_prodgen_seq l : 'P_(sumn l) -> R :=
+  if l is l0 :: l' then
+    fun la : 'P_(sumn (l0 :: l')) =>
+             \sum_(p | la == p.1 +|+ p.2) co p.1 * coeff_prodgen_seq p.2
+  else fun _ => 1.
+ +
+#[local] Notation "''co[' k ]" := (coeff_prodgen_seq k)
+                                 (at level 8, format "''co[' k ]").
+#[local] Notation "''co[' k ]_ l" := (coeff_prodgen_seq (l := l) k)
+                                 (at level 8, only parsing).
+ +
+Definition coeff_prodgen d (la mu : 'P_d) : R :=
+  'co[cast_intpartn (esym (sumn_intpartn la)) mu].
+ +
+Lemma coeff_prodgen_cast l k nu (eqlamu : l = k) (eqsum : sumn l = sumn k) :
+  'co[cast_intpartn eqsum nu] = 'co[nu].
+ +
+Lemma prod_prodgen :
+  (forall d, 'gA_d = \sum_(la : 'P_d) co la *: 'gB[la] :> SF) ->
+  forall d (la : 'P_d),
+    'gA[la] = \sum_(mu : 'P_d)
+               coeff_prodgen la mu *: 'gB[mu] :> SF.
+ +
+Definition prod_syme := prod_gen (@syme n R).
+Definition prod_syme_homog := prod_gen_homog (@syme_homog n R).
+Definition prod_symh := prod_gen (@symh n R).
+Definition prod_symh_homog := prod_gen_homog (@symh_homog n R).
+Definition prod_symp := prod_gen (@symp n R).
+Definition prod_symp_homog := prod_gen_homog (@symp_homog n R).
+ +
+End ProdGen.
+ +
+Notation "''e[' k ]" := (prod_syme _ _ k).
+Notation "''h[' k ]" := (prod_symh _ _ k).
+Notation "''p[' k ]" := (prod_symp _ _ k).
+ +
+
+ +
+Casting the index +
+
+Section Cast.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+Variables R : comNzRingType.
+#[local] Notation SF := {sympoly R[n]}.
+ +
+Variables (d1 d2 : nat) (eq_d : d1 = d2) (la : 'P_d1).
+ +
+Lemma syms_cast : 's[cast_intpartn eq_d la] = 's[la] :> SF.
+ Lemma syme_cast : 'e[cast_intpartn eq_d la] = 'e[la] :> SF.
+ Lemma symh_cast : 'h[cast_intpartn eq_d la] = 'h[la] :> SF.
+ Lemma symp_cast : 'p[cast_intpartn eq_d la] = 'p[la] :> SF.
+ Lemma symm_cast : 'm[cast_intpartn eq_d la] = 'm[la] :> SF.
+ +
+End Cast.
+ +
+
+ +
+

Littlewood-Richardson and Pieri rules

+ +
+
+Section LRrule_Pieri.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+Variables R : comNzRingType.
+#[local] Notation SF := {sympoly R[n]}.
+ +
+Lemma syms_symsM d1 (la : 'P_d1) d2 (mu : 'P_d2) :
+  's[la] * 's[mu] =
+  \sum_(nu : 'P_(d1 + d2) | included la nu)
+     's[nu] *+ LRyam_coeff la mu nu :> SF.
+ +
+Lemma syms_symhM d1 (la : 'P_d1) d2 :
+  's[la] * 'h_d2 = \sum_(nu : 'P_(d1 + d2) | hb_strip la nu) 's[nu] :> SF.
+ +
+Lemma syms_symeM d1 (la : 'P_d1) d2 :
+  's[la] * 'e_d2 = \sum_(nu : 'P_(d1 + d2) | vb_strip la nu) 's[nu] :> SF.
+ +
+End LRrule_Pieri.
+ +
+
+ +
+

Change of scalars

+ +
+
+Section ScalarChange.
+ +
+Variables R S : comNzRingType.
+Variable mor : {rmorphism R -> S}.
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+ +
+Lemma map_mpoly_issym (f : {sympoly R[n]}) : map_mpoly mor f \is symmetric.
+Definition map_sympoly (f : {sympoly R[n]}) : {sympoly S[n]} :=
+           SymPoly (map_mpoly_issym f).
+ +
+Fact map_sympoly_is_zmod_morphism : zmod_morphism map_sympoly.
+ +
+Fact map_sympoly_is_monoid_morphism : monoid_morphism map_sympoly.
+ +
+Lemma map_sympoly_is_scalable : scalable_for (mor \; *:%R) map_sympoly.
+ +
+Lemma map_symm d : map_sympoly 'm[d] = 'm[d].
+ +
+Lemma map_syme d : map_sympoly 'e_d = 'e_d.
+Lemma map_syme_prod d (l : 'P_d) : map_sympoly 'e[l] = 'e[l].
+ +
+Lemma map_symh d : map_sympoly 'h_d = 'h_d.
+Lemma map_symh_prod d (l : 'P_d) : map_sympoly 'h[l] = 'h[l].
+ +
+Lemma map_symp d : map_sympoly 'p_d = 'p_d.
+Lemma map_symp_prod d (l : 'P_d) : map_sympoly 'p[l] = 'p[l].
+ +
+Lemma map_syms d (la : 'P_d) :
+  map_sympoly 's[la] = 's[la].
+ +
+End ScalarChange.
+ +
+
+ +
+

Bases change formulas

+ +
+
+Section ChangeBasis.
+ +
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+Variable R : comNzRingType.
+ +
+#[local] Notation "''Xn'" := 'X_{1.. n}.
+#[local] Notation "''Xn_' m" := 'X_{1.. n < (mdeg m).+1}
+          (at level 10, m at next level, format "''Xn_' m").
+#[local] Notation "''XXn_' m" := 'X_{1.. n < (mdeg m).+1, (mdeg m).+1}
+          (at level 10, m at next level, format "''XXn_' m").
+Implicit Type m : 'Xn.
+#[local] Notation SF := {sympoly R[n]}.
+ +
+
+ +
+

Bases change between homogeneous and elementary

+ +
+
+Lemma sum_symh_syme (d : nat) :
+  d != 0%N ->
+  \sum_(0 <= i < d.+1) (-1) ^+ i *: ('h_i * 'e_(d - i)) = 0 :> SF.
+ +
+Lemma sum_syme_symh (d : nat) :
+  d != 0%N ->
+  \sum_(0 <= i < d.+1) (-1) ^+ i *: ('e_i * 'h_(d - i)) = 0 :> SF.
+ +
+Section HandE.
+ +
+Variable E H : nat -> {sympoly R[n]}.
+ +
+Hypothesis E0 : E 0 = 1.
+Hypothesis H0 : H 0 = 1.
+Hypothesis Hanti : forall d : nat,
+    d != 0%N ->
+    \sum_(0 <= i < d.+1) (-1) ^+ i *: (H i * E (d - i)) = 0.
+ +
+Lemma symHE_rec (d : nat) :
+  d != 0%N ->
+  E d = \sum_(1 <= i < d.+1) H i * ((-1) ^+ i.-1 *: E (d - i)).
+ +
+Lemma symHE_prod_intcomp d :
+  E d = \sum_(c : intcompn d) (-1) ^+ (d - size c) *: (\prod_(i <- c) H i).
+ +
+Lemma symHE_intcompn d :
+  E d = \sum_(c : intcompn d)
+         (-1)^+(d - size c) *: prod_gen H (partn_of_compn c).
+ +
+Definition perm_partn d (la : 'P_d) :=
+  #|[set c : intcompn d | sort geq c == la]|.
+ +
+Lemma symHE_intpartn d :
+  E d = \sum_(la : 'P_d)
+         (-1)^+(d - size la) * (perm_partn la)%:R *: prod_gen H la.
+ +
+End HandE.
+ +
+Lemma syme_symhE (d : nat) :
+  d != 0%N ->
+  'e_d = \sum_(1 <= i < d.+1) 'h_i * ((-1) ^+ i.-1 *: 'e_(d - i)) :> SF.
+ +
+Lemma symh_symeE (d : nat) :
+  d != 0%N ->
+  'h_d = \sum_(1 <= i < d.+1) 'e_i * ((-1) ^+ i.-1 *: 'h_(d - i)) :> SF.
+ +
+Lemma syme_to_symh d :
+  'e_d = \sum_(la : 'P_d)
+          (-1) ^+ (d - size la) * (perm_partn la)%:R *: 'h[la] :> SF.
+ +
+Lemma symh_to_syme d :
+  'h_d = \sum_(la : 'P_d)
+          (-1) ^+ (d - size la) * (perm_partn la)%:R *: 'e[la] :> SF.
+ +
+
+ +
+

Newton formulas

+ +
+
+Lemma mult_symh_U k d i m :
+  (('h_k : {mpoly R[n]}) * 'X_i ^+ d)@_m =
+  ((mdeg m == (k + d)%N) && (m i >= d))%:R.
+ +
+Lemma mult_symh_powersum k d m :
+  ('h_k * 'p_d : SF)@_m =
+  (mdeg m == (k + d)%N)%:R * \sum_(i < n) (m i >= d)%:R.
+ +
+Lemma Newton_symh (k : nat) :
+  k%:R *: 'h_k = \sum_(0 <= i < k) 'h_i * 'p_(k - i) :> SF.
+ +
+Lemma Newton_symh1 (k : nat) :
+  k%:R *: 'h_k = \sum_(1 <= i < k.+1) 'p_i * 'h_(k - i) :> SF.
+ +
+Lemma mult_syme_U k d i m :
+  (('e_k : {mpoly R[n]}) * 'X_i ^+ d)@_m =
+  [&& mdeg m == (k + d)%N, (d <= m i <= d.+1) &
+   [forall j : 'I_n, (j != i) ==> (m j <= 1%N)]]%:R.
+ +
+Lemma mul_ek_p1 (k : nat) :
+  'e_k.+1 * 'p_1 = k.+2%:R *: 'e_k.+2 + 'm[hookpartn k.+2 1] :> SF.
+ +
+Lemma mul_ek_pk (k l : nat) :
+  'e_k.+1 * 'p_l.+2 =
+  'm[hookpartn (k + l).+3 l.+1] + 'm[hookpartn (k + l).+3 l.+2] :> SF.
+ +
+Lemma expri2 i : (-1) ^+ i.+2 = (-1) ^+ i :> R.
+ Lemma Newton_syme1 (k : nat) :
+  k%:R *: 'e_k =
+  \sum_(1 <= i < k.+1) (-1) ^+ i.+1 *: 'p_i * 'e_(k - i) :> SF.
+ +
+Lemma Newton_syme (k : nat) :
+  k%:R *: 'e_k =
+  \sum_(0 <= i < k) (-1) ^+ (k - i).+1 *: 'e_i * 'p_(k - i) :> SF.
+ +
+End ChangeBasis.
+ +
+
+ +
+

Basis change from Schur to monomial

+ +
+ + We start by doing the computation on int using Kostka and KostkaInv +and then tranfer to any commutative ring +
+
+Section SymsSymmInt.
+ +
+Variable (n : nat) (d : nat).
+#[local] Notation SF := {sympoly int[n.+1]}.
+Implicit Type (la mu : 'P_d).
+ +
+Lemma syms_symm_int la :
+  's[la] = \sum_(mu : 'P_d) 'K(la, mu)%:R *: 'm[mu] :> SF.
+ +
+Lemma syms_symm_partdom_int la :
+  's[la] =
+  'm[la] + \sum_(mu : 'PDom_d | (mu < la)%O) 'K(la, mu) *: 'm[mu] :> SF.
+ +
+Lemma symm_syms_int la :
+  'm[la] = \sum_(mu : 'P_d) KostkaInv la mu *: 's[mu] :> SF.
+ +
+Lemma symm_syms_partdom_int la :
+  'm[la] =
+  's[la] + \sum_(mu : 'PDom_d | (mu < la)%O) KostkaInv la mu *:'s[mu] :> SF.
+ +
+End SymsSymmInt.
+ +
+Section SymsSymm.
+ +
+Variable (n : nat) (R : comNzRingType) (d : nat).
+#[local] Notation SF := {sympoly R[n.+1]}.
+Implicit Type (la mu : 'P_d).
+ +
+Lemma syms_symm la :
+  's[la] = \sum_(mu : 'P_d) 'K(la, mu)%:R *: 'm[mu] :> SF.
+ +
+Lemma syms_symm_partdom la :
+  's[la] =
+  'm[la] + \sum_(mu | (mu < la :> 'PDom_d)%O) 'K(la, mu) *: 'm[mu] :> SF.
+ +
+Lemma symm_syms la :
+  'm[la] = \sum_(mu : 'P_d) 'K^-1(la, mu) *: 's[mu] :> SF.
+ +
+Lemma symm_syms_partdom la :
+  'm[la] =
+  's[la] + \sum_(mu | (mu < la :> 'PDom_d)%O) 'K^-1(la, mu) *: 's[mu] :> SF.
+ +
+End SymsSymm.
+ +
+
+ +
+

Basis change from complete and elementary to Schur

+ +
+ + We start by doing the computation on int using Kostka and KostkaInv +and then tranfer to any commutative ring +
+
+Section SymheSymsInt.
+ +
+Variables (n : nat) (d : nat).
+#[local] Notation SF := {sympoly int[n.+1]}.
+Implicit Type la mu : 'P_d.
+ +
+Lemma symh_syms_int mu :
+  'h[mu] = \sum_(la : 'P_d) 'K(la, mu) *: 's[la] :> SF.
+ +
+Lemma symh_syms_partdom_int mu :
+  'h[mu] =
+  's[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K(la, mu) *: 's[la] :> SF.
+ +
+Lemma syms_symh_int mu :
+  's[mu] = \sum_(la : 'P_d) KostkaInv la mu *: 'h[la] :> SF.
+ +
+Lemma syms_symh_partdom_int mu :
+  's[mu] =
+  'h[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) KostkaInv la mu *: 'h[la] :> SF.
+ +
+#[local] Notation "la '^~'" := (conj_intpartn la) (at level 10).
+ +
+Lemma syme_syms_int mu :
+  'e[mu] = \sum_(la : 'P_d) 'K(la, mu) *: 's[la^~] :> SF.
+ +
+Lemma syme_syms_partdom_int mu :
+  'e[mu] =
+  's[mu^~] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K(la, mu) *: 's[la^~] :> SF.
+ +
+Lemma syms_syme_int mu :
+  's[mu^~] = \sum_(la : 'P_d) KostkaInv la mu *: 'e[la] :> SF.
+ +
+Lemma syms_syme_partdom_int mu :
+  's[mu^~] =
+  'e[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) KostkaInv la mu *: 'e[la] :> SF.
+ +
+End SymheSymsInt.
+ +
+Section SymheSyms.
+ +
+Variables (R : comNzRingType) (n : nat) (d : nat).
+#[local] Notation SF := {sympoly R[n.+1]}.
+Implicit Type la mu : 'P_d.
+ +
+Lemma symh_syms mu : 'h[mu] = \sum_(la : 'P_d) 'K(la, mu) *: 's[la] :> SF.
+ +
+Lemma symh_syms_partdom mu :
+  'h[mu] =
+  's[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K(la, mu) *: 's[la] :> SF.
+ +
+Lemma syms_symh mu : 's[mu] = \sum_(la : 'P_d) 'K^-1(la, mu) *: 'h[la] :> SF.
+ +
+Lemma syms_symh_partdom mu :
+  's[mu] =
+  'h[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K^-1(la, mu) *: 'h[la] :> SF.
+ +
+#[local] Notation "la '^~'" := (conj_intpartn la) (at level 10).
+ +
+Lemma syme_syms mu : 'e[mu] = \sum_(la : 'P_d) 'K(la, mu) *: 's[la ^~] :> SF.
+ +
+Lemma syme_syms_partdom mu :
+  'e[mu] =
+  's[mu^~] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K(la, mu) *: 's[la^~] :> SF.
+ +
+Lemma syms_syme mu : 's[mu^~] = \sum_(la : 'P_d) 'K^-1(la, mu) *: 'e[la] :> SF.
+ +
+Lemma syms_syme_partdom mu :
+  's[mu^~] =
+  'e[mu] + \sum_(la | (mu < la :> 'PDom_d)%O) 'K^-1(la, mu) *: 'e[la] :> SF.
+ +
+End SymheSyms.
+ +
+
+ +
+

Basis change from complete to power sums

+ +
+
+Section ChangeBasisSymhPowerSum.
+ +
+Variable nvar0 : nat.
+Variable R : fieldType.
+#[local] Notation nvar := nvar0.+1.
+#[local] Notation SF := {sympoly R[nvar]}.
+ +
+Fixpoint prod_partsum (s : seq nat) :=
+  if s is _ :: s' then (sumn s * prod_partsum s')%N else 1%N.
+ +
+#[local] Notation "\Pi s" := (prod_partsum s)%:R^-1 (at level 0, s at level 2).
+ +
+Lemma symh_to_symp_prod_partsum n :
+  [pchar R] =i pred0 ->
+  'h_n = \sum_(c : intcompn n) \Pi c *: \prod_(i <- c) 'p_i :> SF.
+ +
+Import LeqGeqOrder.
+ +
+Lemma symh_to_symp_intpartn n :
+  [pchar R] =i pred0 ->
+  'h_n = \sum_(l : 'P_n)
+           (\sum_(c : intcompn n | perm_eq l c) \Pi c) *: 'p[l] :> SF.
+ +
+Lemma intcompn_cons_sub_proof i n (c : intcompn (n - i)) :
+  i != 0%N -> i <= n -> is_comp_of_n n (i :: c).
+#[local] Definition intcompn_cons i (Hi : i != 0%N) n (Hin : i <= n) c :=
+  IntCompN (intcompn_cons_sub_proof c Hi Hin).
+ +
+Lemma intcompn_behead_sub_proof i n (c : intcompn n) :
+  i != 0%N -> i <= n ->
+  is_comp_of_n (n - i)%N (if head 0%N c == i then behead c else rowcompn (n-i)).
+#[local] Definition intcompn_behead i (Hi : i != 0%N) n (Hin : i <= n) c :=
+  IntCompN (intcompn_behead_sub_proof c Hi Hin).
+ +
+Lemma part_sumn_count l :
+  is_part l ->
+  (\sum_(i < (sumn l).+1 | val i \in l) i * (count_mem (val i) l))%N
+  = sumn l.
+ +
+Lemma coeff_symh_to_symp n (l : 'P_n) :
+  [pchar R] =i pred0 ->
+  \sum_(c : intcompn n | perm_eq l c) \Pi c = (zcard l)%:R^-1 :> R.
+ +
+Theorem symh_to_symp n :
+  [pchar R] =i pred0 -> 'h_n = \sum_(l : 'P_n) (zcard l)%:R^-1 *: 'p[l] :> SF.
+ +
+End ChangeBasisSymhPowerSum.
+ +
+Section Generators.
+ +
+Variables (n : nat) (R : comNzRingType).
+ +
+Lemma prod_homog nv l (df : 'I_l -> nat) (mf : 'I_l -> {mpoly R[nv]}) :
+  (forall i : 'I_l, mf i \is (df i).-homog) ->
+  \prod_(i < l) mf i \is (\sum_(i < l) df i).-homog.
+ +
+Variable gen : forall nv : nat, nat -> {mpoly R[nv]}.
+Hypothesis gen_homog : forall nv i : nat, gen nv i \is i.-homog.
+#[local] Notation G nv := [tuple gen nv i.+1 | i < n].
+ +
+Lemma homog_X_mPo_gen nv m : 'X_[m] \mPo G nv \is (mnmwgt m).-homog.
+ +
+Lemma pihomog_mPo nv p d :
+  pihomog mdeg d (p \mPo G nv) = (pihomog mnmwgt d p) \mPo G nv.
+ +
+End Generators.
+ +
+
+ +
+

Symmetric polynomials expressed as polynomial in the elementary

+ +
+
+Section MPoESymHomog.
+ +
+Variables (n : nat) (R : comNzRingType).
+#[local] Notation E nv := [tuple mesym nv R i.+1 | i < n].
+ +
+Lemma mwmwgt_homogP (p : {mpoly R[n]}) d :
+  reflect
+    (forall nv, p \mPo E nv \is d.-homog)
+    (p \is d.-homog for mnmwgt).
+ +
+Lemma sym_fundamental_homog (p : {mpoly R[n]}) (d : nat) :
+  p \is symmetric -> p \is d.-homog ->
+  { t | t \mPo (E n) = p /\ t \is d.-homog for mnmwgt }.
+ +
+End MPoESymHomog.
+ +
+Section SymPolF.
+ +
+Variable R : comNzRingType.
+Variable m : nat.
+Implicit Type p : {sympoly R[m]}.
+ +
+#[local] Notation SF p := (sym_fundamental (sympolP p)).
+ +
+Definition sympolyf p := let: exist t _ := SF p in t.
+ +
+Fact sympolyf_is_linear : linear sympolyf.
+ +
+Fact sympolyf_is_monoid_morphism : monoid_morphism sympolyf.
+ +
+
+ +
+

Fundamental theorem of symmetric polynomials

+ +
+
+Lemma sympolyfP p : (sympolyf p) \mPo [tuple sympol 'e_i.+1 | i < m] = p.
+ +
+Definition sympolyf_eval : {mpoly R[m]} -> {sympoly R[m]} :=
+  mmap (GRing.in_alg {sympoly R[m]}) (fun i : 'I_m => 'e_i.+1).
+Lemma sympolyf_evalE (q : {mpoly R[m]}) :
+  q \mPo [tuple sympol 'e_i.+1 | i < m] = sympolyf_eval q.
+ +
+Lemma sympolyfK p : sympolyf_eval (sympolyf p) = p.
+ +
+Lemma sympolyf_evalK q : sympolyf (sympolyf_eval q) = q.
+ +
+Fact esympolyf_eval_is_linear : linear sympolyf_eval.
+ +
+Fact esympolyf_eval_is_monoid_morphism : monoid_morphism sympolyf_eval.
+ +
+Lemma sympolyf_evalX (i : 'I_m) : sympolyf_eval 'X_i = 'e_i.+1.
+ +
+End SymPolF.
+ +
+Section Omega.
+ +
+Variable R : comNzRingType.
+Variable n0 : nat.
+#[local] Notation n := n0.+1.
+Implicit Type p : {sympoly R[n]}.
+#[local] Notation SF p := (sym_fundamental (sympolP p)).
+ +
+Fact omegasf_is_symmetric p :
+  (sympolyf p) \mPo [tuple sympol 'h_i.+1 | i < n] \is @symmetric n R.
+Definition omegasf p : {sympoly R[n]} := SymPoly (omegasf_is_symmetric p).
+ +
+Lemma val_omegasf p :
+  sympol (omegasf p) = (sympolyf p) \mPo [tuple sympol 'h_i.+1 | i < n].
+ +
+Fact omegasf_is_linear : linear omegasf.
+ +
+Fact omegasf_is_monoid_morphism : monoid_morphism omegasf.
+ +
+Lemma omegasf_syme i : i <= n -> omegasf 'e_i = 'h_i.
+ +
+Lemma omegasf_symh i : i <= n -> omegasf 'h_i = 'e_i.
+ +
+Lemma omegasfK : involutive omegasf.
+ +
+Lemma omegasf_symp i : 0 < i <= n -> omegasf 'p_i = (-1) ^+ i.+1 *: 'p_i.
+ +
+Lemma omegasf_homog d :
+  {homo omegasf: p / sympol p \in [in R[n], d.-homog]}.
+ +
+Lemma omegasf_homogE d :
+  {mono omegasf: p / sympol p \in [in R[n], d.-homog]}.
+ +
+Notation S := ([tuple sympol 'h_i.+1 | i < n] : n.-tuple {mpoly R[n]}).
+Notation E := ([tuple sympol 'e_i.+1 | i < n] : n.-tuple {mpoly R[n]}).
+ +
+Lemma msym_fundamental_symh_un (t1 t2 : {mpoly R[n]}) :
+  t1 \mPo S = t2 \mPo S -> t1 = t2.
+ +
+Lemma omegasf_sympolyf_eval q :
+  sympol (omegasf (sympolyf_eval q)) = q \mPo [tuple sympol 'h_i.+1 | i < n].
+ +
+Lemma omegasf_compsymh p q :
+  (sympol p == q \mPo [tuple sympol 'h_i.+1 | i < n]) =
+  (sympol (omegasf p) == q \mPo [tuple sympol 'e_i.+1 | i < n]).
+ +
+Lemma sym_fundamental_symh_homog (p : {mpoly R[n]}) (d : nat) :
+  p \is symmetric -> p \is d.-homog ->
+  { t | t \mPo S = p /\ t \is d.-homog for mnmwgt }.
+ +
+Lemma sym_fundamental_symh (p : {mpoly R[n]}) :
+  p \is symmetric -> { t | t \mPo S = p }.
+ +
+Variable (d : nat).
+Implicit Type (la : 'P_d).
+ +
+Lemma omegasf_prodsyme la : head 0%N la <= n -> omegasf 'e[la] = 'h[la].
+ +
+Lemma omegasf_prodsymh la : head 0%N la <= n -> omegasf 'h[la] = 'e[la].
+ +
+Lemma exp1sumnDsize la :
+  (-1) ^+ (d - size la) = \prod_(i <- la) (-1) ^+ i.+1 :> R.
+ +
+Lemma omegasf_prodsymp la :
+  head 0%N la <= n -> omegasf 'p[la] = (-1) ^+ (d - size la) *: 'p[la].
+ +
+Lemma omegasf_syms la : d <= n -> omegasf 's[la] = 's[conj_intpartn la].
+ +
+End Omega.
+ +
+#[local] Close Scope Combi_scope.
+ +
+
+ +
+

Change of the number of variables

+ +
+
+Section ChangeNVar.
+ +
+Variable R : comNzRingType.
+Variable m0 n0 : nat.
+#[local] Notation m := m0.+1.
+#[local] Notation n := n0.+1.
+#[local] Notation SF p := (sym_fundamental (sympolP p)).
+#[local] Notation E := [tuple sympol 'e_(i.+1) : {mpoly R[n]} | i < m].
+ +
+Lemma cnvarsym_subproof (p : {sympoly R[m]}) : sympolyf p \mPo E \is symmetric.
+ Definition cnvarsym p : {sympoly R[n]} := SymPoly (cnvarsym_subproof p).
+ +
+Fact cnvarsym_is_linear : linear cnvarsym.
+ +
+Fact cnvarsym_is_monoid_morphism : monoid_morphism cnvarsym.
+ +
+Lemma cnvar_leq_symeE i : i <= m -> cnvarsym 'e_i = 'e_i.
+ +
+Lemma cnvar_syme i : (i <= m) || (n <= m) -> cnvarsym 'e_i = 'e_i.
+ +
+Lemma cnvar_symh i : (i <= m) || (n <= m) -> cnvarsym 'h_i = 'h_i.
+ +
+Lemma cnvar_symp i : (i < m) || (n <= m) -> cnvarsym 'p_i.+1 = 'p_i.+1.
+ +
+Section ProdGen.
+ +
+Variable Gen : forall nvar d : nat, {sympoly R[nvar]}.
+Hypothesis Hcnvargen :
+  forall d : nat, (d < m) || (n <= m) -> cnvarsym (Gen m d.+1) = (Gen n d.+1).
+ +
+Lemma cnvar_prodgen d (la : 'P_d) :
+  (d <= m) || (n <= m) ->
+  cnvarsym (prod_gen (Gen m) la) = prod_gen (Gen n) la.
+ +
+End ProdGen.
+ +
+Variable d : nat.
+ +
+Hypothesis Hd : (d <= m) || (n <= m).
+ +
+Lemma cnvar_prodsyme (la : 'P_d) : cnvarsym 'e[la] = 'e[la].
+ +
+Lemma cnvar_prodsymh (la : 'P_d) : cnvarsym 'h[la] = 'h[la].
+ +
+Lemma cnvar_prodsymp (la : 'P_d) : cnvarsym 'p[la] = 'p[la].
+ +
+Lemma cnvar_syms (la : 'P_d) : cnvarsym 's[la] = 's[la].
+ +
+Lemma cnvar_symm (la : 'P_d) : cnvarsym 'm[la] = 'm[la].
+ +
+End ChangeNVar.
+ +
+Section CategoricalSystems.
+ +
+Variable R : comNzRingType.
+ +
+Lemma cnvarsym_id n : @cnvarsym R n n =1 id.
+ +
+Lemma cnvarsym_leq_trans m n p :
+  (m <= n) -> (n <= p) ->
+  @cnvarsym R n p \o @cnvarsym R m n =1 @cnvarsym R m p.
+ +
+Lemma cnvarsym_geq_trans m n p :
+  (m >= n) -> (n >= p) ->
+  @cnvarsym R n p \o @cnvarsym R m n =1 @cnvarsym R m p.
+ +
+End CategoricalSystems.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SSRcomplements.ordcast.html b/combi/1.1.0/Combi.SSRcomplements.ordcast.html new file mode 100644 index 00000000..62c3a6bd --- /dev/null +++ b/combi/1.1.0/Combi.SSRcomplements.ordcast.html @@ -0,0 +1,96 @@ + + + + + +Combi.SSRcomplements.ordcast: Cast between ordinals + + + + +
+ + + +
+ +

Library Combi.SSRcomplements.ordcast: Cast between ordinals

+ +
+
+ +
+ +
+
+
+ +
+

Some complement on casts between ordinals

+ + +
+ +Aside a few basic lemmas, the only new definition is: +
    +
  • cast_set (H : n = m) S == cast S : {set 'I_n} to S : {set 'I_m}. + +
    • +
+ +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype finfun fintype choice seq tuple.
+From mathcomp Require Import finset tuple bigop.
+ +
+Require Import tools.
+ +
+Set Implicit Arguments.
+ +
+Lemma enum_cast_ord m n (H : n = m):
+  enum 'I_m = [seq cast_ord H i | i <- enum 'I_n].
+ +
+Section Casts.
+ +
+Lemma cast_map_cond (T: Type) n m (P : pred 'I_n) (F : 'I_n -> T) (H : m = n) :
+  [seq F i | i <- enum 'I_n & P i] =
+  [seq F (cast_ord H i) | i <- enum 'I_m & P (cast_ord H i) ].
+ +
+Lemma mem_cast m n (H : m = n) (i : 'I_m) (S : {set 'I_m}) :
+  (cast_ord H i) \in [set cast_ord H i | i in S] = (i \in S).
+ +
+Definition cast_set n m (H : n = m) : {set 'I_n} -> {set 'I_m} :=
+  [fun s : {set 'I_n} => (cast_ord H) @: s].
+ +
+Lemma cast_set_inj n m (H : n = m) : injective (cast_set H).
+ +
+Lemma cover_cast m n (P : {set {set 'I_n}}) (H : n = m) :
+  cover (imset (cast_set H) (mem P)) = (cast_set H) (cover P).
+ +
+End Casts.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SSRcomplements.permcomp.html b/combi/1.1.0/Combi.SSRcomplements.permcomp.html new file mode 100644 index 00000000..c327222b --- /dev/null +++ b/combi/1.1.0/Combi.SSRcomplements.permcomp.html @@ -0,0 +1,100 @@ + + + + + +Combi.SSRcomplements.permcomp: Complement on permutations + + + + +
+ + + +
+ +

Library Combi.SSRcomplements.permcomp: Complement on permutations

+ +
+
+ +
+ +
+
+
+ +
+

A few lemmas on permutation

+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice fintype div.
+From mathcomp Require Import finset fingroup perm morphism action.
+ +
+Set Implicit Arguments.
+ +
+Import GroupScope.
+ +
+
+ +
+

Orbit and cycles

+ +
+
+Section PermComp.
+ +
+Variable T : finType.
+Implicit Type (s : {perm T}) (X : {set T}) (P : {set {set T}}).
+ +
+Lemma permKP s : reflect (involutive s) (s^-1 == s).
+ +
+End PermComp.
+ +
+Section SetAct.
+ +
+Variables (aT : finGroupType) (D : {set aT}) (rT : finType) (to : action D rT).
+ +
+
+ +
+TODO: complete setactU, setactI, setactD ... and submit to mathcomp +
+
+Lemma setact0 a : to^* set0 a = set0.
+Lemma setact1 x a : to^* [set x] a = [set to x a].
+Lemma setactI S T a : to^* (S :&: T) a = to^* S a :&: to^* T a.
+ Lemma setactU S T a : to^* (S :|: T) a = to^* S a :|: to^* T a.
+ Lemma setactU1 x T a : to^* (x |: T) a = to x a |: to^* T a.
+ +
+Lemma setactC S a : to^* (~: S) a = ~: to^* S a.
+ +
+End SetAct.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SSRcomplements.sorted.html b/combi/1.1.0/Combi.SSRcomplements.sorted.html new file mode 100644 index 00000000..6bf4c536 --- /dev/null +++ b/combi/1.1.0/Combi.SSRcomplements.sorted.html @@ -0,0 +1,178 @@ + + + + + +Combi.SSRcomplements.sorted: [path] and [sorted] complements + + + + +
+ + + +
+ +

Library Combi.SSRcomplements.sorted: [path] and [sorted] complements

+ +
+
+ +
+ +
+
+
+ +
+

Various Lemmas about path and sorted which are missing in MathComp

+ TODO: these probably should be contributed to path.v +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype fintype choice seq.
+From mathcomp Require Import path order.
+ +
+Set Implicit Arguments.
+ +
+Open Scope N.
+ +
+Module LeqGeqOrder.
+ +
+Definition geq_refl : reflexive geq :=
+  fun x => leqnn x.
+Definition geq_total : total geq :=
+  fun x y => leq_total y x.
+Definition geq_trans : transitive geq :=
+  fun x y z H1 H2 => leq_trans H2 H1.
+Definition anti_geq : antisymmetric geq :=
+  fun x y H => esym (anti_leq H).
+Definition ltn_irr : irreflexive ltn :=
+  fun x => ltnn x.
+Definition gtn_trans : transitive gtn :=
+  fun x y z H1 H2 => ltn_trans H2 H1.
+Definition gtn_irr : irreflexive gtn :=
+  fun x => ltnn x.
+ +
+#[export] Hint Resolve leq_total leq_trans anti_leq : core.
+#[export] Hint Resolve geq_refl geq_total geq_trans anti_geq : core.
+#[export] Hint Resolve ltn_trans ltn_irr gtn_trans gtn_irr : core.
+ +
+End LeqGeqOrder.
+ +
+Import LeqGeqOrder.
+ +
+Section Sorted.
+ +
+Variable T : eqType.
+Variable Z : T.
+Variable R : rel T.
+ +
+Implicit Type l : T.
+Implicit Type r : seq T.
+ +
+#[local] Notation sorted r := (sorted R r).
+#[local] Notation "x <=R y" := (R x y) (at level 70, y at next level).
+ +
+Lemma sorted_consK l r : sorted (cons l r) -> sorted r.
+ +
+Lemma sorted_rconsK l r : sorted (rcons r l) -> sorted r.
+ +
+Lemma sorted_rcons l r : sorted r -> (last l r <=R l) -> sorted (rcons r l).
+ +
+Hypothesis Rtrans : transitive R.
+ +
+Lemma incr_equiv r :
+  (forall (i j : nat), i < j < (size r) -> nth Z r i <=R nth Z r j)
+  <->
+  (forall (i : nat), i.+1 < (size r) -> nth Z r i <=R nth Z r i.+1).
+ +
+Lemma sorted_strictP r :
+  reflect
+    (forall (i j : nat), i < j < (size r) -> (nth Z r i <=R nth Z r j))
+    (sorted r).
+ +
+Hypothesis Rrefl : reflexive R.
+ +
+Lemma non_decr_equiv r :
+  (forall (i j : nat), i <= j < (size r) -> nth Z r i <=R nth Z r j)
+  <->
+  (forall (i : nat), i.+1 < (size r) -> nth Z r i <=R nth Z r i.+1).
+ +
+Lemma sorted2P r :
+  reflect
+    (forall (i j : nat), i <= j < (size r) -> (nth Z r i <=R nth Z r j))
+    (sorted r).
+ +
+Lemma sorted_cons l r : sorted (cons l r) -> (l <=R head l r) /\ sorted r.
+ +
+Lemma sorted_last l r : sorted (rcons r l) -> (last l r <=R l).
+ +
+Lemma head_leq_last_sorted l r : sorted (l :: r) -> (l <=R last l r).
+ +
+Hypothesis Hanti : antisymmetric R.
+ +
+Lemma sorted_lt_by_pos r p q :
+  sorted r -> p < size r -> q < size r ->
+   (nth Z r p != nth Z r q) && (nth Z r p <=R nth Z r q) -> p < q.
+ +
+End Sorted.
+ +
+Require Import tools.
+ +
+Lemma enum_ord_sorted_ltn N :
+  sorted (fun i j : 'I_N => i < j) (enum 'I_N).
+ +
+Lemma enum_ord_sorted N :
+  sorted (fun i j : 'I_N => i <= j) (enum 'I_N).
+ +
+Lemma sorted_ltn_ind s :
+  sorted ltn s -> sumn (iota 0 (size s)) <= sumn s /\ last 0 s >= (size s).-1.
+ +
+Lemma sorted_sumn_iotaE s :
+  sorted ltn s -> sumn s = sumn (iota 0 (size s)) -> s = iota 0 (size s).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SSRcomplements.tools.html b/combi/1.1.0/Combi.SSRcomplements.tools.html new file mode 100644 index 00000000..11dd2a4d --- /dev/null +++ b/combi/1.1.0/Combi.SSRcomplements.tools.html @@ -0,0 +1,357 @@ + + + + + +Combi.SSRcomplements.tools: Missing SSReflect sequence and set lemmas + + + + +
+ + + +
+ +

Library Combi.SSRcomplements.tools: Missing SSReflect sequence and set lemmas

+ +
+
+ +
+ +
+
+
+ +
+

A bunch of lemmas about seqs and sets which are missing in SSReflect

+ + +
+ +No new notions are defined here. + +
+ +TODO: these probably should be contributed to SSReflect itself + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype fintype choice.
+From mathcomp Require Import seq finset bigop path binomial order.
+ +
+Set Implicit Arguments.
+ +
+ +
+#[export] Hint Resolve nth_nil addn0 : core.
+ +
+Lemma leq_addE m1 m2 n1 n2 :
+  m1 <= m2 -> n1 <= n2 -> m1 + n1 = m2 + n2 -> m1 = m2 /\ n1 = n2.
+ +
+
+ +
+

rcons and cons related lemmas

+ +
+
+Section SeqLemmas.
+ +
+Variable (T : eqType).
+Implicit Type s w : seq T.
+Implicit Type a b : T.
+ +
+Lemma drop_nilE s m :
+  (drop m s == [::]) = (size s <= m).
+ +
+Lemma cons_head_behead x s : (s != [::]) -> head x s :: behead s = s.
+ +
+Lemma belast_behead_rcons x l s :
+  belast (head x (rcons s l)) (behead (rcons s l)) = s.
+ +
+Lemma last_behead_rcons x l s :
+  last (head x (rcons s l)) (behead (rcons s l)) = l.
+ +
+Lemma set_head_default b a s : s != [::] -> head a s = head b s.
+ +
+Lemma rcons_set_nth a s l : set_nth a s (size s) l = rcons s l.
+ +
+Lemma set_nth_rcons x0 s x n y :
+  set_nth x0 (rcons s x) n y =
+  if n < size s then rcons (set_nth x0 s n y) x
+  else if n == size s then rcons s y
+       else (rcons s x) ++ ncons (n - size s).-1 x0 [:: y].
+ +
+Lemma rconsK a b u v : rcons u a = rcons v b -> u = v.
+ +
+Lemma rcons_nilF s l : ((rcons s l) == [::]) = false.
+ +
+Lemma cons_in_map_cons a b s w (l : seq (seq T)) :
+  a :: s \in [seq b :: s1 | s1 <- l] -> a == b.
+ +
+Lemma count_rcons w P l :
+  count P (rcons w l) = count P w + P l.
+ +
+
+ +
+

set_nth related lemmas

+ +
+
+Lemma set_nth_non_nil d s n y : set_nth d s n y != [::].
+ +
+Lemma nth_set_nth_expand a b l i c j :
+  (size l <= j < i) -> nth a (set_nth b l i c) j = b.
+ +
+Lemma nth_set_nth_any a b l i c j :
+  nth a (set_nth b l i c) j =
+  if j == i then c else
+    if j < size l then nth a l j else
+      if j <= i then b else a.
+ +
+Lemma eq_from_nth_notin x0 s1 s2 :
+  x0 \notin s1 -> x0 \notin s2 ->
+  (forall i : nat, nth x0 s1 i = nth x0 s2 i) -> s1 = s2.
+ +
+End SeqLemmas.
+ +
+
+ +
+

minn related lemmas

+ +
+
+Lemma minSS i j : minn i.+1 j.+1 = (minn i j).+1.
+ +
+
+ +
+

sumn related lemmas

+ +
+
+Lemma leq_sumn_in (sh : seq nat) i : i \in sh -> i <= sumn sh.
+ +
+Lemma sumn_map_condE (T : Type) (s : seq T) (f : T -> nat) (P : pred T) :
+  sumn [seq f i | i <- s & P i] = \sum_(i <- s | P i) f i.
+ +
+Lemma sumn_mapE (T : Type) (s : seq T) (f : T -> nat) :
+  sumn [seq f i | i <- s] = \sum_(i <- s) f i.
+ +
+Lemma sum_minn s b :
+  \sum_(l <- s) minn l b = sumn s - \sum_(l <- s) (l - b).
+ +
+Lemma sum_take r s F :
+  F 0 = 0 -> \sum_(l <- take r s) F l = \sum_(0 <= i < r) F (nth 0 s i).
+ +
+Lemma sumn_take r s : sumn (take r s) = \sum_(0 <= i < r) nth 0 s i.
+ Lemma sumn_drop r s : sumn (drop r s) = \sum_(r <= i < size s) nth 0 s i.
+ +
+Lemma sumn_nth_le l n :
+  size l <= n -> sumn l = \sum_(0 <= i < n) nth 0 l i.
+ +
+
+ +
+

iota related lemmas

+ +
+
+Lemma binomial_sumn_iota n : 'C(n, 2) = sumn (iota 0 n).
+ +
+Lemma sumn_pred1_iota a b x :
+  sumn [seq ((i == x) : nat) | i <- iota a b] = (a <= x < a + b).
+ +
+Lemma count_mem_iota a b i :
+  count_mem i (iota a b) = (a <= i < a + b).
+ +
+Lemma iota_ltn a b : b <= a -> [seq i <- iota 0 a | i < b] = iota 0 b.
+ +
+Lemma iota_geq a b : [seq i <- iota 0 a | b <= i] = iota b (a - b).
+ +
+Section FinSet.
+ +
+Variable T : finType.
+ +
+Lemma setU1E (x : T) (S : {set T}) : (x \in S) = (x |: S == S).
+ +
+End FinSet.
+ +
+Lemma uniq_sum_count_mem (T : eqType) (P : pred T) l s :
+  uniq s ->
+  \sum_(i <- s | P i) (count_mem i) l = count (predI (mem s) P) l.
+ +
+Lemma sumn_sort l S : sumn (sort S l) = sumn l.
+ +
+Lemma map_filter_comp (T1 T2: Type) (l : seq T1) (PP : pred T2) (F : T1 -> T2) :
+  [seq F i | i <- l & PP (F i)] = [seq i | i <- map F l & PP i ].
+ +
+Lemma set1_disjoint (T : finType) (i j : T) :
+  [set i] != [set j] -> [disjoint [set i] & [set j]].
+ +
+Lemma subset_imsetK (T1 T2 : finType) (f : T1 -> T2) (s t : {set T1}):
+  injective f -> f @: s \subset f @: t -> s \subset t.
+ +
+Section SSRComplFinset.
+ +
+Variables aT rT : finType.
+Variables (f : aT -> rT).
+ +
+Lemma imsetD (A B : {set aT}) :
+  {in A :|: B &, injective f} -> f @: (B :\: A) = (f @: B) :\: (f @: A).
+ +
+End SSRComplFinset.
+ +
+Section ImsetInj.
+ +
+Variables (T T1 T2 : finType) (f : T1 -> T2).
+ +
+Lemma preimset_trivIset (P : {set {set T2}}) :
+  trivIset P -> trivIset ((fun s : {set T2} => f @^-1: s) @: P).
+ +
+Hypothesis (f_inj : injective f).
+ +
+Lemma imset_inj : injective (fun s : {set T1} => imset f (mem s)).
+ +
+Lemma imset_trivIset (P : {set {set T1}}) :
+  trivIset P -> trivIset ((fun s : {set T1} => f @: s) @: P).
+ +
+Lemma disjoint_imset (A B : {set T1}) :
+  [disjoint A & B] -> [disjoint [set f x | x in A] & [set f x | x in B]].
+ +
+End ImsetInj.
+ +
+Lemma uniq_next (T : eqType) (p : seq T) : uniq p -> injective (next p).
+ +
+Lemma mem_takeP (T : eqType) x0 x k (s : seq T) :
+  reflect (exists2 i, i < minn k (size s) & x = nth x0 s i) (x \in take k s).
+ +
+Lemma mem_take_enumI n (i : 'I_n) k : i \in take k (enum 'I_n) = (i < k).
+ +
+Lemma take_enumI n k : take k (enum 'I_n) = filter ((gtn k) \o val) (enum 'I_n).
+ +
+Lemma mem_drop_enumI n (i : 'I_n) k : i \in drop k (enum 'I_n) = (i >= k).
+ +
+Lemma drop_enumI n k : drop k (enum 'I_n) = filter ((leq k) \o val) (enum 'I_n).
+ +
+Lemma enum0 : enum 'I_0 = [::].
+ +
+Section AbelianBigOp.
+ +
+Import Monoid.Theory.
+ +
+Variable R : Type.
+Variable idx : R.
+#[local] Notation "1" := idx.
+Variable op : Monoid.com_law 1.
+#[local] Notation "'*%M'" := op (at level 0).
+#[local] Notation "x * y" := (op x y).
+ +
+Lemma big_seq_sub (T : countType) (s : seq T) F :
+  \big[op/idx]_(x : seq_sub s) F (ssval x) = \big[op/idx]_(x <- undup s) F x.
+ +
+End AbelianBigOp.
+Arguments partition_big [R idx op I s J P] p Q [F].
+ +
+Section SetPartition.
+ +
+Variable T : finType.
+Implicit Types (X : {set T}) (P : {set {set T}}).
+ +
+Lemma triv_part P X : X \in P -> partition P X -> P = [set X].
+ +
+End SetPartition.
+ +
+Notation "#{ x }" := #|(x : {set _})|
+                      (at level 0, x at level 10, format "#{ x }").
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.Frobenius_char.html b/combi/1.1.0/Combi.SymGroup.Frobenius_char.html new file mode 100644 index 00000000..e5115711 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.Frobenius_char.html @@ -0,0 +1,499 @@ + + + + + +Combi.SymGroup.Frobenius_char: Frobenius characteristic + + + + +
+ + + +
+ +

Library Combi.SymGroup.Frobenius_char: Frobenius characteristic

+ +
+
+ +
+ +
+
+
+ +
+

Frobenius / Schur character theory for the symmetric groups.

+ + +
+ +We develop the theory of Frobenius characteristic associated to a class +function of 'SG_n, it is an isometry from class function to symmetric +functions, mapping +
    +
  • 1%g to 'HH[rowpartn n] (this is Fchar_triv); + +
    • +
  • 1z_l to 'hpl (this is Fchar_ncfuniCT); + +
    • +
  • induction product to product Fchar_ind_morph; + +
    • +
  • omega involution to multiplication by the sign character omega_Fchar. + +
    • +
+ +
+ +We define the following notions and notations: + +
+ +
    +
  • Fchar f == the Frobenius characteristic of the class function f. + the number of variable is inferred from the context. + +
    • +
  • Fchar_inv f == the inverse Frobenius characteristic of the + homogeneous symmetric polynomial f. + +
    + + +
    • +
  • 'M[la] == the Young character for 'SG_n associated to the + partition la of n. If la = (l1, ..., lk) it is the + character induced from the trivial representations of the + group 'SG_l1 * ... * SG_lk. + +
    • +
  • 'irrSG[la] == the irreducible character for 'SG_n associated to the + partition la of n. + +
    • +
+ +
+ +Here is a list of fundamental results: + +
+ +
    +
  • Fchar_isometry : The Frobenius characteristic is an isometry. + +
    • +
  • Young_rule : The Young's rule for character of 'SG_n. + +
    • +
  • irrSGP : The 'irrSG[la] | la : 'P_n forms a complete set of + irreducible characters for 'SG_n. + +
    • +
  • Frobenius_char : Frobenius character formula for 'SG_n. + +
    • +
  • Murnaghan_Nakayama_char : Murnaghan-Nakayama character formula for 'SG_n. + +
    • +
  • dim_cfReprSG : the dimension of irreducible representation of 'SG_n. + +
    • +
  • LR_rule_irrSG : Littlewood-Richardson rule for characters of 'SG_n. + + +
    • +
+
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import fingroup perm morphism gproduct.
+From mathcomp Require Import rat ssralg ssrint ssrnum algC vector archimedean.
+From mathcomp Require Import mxrepresentation classfun character.
+From mathcomp Require Import sesquilinear.
+From mathcomp Require Import mpoly.
+ +
+Require Import sorted ordtype tools partition antisym sympoly homogsym Cauchy
+        Schur_altdef stdtab therule.
+Require Import permcomp cycletype towerSn permcent reprSn unitriginv.
+Require Import MurnaghanNakayama.
+ +
+Require ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import LeqGeqOrder.
+Import GroupScope GRing.Theory Num.Theory.
+Open Scope ring_scope.
+ +
+Reserved Notation "''irrSG[' l ']'"
+         (at level 0, l at level 2, format "''irrSG[' l ]").
+Reserved Notation "''M[' l ']'"
+         (at level 0, l at level 2, format "''M[' l ]").
+ +
+#[local] Lemma pchar0_rat : [pchar rat] =i pred0.
+ #[local] Lemma pchar0_algC : [pchar algC] =i pred0.
+ #[local] Hint Resolve pchar0_algC pchar0_rat : core.
+ +
+
+ +
+TODO: contribute to mathcomp +
+
+Section CharDotProduct.
+ +
+Variable (gT : finGroupType) (G : {group gT}).
+ +
+#[local] Notation cfdot := (cfdot (B := G)).
+ +
+Fact cfdot_is_bilinear :
+  bilinear_for *%R (Num.conj \; *%R) cfdot.
+ +
+Fact cfdot_is_hermitian (phi psi : 'CF(G)) :
+  '[phi, psi] = (-1) ^+ false * '[psi, phi]^*.
+ +
+Fact cfdot_is_dot (phi : 'CF(G)) : phi != 0 -> 0 < '[phi].
+ +
+End CharDotProduct.
+ +
+
+ +
+

Definition and basic properties

+ +
+
+Section NVar.
+ +
+Variable nvar0 : nat.
+#[local] Notation nvar := nvar0.+1.
+ +
+Section Defs.
+ +
+Variable n : nat.
+#[local] Notation HS := {homsym algC[nvar, n]}.
+ +
+Definition Fchar (f : 'CF('SG_n)) : HS :=
+  locked (\sum_(la : 'P_n) (f (permCT la) / 'z_la) *: 'hp[la]).
+ +
+Definition Fchar_inv (p : HS) : 'CF('SG_n) :=
+  locked (\sum_(la : 'P_n) (coord 'hp (enum_rank la) p) *: '1z_[la]).
+ +
+Lemma FcharE (f : 'CF('SG_n)) :
+  Fchar f = \sum_(la : 'P_n) (f (permCT la) / 'z_la) *: 'hp[la].
+ +
+Lemma Fchar_invE (p : HS) :
+  Fchar_inv p = \sum_(la : 'P_n) (coord 'hp (enum_rank la) p) *: '1z_[la].
+ +
+Lemma Fchar_is_linear : linear Fchar.
+ +
+Lemma Fchar_ncfuniCT (l : 'P_n) : Fchar '1z_[l] = 'hp[l].
+ +
+Lemma Fchar_inv_is_linear : linear Fchar_inv.
+ +
+Hypothesis Hn : (n <= nvar)%N.
+ +
+Lemma FcharK : cancel Fchar Fchar_inv.
+ +
+Lemma Fchar_invK : cancel Fchar_inv Fchar.
+ +
+Lemma Fchar_triv : Fchar 1 = 'hh[rowpartn n].
+ +
+Lemma Fchar_inv_homsymp (l : 'P_n) : Fchar_inv 'hp[l] = '1z_[l].
+ +
+
+ +
+

Frobenius Characteristic and omega involution

+ +
+
+Lemma omega_Fchar_inv (p : HS) :
+  Fchar_inv (omegahomsym p) = sign_char * (Fchar_inv p).
+ +
+Lemma omega_Fchar f : omegahomsym (Fchar f) = Fchar (sign_char * f).
+ +
+Lemma Fchar_sign : Fchar sign_char = 'he[rowpartn n].
+ +
+
+ +
+

The Frobenius Characteristic is an isometry

+ +
+
+Corollary Fchar_isometry : isometry homsymdot cfdot Fchar.
+ +
+Theorem Fchar_inv_isometry p q : '[Fchar_inv p, Fchar_inv q] = '[p | q].
+ +
+End Defs.
+ +
+
+ +
+

The Frobenius Characteristic is a graded ring morphism

+ +
+ +This cannot be written as a SSReflect {morph Fchar : f g / ... >-> ... } +because the dependency of Fchar on the degree n. The three Fchar below are +actually three different functions. + +
+ +Note: this can be solved using a dependant record {n; 'CF('S_n)} with a +dependent equality but I'm not sure this is really needed. + +
+
+Theorem Fchar_ind_morph m n (f : 'CF('SG_m)) (g : 'CF('SG_n)) :
+  Fchar ('Ind['SG_(m + n)] (f \o^ g)) = Fchar f *h Fchar g.
+ +
+
+ +
+

Combinatorics of characters of the symmetric groups

+ +
+ +

Young characters and Young's Rule

+ +
+
+Section Character.
+ +
+Import LeqGeqOrder.
+ +
+Variable n : nat.
+Hypothesis Hn : (n <= nvar)%N.
+#[local] Notation HS := {homsym algC[nvar, n]}.
+ +
+Lemma homsymh_character (la : 'P_n) : Fchar_inv 'hh[la] \is a character.
+ +
+Lemma homsyme_character (la : 'P_n) : Fchar_inv 'he[la] \is a character.
+ +
+End Character.
+ +
+End NVar.
+#[local] Hint Resolve leqSpred : core.
+ +
+Arguments Fchar nvar0 [n] f.
+Arguments Fchar_inv nvar0 [n] p.
+ +
+Lemma FcharNvar (nvar0 nvar1 n : nat) (f : 'CF('SG_n)) :
+  (n <= nvar0.+1)%N || (nvar1 < nvar0.+1)%N ->
+  cnvarhomsym nvar1 (Fchar nvar0 f) = (Fchar nvar1 f).
+ +
+Section YoungIrrDef.
+ +
+Variable (n : nat).
+Implicit Type (la mu : 'P_n).
+ +
+Definition YoungSG la : 'CF('SG_n) := Fchar_inv (n.-1) 'hh[la].
+Definition irrSG la : 'CF('SG_n) := Fchar_inv (n.-1) 'hs[la].
+ +
+Notation "''M[' l ']'" := (YoungSG l).
+Notation "''irrSG[' l ']'" := (irrSG l).
+ +
+Lemma Fchar_irrSGE nvar0 la : Fchar nvar0 'irrSG[la] = 'hs[la].
+ +
+Lemma Young_char la : 'M[la] \is a character.
+ +
+Lemma Young_rule la : 'M[la] = \sum_(mu : 'P_n) 'K(mu, la) *: 'irrSG[mu].
+ +
+Lemma Young_rule_partdom la :
+  'M[la] =
+  'irrSG[la] + \sum_(mu | (la < mu :> 'PDom_n)%O) 'K(mu, la) *: 'irrSG[mu].
+ +
+
+ +
+

Irreducible character

+ +
+ + Substracting characters +
+
+Lemma rem_irr1 (xi phi : 'CF('SG_n)) :
+  xi \in irr 'SG_n -> phi \is a character -> '[phi, xi] != 0 ->
+     phi - xi \is a character.
+ +
+Lemma rem_irr (xi phi : 'CF('SG_n)) :
+  xi \in irr 'SG_n -> phi \is a character ->
+     phi - '[phi, xi] *: xi \is a character.
+ +
+Lemma irrSG_orthonormal la mu :
+  '['irrSG[la], 'irrSG[mu]] = (la == mu)%:R.
+ +
+Theorem irrSG_irr la : 'irrSG[la] \in irr 'SG_n.
+ +
+
+ +
+The 'irrSG[la] forms a complete set of irreducible character +
+
+Theorem irrSGP : perm_eq [seq 'irrSG[la] | la : 'P_n] (irr 'SG_n).
+ +
+
+ +
+

Frobenius character formula for the symmetric group 'SG_n

+ +
+ + The value of the irreducible character 'irrSG[la] using scalar product of + symmetric function +
+
+Theorem Frobenius_char_homsymdot la (sigma : 'S_n) :
+  'irrSG[la] sigma = '[ 'hs[la] | 'hp[cycle_typeSn sigma] ] _(n.-1, n).
+ +
+Theorem Frobenius_char_coord la (sigma : 'S_n) :
+  'irrSG[la] sigma =
+  coord 'hs (enum_rank la) ('hp[cycle_typeSn sigma] : {homsym algC[n.-1.+1, n]}).
+ +
+End YoungIrrDef.
+Notation "''M[' l ']'" := (YoungSG l).
+Notation "''irrSG[' l ']'" := (irrSG l).
+ +
+
+ +
+Frobenius character formula for 'SG_n +
+
+Theorem Frobenius_char n (la mu : 'P_n) :
+  'irrSG[la] (permCT mu) =
+  (Vanprod * \prod_(d <- mu) (symp_pol n algC d))@_(mpart la + rho n).
+ +
+
+ +
+The Murnaghan Nakayama rule for irreducible character 'irrSG[la] +
+
+Theorem Murnaghan_Nakayama_char n la (sigma : 'S_n) :
+  'irrSG[la] sigma = (MN_coeff la (cycle_typeSn sigma))%:~R.
+Corollary Murnaghan_NakayamaCT n (la mu : 'P_n) :
+  'irrSG[la] (permCT mu) = (MN_coeff la mu)%:~R.
+ +
+Corollary irrSG_char_int n (la mu : 'P_n) : 'irrSG[la] (permCT mu) \in Num.int.
+ +
+Example Wikipedia_Murnaghan_Nakayama :
+  let p521 := @IntPartN 8 [:: 5; 2; 1]%N is_true_true in
+  let p3311 := @IntPartN 8 [:: 3; 3; 1; 1]%N is_true_true in
+  'irrSG[p521] (permCT p3311) = - 2%:~R.
+ +
+
+ +
+The dimension of the irreducible character 'irrSG[la] is the number of + standard tableau of shape la +
+
+Theorem dim_irrSG n (la : 'P_n) : 'irrSG[la] 1%g = #|{: stdtabsh la}|%:R.
+ +
+Theorem dim_cfReprSG n (la : 'P_n) d (rG : mx_representation algC 'SG_n d) :
+  cfRepr rG = 'irrSG[la] -> d = #|{: stdtabsh la}|.
+ +
+
+ +
+

Littlewood-Richardson rule for irreducible characters

+ +
+
+Theorem LR_rule_irrSG c d (la : 'P_c) (mu : 'P_d) :
+  'Ind['SG_(c + d)] ('irrSG[la] \o^ 'irrSG[mu]) =
+  \sum_(nu : 'P_(c + d) | included la nu) 'irrSG[nu] *+ LRyam_coeff la mu nu.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.cycles.html b/combi/1.1.0/Combi.SymGroup.cycles.html new file mode 100644 index 00000000..a96a9414 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.cycles.html @@ -0,0 +1,383 @@ + + + + + +Combi.SymGroup.cycles: The Cycle Decomposition of a Permutation + + + + +
+ + + +
+ +

Library Combi.SymGroup.cycles: The Cycle Decomposition of a Permutation

+ +
+
+ +
+ +
+
+
+ +
+

The Cycle Decomposition of a Permutation

+ + +
+ +This files deals with decomposition of permutation into cycles. We define the +following notions, where s is a permutation over a finite type T: + +
+ +
    +
  • psupport s == the set of non fixed points for s + +
    • +
  • porbit_set s == the set of the supports of the cycles of s + +
    • +
  • s \is cyclic == s is a cyclic permutation + +
    • +
  • perm_dec E s == for E : {set {set T}} the set of the restriction of s + to the elements of E. + +
    • +
  • cycle_dec s == the decomposition of s in cyclic permutations. + +
    • +
  • disjoint_supports A == For A : {set {perm T}}, the element of A + have pairwise dijoint supports. + +
    • +
  • cycle_dec_spec s A == A is a decomposition of s into disjoint cycles. + +
    • +
+ +
+ +The main result is Theorem cycle_decP which asserts that cycle_dec s is +the unique decomposition of s into disjoint cycles: + +
+ + unique (cycle_dec_spec s) (cycle_dec s). + +
+ + +
+
+ +
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq fintype.
+From mathcomp Require Import tuple path bigop finset div.
+From mathcomp Require Import fingroup perm action ssralg.
+From mathcomp Require finmodule.
+ +
+Require Import tools permcomp.
+ +
+Set Implicit Arguments.
+ +
+Import GroupScope.
+ +
+#[export] Hint Resolve porbit_id : core.
+ +
+Section PermCycles.
+Variable T : finType.
+Implicit Type (s : {perm T}).
+Implicit Type (X : {set T}).
+Implicit Type (A : {set {perm T}}).
+ +
+
+ +
+

Support of a permutation

+ +
+
+Definition psupport s := ~: 'Fix_('P)([set s]).
+ +
+Lemma in_psupport s x : (x \in psupport s) = (s x != x).
+ +
+Lemma psupport_expg s n : psupport (s ^+ n) \subset psupport s.
+ +
+Lemma psupport_perm_on S s : (perm_on S s) = (psupport s \subset S).
+ +
+Lemma psupport1 : psupport (perm_one T) = set0.
+ +
+Lemma psupport_eq0 s : (s == 1) = (psupport s == set0).
+ +
+Lemma psupport_stable s x : (x \in psupport s) = (s x \in psupport s).
+ +
+Lemma card_psupport_noteq1 s : #|psupport s| != 1%N.
+ +
+Lemma psupport_card_porbit s x : (#|porbit s x| != 1%N) = (x \in psupport s).
+ +
+
+ +
+Complement on porbit +
+
+Lemma porbit_fix s x : (s x == x) = (porbit s x == [set x]).
+ +
+Lemma porbit_mod s x i :
+  (s ^+ i)%g x = (s ^+ (i %% #|porbit s x|))%g x.
+ +
+Lemma eq_in_porbit s x i j :
+  ((s ^+ i)%g x == (s ^+ j)%g x) = (i == j %[mod #|porbit s x|]).
+ +
+
+ +
+

porbit_set of a permutation

+ +
+
+Definition porbit_set s : {set {set T}} := [set x in porbits s | #|x| != 1%N].
+ +
+Lemma in_porbit_setP s X x:
+  reflect (exists2 X, X \in porbit_set s & x \in X) (s x != x).
+ +
+Lemma partition_porbits s : partition (porbits s) setT.
+ +
+Lemma partition_psupport s : partition (porbit_set s) (psupport s).
+ +
+Lemma porbit_set_eq0 s : (s == perm_one T) = (porbit_set s == set0).
+ +
+Lemma porbit_set_astabs s X : X \in porbit_set s -> s \in 'N(X | 'P).
+ +
+
+ +
+

Cyclic permutations

+ +
+
+Definition cyclic := [qualify s | #|porbit_set s| == 1%N].
+ +
+Lemma cyclicP c :
+  reflect (exists2 x, x \in psupport c & psupport c = porbit c x)
+          (c \is cyclic).
+ +
+Lemma cycle_cyclic t :
+  t \is cyclic -> cycle t = [set t ^+ i | i : 'I_#|psupport t|].
+ +
+Lemma order_cyclic t : t \is cyclic -> #[t] = #|psupport t|.
+ +
+
+ +
+Complement about restr_perm +
+
+Lemma psupport_restr_perm_incl X s :
+  psupport (restr_perm X s) \subset X.
+ +
+Lemma restr_perm_neq X s x :
+  restr_perm X s x != x -> restr_perm X s x = s x.
+ +
+Lemma psupport_restr_perm X s :
+  X \in porbit_set s -> psupport (restr_perm X s) = X.
+ +
+Lemma restr_perm_psupportE X s :
+  restr_perm (psupport (restr_perm X s)) s = restr_perm X s.
+ +
+Lemma porbit_restr_perm s x :
+  porbit (restr_perm (porbit s x) s) x = porbit s x.
+ +
+Lemma porbit_set_restr s X :
+  X \in porbit_set s -> porbit_set (restr_perm X s) = [set X].
+ +
+
+ +
+

Decomposition of a permutation by restriction to disjoint stable subsets

+ +
+
+Definition perm_dec (S : {set {set T}}) s : {set {perm T}} :=
+  [set restr_perm X s | X in S].
+Definition cycle_dec s : {set {perm T}} := perm_dec (porbit_set s) s.
+ +
+Lemma cyclic_dec s : {in (cycle_dec s), forall C, C \is cyclic}.
+ +
+Lemma psupport_cycle_dec s :
+  [set psupport C | C in cycle_dec s] = porbit_set s.
+ +
+
+ +
+

Disjoint psupport and commutation

+ +
+
+Definition disjoint_psupports A :=
+  trivIset [set psupport C| C in A] /\ {in A &, injective psupport}.
+ +
+Lemma disjoint_psupport_subset (S1 S2 : {set {perm T}}) :
+  S1 \subset S2 -> disjoint_psupports S2 -> disjoint_psupports S1.
+ +
+Lemma disjoint_perm_dec S s :
+  trivIset S -> disjoint_psupports (perm_dec S s).
+ +
+Lemma disjoint_cycle_dec s :
+  disjoint_psupports (cycle_dec s).
+ +
+Lemma psupport_disjointC s t :
+  [disjoint psupport s & psupport t] -> commute s t.
+ +
+Lemma abelian_disjoint_psupports A : disjoint_psupports A -> abelian <<A>>.
+ +
+Lemma abelian_perm_dec S s : trivIset S -> abelian <<perm_dec S s>>.
+ +
+Lemma abelian_cycle_dec s : abelian <<cycle_dec s>>.
+ +
+Lemma restr_perm_inj s :
+  {in porbit_set s &, injective ((restr_perm (T:=T))^~ s)}.
+ +
+Lemma out_perm_prod A x :
+  {in A, forall C, x \notin psupport C} -> (\prod_(C in A) C) x = x.
+ +
+Import finmodule.FiniteModule morphism.
+ +
+Lemma prod_of_disjoint A C x:
+  disjoint_psupports A -> C \in A ->
+  x \in psupport C -> (\prod_(C0 in A) C0) x = C x.
+ +
+Lemma expg_prod_of_disjoint A C x i:
+  disjoint_psupports A -> C \in A ->
+  x \in psupport C -> ((\prod_(C0 in A) C0) ^+ i) x = (C ^+ i) x.
+ +
+Lemma psupport_of_disjoint A :
+  disjoint_psupports A ->
+  psupport (\prod_(C0 in A) C0) = \bigcup_(C0 in A) psupport C0.
+ +
+Lemma porbit_set_of_disjoint A :
+  disjoint_psupports A ->
+  porbit_set (\prod_(C0 in A) C0) = \bigcup_(C0 in A) porbit_set C0.
+ +
+Lemma perm_decE S s :
+  trivIset S -> psupport s \subset cover S ->
+  s \in 'C(S | ('P)^* ) ->
+  \prod_(C in perm_dec S s) C = s.
+ +
+Lemma cycle_decE s : \prod_(C in cycle_dec s) C = s.
+ +
+Lemma disjoint_psupports_of_decomp A B :
+  disjoint_psupports A -> disjoint_psupports B ->
+    \prod_(C in A) C = \prod_(C in B) C ->
+    {in A & B, forall c1 c2, psupport c1 = psupport c2 -> c1 = c2}.
+ +
+Lemma disjoint_psupports_cycles A :
+  {in A, forall C, C \is cyclic} ->
+  disjoint_psupports A ->
+  {in A, forall C, psupport C \in porbits (\prod_(C in A) C)}.
+ +
+Lemma disjoint_psupports_porbits A :
+  {in A, forall C, C \is cyclic} ->
+  disjoint_psupports A ->
+  {in porbit_set (\prod_(C in A) C),
+    forall X, exists2 C, C \in A & psupport C = X}.
+ +
+
+ +
+

Cycle decomposition of a permutation

+ +
+
+Variant cycle_dec_spec s A : Prop :=
+  CycleDecSpec of
+    {in A, forall C, C \is cyclic} &
+    disjoint_psupports A &
+    \prod_(C in A) C = s : cycle_dec_spec s A.
+ +
+Theorem cycle_decP s : unique (cycle_dec_spec s) (cycle_dec s).
+ +
+End PermCycles.
+ +
+Arguments cyclic {T}.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.cycletype.html b/combi/1.1.0/Combi.SymGroup.cycletype.html new file mode 100644 index 00000000..4d18bfee --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.cycletype.html @@ -0,0 +1,637 @@ + + + + + +Combi.SymGroup.cycletype: The Cycle Type of a Permutation + + + + +
+ + + +
+ +

Library Combi.SymGroup.cycletype: The Cycle Type of a Permutation

+ +
+
+ +
+ +
+
+
+ +
+

Cycle type and conjugacy classes in the symmetric groups

+ + +
+ +This files deals with cycle type of permutations. We define the following +notions, where s is a permutation over a finite type T: + +
+ +
    +
  • canporbit s x == a chosen canonical element in the s-cycle of x + +
    • +
  • indporbit s x == the position of x in its cycle wrt canporbit s x + +
    • +
+ +
+ +Cycle maps: + +
+ +
    +
  • porbits_map s t == if s and t are respectively two permutations of + two fintype U and V, then porbits_map s t is a record for + maps {set U} -> {set V}, sending cycles of s of a given size + to cycles of t of the same size. + +
    • +
  • cymap P == if P : porbits_map s t then cymap P is a map U -> V + which commute with s and t: + +
    + + Lemma cymapP x : cymap (s x) = t (cymap x). + +
    + + moreover, cymap P sends canonical to canonical canporbit_cymap. + +
    • +
+ +
+ +Cycle Type: + +
+ +
    +
  • cycle_type s == the cycle typle of s as a 'P_#|T|. + +
    • +
  • conjbij s t pf == if pf : cycle_type s = cycle_type t, a bijection U -> V + which commute with s and t + +
    • +
+ +
+ +The main results here is Theorem conj_permP which says that +two permutations are conjugate if and only if they have the same cycle type: + +
+ + reflect (exists c, t = (s ^ c)%g) (cycle_type s == cycle_type t). + +
+ +We moreover show that there exists permutations in each partitions: + +
+ +
    +
  • perm_of_porbit C == a cyclic permutation with support C + +
    • +
  • perm_of_setpart P == a permutation whose cycle supports are the part of + the set partition P + +
    • +
  • permCT p == a permutation of cycle type p : P_#|T|. + +
    • +
  • classCT p == the set of permutations of cycle type p : P_#|T|. + +
    • +
+ +
+ +Central functions: + +
+ +
    +
  • '1_[p] == the class function associated to p : P_#|T|. + +
    • +
  • partnCT p == cast a p : 'P_#|'I_n| to a 'P_n. + +
    • +
  • cycle_typeSn s == the cycle type in 'P_n of a permutations in 'S_n. + +
    • +
+ +
+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice fintype.
+From mathcomp Require Import seq tuple finfun finset path bigop.
+From mathcomp Require Import fingroup perm action.
+From mathcomp Require Import ssralg matrix mxalgebra algC classfun.
+ +
+Require Import partition tools sorted.
+Require Import permcomp fibered_set cycles.
+ +
+Set Implicit Arguments.
+ +
+Import LeqGeqOrder.
+ +
+#[local] Hint Resolve porbit_id : core.
+ +
+Reserved Notation "''1_[' G ]"
+         (at level 0, G at level 2, format "''1_[' G ]").
+ +
+Section CanPorbit.
+ +
+Variable T : finType.
+Variable s : {perm T}.
+ +
+Definition canporbit x := odflt x [pick y in porbit s x].
+ +
+Lemma canporbitP x : x \in porbit s (canporbit x).
+ +
+Lemma canporbitE x y :
+  (porbit s x == porbit s y) = (canporbit x == canporbit y).
+ +
+Lemma porbitPb x y :
+  y \in porbit s x -> exists i, y == (s ^+ i)%g x.
+ +
+Definition indporbit (x : T) : nat := ex_minn (porbitPb (canporbitP x)).
+ +
+Lemma indporbitP x : ((s ^+ (indporbit x)) (canporbit x))%g = x.
+ +
+End CanPorbit.
+ +
+
+ +
+

Cycle maps

+ +
+
+Section PorbitBijection.
+ +
+Variables (U V : finType).
+Variables (s : {perm U}) (t : {perm V}).
+ +
+Record porbits_map := PorbitMap {
+                          fs :> {set U} -> {set V};
+                          fs_stab : fs @: porbits s \subset porbits t;
+                          fs_homog : {in porbits s, forall C, #|fs C| = #|C| }
+                        }.
+ +
+Variable CM : porbits_map.
+ +
+Fact cymapcan_aux (x : U) : V.
+ +
+#[local] Definition cymapcan x :=
+  odflt (cymapcan_aux x) [pick y in CM (porbit s x)].
+Definition cymap x := ((t ^+ (indporbit s x)) (cymapcan x))%g.
+ +
+Lemma fs_porbitP x : CM (porbit s x) \in porbits t.
+ +
+Lemma porbit_cymapcan x : porbit t (cymapcan x) = CM (porbit s x).
+ +
+Lemma porbit_cymap x : porbit t (cymap x) = CM (porbit s x).
+ +
+Lemma cymapcan_perm i x : cymapcan ((s ^+ i)%g x) = cymapcan x.
+ +
+Lemma cymapP x : cymap (s x) = t (cymap x).
+ +
+Lemma canporbit_cymap x : canporbit t (cymap x) = cymapcan x.
+ +
+Lemma indporbit_cymap x : indporbit t (cymap x) = indporbit s x.
+ +
+End PorbitBijection.
+ +
+Lemma cymapE (U V : finType) (s : {perm U}) (t : {perm V})
+      (CM1 CM2 : porbits_map s t) :
+   {in porbits s, CM1 =1 CM2} -> cymap CM1 =1 cymap CM2.
+ +
+Lemma cymap_id (U : finType) (s : {perm U}) (CM : porbits_map s s) :
+  {in porbits s, CM =1 id} -> cymap CM =1 id.
+ +
+Lemma cymap_comp (U V W: finType)
+      (u : {perm U}) (v : {perm V}) (w : {perm W})
+      (CMuv : porbits_map u v) (CMvw : porbits_map v w) (CMuw : porbits_map u w) :
+  {in porbits u, CMvw \o CMuv =1 CMuw} ->
+  (cymap CMvw) \o (cymap CMuv) =1 (cymap CMuw).
+ +
+Lemma cymapK (U V : finType)
+      (u : {perm U}) (v : {perm V})
+      (CM : porbits_map u v) (CMi : porbits_map v u) :
+  {in porbits u, cancel CM CMi} -> cancel (cymap CM) (cymap CMi).
+ +
+
+ +
+

Cycle type and conjugacy classes

+ +
+
+Section CycleTypeConj.
+ +
+Variable T : finType.
+Implicit Types (a b s t : {perm T}) (n : nat).
+ +
+Fact cycle_type_subproof s : is_part_of_n #|T| (setpart_shape (porbits s)).
+Definition cycle_type s := IntPartN (cycle_type_subproof s).
+ +
+Lemma size_cycle_type s : size (cycle_type s) = #|porbits s|.
+ +
+Lemma odd_cycle_type s :
+  odd_perm s = odd (sumn (cycle_type s) - size (cycle_type s)).
+ +
+Lemma cycle_type1 : cycle_type 1%g = colpartn #|T|.
+ +
+Lemma cycle_typeV s : cycle_type s^-1 = cycle_type s.
+ +
+Lemma conjg_cycle s a : (<[s]> :^ a = <[s ^ a]>)%g.
+ +
+Lemma porbit_conjg s a x :
+  porbit ((s ^ a)%g) (a x) = [set a y | y in porbit s x].
+ +
+Lemma porbits_conjg s a :
+  porbits (s ^ a)%g = [set [set a y | y in (X : {set T})] | X in porbits s].
+ +
+Lemma cycle_type_conjg s a : cycle_type (s ^ a)%g = cycle_type s.
+ +
+Lemma card_pred_card_porbits s (P : pred nat) :
+  #|[set x in porbits s | P #|x| ]| = count P (cycle_type s).
+ +
+Lemma cycle_type_cyclic s :
+  (s \is cyclic) = (count (fun x => x != 1) (cycle_type s) == 1).
+ +
+Lemma cyclic_conjg s a : s \is cyclic -> (s ^ a)%g \is cyclic.
+ +
+Lemma psupport_conjg s a : psupport (s ^ a) = [set a x | x in psupport s].
+ +
+Lemma card_psupport_conjg s a : #|psupport s| = #|psupport (s ^ a)%g|.
+ +
+Lemma disjoint_psupports_conjg (A : {set {perm T}}) a :
+  disjoint_psupports A -> disjoint_psupports [set (s ^ a)%g | s in A].
+ +
+Import finmodule.FiniteModule morphism.
+ +
+Lemma cycle_dec_conjg s a:
+  [set (c ^ a)%g | c in cycle_dec s] = cycle_dec (s ^ a)%g.
+ +
+End CycleTypeConj.
+ +
+#[local] Definition slporbits (T : finType) (s : {perm T}) :=
+  FiberedSet set0 (porbits s) (fun x => #{x}).
+ +
+Lemma fiber_slporbitE (T : finType) (s : {perm T}) i :
+  #|fiber (slporbits s) i| = count_mem i (cycle_type s).
+ +
+Section DefsFiber.
+ +
+Variables (U V : finType).
+Variables (s : {perm U}) (t : {perm V}).
+Hypothesis eqct : cycle_type s = cycle_type t :> seq nat.
+ +
+Lemma cycle_type_eq :
+  forall i, #|fiber (slporbits s) i| = #|fiber (slporbits t) i|.
+ +
+Fact conjg_porbits_stab :
+  [set fbbij (slporbits t) x | x in (slporbits s)] \subset slporbits t.
+Fact conjg_porbits_homog :
+  {in porbits s, forall C, #|fbbij (U := slporbits s) (slporbits t) C| = #|C| }.
+#[local] Definition CMbij := PorbitMap conjg_porbits_stab conjg_porbits_homog.
+Definition conjbij := cymap CMbij.
+ +
+Lemma conjbijP x : conjbij (s x) = t (conjbij x).
+ +
+End DefsFiber.
+ +
+Lemma conjbijK
+      (U V : finType) (s : {perm U}) (t : {perm V})
+      (eqct : cycle_type s = cycle_type t :> seq nat) :
+  cancel (conjbij eqct) (conjbij (esym eqct)).
+ +
+Section CycleType.
+ +
+Variable T : finType.
+Implicit Types (s t : {perm T}) (C : {set T}) (P : {set {set T}}).
+ +
+Theorem conj_permP s t :
+  reflect (exists c, t = (s ^ c)%g) (cycle_type s == cycle_type t).
+ +
+Lemma classes_of_permP s t :
+  (t \in (s ^: [set: {perm T}])%g) = (cycle_type s == cycle_type t).
+ +
+Section Permofcycletype.
+ +
+Implicit Types (l : nat) (ct : 'P_#|T|).
+ +
+Fact perm_of_porbit_subproof C : injective (next (enum C)).
+Definition perm_of_porbit C := perm (@perm_of_porbit_subproof C).
+ +
+Lemma psupport_perm_of_porbit C :
+  #|C| > 1 -> psupport (perm_of_porbit C) = C.
+ +
+Lemma perm_of_porbitE C x :
+  x \in C -> C = porbit (perm_of_porbit C) (head x (enum C)).
+ +
+Lemma porbits_of_set C : #|C| > 1 -> C \in porbits (perm_of_porbit C).
+ +
+Lemma porbit_set_of_set C : #|C| > 1 -> porbit_set (perm_of_porbit C) = [set C].
+ +
+Lemma isperm_of_porbit C : #|C| > 1 -> perm_of_porbit C \is cyclic.
+ +
+Definition perm_of_setpart P : {perm T} :=
+  (\prod_(C in [set perm_of_porbit s | s in [set X in P |#|X|>1]]) C)%g.
+ +
+Lemma psupports_perm_of_porbit P :
+  [set psupport (perm_of_porbit s) | s in [set X in P | 1 < #|X| ]] =
+  [set X in P | 1 < #|X|].
+ +
+Lemma disj_perm_of_setpart P :
+  partition P [set: T] ->
+  disjoint_psupports [set perm_of_porbit s| s in [set X0 in P | 1 < #|X0|]].
+ +
+Lemma porbits_perm_of_setpart P :
+  partition P [set: T] -> porbits (perm_of_setpart P) = P.
+ +
+Lemma perm_of_setpartE P :
+  partition P [set: T] ->
+  cycle_type (perm_of_setpart P) = setpart_shape P :> seq nat.
+ +
+End Permofcycletype.
+ +
+Section TPerm.
+ +
+Implicit Types (x y z : T).
+ +
+Lemma porbit_tpermD x y z :
+  x != z -> y != z -> porbit (tperm x y) z = [set z].
+ +
+Lemma porbit_tpermL x y : porbit (tperm x y) x = [set x; y].
+Lemma porbit_tpermR x y : porbit (tperm x y) y = [set x; y].
+Lemma porbits_tperm x y :
+  porbits (tperm x y) =
+  [set x; y] |: [set [set z] | z : T & (x != z) && (y != z)].
+ +
+Lemma cycle_type_tperm x y :
+  x != y -> cycle_type (tperm x y) = hookpartn #|T| 1.
+ +
+Lemma tperm_conj x1 y1 x2 y2 :
+  x1 != y1 -> x2 != y2 -> exists t, tperm x2 y2 = ((tperm x1 y1) ^ t)%g.
+ +
+Lemma cycle_type_tpermP (s : {perm T}) :
+  #|T| > 1 ->
+  cycle_type s = hookpartn #|T| 1 ->
+  exists x y, x != y /\ s = tperm x y.
+ +
+End TPerm.
+ +
+Section Classes.
+ +
+Variable ct : 'P_#|T|.
+ +
+Lemma permCT_exists : {s | cycle_type s == ct}.
+Definition permCT := val permCT_exists.
+Lemma permCTP : cycle_type permCT = ct.
+ +
+Definition classCT : {set {perm T}} := class permCT [set: {perm T}].
+ +
+Lemma classCTP s :
+  (cycle_type s == ct) = (s \in classCT).
+ +
+Lemma card_classCT_neq0 : #|classCT| != 0%N.
+ +
+End Classes.
+ +
+Lemma permCT_colpartn_card : permCT (colpartn #|T|) = 1%g.
+ +
+Lemma classCT_inj : injective classCT.
+ +
+Lemma imset_classCT :
+  [set classCT p | p in setT] = classes [set: {perm T}].
+ +
+Lemma card_classes_perm :
+  #|classes [set: {perm T}]| = #|{: 'P_#|T| }|.
+ +
+Import GroupScope GRing.Theory.
+#[local] Open Scope ring_scope.
+ +
+
+ +
+

Cycle indicator

+ +
+
+Section CFunIndicator.
+ +
+Variable ct : 'P_#|T|.
+ +
+Definition cfuniCT :=
+  cfun_indicator [set: {perm T}] (classCT ct).
+ +
+#[local] Notation "''1_[' p ]" := (cfuniCT p) : ring_scope.
+ +
+Lemma cfuniCTE s :
+  ('1_[s]) = (cycle_type s == ct)%:R.
+ +
+End CFunIndicator.
+End CycleType.
+ +
+Notation "''1_[' p ]" := (cfuniCT p) : ring_scope.
+ +
+Coercion CTpartn n := cast_intpartn (esym (card_ord n)).
+ +
+Import GroupScope GRing.Theory.
+#[local] Open Scope ring_scope.
+ +
+
+ +
+

Central function for 'S_n

+ +
+
+Section Sn.
+ +
+Variable n : nat.
+ +
+Definition partnCT : 'P_#|'I_n| -> 'P_n := cast_intpartn (card_ord n).
+Definition cycle_typeSn (s : 'S_n) : 'P_n := partnCT (cycle_type s).
+ +
+Lemma CTpartnK (p : 'P_n) : partnCT p = p.
+ +
+Lemma partnCTE p1 p2 : (p1 == p2) = (partnCT p1 == partnCT p2).
+ +
+Lemma partnCTK (p : 'P_#|'I_n|) : p = partnCT p.
+ +
+Lemma partnCT_congr p1 (p2 : 'P_n) : (partnCT p1 == p2) = (p1 == p2).
+ +
+Lemma CTpartn_colpartn : CTpartn (colpartn n) = colpartn #|'I_n|.
+ +
+Lemma cycle_typeSn1 : cycle_typeSn 1%g = colpartn n.
+ +
+Lemma permCT_colpartn : permCT (colpartn n) = 1%g.
+ +
+Lemma cfuniCTnE (ct : 'P_n) (s : 'S_n) :
+  '1_[ct] s = (cycle_typeSn s == ct)%:R.
+ +
+Lemma cycle_typeSn_permCT (ct : 'P_n) : cycle_typeSn (permCT ct) = ct.
+ +
+End Sn.
+ +
+Lemma cast_cycle_typeSN m n (s : 'S_m) (eq_mn : m = n) :
+  cycle_typeSn (cast_perm eq_mn s) = cast_intpartn eq_mn (cycle_typeSn s).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.permcent.html b/combi/1.1.0/Combi.SymGroup.permcent.html new file mode 100644 index 00000000..3fed4e1e --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.permcent.html @@ -0,0 +1,396 @@ + + + + + +Combi.SymGroup.permcent: The Centralizer of a Permutation + + + + +
+ + + +
+ +

Library Combi.SymGroup.permcent: The Centralizer of a Permutation

+ +
+
+ +
+ +
+
+
+ +
+

The Centralizer of a Permutation

+ + +
+ +The main goal is to understand the structure of the centralizer group of a +given permutation s, compute its cardinality and deduce the cardinality +of the conjugacy class of s. + +
+ +Here are the notion defined is this file, where s is a fixed permutation on +a finite type T: + +
+ +
    +
  • 'CC(s) == the group of permutations of T which commute with s and + stabilize each cycle of s. + +
    • +
  • stab_iporbits s == the group of permutations of T which only move + the cycles of s and sends any cycle to a cycle of the same + length. + +
    • +
  • inporbits s t == the permutation induced by t on the set of the cycles + of s. inporbits s is a morphism from {perm T} to + stab_iporbits s. + +
    • +
  • permcycles s P == a right inverse morphism from stab_iporbits s to + {perm T}. If P belongs to stab_iporbits s then + permcycles s P si a compatible lifting of P in {perm T}, + otherwise the identity. + +
    + + +
    • +
  • zcard l == where is the number of + occurrence of in l. + +
    • +
+ +
+ +Here are the main results: + +
+ +
    +
  • 'CC(s)' is the direct product of the group generated by the cycles of s. + Lemma porbitgrpE: + +
    + + 'CC(s) = \big[dprod/1]_(c in cycle_dec s) <[c]>. + +
    + + +
    • +
  • stab_iporbits s is the direct product over i of the group permuting + the set of the cycles of size i. Theorem stab_iporbitsE: + +
    + + + stab_iporbits s = + \big[dprod/1]_(i < #|T|.+1) Sym (porbits s :&: 'SC_i). + + +
    + + +
    • +
  • The centralizer 'C[s] is the semidirect product of 'CC(s)' and the + lifting of stab_iporbits s. Theorem cent1_permE: + +
    + + 'C[s] = 'CC(s) ><| (permcycles s) @* (stab_iporbits s). + +
    + + +
    • +
  • The cardinality of the centralizer of s given by zcard. + Corollary card_cent1_perm: + +
    + + #|'C[s]| = zcard (cycle_type s). + +
    + + +
    • +
  • The cardinality of the conjugacy class associated to the partition l + of an integer n is given by Theorem card_class_of_part: + +
    + + #|classCT l| = n`! %/ zcard l. + +
    • +
+ +
+ + +
+
+From Corelib Require Import Setoid.
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq fintype.
+From mathcomp Require Import tuple path bigop finset div.
+From mathcomp Require Import fingroup perm action gproduct morphism.
+ +
+Require Import tools partition permcomp cycles cycletype.
+ +
+Import GroupScope.
+ +
+Set Implicit Arguments.
+ +
+#[local] Hint Resolve porbit_id : core.
+ +
+#[local] Notation "''SC_' i " := (finset (fun x => #{x} == i))
+    (at level 0).
+ +
+
+ +
+

Support and cycle in the centralizer

+ +
+
+Section PermCycles.
+ +
+Variable T : finType.
+Implicit Type (s t c : {perm T}).
+ +
+Lemma disjoint_psupport_dprodE (S : {set {perm T}}) :
+  disjoint_psupports S ->
+  (\big[dprod/1]_(s in S) <[s]>) = (\prod_(s in S) <[s]>)%G.
+ +
+Lemma cent1_act_porbit s t x :
+  t \in 'C[s] -> ('P)^* (porbit s x) t = porbit s (t x).
+ +
+Lemma cent1_act_on_porbits s : [acts 'C[s], on porbits s | ('P)^*].
+ +
+Lemma cent1_act_on_iporbits s i :
+  [acts 'C[s], on porbits s :&: 'SC_i | ('P)^*].
+ +
+Lemma commute_cyclic c t :
+  c \is cyclic -> t \in 'C[c] -> perm_on (psupport c) t -> exists i, t = c ^+ i.
+ +
+
+ +
+

The cyclic centralizer

+ +
+
+Notation "''CC' ( s )" :=
+  'C_('C[s])(porbits s | ('P)^* ) (format "''CC' ( s )") : group_scope.
+ +
+Lemma restr_perm_genC C s t :
+  C \in porbit_set s -> t \in 'CC(s) -> restr_perm C t \in <[restr_perm C s]>%G.
+ +
+Lemma stab_porbit S s :
+  s \in 'N(S | 'P) -> forall x, (x \in S) = (porbit s x \subset S).
+ +
+Lemma restr_perm_porbits S s :
+  restr_perm S s \in 'C(porbits s | ('P)^* ).
+ +
+Lemma porbitgrpE s : 'CC(s) = \big[dprod/1]_(c in cycle_dec s) <[c]>.
+ +
+Lemma card_porbitgrpE s : #|'CC(s)| = (\prod_(i <- cycle_type s) i)%N.
+ +
+
+ +
+

Permuting the cycles among themselves

+ +
+
+Definition stab_iporbits s : {set {perm {set T}}} :=
+  Sym (porbits s) :&:
+    \bigcap_(i : 'I_#|T|.+1) 'N(porbits s :&: 'SC_i | 'P).
+ +
+Definition inporbits s : {perm T} -> {perm {set T}} :=
+  restr_perm (porbits s) \o actperm 'P^*.
+ +
+Section CM.
+ +
+Variable s : {perm T}.
+Implicit Type P : {perm {set T}}.
+ +
+Lemma stab_iporbits_stab P :
+  (if P \in stab_iporbits s then P else 1) @: porbits s \subset porbits s.
+ +
+Lemma stab_iporbits_homog P :
+  {in porbits s, forall C : {set T},
+       #|(if P \in stab_iporbits s then P else 1) C| = #|C| }.
+ +
+#[local] Definition stab_iporbits_porbitmap P :=
+  PorbitMap (stab_iporbits_stab P) (stab_iporbits_homog P).
+#[local] Definition stab_iporbits_map P := cymap (stab_iporbits_porbitmap P).
+ +
+Lemma stab_iporbits_map_inj P : injective (stab_iporbits_map P).
+Definition permcycles P := perm (@stab_iporbits_map_inj P).
+ +
+Lemma permcyclesC P : commute (permcycles P) s.
+ +
+Lemma permcyclesP P : (permcycles P) \in 'C[s].
+ +
+Lemma porbit_permcycles P x :
+  P \in stab_iporbits s -> porbit s (permcycles P x) = P (porbit s x).
+ +
+End CM.
+ +
+Lemma permcyclesM s :
+  {in stab_iporbits s &, {morph permcycles s : P Q / P * Q}}.
+Canonical permcycles_morphism s := Morphism (permcyclesM (s := s)).
+ +
+Lemma permcyclesK s :
+  {in stab_iporbits s, cancel (permcycles s) (inporbits s)}.
+ +
+Lemma permcycles_inj s : 'injm (permcycles s).
+ +
+Lemma inporbits_im s : inporbits s @: 'C[s] = stab_iporbits s.
+ +
+Lemma trivIset_iporbits s : trivIset [set porbits s :&: 'SC_i | i : 'I_#|T|.+1].
+ +
+Lemma cover_iporbits s :
+  cover [set porbits s :&: 'SC_i | i : 'I_#|T|.+1] = porbits s.
+ +
+Lemma stab_iporbitsE_prod s :
+  stab_iporbits s =
+  (\prod_(i < #|T|.+1) Sym_group (porbits s :&: 'SC_i))%G.
+ +
+Theorem stab_iporbitsE s :
+  stab_iporbits s = \big[dprod/1]_(i < #|T|.+1) Sym (porbits s :&: 'SC_i).
+ +
+Lemma card_stab_iporbits s :
+  #|stab_iporbits s| =
+    (\prod_(i < #|T|.+1) (count_mem (nat_of_ord i) (cycle_type s))`!)%N.
+ +
+Lemma conj_porbitgrp s y z :
+  y \in 'C[s] -> z \in 'CC(s) -> z ^ y \in 'CC(s).
+ +
+Lemma inporbits1 s t : t \in 'C(porbits s | ('P)^* ) -> inporbits s t = 1.
+ +
+Lemma cent1_stab_iporbit_porbitgrpS s :
+  'C[s] \subset permcycles s @* stab_iporbits s * 'CC(s).
+ +
+
+ +
+

Main theorem

+ +
+
+Theorem cent1_permE s :
+  'C[s] = 'CC(s) ><| (permcycles s) @* (stab_iporbits s).
+ +
+#[local] Open Scope nat_scope.
+ +
+
+ +
+

Conjugacy class cardinality

+ +
+
+Definition zcard l :=
+  \prod_(i <- l) i * \prod_(i < (sumn l).+1) (count_mem (i : nat) l)`!.
+ +
+Lemma zcard_nil : zcard [::] = 1.
+ +
+Lemma zcard_any l b :
+  (sumn l < b) ->
+  \prod_(i <- l) i * \prod_(i < b) (count_mem (i : nat) l)`! = zcard l.
+ +
+Lemma zcard_rem i l :
+  i != 0 -> i \in l -> i * (count_mem i l) * (zcard (rem i l)) = zcard l.
+ +
+Corollary card_cent1_perm s : #|'C[s]| = zcard (cycle_type s).
+ +
+Theorem card_class_perm s :
+  #|class s [set: {perm T}]| = #|T|`! %/ zcard (cycle_type s).
+ +
+End PermCycles.
+ +
+Lemma dvdn_zcard_fact n (l : 'P_n) : zcard l %| n`!.
+ +
+Lemma neq0zcard n (l : 'P_n) : zcard l != 0.
+ +
+Theorem card_class_of_part n (l : 'P_n) : #|classCT l| = n`! %/ zcard l.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.presentSn.html b/combi/1.1.0/Combi.SymGroup.presentSn.html new file mode 100644 index 00000000..6eace5e6 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.presentSn.html @@ -0,0 +1,1454 @@ + + + + + +Combi.SymGroup.presentSn: The Coxeter Presentation of the Symmetric Group + + + + +
+ + + +
+ +

Library Combi.SymGroup.presentSn: The Coxeter Presentation of the Symmetric Group

+ +
+
+ +
+ +
+
+
+ +
+

The Coxeter Presentation of the Symmetric Group

+ + +
+ +The main goal is to show that elementary transpositions generate the symmetric +groups as a Coxeter group. We follow the proofs from Alain Lascoux "The +Symmetric Group", unfinished notes. It is not the shortest nor the simplest, +but it is fully explicit and algorithmic. In particular, it goes through two +algorithms to +
    +
  • compute the size-lexicographically minimal reduced word of a permutation + +
    • +
  • straighten a word to the previous reduced word. + +
    • +
+ +
+ +Here are the notion defined is this file: + +
+ +Elementary tranpositions + +
+ +
    +
  • 's_i == the i-th elementary transposition. It is of type 'S_n.+1 + where n is inferred from the context. i is an integer + smaller than n (otherwise eltr i is the identity) + +
    • +
+ +
+ +Inversion sets + +
+ +
    +
  • invset s == the set of inversion of s + +
    • +
  • Delta == the set of pair (i, j) such that 0 <= i < j < n.+1; + n is infered from the context + +
    • +
  • is_invset IS == IS is the inversion set of a permutation, that is + subdiagonal transitive and co-transitive + +
    • +
  • length s == the number of inversion of s we show latter that this is the + Coxeter length of s + +
    • +
  • rsymrel IS == the reflexive and antisymmetric closure of the binary + relation associated to the inversion set IS + +
    • +
  • perm_of_invset IS == the permutation whose inversion set is IS + +
    • +
+ +
+ +Notion of code + +
+ +
    +
  • c \is a code == c is a list such that c_i <= i. + +
    • +
  • wordcd c == for a code c the list 0, 1-c1..1, 2-c2..2, .. + +
    + + +
    • +
  • is_code_of_size c n == the predicate c is a code of size n + +
    • +
  • enum_codesz n == a list enumerating all the code of size n + +
    • +
  • codesz n == sigma type for codes of size n, canonically a finType + +
    • +
+ +
+ +Inverse Lehmer code + +
+ +
    +
  • cocode s == the recursively defined Lehmer code of s^-1 + +
    • +
  • canword s == the canonical reduced word for s as a seq 'I_n + +
    • +
  • prods_codesz c == the product eltr i associated to the code c that + is for i <- wordcd c. Lemma prods_codesz_bij shows that it + is a bijection from codesz n.+1 to 'S_n.+1. + +
    • +
+ +
+ +Reduced words + +
+ +
    +
  • w \is reduced == w is a word of size length 's_[w]. + +
    • +
+ +
+ +Braid relations + +
+ +
    +
  • braid_aba == the braid rewriting rule a a.+1 a == a.+1 a a.+1 + +
    • +
  • braidC == the braid rewriting rule a b == b a if |b - a| > 1 + +
    • +
  • braidrule == the union of the two previous rules + +
    • +
  • braidcongr == the braid monoid congruence generated by braidrule + +
    • +
  • braidclass s == the braid monoid class of s + +
    • +
  • a =Br b == a and b are equivalent under the braid congruence + +
    + + +
    • +
  • reduce u v == u reduces to v using the rule [:: i i] -> [::] + +
    • +
  • braidred u v == either u =Br v or reduce u v + +
    • +
+ +
+ +The algorithms + +
+ +
    +
  • inscode c i == the insertion the letter i in the code c. If c is the + code for the permutation s then it is the code for the + permutation s * 's_i + +
    • +
  • path_catl p w == append w to all the sequences of p that is + [seq v ++ w | v <- p] + +
    • +
  • straighten w == the straigtened reduced word equivalent to w + +
    • +
+ +
+ +The presentation + +
+ +
    +
  • relat_Sn n g == if g : nat -> gT for a finGroupType gt. A + proposition which asserts that the relation of the symmetric + group holds for the g i denoted 'g_i. Namely + +
    + +
      +
    • forall i, i < n -> 'g_i^+2 = 1 + +
      • +
    • forall i, i.+1 < n -> 'g_i * 'g_i.+1 * 'g_i = 'g_i.+1 * 'g_i * 'g_i.+1 + +
      • +
    • forall i j, i.+1 < j < n -> 'g_i * 'g_j = 'g_j * 'g_i + +
      • +
    + +
    • +
+ +
+ +The main result is thus Theorem presentation_Sn_eltr: + +
+ + + relat_Sn -> + exists f : {morphism 'SG_n.+1 >-> gT}, forall i, i < n -> f 's_i = 'g_i. + + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import fingroup perm morphism presentation.
+From mathcomp Require Import ssralg poly ssrint.
+ +
+Require Import permcomp tools permuted combclass congr.
+ +
+Set Implicit Arguments.
+ +
+Section SRel.
+ +
+Variable T : finType.
+Implicit Type (S A B : {set T * T}).
+Definition srel S := [rel x y : T | (x, y) \in S].
+ +
+Lemma srelE S1 S2 : srel S1 =2 srel S2 -> S1 = S2.
+ +
+Definition tclosure A : {set T * T} :=
+  [set p | (p.1 != p.2) && (connect (srel A) p.1 p.2)].
+ +
+Lemma tclosure_sub A B :
+  A \subset B -> transitive (srel B) -> tclosure A \subset B.
+ +
+End SRel.
+ +
+Notation "''SG_' n" := [set: 'S_n]
+  (at level 8, n at level 2, format "''SG_' n").
+ +
+Reserved Notation "''s_' i"
+      (at level 8, i at level 2, format "''s_' i").
+Reserved Notation "''s_[' w ] "
+      (at level 0, w at level 100, format "''s_[' w ]").
+ +
+#[local] Reserved Notation "''II_' n" (at level 8, n at level 2).
+#[local] Reserved Notation "a =Br b" (at level 70).
+#[local] Reserved Notation "''I[' a '..' b ']'" (at level 0, a, b at level 2).
+ +
+Lemma ieqi1F i : (i == i.+1) = false.
+Lemma ieqi2F i : (i == i.+2) = false.
+Lemma i1eqiF i : (i.+1 == i) = false.
+Lemma i2eqiF i : (i.+2 == i) = false.
+ +
+Lemma inordi n i : i < n -> (@inord n i = i :> nat).
+ +
+Lemma inordi1 n i : i < n -> (@inord n i.+1 = i.+1 :> nat).
+ +
+Lemma inord1i n i : i.+1 < n -> (@inord n i = i :> nat).
+ +
+Lemma inordi_neq_i1 n i : i < n -> (@inord n i != @inord n i.+1).
+ +
+Definition trivSimpl := (eq_refl, eqSS, ieqi1F, ieqi2F, i1eqiF, i2eqiF).
+ +
+
+ +
+

Codes for permutations

+ +
+
+Section Codes.
+ +
+Definition code := [qualify a c : seq nat |
+  all (fun i => nth 0 c i <= i) (iota 0 (size c)) ].
+ +
+Definition wordcd (c : seq nat) : seq nat :=
+  flatten [seq rev (iota (i - nth 0 c i) (nth 0 c i)) | i <- iota 0 (size c)].
+ +
+Lemma size_wordcd c : size (wordcd c) = sumn c.
+ +
+Lemma is_codeP c :
+  reflect (forall i, i < size c -> nth 0 c i <= i) (c \is a code).
+ +
+Lemma is_code_rcons c i : i <= size c -> c \is a code -> rcons c i \is a code.
+ +
+Lemma is_code_rconsK c i : rcons c i \is a code -> c \is a code.
+ +
+Lemma code_ltn_size c : c \is a code -> all (gtn (size c)) c.
+ +
+Lemma wordcd_ltn c :
+  c \is a code -> all (gtn (size c).-1) (wordcd c).
+ +
+Lemma insub_wordcdK n c :
+  c \is a code -> size c <= n.+1 ->
+  [seq (i : nat) | i : 'I_n <- pmap insub (wordcd c)] = wordcd c.
+ +
+Definition is_code_of_size n c := (c \is a code) && (size c == n).
+ +
+Fixpoint enum_codesz n :=
+  if n is n'.+1 then
+    flatten [seq [seq rcons c i | c <- enum_codesz n'] | i <- iota 0 n]
+  else [:: [::]].
+ +
+Lemma enum_codeszP n : all (is_code_of_size n) (enum_codesz n).
+ +
+Lemma enum_codesz_countE n c :
+  is_code_of_size n c -> count_mem c (enum_codesz n) = 1.
+ +
+
+ +
+

Fintype for codes for permutations of size n

+ +
+
+Section FinType.
+ +
+Variable n : nat.
+ +
+Structure codesz : Set :=
+  CodeSZ {cdval :> seq nat; _ : (cdval \is a code) && (size cdval == n)}.
+ +
+Implicit Type (c : codesz).
+ +
+Lemma codeszP c : val c \is a code.
+ +
+Lemma size_codesz c : size c = n.
+ +
+Lemma enum_codeszE : map val (enum {: codesz}) = enum_codesz n.
+ +
+End FinType.
+ +
+Lemma card_codesz n : #|{: codesz n}| = n`!.
+ +
+End Codes.
+#[export] Hint Resolve codeszP : core.
+ +
+Import GroupScope.
+ +
+
+ +
+

Elementary transpositions

+ +
+
+Section Transp.
+ +
+Variable T : finType.
+Implicit Types (x y z t : T).
+ +
+Lemma tperm_braid x y z :
+  x != y -> y != z ->
+  tperm x y * tperm y z * tperm x y = tperm y z * tperm x y * tperm y z.
+ +
+Lemma tpermC x y a b :
+  x != a -> y != a -> x != b -> y != b ->
+  tperm x y * tperm a b = tperm a b * tperm x y.
+ +
+End Transp.
+ +
+Section MaxPerm.
+ +
+Context {n : nat}.
+Definition maxperm : 'S_n := perm (@rev_ord_inj n).
+ +
+Lemma maxpermK : involutive maxperm.
+ Lemma maxpermV : maxperm^-1 = maxperm.
+ +
+End MaxPerm.
+ +
+#[local] Notation "''II_' n" := ('I_n * 'I_n)%type.
+ +
+
+ +
+

Inversion set of a permutation

+ +
+
+Section InvSet.
+ +
+Context {n : nat}.
+ +
+Implicit Type (p : 'II_n) (IS S : {set 'II_n}).
+Implicit Type s t : 'S_n.
+ +
+Definition Delta : {set 'II_n} := [set p : 'II_n | (p.1 < p.2)].
+Definition invset s : {set 'II_n} :=
+  [set p : 'II_n | (p.1 < p.2) && (s p.1 > s p.2) ].
+Definition rsymrel IS :=
+  [rel i j : 'I_n |
+   [|| (i == j), (j, i) \in IS | (i < j) && ((i, j) \notin IS)]].
+Definition length s := #|invset s|.
+ +
+Variant is_invset IS : Prop :=
+  IsInvset of IS \subset Delta
+  & transitive (srel IS)
+  & transitive (srel (Delta :\: IS)).
+ +
+Lemma mem_Delta i j : (i, j) \in Delta = (i < j).
+ Lemma DeltaP i j : (i, j) \in Delta -> (i < j).
+ +
+Lemma card_Delta : #|Delta| = 'C(n, 2).
+ +
+Lemma is_invset_Delta IS : is_invset IS -> IS \subset Delta.
+ +
+Lemma transitive_DeltaI1 IS :
+  (transitive (srel (Delta :\: IS))) <->
+  (forall i j k : 'I_n,
+      i < j < k -> (i, k) \in IS -> ((i, j) \in IS \/ (j, k) \in IS)).
+ +
+Lemma invset_Delta s : invset s \subset Delta.
+ +
+Lemma invset_permV s :
+  invset s^-1 = [set (s (p.2), s (p.1)) | p in invset s].
+ +
+Lemma invsetP s : is_invset (invset s).
+ +
+Lemma invset_maxperm : invset (@maxperm n) = Delta.
+ +
+Lemma invset_maxpermMr s : invset (s * maxperm) = Delta :\: invset s.
+Lemma invset_maxpermMl s :
+  invset (maxperm * s) =
+  Delta :\: [set (maxperm p.2, maxperm p.1) | p in invset s].
+ +
+Lemma rsymrel_refl IS : reflexive (rsymrel IS).
+ Lemma rsymrel_anti IS :
+  IS \subset Delta -> antisymmetric (rsymrel IS).
+Lemma rsymrel_trans IS : is_invset IS -> transitive (rsymrel IS).
+Lemma rsymrel_total (IS : {set 'II_n}) :
+  IS \subset Delta -> total (rsymrel IS).
+ +
+Lemma rsym_invset_refl s : reflexive (rsymrel (invset s)).
+ Lemma rsym_invset_anti s : antisymmetric (rsymrel (invset s)).
+ Lemma rsym_invset_trans s : transitive (rsymrel (invset s)).
+ Lemma rsym_invset_total s : total (rsymrel (invset s)).
+ +
+Lemma perm_of_relP (r : rel 'I_n) :
+  injective (fun i : 'I_n => nth i (sort r (enum 'I_n)) i).
+Definition perm_of_invset IS :=
+  (perm (@perm_of_relP (rsymrel IS)))^-1.
+ +
+Lemma rsym_invsetP s :
+  sorted (rsymrel (invset s)) [seq s^-1 i | i : 'I_n].
+ +
+Lemma invsetK : cancel invset perm_of_invset.
+ +
+Theorem invset_inj : injective invset.
+ +
+Theorem perm_of_invsetK (IS : {set 'II_n}) :
+  is_invset IS -> invset (perm_of_invset IS) = IS.
+ +
+Lemma length1 : length 1 = 0.
+ +
+Lemma lengthV s : length s^-1 = length s.
+ +
+Lemma length_max s : length s <= 'C(n, 2).
+ +
+Lemma length_maxperm : length (@maxperm n) = 'C(n, 2).
+ Lemma length_maxpermE s : length s = 'C(n, 2) -> s = (@maxperm n).
+ +
+Lemma length_maxpermMr s : length (s * maxperm) = 'C(n, 2) - length s.
+Lemma length_maxpermMl s : length (maxperm * s) = 'C(n, 2) - length s.
+ +
+End InvSet.
+ +
+Section ElemTransp.
+ +
+Variable n0 : nat.
+Notation n := n0.+1.
+ +
+Definition eltr i : 'S_n0.+1 := tperm (inord i) (inord i.+1).
+ +
+Notation "''s_' i" := (eltr i).
+Notation "''s_[' w ]" := (\prod_(i <- w) 's_i).
+ +
+Implicit Type s t : 'S_n.
+ +
+Lemma eltrV i : ('s_i)^-1 = 's_i.
+Lemma eltrK i : involutive 's_i.
+Lemma eltr2 i : 's_i * 's_i = 1.
+ +
+Lemma eltr_braid i :
+  i.+1 < n0 -> 's_i * 's_i.+1 * 's_i = 's_i.+1 * 's_i * 's_i.+1.
+ +
+Lemma eltr_comm i j :
+  i.+1 < j < n0 -> 's_i * 's_j = 's_j * 's_i.
+ +
+#[local] Lemma eltrL_ord (i : 'I_n) : 's_i i = inord i.+1.
+ #[local] Lemma eltrR_ord (i : 'I_n) : 's_i (inord i.+1) = i.
+ #[local] Lemma eltrD_ord (i j : 'I_n) : i != j -> inord i.+1 != j -> 's_i j = j.
+ +
+Definition eltrL := (eltrL_ord, tpermL).
+Definition eltrR := (eltrR_ord, tpermR).
+Definition eltrD := (eltrD_ord, tpermD).
+ +
+Lemma prods_iota_mi (m : 'I_n) i :
+  i <= m -> 's_[iota (m - i) i] (inord (m - i)) = m.
+ +
+Lemma prods_iota_ltmi i (m u : 'I_n) :
+  u < m - i -> 's_[(iota (m - i) i)] u = u.
+ +
+Lemma cycleij_j (i j : 'I_n) :
+  i <= j -> 's_[index_iota i j] i = j.
+ +
+Lemma cycleij_lt (i j k : 'I_n) :
+  i <= j -> k < i -> 's_[index_iota i j] k = k.
+ +
+Lemma cycleij_gt (i j : nat) (k : 'I_n) :
+  j < k -> 's_[index_iota i j] k = k.
+ +
+Lemma cycleij_inS i (j k : 'I_n) :
+  i <= k < j -> 's_[index_iota i j] (inord k.+1) = k.
+ +
+Lemma cycleij_in i (j k : 'I_n) :
+  i < k <= j -> 's_[index_iota i j] k = (inord k.-1).
+ +
+Lemma prodsK w : 's_[w] * 's_[rev w] = 1.
+ +
+Lemma prodsV w : 's_[rev w] = 's_[w]^-1.
+ +
+Lemma odd_eltr i : (i < n0)%N -> odd_perm 's_i.
+ +
+End ElemTransp.
+ +
+Section PermOfInvSetEltr.
+ +
+Variable n0 : nat.
+Local Notation n := n0.+1.
+Implicit Type s t : 'S_n.
+ +
+Notation "''s_' i" := (eltr _ i).
+Notation "''s_[' w ]" := (\prod_(i <- w) 's_i).
+ +
+Lemma eltr_exchange i (a b : 'I_n) :
+  i < n0 -> a < b -> 's_i a < 's_i b = (i != a) || (i.+1 != b).
+ +
+Lemma invset_eltrL s (i : 'I_n) :
+  i < n0 -> s i < s (inord i.+1) ->
+  invset ('s_i * s) =
+  (i, inord i.+1) |: [set ('s_i p.1, 's_i p.2) | p in invset s].
+ +
+Lemma invset_eltrR s (i : 'I_n) :
+  i < n0 -> s^-1 i < s^-1 (inord i.+1) ->
+  invset (s * 's_i) = (s^-1 i, s^-1 (inord i.+1)) |: invset s.
+ +
+End PermOfInvSetEltr.
+Arguments Delta {n}.
+ +
+Section Length.
+ +
+Variable n0 : nat.
+Notation n := n0.+1.
+ +
+Notation "''s_' i" := (eltr n0 i).
+Notation "''s_[' w ]" := (\prod_(i <- w) 's_i).
+ +
+Implicit Type s t u : 'S_n.
+ +
+
+ +
+

Length of a permutation

+ the length was defined with invset +
+
+ +
+Lemma length_add1L s (i : 'I_n) :
+  i < n0 -> s i < s (inord i.+1) -> length ('s_i * s) = (length s).+1.
+ +
+Lemma length_sub1L s (i : 'I_n) :
+  i < n0 -> s i > s (inord i.+1) -> length s = (length ('s_i * s)).+1.
+ +
+Lemma length_descL s (i : 'I_n) :
+  i < n0 -> (s i < s (inord i.+1)) = (length ('s_i * s) > length s).
+ +
+Lemma length_add1R s (i : 'I_n) :
+  i < n0 -> s^-1 i < s^-1 (inord i.+1) -> length (s * 's_i) = (length s).+1.
+ +
+Lemma length_sub1R s (i : 'I_n) :
+  i < n0 -> s^-1 i > s^-1 (inord i.+1) -> length s = (length (s * 's_i)).+1.
+ +
+Lemma length_descR s (i : 'I_n) :
+  i < n0 -> (s^-1 i < s^-1 (inord i.+1)) = (length (s * 's_i) > length s).
+ +
+Lemma length_eltr (i : 'I_n0) : length 's_i = 1%N.
+ +
+Lemma length_prods (w : seq 'I_n0) : length 's_[w] <= size w.
+ +
+
+ +
+

The canonical reduced word of a permutation

+ +
+
+Fixpoint cocode_rec m c (s : 'S_n) : seq nat :=
+  if m is m'.+1 then
+    let mo := inord m' in
+    cocode_rec m' (mo - s mo :: c) (s * 's_[iota (s mo) (mo - s mo)])
+  else c.
+Definition cocode s := cocode_rec n [::] s.
+Definition canword s : seq 'I_n0 := pmap insub (wordcd (cocode s)).
+ +
+ +
+
+ +
+

Properties of the dual code

+ +
+
+Lemma cocode_rec_cat m c s : cocode_rec m c s = (cocode_rec m [::] s ++ c).
+ +
+Lemma wordcdE c :
+  's_[wordcd c] =
+  \prod_(i <- iota 0 (size c)) 's_[rev (iota (i - nth 0 c i) (nth 0 c i))].
+ +
+Lemma size_cocode_rec m s c : size (cocode_rec m c s) = m + size c.
+ +
+Lemma size_cocode s : size (cocode s) = n.
+ +
+
+ +
+

Partial codes

+ +
+
+Section PartCode.
+ +
+Let is_partcode m c :=
+  forall i, i < size c -> nth 0 c i <= i + m.
+Let word_of_partcocode m c : seq nat :=
+  flatten [seq rev (iota (m + i - nth 0 c i) (nth 0 c i)) |
+           i <- iota 0 (size c)].
+ +
+Lemma perm_on_cocode_recP m c s0 s :
+  m <= n ->
+  is_partcode m c ->
+  s0 = s * 's_[word_of_partcocode m c] ->
+  perm_on [set k : 'I_n | k < m] s ->
+  let cf := cocode_rec m c s in cf \is a code /\ s0 = 's_[wordcd cf].
+ +
+End PartCode.
+ +
+Lemma perm_on_prods c m :
+  c \is a code -> m <= size c -> m <= n ->
+  perm_on [set k : 'I_n | k < m]
+          (\prod_(i <- iota 0 m) 's_[(rev (iota (i - nth 0 c i) (nth 0 c i)))]).
+ +
+Lemma perm_on_prods_length_ord s i (m : 'I_n) :
+  i <= m -> perm_on [set k : 'I_n | k < m] s ->
+  length (s * 's_[rev (iota (m - i) i)]) = length s + i.
+ +
+Lemma perm_on_prods_length s i m :
+  m < n -> i <= m -> perm_on [set k : 'I_n | k < m] s ->
+  length (s * 's_[(rev (iota (m - i) i))]) = length s + i.
+ +
+Lemma length_permcd c :
+  c \is a code -> size c <= n -> length 's_[wordcd c] = sumn c.
+ +
+Lemma cocode2P s :
+  let c := cocode s in c \is a code /\ s = 's_[wordcd c].
+ +
+Lemma cocodeP s : cocode s \is a code.
+ +
+Lemma cocodeE s : s = 's_[wordcd (cocode s)].
+ +
+
+ +
+

Properties of canword s

+ + +
+ +We show that canword s meets it specification : The canonical reduced word +of s is indeed a word whose length is the size of s, that is a reduced +word (defined later). + +
+
+ +
+Lemma canwordE s :
+  [seq (i : nat) | i : 'I_n0 <- canword s] = wordcd (cocode s).
+ +
+Theorem canwordP s : 's_[canword s] = s.
+ +
+
+ +
+As a consequence 'SG_n is generated by the elementary transpositions +
+
+Corollary eltr_genSn : 'SG_n = <<[set 's_(i : 'I_n0) | i in 'I_n0]>>%G.
+ +
+
+ +
+The corresponding induction theorem +
+
+Theorem eltr_ind (P : 'S_n -> Type) :
+  P 1 -> (forall s i, i < n0 -> P s -> P ('s_i * s)) ->
+  forall s, P s.
+ +
+Corollary morph_eltr (gT : finGroupType)
+          (f g : {morphism 'SG_n >-> gT}) :
+  (forall i : 'I_n0, f 's_i = g 's_i) -> f =1 g.
+ +
+
+ +
+A simple application +
+
+Lemma odd_size_permE ts :
+  all (gtn n0) ts -> odd (size ts) = odd_perm 's_[ts].
+ +
+
+ +
+

Various properties of the length

+ +
+
+Lemma sumn_cocode s : sumn (cocode s) = length s.
+Theorem size_canword s : size (canword s) = length s.
+ +
+Corollary length_eq0 s : length s = 0 -> s = 1.
+ +
+Corollary length_eq1 s : length s = 1%N -> exists i : 'I_n0, s = 's_i.
+ +
+Corollary lengthM s t : length (s * t) <= length s + length t.
+ +
+Corollary lengthME s t :
+  length s + length t <= length (s * t) -> length (s * t) = length s + length t.
+ +
+Corollary lengthKR s t u :
+  length (s * t * u) = length s + length t + length u ->
+  length (t * u) = length t + length u.
+ +
+Corollary lengthKL s t u :
+  length (s * t * u) = length s + length t + length u ->
+  length (s * t) = length s + length t.
+ +
+Definition prods_codesz (c : codesz n) : 'S_n := 's_[wordcd c].
+ +
+Lemma prods_codesz_bij : bijective prods_codesz.
+ +
+Lemma prods_wordcd_inj c1 c2 :
+  c1 \is a code -> c2 \is a code -> size c1 = n -> size c2 = n ->
+  's_[wordcd c1] = 's_[wordcd c2] -> c1 = c2.
+ +
+End Length.
+#[export] Hint Resolve cocodeP : core.
+#[export] Hint Resolve codeszP : core.
+ +
+
+ +
+

Let's do some real combinatorics !!!

+ + +
+ +The generating polynomial for permutations counted by their length. + +
+
+ +
+Section Combi.
+ +
+Import GRing.Theory.
+Open Scope ring_scope.
+ +
+Corollary genfun_length n :
+  \sum_(s : 'S_n) 'X^(length s) =
+  \prod_(0 <= i < n) \sum_(0 <= j < i.+1) 'X^j :> {poly int}.
+ +
+End Combi.
+ +
+
+ +
+

Reduced words

+ +
+
+Section Reduced.
+ +
+Variable n : nat.
+Implicit Type u v w : seq 'I_n.
+ +
+Notation "''s_' i" := (eltr n i).
+Notation "''s_[' w ]" := (\prod_(i <- w) 's_i).
+ +
+Definition reduced_word := [qualify w : seq 'I_n | length 's_[w] == size w ].
+Notation reduced := reduced_word.
+ +
+Lemma reducedP w : reflect (length 's_[w] = size w) (w \is reduced).
+ +
+Lemma reduced_nil : [::] \is reduced.
+ +
+Hint Resolve reduced_nil : core.
+ +
+Lemma reduced_iiF i : [:: i; i] \is reduced = false.
+ +
+Lemma reduced_rev w : w \is reduced -> rev w \is reduced.
+ +
+Lemma reduced_revE w : (w \is reduced) = (rev w \is reduced).
+ +
+Lemma reduced_sprod_code c :
+  c \is a code -> size c <= n.+1 -> pmap insub (wordcd c) \is reduced.
+ +
+
+ +
+

canword s is reduced

+ +
+
+Theorem canword_reduced s : canword s \is reduced.
+ +
+Lemma reduced_catr u v : u ++ v \is reduced -> v \is reduced.
+ +
+Lemma reduced_catl u v : u ++ v \is reduced -> u \is reduced.
+ +
+Lemma reduced_consK i u : i :: u \is reduced -> u \is reduced.
+ +
+Lemma reduced_rconsK u i : rcons u i \is reduced -> u \is reduced.
+ +
+Lemma reducedM (s t : 'S_(n.+1)) :
+  length (s * t) = length s + length t -> canword s ++ canword t \is reduced.
+ +
+Lemma canword1 : canword (1 : 'S_n.+1) = [::].
+ +
+
+ +
+

Braid monoid relations

+ +
+
+Definition braid_aba :=
+  fun s : seq 'I_n => match s with
+             | [:: a; b; c] => if (a == c) && ((a.+1 == b) || (b.+1 == a))
+                               then [:: [:: b; a; b]] else [::]
+             | _ => [::]
+           end.
+ +
+Definition braidC :=
+  fun s : seq 'I_n => match s with
+             | [:: a; b] => if (a.+1 < b) || (b.+1 < a)
+                            then [:: [:: b; a]] else [::]
+             | _ => [::]
+           end.
+ +
+Definition braidrule := [fun s => braid_aba s ++ braidC s].
+ +
+Lemma braid_abaP (u v : seq 'I_n) :
+  reflect (exists a b : 'I_n,
+              [/\ ((a.+1 == b) || (b.+1 == a)),
+               u = [:: a; b; a] & v = [:: b; a; b] ] )
+          (v \in braid_aba u).
+ +
+Lemma braidCP (u v : seq 'I_n) :
+  reflect (exists a b : 'I_n,
+             [/\ ((a.+1 < b) || (b.+1 < a)), u = [:: a; b] & v = [:: b; a] ] )
+          (v \in braidC u).
+ +
+Lemma braidrule_sym (u v : seq 'I_n) :
+  v \in (braidrule u) -> u \in (braidrule v).
+ +
+Lemma braidrule_homog (u : seq 'I_n) :
+  all [pred v | size v == size u] (braidrule u).
+ +
+Definition braidcongr := gencongr_hom braidrule_homog.
+Definition braidclass := genclass_hom braidrule_homog.
+ +
+#[local] Notation "a =Br b" := (braidcongr a b).
+ +
+Lemma braid_equiv : equivalence_rel braidcongr.
+ +
+Lemma braid_refl : reflexive braidcongr.
+ +
+Lemma braidww w : braidcongr w w.
+ +
+Lemma braid_sym : symmetric braidcongr.
+ +
+Lemma braid_ltrans : left_transitive braidcongr.
+ +
+Lemma braid_trans : transitive braidcongr.
+ +
+Lemma braid_is_congr : congruence_rel braidcongr.
+ +
+Definition braid_cons := congr_cons braid_is_congr.
+Definition braid_rcons := congr_rcons braid_is_congr.
+Definition braid_catl := congr_catl braid_is_congr.
+Definition braid_catr := congr_catr braid_is_congr.
+Definition braid_cat := congr_cat braid_is_congr braid_equiv.
+ +
+Lemma size_braid u v : u =Br v -> size u = size v.
+ +
+Lemma braid_rev u v : u =Br v -> rev u =Br rev v.
+ +
+Lemma class_braid1 (i : 'I_n) u : u =Br [:: i] -> u = [:: i].
+ +
+Theorem braid_prods v w : v =Br w -> 's_[v] = 's_[w].
+ +
+Corollary braid_reduced (v w : seq 'I_n) :
+  v =Br w -> v \is reduced -> w \is reduced.
+ +
+
+ +
+

reducing words for permutation

+ +
+
+Fixpoint reduces (u v : seq 'I_n) :=
+  match u with
+    | [::] | [:: _] => false
+    | [:: a, b & l] =>
+      ((a == b) && (l == v))
+        || if v is v0 :: v' then (a == v0) && reduces (b :: l) v'
+           else false
+  end.
+ +
+Lemma reducesP (u v : seq 'I_n) :
+  reflect
+    (exists x i y, u = x ++ [:: i; i] ++ y /\ v = x ++ y)
+    (reduces u v).
+ +
+Lemma reduces_catl u v w : reduces u v -> reduces (u ++ w) (v ++ w).
+ +
+Lemma prods_reducesE u v : reduces u v -> 's_[u] = 's_[v].
+ +
+Definition braid_reduces (u v : seq 'I_n) := (u =Br v) || (reduces u v).
+ +
+Lemma braidred_catl (u v w : seq 'I_n) :
+  braid_reduces u v -> braid_reduces (u ++ w) (v ++ w).
+ +
+Lemma braidredE u v : braid_reduces u v -> 's_[u] = 's_[v].
+ +
+End Reduced.
+Arguments reducedP {n w}.
+ +
+Notation reduced := (reduced_word _).
+Notation braidred := (@braid_reduces _).
+ +
+#[export] Hint Resolve braidww : core.
+ +
+
+ +
+

The cocode insertion algorithm

+ +
+
+Section CanWord.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+#[local] Notation "''s_' i" := (eltr n i) : group_scope.
+#[local] Notation "''s_[' w ]" := (\prod_(i <- w) 's_i).
+#[local] Notation "a =Br b" := (braidcongr a b).
+ +
+Fixpoint inscode (c : seq nat) (i : 'I_n) :=
+  if c is c0 :: c' then
+    let m := size c' in
+    if i > m - c0 then c0 :: inscode c' (inord i.-1)
+    else if i == m - c0 :> nat then c0.-1 :: c'
+    else if i.+1 == m - c0 :> nat then c0.+1 :: c'
+    else c0 :: inscode c' i
+  else [::].
+ +
+Lemma size_inscode c i : size (inscode c i) = size c.
+ +
+Lemma head_revcode c0 c : rev (c0 :: c) \is a code -> c0 <= size c.
+ +
+Lemma inord_predS (i : 'I_n) a b :
+  a < i -> i < b -> (inord (n' := n0) i.-1).+1 < b.
+ +
+Lemma inscodeP c (i : 'I_n) :
+  rev c \is a code -> i.+1 < size c -> rev (inscode c i) \is a code.
+ +
+Definition wcord c : seq 'I_n := map inord (wordcd (rev c)).
+ +
+Lemma wcordE c :
+  rev c \is a code -> size c <= n.+1 -> wcord c = pmap insub (wordcd (rev c)).
+ +
+Lemma reduced_wcord c :
+  rev c \is a code -> size c <= n.+1 -> wcord c \is reduced.
+ +
+Lemma wcord_cons c i :
+  wcord (i :: c) = wcord c ++ map inord (rev (iota (size c - i) i)).
+ +
+Lemma ltn_braidC (s : seq 'I_n) (i : 'I_n) :
+  (forall u, u \in s -> u.+1 < i) -> [:: i] ++ s =Br s ++ [:: i].
+ +
+Lemma gtn_braidC (s : seq 'I_n) (i : 'I_n) :
+  (forall u, u \in s -> i.+1 < u) -> [:: i] ++ s =Br s ++ [:: i].
+ +
+Lemma iota_cut_i (l b : nat) (i : 'I_n) :
+  l <= b -> b - l < i -> i < b -> iota (b - l) l =
+  (iota (b - l) (i.-1 - (b - l))) ++ [:: i.-1; i : nat] ++ (iota i.+1 (b - i.+1)).
+ +
+#[local] Notation "''I[' a '..' b ']'" :=
+  [seq inord i | i <- rev (iota (b - a) a)].
+ +
+Lemma braid_pred_lineC (i : 'I_n) (sz c : nat) :
+  sz <= n -> i < sz -> c <= sz -> sz - c < i ->
+  ([:: inord i.-1] ++ 'I[c .. sz]) =Br ('I[c .. sz] ++ [:: i]).
+ +
+Lemma braid_ltn_lineC (i : 'I_n) (sz c : nat) :
+  sz <= n -> i.+1 < sz - c -> c <= sz ->
+  [:: i] ++ 'I[c .. sz] =Br 'I[c .. sz] ++ [:: i].
+ +
+Implicit Type (u v w : seq 'I_n).
+ +
+Lemma path_braidred_catl p u w :
+  path braidred u p -> path braidred (u ++ w) [seq v ++ w | v <- p].
+ +
+Lemma braidred_inscode_path c (i : 'I_n) :
+  rev c \is a code -> size c <= n.+1 -> i.+1 < size c ->
+  { p | path braidred ((wcord c) ++ [:: i]) p /\
+        last ((wcord c) ++ [:: i]) p = wcord (inscode c i) }.
+ +
+
+ +
+

Straigthening a word

+ +
+
+Fixpoint straighten_rev w :=
+  if w is w0 :: w then inscode (straighten_rev w) w0 else (nseq n.+1 0).
+Definition straighten w := straighten_rev (rev w).
+ +
+Lemma size_straighten w : size (straighten w) = n.+1.
+ +
+Lemma is_code_straighten w : rev (straighten w) \is a code.
+ +
+Lemma straighten_path_npos w :
+  { p | path braidred w p /\ last w p = wcord (straighten w) }.
+ +
+Theorem prods_straighten w : 's_[wcord (straighten w)] = 's_[w].
+ +
+Corollary cocode_straightenE w : rev (straighten w) = cocode 's_[w].
+ +
+Corollary canword_straightenE w : wcord (straighten w) = canword 's_[w].
+ +
+Corollary canword_path_npos w :
+  { p | path braidred w p /\ last w p = canword 's_[w] }.
+ +
+End CanWord.
+ +
+Notation "''s_' i" := (eltr _ i) : group_scope.
+Notation "''s_[' w ']'" := (\prod_(i <- w) 's_i) : group_scope.
+ +
+#[local] Notation "a =Br b" := (braidcongr a b) : bool_scope.
+ +
+Section BraidRed.
+ +
+Variable n : nat.
+Implicit Types (u v w : seq 'I_n).
+ +
+Theorem braidred_to_canword w :
+  { p | path braidred w p /\ last w p = canword 's_[w] }.
+ +
+Lemma braidred_size_decr w p : path braidred w p -> size w >= size (last w p).
+ +
+Theorem braid_to_canword w : w \is reduced -> w =Br canword 's_[w].
+ +
+Theorem reduceP u : u \isn't reduced -> exists v w, u =Br v /\ reduces v w.
+ +
+
+ +
+Matsumoto Theorem +
+
+Corollary reduced_braid v w :
+  v \is reduced -> w \is reduced -> ('s_[v] == 's_[w] :> 'S_n.+1) = (v =Br w).
+ +
+Lemma canword_eltr (i : 'I_n) : canword 's_i = [:: i].
+ +
+End BraidRed.
+ +
+
+ +
+

The presentation of the symmetric groups

+ +
+
+Section PresentationSn.
+ +
+Variable n : nat.
+Variable gT : finGroupType.
+Variable eltrG : nat -> gT.
+ +
+#[local] Notation "''g_' i" :=
+  (eltrG i) (at level 8, i at level 2, format "''g_' i").
+ +
+Variant relat_Sn : Prop :=
+  RelatSn of
+    (forall i, i < n -> 'g_i^+2 = 1) &
+    (forall i, i.+1 < n -> 'g_i * 'g_i.+1 * 'g_i = 'g_i.+1 * 'g_i * 'g_i.+1) &
+    (forall i j, i.+1 < j < n -> 'g_i * 'g_j = 'g_j * 'g_i)
+  : relat_Sn.
+ +
+Theorem presentation_Sn_eltr :
+  relat_Sn ->
+  exists f : {morphism 'SG_n.+1 >-> gT}, forall i, i < n -> f 's_i = 'g_i.
+ +
+End PresentationSn.
+ +
+Lemma joingU1 (gt : finGroupType) (a : gt) (S : {set gt}) :
+  <[a]> <*> <<S>> = << a |: S >>.
+ +
+Lemma presentation_S2 :
+  'SG_2 \isog Grp ( s0 : s0^+2 ).
+ +
+Lemma presentation_S3 :
+  'SG_3 \isog Grp ( s0 : s1 : (s0^+2 = s1^+2 = 1, s0*s1*s0 = s1*s0*s1) ).
+ +
+Lemma presentation_S4 :
+  'SG_4 \isog Grp (
+          s0 : s1 : s2 :
+            (s0^+2, s1^+2, s2^+2,
+             s0*s1*s0 = s1*s0*s1, s1*s2*s1 = s2*s1*s2,
+             s0*s2 = s2*s0
+        ) ).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.reprSn.html b/combi/1.1.0/Combi.SymGroup.reprSn.html new file mode 100644 index 00000000..b7240a34 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.reprSn.html @@ -0,0 +1,309 @@ + + + + + +Combi.SymGroup.reprSn: Basic representations of the Symmetric Groups + + + + +
+ + + +
+ +

Library Combi.SymGroup.reprSn: Basic representations of the Symmetric Groups

+ +
+
+ +
+ +
+
+
+ +
+

Basic representation of the Symmetric Groups

+ + +
+ +We define the two representations of dimension 1 together with the natural +representation, and show that 'S_2 has only 2 representations. More precisely, +for a fixed n and g : 'S_n: + +
+ +
    +
  • triv_mx g == the 1x1 identity matrix. + +
    • +
  • sign_mx g == the 1x1 scalar matrix equal to the sign of g. + +
    • +
  • signed_mx R g == given a representation R the matrix sign g * R g + +
    • +
  • nat_mx g == the permutation matrix associated to g + +
    • +
+ +
+ +These four matrices are actually representation matrices: + +
+ +
    +
  • triv_repr == the trivial representation + +
    • +
  • sign_repr == the sign representation + +
    • +
  • signed_repr R == the representation R times the sign + +
    • +
  • nat_repr n == the natural representation of degree n + +
    • +
+ +
+ +We show in Lemmas repr1 and repr_S2 that triv_repr and sign_repr +exhausts all representations of 'S_n of degree 1 and all irreducible +representations of of 'S_2. + +
+
+From mathcomp Require Import all_boot.
+From mathcomp Require Import fingroup perm ssralg morphism perm action.
+From mathcomp Require Import zmodp. From mathcomp Require Import vector matrix mxalgebra ssrnum algC.
+From mathcomp Require Import mxrepresentation classfun character.
+ +
+Require Import permcomp tools partition congr cycles cycletype presentSn.
+ +
+Set Implicit Arguments.
+ +
+Import GroupScope GRing.Theory Num.Theory.
+#[local] Open Scope ring_scope.
+ +
+#[local] Notation reprS n d := (mx_representation algC 'SG_n d).
+ +
+Section TcastVal.
+Variable (T : eqType).
+ +
+Lemma tval_tcastE m n (eq_mn : m = n) (t : m.-tuple T) :
+  tcast eq_mn t = t :> seq T.
+ +
+End TcastVal.
+ +
+Section LinRepr.
+ +
+Variables (gT : finGroupType) (G : {group gT}).
+ +
+Lemma cfRepr1_lin_char (rG : mx_representation algC G 1) :
+  cfRepr rG \is a linear_char.
+ +
+Lemma lin_char_reprP xi :
+  reflect (exists rG : mx_representation algC G 1, xi = cfRepr rG)
+          (xi \is a linear_char).
+ +
+End LinRepr.
+ +
+Lemma NirrSn n : Nirr 'SG_n = #|{: 'P_n}|.
+ +
+Section EltrConj.
+ +
+Variable n : nat.
+ +
+Lemma cycle_type_eltr i :
+  (i < n)%N -> cycle_typeSn (eltr n i) = hookpartn n.+1 1.
+ +
+Lemma eltr_conj i j :
+  (i < n)%N -> (j < n)%N -> exists t, eltr n i = ((eltr n j) ^ t)%g.
+ +
+End EltrConj.
+ +
+
+ +
+

Representation of dimension 1 and natural representation

+ +
+
+Section DefTrivSign.
+ +
+Context {n : nat} (d : nat).
+ +
+Definition triv_mx (g : 'S_n) : 'M[algC]_1 := 1.
+Definition sign_mx (g : 'S_n) : 'M[algC]_1 := (-1) ^+ (odd_perm g).
+Definition signed_mx (rho : reprS n d) (g : 'S_n) : 'M[algC]_d :=
+  (-1) ^+ (odd_perm g) *: rho g.
+ +
+Definition nat_mx (g : 'S_n) : 'M[algC]_n := perm_mx g.
+ +
+Lemma triv_mx_repr : mx_repr 'SG_n triv_mx.
+ Canonical triv_repr : reprS n 1 := MxRepresentation triv_mx_repr.
+ +
+Lemma triv_irr : mx_irreducible triv_repr.
+ +
+Lemma sign_mx_repr : mx_repr 'SG_n sign_mx.
+Canonical sign_repr : reprS n 1 := MxRepresentation sign_mx_repr.
+ +
+Lemma sign_irr : mx_irreducible sign_repr.
+ +
+Lemma signed_mx_repr rho : mx_repr 'SG_n (signed_mx rho).
+Canonical signed_repr rho : reprS n d := MxRepresentation (signed_mx_repr rho).
+ +
+Lemma nat_mx_repr : mx_repr 'SG_n nat_mx.
+ Canonical nat_repr : reprS n n := MxRepresentation nat_mx_repr.
+ +
+Lemma cfRepr_triv : cfRepr triv_repr = 1.
+ +
+Lemma cfRepr_trivE : cfRepr triv_repr = 'chi_0.
+Lemma triv_Chi : mx_rsim triv_repr 'Chi_0.
+ +
+Lemma sign_char_subproof :
+  is_class_fun <<'SG_n>> [ffun g => (-1) ^+ (odd_perm g)].
+Definition sign_char := Cfun 0 sign_char_subproof.
+ +
+Lemma cfRepr_sign : cfRepr sign_repr = sign_char.
+Lemma sign_charP : sign_char \is a linear_char.
+ +
+Lemma cfRepr_signed (rho : reprS n d) :
+  cfRepr (signed_repr rho) = sign_char * cfRepr rho.
+ +
+End DefTrivSign.
+ +
+
+ +
+

Representations of the symmetric Group for n = 0 and 1

+ +
+
+ +
+Lemma row_free1 : row_free (1 : 'M[algC]_1).
+ +
+Lemma charSG0 X : X \in irr 'SG_0 -> X = 1.
+ +
+Lemma charSG1 X : X \in irr 'SG_1 -> X = 1.
+ +
+Lemma repr1_S0 (rho : reprS 0 1) : mx_rsim rho triv_repr.
+ +
+Lemma repr1_S1 (rho : reprS 1 1) : mx_rsim rho triv_repr.
+ +
+
+ +
+

Representations of dimension 1 the symmetric Group for n > 1

+ +
+
+ +
+Lemma triv_sign_neq n : (n > 1)%N -> 1 != sign_char :> 'CF('SG_n).
+ +
+Lemma triv_sign_not_sim n :
+  (n > 1)%N -> ~ mx_rsim (G := [group of 'SG_n]) triv_repr sign_repr.
+ +
+Lemma lin_char_Sn n (xi : 'CF('SG_n)) :
+  xi \is a linear_char -> xi = 1 \/ xi = sign_char.
+ +
+Lemma repr1 n (rho : reprS n 1) :
+  mx_rsim rho triv_repr \/ mx_rsim rho sign_repr.
+ +
+
+ +
+

Representations of the symmetric Group for n=2

+ +
+
+ +
+Lemma NirrS2 : Nirr 'SG_2 = 2.
+ +
+Lemma cast_IirrS2 (i : Iirr 'SG_2) :
+  i != 0 -> i = cast_ord (esym NirrS2) 1.
+ +
+Lemma sign_char2 : sign_char = 'chi_(cast_ord (esym NirrS2) 1).
+ +
+Lemma cfRepr_sign2 : cfRepr sign_repr = 'chi_(cast_ord (esym NirrS2) 1).
+ +
+Lemma sign_Chi2 : mx_rsim sign_repr 'Chi_(cast_ord (esym NirrS2) 1).
+ +
+Lemma irr_S2 : irr 'SG_2 = tcast (esym NirrS2) [tuple 1; sign_char].
+ +
+Lemma repr_S2 (rho : representation algC [group of 'SG_2]) :
+  mx_irreducible rho -> mx_rsim rho triv_repr \/ mx_rsim rho sign_repr.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.towerSn.html b/combi/1.1.0/Combi.SymGroup.towerSn.html new file mode 100644 index 00000000..6fdb6be9 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.towerSn.html @@ -0,0 +1,534 @@ + + + + + +Combi.SymGroup.towerSn: The Tower of the Symmetric Groups + + + + +
+ + + +
+ +

Library Combi.SymGroup.towerSn: The Tower of the Symmetric Groups

+ +
+
+ +
+ +
+
+
+ +
+

The Tower of the Symmetric Groups

+ + +
+ +External product of class functions: + +
+ +We fix two groups G and H. Then we define: + +
+ +
    +
  • f \ox g = cfextprod g h == the external product of class function, + defined by f \ox g (u, v) := f u * f v for (u, v) in G * H. + It is of of type 'CF(setX G H) when g : 'CF(G) and + h : 'CF(H). + +
    • +
  • cfextprodr g h == cfextprod h g + +
    • +
  • extprod_repr rG rH == the external product (tensor product) of matrix + representation. Its character is the external product of the + character of rG and rH. + +
    • +
+ +
+ + +
+ +The tower of the symmetric groups + +
+ +
    +
  • tinj == the tower injection morphism : 'S_m * 'S_n -> 'S_(m + n) + +
    • +
  • f \o^ g == the image along tinj of the external product of f and g. + +
    • +
+ +
+ +Induction and restriction of class functions + +
+ +
    +
  • 'z_p == zcoeff p == The cardinality of the centralizer of any + permutation of cycle type p in algC, that is + #|'S_k| / #|class p|. + +
    • +
  • '1z_p = ncfuniCT p == the normalized cycle indicator class function + for cycle type p == 'z_p *: '1_[p]. + +
    • +
+ +
+ +The two main results are: + +
+ +
    +
  • Theorem cfuni_Res which expands the restriction to 'S_m * 'S_n + of the cycle indicator class function '1_[l]: +
    +    'Res[tinj @* ('dom tinj)] '1_[l] =
    +      \sum_(pp | l == pp.1 +|+ pp.2) '1_[pp.1] \o^ '1_[pp.2]. +
    + +     by Frobenius duality it implies: + +
    + + +
    • +
  • Theorem ncfuniCT_Ind which expands the induction of two normalized + cycle indicator class functions: +
    +    'Ind['SG_(m + n)] ('1z_[p] \o^ '1z_[q]) = '1z_[p +|+ q]. +
    + +     +
    • +
+ +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot.
+From mathcomp Require Import ssralg fingroup morphism perm gproduct.
+From mathcomp Require Import ssrnum matrix vector mxalgebra algC.
+From mathcomp Require Import classfun character mxrepresentation.
+From mathcomp Require Import sesquilinear.
+ +
+Require Import tools ordcast partition sorted.
+Require Import permcomp cycles cycletype permcent.
+ +
+Notation "''SG_' n" := [set: 'S_n]
+  (at level 8, n at level 2, format "''SG_' n").
+ +
+Import LeqGeqOrder.
+ +
+Set Implicit Arguments.
+ +
+Import GroupScope GRing.Theory.
+#[local] Open Scope Combi_scope.
+ +
+Section classGroup.
+ +
+Variables gT aT: finGroupType.
+Variable G H : {group gT}.
+ +
+Lemma class_disj x y :
+  y \notin x ^: G -> x ^: G :&: y ^: G = set0.
+ +
+Lemma prod_conjg (x y : gT * aT) : x ^ y = (x.1 ^ y.1, x.2 ^ y.2).
+ +
+End classGroup.
+ +
+
+ +
+

External product of class functions

+ +
+
+Section CFExtProdDefs.
+ +
+Variables (gT aT : finGroupType).
+Variables (G : {group gT}) (H : {group aT}).
+ +
+#[local] Open Scope ring_scope.
+ +
+
+ +
+One could use the following alternative definition +
+Definition cast_cf (G H : {set gT}) (eq_GH : G = H) (f : 'CF(G)) :=
+  let: erefl in _ = H := eq_GH return 'CF(H) in f.
+Definition cfextprod (g : 'CF(G)) (h : 'CF(H)) : 'CF(setX G H) :=
+  (cfMorph (cast_cf (esym (@morphim_fstX _ _ G H)) g)) *
+  (cfMorph (cast_cf (esym (@morphim_sndX _ _ G H)) h)). +
+ +and prove the equivalence: +
+Lemma cfextprodE g h x y : (g \ox h) (x, y) = (g x) * (h y). +
+ +However, the more direct definition below leads to simpler proofs, at least +once we have proved that it is indeed a class function. + +
+
+Fact cfextprod_subproof (g : 'CF(G)) (h : 'CF(H)) :
+  is_class_fun <<setX G H>> [ffun x => g x.1 * h x.2].
+Definition cfextprod g h := Cfun 0 (cfextprod_subproof g h).
+ +
+End CFExtProdDefs.
+ +
+Notation "f \ox g" := (cfextprod f g) (at level 40, left associativity).
+ +
+Section CFExtProdTheory.
+ +
+Variables (gT aT : finGroupType).
+Variables (G : {group gT}) (H : {group aT}).
+Implicit Type (g : 'CF(G)) (h : 'CF(H)).
+ +
+#[local] Open Scope ring_scope.
+ +
+Fact cfextprod_is_bilinear :
+  bilinear_for *:%R *:%R (cfextprod (G := G) (H := H)).
+ +
+Lemma cfextprod0r g : g \ox (0 : 'CF(H)) = 0.
+Lemma cfextprodNr g h : g \ox - h = - (g \ox h).
+Lemma cfextprodDr g h1 h2 : g \ox (h1 + h2) = g \ox h1 + g \ox h2.
+Lemma cfextprodBr g h1 h2 : g \ox (h1 - h2) = g \ox h1 - g \ox h2.
+Lemma cfextprodMnr h g n : g \ox (h *+ n) = (g \ox h) *+ n.
+Lemma cfextprod_sumr g I r (P : pred I) (phi : I -> 'CF(H)) :
+  g \ox (\sum_(i <- r | P i) phi i) = \sum_(i <- r | P i) g \ox phi i.
+Lemma cfextprodZr g a h : g \ox (a *: h) = a *: (g \ox h).
+ +
+Lemma cfextprod0l h : (0 : 'CF(G)) \ox h = 0.
+Lemma cfextprodNl h g : - g \ox h = - g \ox h.
+Lemma cfextprodDl h g1 g2 : (g1 + g2) \ox h = g1 \ox h + g2 \ox h.
+Lemma cfextprodBl h g1 g2 : (g1 - g2) \ox h = g1 \ox h - g2 \ox h.
+Lemma cfextprodMnl h g n : g *+ n \ox h = g \ox h *+ n.
+Lemma cfextprod_suml h I r (P : pred I) (phi : I -> 'CF(G)) :
+  (\sum_(i <- r | P i) phi i) \ox h = \sum_(i <- r | P i) phi i \ox h.
+Lemma cfextprodZl h a g : (a *: g) \ox h = a *: (g \ox h).
+ +
+Section ReprExtProd.
+ +
+Variables (n1 n2 : nat).
+Variables (rG : mx_representation algC G n1)
+          (rH : mx_representation algC H n2).
+ +
+Fact extprod_mx_repr_subproof :
+  mx_repr (setX G H) (fun x => tprod (rG x.1) (rH x.2)).
+Definition extprod_mx_repr := MxRepresentation extprod_mx_repr_subproof.
+ +
+Lemma cfRepr_extprod : cfRepr extprod_mx_repr = cfRepr rG \ox cfRepr rH.
+ +
+End ReprExtProd.
+ +
+Lemma cfextprod_char g h :
+  g \is a character -> h \is a character -> g \ox h \is a character.
+ +
+End CFExtProdTheory.
+ +
+
+ +
+

Injection morphism of the tower of the symmetric groups

+ +
+
+Section TowerMorphism.
+ +
+Variables m n : nat.
+ +
+#[local] Notation ct := cycle_typeSn.
+#[local] Notation SnXm := (setX 'SG_m 'SG_n).
+ +
+Definition tinjval (s : 'S_m * 'S_n) :=
+  fun (x : 'I_(m + n)) => match split x with
+  | inl a => unsplit (inl (s.1 a))
+  | inr a => unsplit (inr (s.2 a))
+  end.
+ +
+Fact tinjval_inj s : injective (tinjval s).
+Definition tinj s : 'S_(m + n) := perm (@tinjval_inj s).
+ +
+Fact tinj_morphM : {morph tinj : x y / x * y >-> x * y}.
+Canonical morph_of_tinj := Morphism (D := SnXm) (in2W tinj_morphM).
+ +
+Lemma isom_tinj : isom SnXm (tinj @* ('dom tinj)) tinj.
+ +
+Lemma expg_tinj_lshift s a i :
+ (tinj s ^+ i) (lshift n a) = lshift n ((s.1 ^+ i) a).
+ +
+Lemma expg_tinj_rshift s a i :
+  (tinj s ^+ i) (rshift m a) = rshift m ((s.2 ^+ i) a).
+ +
+Lemma porbit_tinj_lshift s a :
+  porbit (tinj s) (lshift n a) = [set @lshift m n x | x in porbit s.1 a].
+ +
+Lemma porbit_tinj_rshift s a :
+  porbit (tinj s) (rshift m a) = [set @rshift m n x | x in porbit s.2 a].
+ +
+Lemma porbits_tinj s :
+  porbits (tinj s) =
+  [set (@lshift m n) @: x | x : {set 'I_m} in porbits s.1]
+    :|:
+    [set (@rshift m n) @: x | x : {set 'I_n} in porbits s.2].
+ +
+Lemma cycle_type_tinj s : ct (tinj s) = ct s.1 +|+ ct s.2.
+ +
+End TowerMorphism.
+ +
+Arguments tinj {m n} s.
+ +
+
+ +
+

The tower is associative (upto isomorphism) with unit

+ +
+
+Section Assoc.
+ +
+Variables m n p : nat.
+ +
+Lemma cast_rshift j : cast_ord (esym (add0n n)) j = rshift 0 j.
+ +
+Lemma cast_lshift j : cast_ord (esym (addn0 n)) j = lshift 0 j.
+ +
+Lemma tinj1E (t : 'S_n) : tinj (1%g, t) = cast_perm (esym (add0n n)) t.
+ +
+Lemma tinjE1 (t : 'S_n) : tinj (t, 1%g) = cast_perm (esym (addn0 n)) t.
+ +
+Lemma tinjA (s : 'S_m) (t : 'S_n) (u : 'S_p) :
+  tinj (tinj(s, t), u) = cast_perm (addnA m n p) (tinj (s, tinj (t, u))).
+ +
+Lemma cycle_type_tinjC (s : 'S_m) (t : 'S_n) :
+  cycle_typeSn (tinj (s, t)) =
+  cycle_typeSn (cast_perm (esym (addnC m n)) (tinj (t, s))).
+ +
+End Assoc.
+ +
+Notation "f \o^ g" := (cfIsom (isom_tinj _ _) (f \ox g)) (at level 40).
+ +
+#[local] Open Scope ring_scope.
+ +
+
+ +
+

Restriction formula

+ +
+
+Section Restriction.
+ +
+Variables m n : nat.
+#[local] Notation ct := cycle_typeSn.
+ +
+Lemma cfuni_tinj s (l : 'P_(m + n)) :
+  '1_[l] (tinj s) = (l == ct s.1 +|+ ct s.2)%:R.
+ +
+Theorem cfuni_Res (l : 'P_(m + n)) :
+  'Res[tinj @* ('dom tinj)] '1_[l] =
+  \sum_(pp | l == pp.1 +|+ pp.2) '1_[pp.1] \o^ '1_[pp.2].
+ +
+End Restriction.
+ +
+
+ +
+

Induction formula

+ +
+
+Section Induction.
+ +
+Variables m n : nat.
+ +
+Implicit Types (p : 'P_m) (q : 'P_n).
+ +
+#[local] Notation ct := cycle_typeSn.
+#[local] Notation SnXm := (setX 'SG_m 'SG_n).
+#[local] Notation classX p q := ((permCT p, permCT q) ^: SnXm).
+ +
+Import GroupScope GRing.Theory Num.Theory.
+#[local] Open Scope ring_scope.
+ +
+Lemma classXE p q : classX p q = setX (classCT p) (classCT q).
+ +
+Lemma cfextprod_cfuni p q : '1_[p] \ox '1_[q] = '1_(classX p q).
+ +
+Lemma cfdot_classfun_part p1 p2 :
+  '[ '1_[p1], '1_[p2] ] = #|'SG_m|%:R^-1 * #|classCT p1|%:R * (p1 == p2)%:R.
+ +
+Lemma decomp_cf_triv : \sum_(p : 'P_n) '1_[p] = 1.
+ +
+Lemma classXI p1 p2 q1 q2 :
+  (p2, q2) != (p1, q1) -> (classX p1 q1) :&: (classX p2 q2) = set0.
+ +
+
+ +
+The normalized cycle type indicator basis +
+
+Definition zcoeff (k : nat) (p : 'P_k) : algC :=
+  #|'SG_k|%:R / #|classCT p|%:R.
+ +
+Notation "''z_' p" := (zcoeff p) (at level 2, format "''z_' p").
+ +
+Lemma zcoeffE k (l : 'P_k) : zcoeff l = (zcard l)%:R.
+ +
+Lemma neq0zcoeff (k : nat) (p : 'P_k) : 'z_p != 0.
+ +
+Hint Resolve neq0zcoeff : core.
+ +
+Definition ncfuniCT (k : nat) (p : 'P_k) := 'z_p *: '1_[p].
+ +
+Notation "''1z_[' p ]" := (ncfuniCT p) (format "''1z_[' p ]").
+ +
+Lemma ncfuniCT_gen k (f : 'CF('SG_k)) :
+  f = \sum_(p : 'P_k) f (permCT p) / 'z_p *: '1z_[p].
+ +
+Lemma cfdotr_ncfuniCT k (f : 'CF('SG_k)) (s : 'S_k) : (f s) = '[f, '1z_[ct s]].
+ +
+
+ +
+Application of Frobenius duality : cfdot_Res_r +
+
+Lemma cfdot_Ind_cfuniCT p q (l : 'P_(m + n)) :
+  '[ 'Ind['SG_(m + n)] ('1_[p] \o^ '1_[q]), '1_[l] ] =
+  (p +|+ q == l)%:R / 'z_p / 'z_q.
+ +
+Lemma cfdot_Ind_ncfuniCT p q (l : 'P_(m + n)) :
+  '[ 'Ind['SG_(m + n)] ('1z_[p] \o^ '1z_[q]), '1z_[l] ] =
+  (p +|+ q == l)%:R * 'z_l.
+ +
+Theorem ncfuniCT_Ind p q :
+  'Ind['SG_(m + n)] ('1z_[p] \o^ '1z_[q]) = '1z_[p +|+ q].
+ +
+End Induction.
+ +
+Notation "''z_' p" := (zcoeff p) (at level 2, format "''z_' p").
+Notation "''1z_[' p ]" := (ncfuniCT p) (format "''1z_[' p ]").
+ +
+#[export] Hint Resolve neq0zcoeff : core.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/Combi.SymGroup.weak_order.html b/combi/1.1.0/Combi.SymGroup.weak_order.html new file mode 100644 index 00000000..fc4f9062 --- /dev/null +++ b/combi/1.1.0/Combi.SymGroup.weak_order.html @@ -0,0 +1,313 @@ + + + + + +Combi.SymGroup.weak_order: The weak order on the Symmetric Group + + + + +
+ + + +
+ +

Library Combi.SymGroup.weak_order: The weak order on the Symmetric Group

+ +
+
+ +
+ +
+
+
+ +
+

The weak order on the Symmetric Group

+ + +
+ +We define the right (with the mathcomp convention of product of permutation) +weak order on the symmetric group. We show that it is equivalent to inclusion +of sets of inversions and that it is a lattice. + +
+ +We define the following notations: + +
+ +
    +
  • s <=R t == s is smaller than t for the right weak order. + +
    • +
  • s <R t == s is strictly smaller than t for the right weak order. + +
    • +
  • s \/R t == the meet of s and t for the right weak order. + +
    • +
  • s /\R t == the join of s and t for the right weak order. + +
    • +
+ +
+ + +
+
+From HB Require Import structures.
+From mathcomp Require Import all_boot order.
+From mathcomp Require Import fingroup perm morphism presentation.
+ +
+Require Import permcomp tools permuted combclass congr presentSn ordtype.
+ +
+Set Implicit Arguments.
+ +
+Import GroupScope.
+Import Order.Theory.
+ +
+#[local] Open Scope Combi_scope.
+ +
+Reserved Notation "s '<=R' t" (at level 70, t at next level).
+Reserved Notation "s '<R' t" (at level 70, t at next level).
+Reserved Notation "s '/\R' t" (at level 70, t at next level).
+Reserved Notation "s '\/R' t" (at level 70, t at next level).
+ +
+Fact perm_display : Order.disp_t.
+ +
+Module WeakOrder.
+Section Def.
+ +
+Context {n0 : nat}.
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n).
+ +
+Definition leperm s t :=
+  [exists u, (t == s * u) && (length t == length s + length u)].
+ +
+#[local] Notation "s '<=R' t" := (leperm s t).
+ +
+Fact lepermP s t :
+  reflect (exists2 u, t = s * u & length t = length s + length u)
+          (s <=R t).
+ +
+Fact leperm_length s t : s <=R t -> length s <= length t.
+ +
+Fact leperm_lengthE s t : s <=R t -> length s = length t -> s = t.
+ +
+Fact leperm_refl s : s <=R s.
+ +
+Fact leperm_trans : transitive leperm.
+ +
+Fact leperm_anti : antisymmetric leperm.
+ +
+#[export] HB.instance Definition _ := Finite.on 'S_n.
+#[export] HB.instance Definition _ :=
+  Order.Le_isPOrder.Build perm_display 'S_n
+    leperm_refl leperm_anti leperm_trans.
+ +
+End Def.
+ +
+Module Exports.
+ +
+Notation "x <=R y" := (@Order.le perm_display _ (x : 'S__) y) : Combi_scope.
+Notation "x <R y" := (@Order.lt perm_display _ (x : 'S__) y) : Combi_scope.
+Notation "x /\R y" := (@Order.meet perm_display _ (x : 'S__) y) : Combi_scope.
+Notation "x \/R y" := (@Order.join perm_display _ (x : 'S__) y) : Combi_scope.
+ +
+Section WeakOrder.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n).
+ +
+Definition lepermP s t :
+  reflect
+    (exists2 u : 'S_n, t = s * u & length t = length s + length u)
+    (s <=R t)
+  := lepermP s t.
+Definition leperm_length : forall s t, s <=R t -> length s <= length t
+  := leperm_length.
+Definition leperm_lengthE : forall s t, s <=R t -> length s = length t -> s = t
+  := leperm_lengthE.
+End WeakOrder.
+End Exports.
+End WeakOrder.
+ +
+Section LEPermTheory.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n).
+ +
+Lemma ltperm_length s t : s <R t -> length s < length t.
+ +
+Lemma leperm1p s : (1%g : 'S_n) <=R s.
+ +
+Lemma leperm_maxpermMl s t : (maxperm * t <=R maxperm * s) = (s <=R t).
+ +
+Lemma leperm_maxperm s : s <=R maxperm.
+ +
+Lemma leperm_factorP s t :
+  reflect (exists2 w : seq 'I_n0, w \is reduced &
+              exists2 l : nat, t = 's_[w] & s = 's_[take l w])
+          (s <=R t).
+ +
+Lemma leperm_succ s t :
+  s <R t -> exists2 i : 'I_n0, (s <R s * 's_i) & (s * 's_i <=R t).
+ +
+Lemma covers_permP s t :
+  reflect (exists2 i : 'I_n0, s <R s * 's_i & t = s * 's_i) (covers s t).
+ +
+Theorem leperm_invset s t : (s <=R t) = (invset s \subset invset t).
+ +
+Corollary ltperm_invset s t : (s <R t) = (invset s \proper invset t).
+ +
+End LEPermTheory.
+ +
+Section TClosureInvset.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n) (A B : {set 'I_n * 'I_n}).
+ +
+Lemma tclosure_Delta A : A \subset Delta -> tclosure A \subset Delta.
+ +
+Lemma tclosureP A : A \subset Delta -> transitive (srel (tclosure A)).
+ +
+Lemma is_invset_tclosureU A B :
+  is_invset A -> is_invset B -> is_invset (tclosure (A :|: B)).
+ +
+End TClosureInvset.
+ +
+Module PermLattice.
+Section PermLattice.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n) (A B : {set 'I_n * 'I_n}).
+ +
+Definition supperm s t : 'S_n :=
+  perm_of_invset (tclosure (invset s :|: invset t)).
+Definition infperm s t : 'S_n :=
+  maxperm * (supperm (maxperm * s) (maxperm * t)).
+ +
+Lemma invset_supperm s t :
+  invset (supperm s t) = tclosure (invset s :|: invset t).
+ +
+Lemma suppermC s t : supperm s t = supperm t s.
+ +
+Lemma suppermPr s t : s <=R (supperm s t).
+ +
+Lemma suppermPl s t : t <=R (supperm s t).
+ +
+Fact suppermP s t w : s <=R w -> t <=R w -> (supperm s t) <=R w.
+ +
+Fact supperm_is_join x y z : (supperm x y <=R z) = (x <=R z) && (y <=R z).
+Fact infperm_is_meet x y z : (x <=R infperm y z) = (x <=R y) && (x <=R z).
+ +
+#[export] HB.instance Definition _ :=
+  Order.POrder_MeetJoin_isLattice.Build perm_display ('S_n)
+    infperm_is_meet supperm_is_join.
+#[export] HB.instance Definition _ :=
+  Order.hasBottom.Build perm_display ('S_n) (@leperm1p n0).
+#[export] HB.instance Definition _ :=
+  Order.hasTop.Build perm_display ('S_n) (@leperm_maxperm n0).
+ +
+End PermLattice.
+ +
+Module Exports.
+Section PermLattice.
+ +
+Variable (n0 : nat).
+#[local] Notation n := n0.+1.
+Implicit Type (s t u v : 'S_n) (A B : {set 'I_n * 'I_n}).
+ +
+Lemma bottom_perm : Order.bottom = (1 : 'S_n).
+Lemma top_perm : Order.top = @maxperm n.
+ +
+Lemma invset_join s t : invset (s \/R t) = tclosure (invset s :|: invset t).
+ +
+Lemma perm_join_meetE s t :
+  s /\R t = maxperm * (maxperm * s \/R maxperm * t).
+ +
+End PermLattice.
+End Exports.
+End PermLattice.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/coqdoc.css b/combi/1.1.0/coqdoc.css new file mode 100644 index 00000000..48096e55 --- /dev/null +++ b/combi/1.1.0/coqdoc.css @@ -0,0 +1,338 @@ +body { padding: 0px 0px; + margin: 0px 0px; + background-color: white } + +#page { display: block; + padding: 0px; + margin: 0px; + padding-bottom: 10px; } + +#header { display: block; + position: relative; + padding: 0; + margin: 0; + vertical-align: middle; + border-bottom-style: solid; + border-width: thin } + +#header h1 { padding: 0; + margin: 0;} + + +/* Contents */ + +#main{ display: block; + padding: 10px; + font-family: sans-serif; + font-size: 100%; + line-height: 100% } + +#main h1 { line-height: 95% } /* allow for multi-line headers */ + +#main a.idref:visited {color : #416DFF; text-decoration : none; } +#main a.idref:link {color : #416DFF; text-decoration : none; } +#main a.idref:hover {text-decoration : none; } +#main a.idref:active {text-decoration : none; } + +#main 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Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(8361 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(178 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(68 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(910 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(57 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3352 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(181 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(17 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(363 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(67 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(261 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(2820 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(68 entries)
+
+

Global Index

+

A

+AbelianBigOp [section, in Combi.SSRcomplements.tools]
+AbelianBigOp.idx [variable, in Combi.SSRcomplements.tools]
+AbelianBigOp.op [variable, in Combi.SSRcomplements.tools]
+AbelianBigOp.R [variable, in Combi.SSRcomplements.tools]
+_ * _ [notation, in Combi.SSRcomplements.tools]
+*%M [notation, in Combi.SSRcomplements.tools]
+1 [notation, in Combi.SSRcomplements.tools]
+abelian_cycle_dec [lemma, in Combi.SymGroup.cycles]
+abelian_perm_dec [lemma, in Combi.SymGroup.cycles]
+abelian_disjoint_psupports [lemma, in Combi.SymGroup.cycles]
+ActOnTuple [section, in Combi.Combi.permuted]
+ActOnTuple.n [variable, in Combi.Combi.permuted]
+ActOnTuple.T [variable, in Combi.Combi.permuted]
+ActOnTuple.w [variable, in Combi.Combi.permuted]
+_ # _ [notation, in Combi.Combi.permuted]
+add [definition, in Combi.LRrule.implem]
+addE [lemma, in Combi.LRrule.implem]
+add_corner_decr_nth [lemma, in Combi.Combi.partition]
+add_mesymK [lemma, in Combi.MPoly.Schur_altdef]
+add_mesymE [lemma, in Combi.MPoly.Schur_altdef]
+add_mesym [definition, in Combi.MPoly.Schur_altdef]
+add_mpart_mesymP [lemma, in Combi.MPoly.Schur_altdef]
+add_ribbon_intpartnP [lemma, in Combi.Combi.skewpart]
+add_ribbon_intpartnE [lemma, in Combi.Combi.skewpart]
+add_ribbon_intpartn [definition, in Combi.Combi.skewpart]
+add_ribbon_intpartn_spec [lemma, in Combi.Combi.skewpart]
+add_ribbonP [lemma, in Combi.Combi.skewpart]
+add_ribbon_onP [lemma, in Combi.Combi.skewpart]
+add_ribbon_height [lemma, in Combi.Combi.skewpart]
+add_ribbon_on_remP [lemma, in Combi.Combi.skewpart]
+add_ribbon [definition, in Combi.Combi.skewpart]
+add_ribbon_on [definition, in Combi.Combi.skewpart]
+admissible [definition, in ALEA.Ccpo]
+admissible2 [definition, in ALEA.Ccpo]
+allLeq [definition, in Combi.Basic.ordtype]
+allLeqCons [lemma, in Combi.Basic.ordtype]
+allLeqConsE [lemma, in Combi.Basic.ordtype]
+allLeqE [lemma, in Combi.Basic.ordtype]
+AllLeqLtn [section, in Combi.Basic.ordtype]
+AllLeqLtn.disp [variable, in Combi.Basic.ordtype]
+AllLeqLtn.T [variable, in Combi.Basic.ordtype]
+allLeqP [lemma, in Combi.Basic.ordtype]
+allLeq_to_word_tl [lemma, in Combi.Combi.stdtab]
+allLeq_to_word_hd [lemma, in Combi.Combi.stdtab]
+allLeq_is_row_rcons [lemma, in Combi.LRrule.plactic]
+allLeq_posbig [lemma, in Combi.Basic.ordtype]
+allLeq_last [lemma, in Combi.Basic.ordtype]
+allLeq_rconsK [lemma, in Combi.Basic.ordtype]
+allLeq_rev [lemma, in Combi.Basic.ordtype]
+allLeq_catE [lemma, in Combi.Basic.ordtype]
+allLeq_consK [lemma, in Combi.Basic.ordtype]
+allLtn [definition, in Combi.Basic.ordtype]
+allLtnCons [lemma, in Combi.Basic.ordtype]
+allLtnConsE [lemma, in Combi.Basic.ordtype]
+allLtnW [lemma, in Combi.Basic.ordtype]
+allLtn_std_rec [lemma, in Combi.Combi.std]
+allLtn_posbig [lemma, in Combi.Basic.ordtype]
+allLtn_last [lemma, in Combi.Basic.ordtype]
+allLtn_rconsK [lemma, in Combi.Basic.ordtype]
+allLtn_rev [lemma, in Combi.Basic.ordtype]
+allLtn_catE [lemma, in Combi.Basic.ordtype]
+allLtn_consK [lemma, in Combi.Basic.ordtype]
+allLtn_notin [lemma, in Combi.Basic.ordtype]
+all_leqzip_refl [lemma, in Combi.Combi.bintree]
+all_in_shuffler [lemma, in Combi.LRrule.shuffle]
+all_in_shufflel [lemma, in Combi.LRrule.shuffle]
+all_size_shuffle [lemma, in Combi.LRrule.shuffle]
+all_allLtn_cat [lemma, in Combi.Combi.skewtab]
+all_ltn_nth_tabsh [lemma, in Combi.Combi.tableau]
+all_permuted [lemma, in Combi.Combi.permuted]
+all_unionP [lemma, in Combi.Basic.combclass]
+all_subenum [lemma, in Combi.Basic.combclass]
+all_partsums [lemma, in Combi.Combi.composition]
+Alternant [section, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr [section, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.add_mpart_mesym [variable, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.d [variable, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.h [variable, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.k [variable, in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.la [variable, in Combi.MPoly.Schur_altdef]
+Alternant.n [variable, in Combi.MPoly.Schur_altdef]
+Alternant.R [variable, in Combi.MPoly.Schur_altdef]
+'a_ _ [notation, in Combi.MPoly.Schur_altdef]
+'e_ _ [notation, in Combi.MPoly.Schur_altdef]
+AlternIDomain [section, in Combi.MPoly.antisym]
+AlternIDomain.Hchar [variable, in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial [section, in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hpanti [variable, in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hphomog [variable, in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hpn0 [variable, in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.p [variable, in Combi.MPoly.antisym]
+AlternIDomain.n [variable, in Combi.MPoly.antisym]
+AlternIDomain.R [variable, in Combi.MPoly.antisym]
+_ # _ [notation, in Combi.MPoly.antisym]
+'a_ _ [notation, in Combi.MPoly.antisym]
+alternpol [definition, in Combi.MPoly.antisym]
+AlternStraighten [section, in Combi.MPoly.Schur_altdef]
+AlternStraighten.n0 [variable, in Combi.MPoly.Schur_altdef]
+AlternStraighten.R [variable, in Combi.MPoly.Schur_altdef]
+_ # _ [notation, in Combi.MPoly.Schur_altdef]
+'a_ _ [notation, in Combi.MPoly.Schur_altdef]
+alt_add1_0 [lemma, in Combi.MPoly.antisym]
+alt_alternate [lemma, in Combi.MPoly.antisym]
+alt_rho_non0 [lemma, in Combi.MPoly.antisym]
+alt_uniq_non0 [lemma, in Combi.MPoly.antisym]
+alt_detE [lemma, in Combi.MPoly.antisym]
+alt_anti [lemma, in Combi.MPoly.antisym]
+alt_homog [lemma, in Combi.MPoly.antisym]
+alt_straight_add_ribbon [lemma, in Combi.MPoly.Schur_altdef]
+alt_straight_add_ribbon0 [lemma, in Combi.MPoly.Schur_altdef]
+alt_straight_step [lemma, in Combi.MPoly.Schur_altdef]
+alt_uniq [lemma, in Combi.MPoly.Schur_altdef]
+alt_SchurE [lemma, in Combi.MPoly.Schur_altdef]
+alt_mpart_syme [lemma, in Combi.MPoly.Schur_altdef]
+alt_syme [lemma, in Combi.MPoly.Schur_altdef]
+antim [definition, in Combi.MPoly.antisym]
+antisym [definition, in Combi.MPoly.antisym]
+antisym [library]
+antisym_submod_closed [lemma, in Combi.MPoly.antisym]
+antisym_zmod [lemma, in Combi.MPoly.antisym]
+antisym_smalln [lemma, in Combi.MPoly.antisym]
+antisym_pchar2 [lemma, in Combi.MPoly.antisym]
+antisym_keyed [definition, in Combi.MPoly.antisym]
+antisym_key [lemma, in Combi.MPoly.antisym]
+anti_anti [lemma, in Combi.MPoly.antisym]
+AppendNth [section, in Combi.Combi.stdtab]
+AppendNth.disp [variable, in Combi.Combi.stdtab]
+AppendNth.T [variable, in Combi.Combi.stdtab]
+append_nth_conj_tab [lemma, in Combi.Combi.stdtab]
+append_nth_injl [lemma, in Combi.Combi.stdtab]
+append_nth_remn [lemma, in Combi.Combi.stdtab]
+append_nth [definition, in Combi.Combi.stdtab]
+app_mon [instance, in ALEA.Qmeasure]
+app2 [definition, in ALEA.Ccpo]
+app2_simpl [lemma, in ALEA.Ccpo]
+arm_length_incr_nth_nrow [lemma, in Combi.HookFormula.hook]
+arm_length_incr_nth_row [lemma, in Combi.HookFormula.hook]
+arm_length_corner_box [lemma, in Combi.HookFormula.hook]
+arm_length_ltl [lemma, in Combi.HookFormula.hook]
+arm_length_ler [lemma, in Combi.HookFormula.hook]
+arm_length [definition, in Combi.HookFormula.hook]
+Assoc [section, in Combi.SymGroup.towerSn]
+Assoc.m [variable, in Combi.SymGroup.towerSn]
+Assoc.n [variable, in Combi.SymGroup.towerSn]
+Assoc.p [variable, in Combi.SymGroup.towerSn]
+auxbij [definition, in Combi.LRrule.Yam_plact]
+auxbijP [lemma, in Combi.LRrule.Yam_plact]
+auxbij_inv [definition, in Combi.LRrule.Yam_plact]
+auxbij_inj [lemma, in Combi.LRrule.Yam_plact]
+

B

+BalToDyck [section, in Combi.Combi.Dyckword]
+BalToDyck.Hbal1 [variable, in Combi.Combi.Dyckword]
+BalToDyck.w [variable, in Combi.Combi.Dyckword]
+bal_of_DyckK [lemma, in Combi.Combi.Dyckword]
+bal_of_DyckP [lemma, in Combi.Combi.Dyckword]
+bal_of_Dyck [definition, in Combi.Combi.Dyckword]
+bal_hsz [definition, in Combi.Combi.Dyckword]
+Bases [section, in Combi.MPoly.sympoly]
+Bases.n [variable, in Combi.MPoly.sympoly]
+Bases.R [variable, in Combi.MPoly.sympoly]
+_ # _ [notation, in Combi.MPoly.sympoly]
+'e_ _ [notation, in Combi.MPoly.sympoly]
+'h_ _ [notation, in Combi.MPoly.sympoly]
+'m[ _ ] [notation, in Combi.MPoly.sympoly]
+'p_ _ [notation, in Combi.MPoly.sympoly]
+Basis [abbreviation, in Combi.MPoly.homogsym]
+basis_homsym [lemma, in Combi.MPoly.homogsym]
+behead_hookpartn [lemma, in Combi.Combi.partition]
+behead_incr_nth [lemma, in Combi.LRrule.Schensted]
+belast_behead_rcons [lemma, in Combi.SSRcomplements.tools]
+beq_nat_neq [lemma, in ALEA.Misc]
+bigcup_seq_cover [lemma, in Combi.LRrule.Greene]
+Bigop [section, in Combi.Combi.subseq]
+bigop_trivIseq [lemma, in Combi.LRrule.Greene]
+Bigop.idx [variable, in Combi.Combi.subseq]
+Bigop.op [variable, in Combi.Combi.subseq]
+Bigop.R [variable, in Combi.Combi.subseq]
+Bigop.T [variable, in Combi.Combi.subseq]
+_ * _ [notation, in Combi.Combi.subseq]
+*%M [notation, in Combi.Combi.subseq]
+1 [notation, in Combi.Combi.subseq]
+Bigsums [section, in ALEA.Qmeasure]
+Bigsums.A [variable, in ALEA.Qmeasure]
+BigTrivISeq [section, in Combi.LRrule.Greene]
+BigTrivISeq.idx [variable, in Combi.LRrule.Greene]
+BigTrivISeq.op [variable, in Combi.LRrule.Greene]
+BigTrivISeq.R [variable, in Combi.LRrule.Greene]
+BigTrivISeq.T [variable, in Combi.LRrule.Greene]
+big_mxvec_index [lemma, in Combi.MPoly.Cauchy]
+big_seq_sub [lemma, in Combi.SSRcomplements.tools]
+big_enum_box_in [lemma, in Combi.Combi.partition]
+big_enum_box_skew [lemma, in Combi.Combi.partition]
+big_box_skew2 [lemma, in Combi.Combi.partition]
+big_box_skew [lemma, in Combi.Combi.partition]
+big_subseqs_undup_cond [lemma, in Combi.Combi.subseq]
+big_subseqs_undup [lemma, in Combi.Combi.subseq]
+big_subseqs_cons_cond [lemma, in Combi.Combi.subseq]
+big_subseqs_cons [lemma, in Combi.Combi.subseq]
+big_subseqs0 [lemma, in Combi.Combi.subseq]
+big_subseqs_cond [lemma, in Combi.Combi.subseq]
+big_subseqs [lemma, in Combi.Combi.subseq]
+BijBinTrees [section, in Combi.Combi.Dyckword]
+Bijection [section, in Combi.Combi.stdtab]
+BijectionExtTab [section, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.Hmu [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.Hstrip [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.Hsz [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.la [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.m [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.mu [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.n [variable, in Combi.MPoly.Schur_altdef]
+BijectionExtTab.s [variable, in Combi.MPoly.Schur_altdef]
+Bijection.StdTabInd [section, in Combi.Combi.stdtab]
+BijFiberedSet [section, in Combi.Combi.fibered_set]
+BijFiberedSet.Defs [section, in Combi.Combi.fibered_set]
+BijFiberedSet.Defs.HcardEq [variable, in Combi.Combi.fibered_set]
+BijFiberedSet.Defs.U [variable, in Combi.Combi.fibered_set]
+BijFiberedSet.Defs.V [variable, in Combi.Combi.fibered_set]
+BijFiberedSet.I [variable, in Combi.Combi.fibered_set]
+bijLR [definition, in Combi.LRrule.therule]
+bijLRyam [definition, in Combi.LRrule.therule]
+bijLRyamP [lemma, in Combi.LRrule.therule]
+bijLR_image [lemma, in Combi.LRrule.therule]
+bijLR_inj [lemma, in Combi.LRrule.therule]
+bijLR_surj [lemma, in Combi.LRrule.therule]
+bijLR_LRsupport [lemma, in Combi.LRrule.therule]
+bijRS [lemma, in Combi.LRrule.Schensted]
+bijRStab [lemma, in Combi.LRrule.Schensted]
+bij_LRsupport [definition, in Combi.LRrule.freeSchur]
+bij_LRsupportP [lemma, in Combi.LRrule.freeSchur]
+BinLeaf [constructor, in Combi.Combi.bintree]
+BinNode [constructor, in Combi.Combi.bintree]
+binomial_sumn_iota [lemma, in Combi.SSRcomplements.tools]
+bintree [inductive, in Combi.Combi.bintree]
+bintree [library]
+bintreesz [record, in Combi.Combi.bintree]
+bintreeszP [lemma, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__fintype_SubFinite [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__fintype_Finite [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_SubCountable [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_Countable [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_SubChoice [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_Choice [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_SubEquality [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_Equality [definition, in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_SubType [definition, in Combi.Combi.bintree]
+bintree_bintree__canonical__choice_Countable [definition, in Combi.Combi.bintree]
+bintree_bintree__canonical__choice_Choice [definition, in Combi.Combi.bintree]
+bintree_bintree__canonical__eqtype_Equality [definition, in Combi.Combi.bintree]
+bintree_sind [definition, in Combi.Combi.bintree]
+bintree_rec [definition, in Combi.Combi.bintree]
+bintree_ind [definition, in Combi.Combi.bintree]
+bintree_rect [definition, in Combi.Combi.bintree]
+bintree_of_join_Dyck [lemma, in Combi.Combi.Dyckword]
+bintree_of_nil_Dyck [lemma, in Combi.Combi.Dyckword]
+bintree_of_DyckK [lemma, in Combi.Combi.Dyckword]
+bintree_of_Dyck [definition, in Combi.Combi.Dyckword]
+bintree_of_Dyck_spec [lemma, in Combi.Combi.Dyckword]
+bin_to_ordtreeK [lemma, in Combi.Combi.ordtree]
+bin_to_forestK [lemma, in Combi.Combi.ordtree]
+bin_to_ordtree [definition, in Combi.Combi.ordtree]
+bin_to_forest [definition, in Combi.Combi.ordtree]
+bisimul_instab [lemma, in Combi.LRrule.plactic]
+boolRP [lemma, in Combi.MPoly.sympoly]
+bool_of_braceK [lemma, in Combi.Combi.Dyckword]
+bool_of_brace [definition, in Combi.Combi.Dyckword]
+bound [projection, in Combi.Basic.congr]
+box [abbreviation, in Combi.Combi.skewpart]
+box [abbreviation, in Combi.Combi.skewpart]
+box [abbreviation, in Combi.Combi.skewpart]
+boxex [abbreviation, in Combi.Combi.skewpart]
+BoxIn [definition, in Combi.Combi.partition]
+BoxInSkew [section, in Combi.Combi.partition]
+BoxInSkew.inner [variable, in Combi.Combi.partition]
+BoxInSkew.outer [variable, in Combi.Combi.partition]
+BoxIn_subproof [lemma, in Combi.Combi.partition]
+box_in_incr_nth [lemma, in Combi.Combi.partition]
+box_inP [lemma, in Combi.Combi.partition]
+box_in [abbreviation, in Combi.Combi.partition]
+box_skewP [lemma, in Combi.Combi.partition]
+box_skewval [projection, in Combi.Combi.partition]
+box_skew [record, in Combi.Combi.partition]
+brace [inductive, in Combi.Combi.Dyckword]
+brace_of_boolK [lemma, in Combi.Combi.Dyckword]
+brace_of_bool [definition, in Combi.Combi.Dyckword]
+brace_sind [definition, in Combi.Combi.Dyckword]
+brace_rec [definition, in Combi.Combi.Dyckword]
+brace_ind [definition, in Combi.Combi.Dyckword]
+brace_rect [definition, in Combi.Combi.Dyckword]
+braidC [definition, in Combi.SymGroup.presentSn]
+braidclass [definition, in Combi.SymGroup.presentSn]
+braidcongr [definition, in Combi.SymGroup.presentSn]
+braidCP [lemma, in Combi.SymGroup.presentSn]
+BraidRed [section, in Combi.SymGroup.presentSn]
+braidred [abbreviation, in Combi.SymGroup.presentSn]
+braidredE [lemma, in Combi.SymGroup.presentSn]
+braidred_size_decr [lemma, in Combi.SymGroup.presentSn]
+braidred_to_canword [lemma, in Combi.SymGroup.presentSn]
+braidred_inscode_path [lemma, in Combi.SymGroup.presentSn]
+braidred_catl [lemma, in Combi.SymGroup.presentSn]
+BraidRed.n [variable, in Combi.SymGroup.presentSn]
+braidrule [definition, in Combi.SymGroup.presentSn]
+braidrule_homog [lemma, in Combi.SymGroup.presentSn]
+braidrule_sym [lemma, in Combi.SymGroup.presentSn]
+braidww [lemma, in Combi.SymGroup.presentSn]
+braid_to_canword [lemma, in Combi.SymGroup.presentSn]
+braid_ltn_lineC [lemma, in Combi.SymGroup.presentSn]
+braid_pred_lineC [lemma, in Combi.SymGroup.presentSn]
+braid_reduces [definition, in Combi.SymGroup.presentSn]
+braid_reduced [lemma, in Combi.SymGroup.presentSn]
+braid_prods [lemma, in Combi.SymGroup.presentSn]
+braid_rev [lemma, in Combi.SymGroup.presentSn]
+braid_cat [definition, in Combi.SymGroup.presentSn]
+braid_catr [definition, in Combi.SymGroup.presentSn]
+braid_catl [definition, in Combi.SymGroup.presentSn]
+braid_rcons [definition, in Combi.SymGroup.presentSn]
+braid_cons [definition, in Combi.SymGroup.presentSn]
+braid_is_congr [lemma, in Combi.SymGroup.presentSn]
+braid_trans [lemma, in Combi.SymGroup.presentSn]
+braid_ltrans [lemma, in Combi.SymGroup.presentSn]
+braid_sym [lemma, in Combi.SymGroup.presentSn]
+braid_refl [lemma, in Combi.SymGroup.presentSn]
+braid_equiv [lemma, in Combi.SymGroup.presentSn]
+braid_abaP [lemma, in Combi.SymGroup.presentSn]
+braid_aba [definition, in Combi.SymGroup.presentSn]
+Builders_1.Builders_Export_5 [module, in Combi.Basic.ordtype]
+Builders_1.HB_unnamed_factory_3 [definition, in Combi.Basic.ordtype]
+Builders_1.x [abbreviation, in Combi.Basic.ordtype]
+Builders_1.Builders_1.fresh_name_2 [variable, in Combi.Basic.ordtype]
+Builders_1.Builders_1.T [variable, in Combi.Basic.ordtype]
+Builders_1.Builders_1.Builders_1 [section, in Combi.Basic.ordtype]
+Builders_1.Super [module, in Combi.Basic.ordtype]
+Builders_1 [module, in Combi.Basic.ordtype]
+Builders_6.Builders_Export_10 [module, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_TPOrder [definition, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_TPreorder [definition, in Combi.Combi.composition]
+Builders_6.HB_unnamed_factory_8 [definition, in Combi.Combi.composition]
+Builders_6.isotop [lemma, in Combi.Combi.composition]
+Builders_6.top [definition, in Combi.Combi.composition]
+Builders_6.f_mono [abbreviation, in Combi.Combi.composition]
+Builders_6.f'_can [abbreviation, in Combi.Combi.composition]
+Builders_6.f_can [abbreviation, in Combi.Combi.composition]
+Builders_6.f' [abbreviation, in Combi.Combi.composition]
+Builders_6.f [abbreviation, in Combi.Combi.composition]
+Builders_6.T' [abbreviation, in Combi.Combi.composition]
+Builders_6.disp' [abbreviation, in Combi.Combi.composition]
+Builders_6.Builders_6.fresh_name_7 [variable, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_POrder [definition, in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_Order_Preorder_isDuallyPOrder [variable, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_Preorder [definition, in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_Order_isDuallyPreorder [variable, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__choice_Choice [definition, in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_eqtype_hasDecEq [variable, in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_choice_hasChoice [variable, in Combi.Combi.composition]
+Builders_6.Builders_6.T [variable, in Combi.Combi.composition]
+Builders_6.Builders_6.disp [variable, in Combi.Combi.composition]
+Builders_6.Builders_6.Builders_6 [section, in Combi.Combi.composition]
+Builders_6.Super [module, in Combi.Combi.composition]
+Builders_6 [module, in Combi.Combi.composition]
+Builders_1.Builders_Export_5 [module, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_BPOrder [definition, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_BPreorder [definition, in Combi.Combi.composition]
+Builders_1.HB_unnamed_factory_3 [definition, in Combi.Combi.composition]
+Builders_1.isobottom [lemma, in Combi.Combi.composition]
+Builders_1.bottom [definition, in Combi.Combi.composition]
+Builders_1.f_mono [abbreviation, in Combi.Combi.composition]
+Builders_1.f'_can [abbreviation, in Combi.Combi.composition]
+Builders_1.f_can [abbreviation, in Combi.Combi.composition]
+Builders_1.f' [abbreviation, in Combi.Combi.composition]
+Builders_1.f [abbreviation, in Combi.Combi.composition]
+Builders_1.T' [abbreviation, in Combi.Combi.composition]
+Builders_1.disp' [abbreviation, in Combi.Combi.composition]
+Builders_1.Builders_1.fresh_name_2 [variable, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_POrder [definition, in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_Order_Preorder_isDuallyPOrder [variable, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_Preorder [definition, in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_Order_isDuallyPreorder [variable, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__choice_Choice [definition, in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_eqtype_hasDecEq [variable, in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_choice_hasChoice [variable, in Combi.Combi.composition]
+Builders_1.Builders_1.T [variable, in Combi.Combi.composition]
+Builders_1.Builders_1.disp [variable, in Combi.Combi.composition]
+Builders_1.Builders_1.Builders_1 [section, in Combi.Combi.composition]
+Builders_1.Super [module, in Combi.Combi.composition]
+Builders_1 [module, in Combi.Combi.composition]
+bump [abbreviation, in Combi.LRrule.Schensted]
+bump [definition, in Combi.LRrule.Schensted]
+bumped [definition, in Combi.LRrule.Schensted]
+bumped_lt [lemma, in Combi.LRrule.plactic]
+bumpRow [abbreviation, in Combi.LRrule.Schensted]
+bumprow [definition, in Combi.LRrule.Schensted]
+bumprowinvK [lemma, in Combi.LRrule.Schensted]
+bumprow_rcons [lemma, in Combi.LRrule.Schensted]
+bumprow_count [lemma, in Combi.LRrule.Schensted]
+bumprow_size [lemma, in Combi.LRrule.Schensted]
+bump_dominateK [lemma, in Combi.LRrule.Schensted]
+bump_dominate [lemma, in Combi.LRrule.Schensted]
+bump_bumprowE [lemma, in Combi.LRrule.Schensted]
+bump_tail [lemma, in Combi.LRrule.Schensted]
+bump_nil [lemma, in Combi.LRrule.Schensted]
+bump_size_ins [lemma, in Combi.LRrule.Schensted]
+bump_inspos_lt_size [lemma, in Combi.LRrule.Schensted]
+bump_insposE [lemma, in Combi.LRrule.Schensted]
+bump_mininspredE [lemma, in Combi.LRrule.Schensted]
+bump_bumprow_rconsE [lemma, in Combi.LRrule.plactic]
+

C

+canporbit [definition, in Combi.SymGroup.cycletype]
+CanPorbit [section, in Combi.SymGroup.cycletype]
+canporbitE [lemma, in Combi.SymGroup.cycletype]
+canporbitP [lemma, in Combi.SymGroup.cycletype]
+canporbit_cymap [lemma, in Combi.SymGroup.cycletype]
+CanPorbit.s [variable, in Combi.SymGroup.cycletype]
+CanPorbit.T [variable, in Combi.SymGroup.cycletype]
+CanWord [section, in Combi.SymGroup.presentSn]
+canword [definition, in Combi.SymGroup.presentSn]
+canwordE [lemma, in Combi.SymGroup.presentSn]
+canwordP [lemma, in Combi.SymGroup.presentSn]
+canword_eltr [lemma, in Combi.SymGroup.presentSn]
+canword_path_npos [lemma, in Combi.SymGroup.presentSn]
+canword_straightenE [lemma, in Combi.SymGroup.presentSn]
+canword_reduced [lemma, in Combi.SymGroup.presentSn]
+CanWord.n0 [variable, in Combi.SymGroup.presentSn]
+'s_ _ (group_scope) [notation, in Combi.SymGroup.presentSn]
+_ =Br _ [notation, in Combi.SymGroup.presentSn]
+'I[ _ .. _ ] [notation, in Combi.SymGroup.presentSn]
+'s_[ _ ] [notation, in Combi.SymGroup.presentSn]
+canword1 [lemma, in Combi.SymGroup.presentSn]
+card_class_of_part [lemma, in Combi.SymGroup.permcent]
+card_class_perm [lemma, in Combi.SymGroup.permcent]
+card_cent1_perm [lemma, in Combi.SymGroup.permcent]
+card_stab_iporbits [lemma, in Combi.SymGroup.permcent]
+card_porbitgrpE [lemma, in Combi.SymGroup.permcent]
+card_stdwordn [lemma, in Combi.Combi.std]
+card_box_in [lemma, in Combi.Combi.partition]
+card_box_skew [lemma, in Combi.Combi.partition]
+card_intpartn [lemma, in Combi.Combi.partition]
+card_bintreesz [lemma, in Combi.Combi.bintree]
+card_seq [lemma, in Combi.LRrule.Greene]
+card_eq_eval [lemma, in Combi.MPoly.Schur_altdef]
+card_setdiff [lemma, in Combi.MPoly.Schur_altdef]
+card_stdtabsh_rat_rec [lemma, in Combi.HookFormula.hook]
+card_yam_stdtabE [lemma, in Combi.HookFormula.hook]
+card_yama0 [lemma, in Combi.HookFormula.hook]
+card_yama_rec [lemma, in Combi.HookFormula.hook]
+card_LRtab_set_shapeE [lemma, in Combi.LRrule.freeSchur]
+card_LRtab_set_leq [lemma, in Combi.LRrule.freeSchur]
+card_Dyck_hsz [lemma, in Combi.Combi.Dyckword]
+card_bal_Dyck_hsz [lemma, in Combi.Combi.Dyckword]
+card_preim_Dyck_of_bal [lemma, in Combi.Combi.Dyckword]
+card_bal_hsz [lemma, in Combi.Combi.Dyckword]
+card_bintreesz_dyck [lemma, in Combi.Combi.Dyckword]
+card_preim_nth [lemma, in Combi.Combi.Dyckword]
+card_preim_partition [lemma, in Combi.Combi.Dyckword]
+card_stdtabsh_conj_part [lemma, in Combi.Combi.stdtab]
+card_classes_perm [lemma, in Combi.SymGroup.cycletype]
+card_classCT_neq0 [lemma, in Combi.SymGroup.cycletype]
+card_psupport_conjg [lemma, in Combi.SymGroup.cycletype]
+card_pred_card_porbits [lemma, in Combi.SymGroup.cycletype]
+card_psupport_noteq1 [lemma, in Combi.SymGroup.cycles]
+card_ordtreesz [lemma, in Combi.Combi.ordtree]
+card_stpn_shape_hook [lemma, in Combi.HookFormula.Frobenius_ident]
+card_stpn_shape [lemma, in Combi.HookFormula.Frobenius_ident]
+card_preim_part_of_compn [lemma, in Combi.Combi.permuted]
+card_permuted_multinomial_subset [lemma, in Combi.Combi.permuted]
+card_permuted_multinomial [lemma, in Combi.Combi.permuted]
+card_permuted [lemma, in Combi.Combi.permuted]
+card_permuted_seq_sub [lemma, in Combi.Combi.permuted]
+card_permuted_prod [lemma, in Combi.Combi.permuted]
+card_stab_tuple [lemma, in Combi.Combi.permuted]
+card_Delta [lemma, in Combi.SymGroup.presentSn]
+card_codesz [lemma, in Combi.SymGroup.presentSn]
+card_unionE [lemma, in Combi.Basic.combclass]
+card_subE [lemma, in Combi.Basic.combclass]
+card_descset [lemma, in Combi.Combi.composition]
+card_intcompn [lemma, in Combi.Combi.composition]
+carrier [projection, in Combi.Combi.fibered_set]
+Cast [section, in Combi.LRrule.Greene]
+Cast [section, in Combi.MPoly.sympoly]
+Casts [section, in Combi.SSRcomplements.ordcast]
+cast_conj_inpart [lemma, in Combi.Combi.partition]
+cast_intpartn_bij [lemma, in Combi.Combi.partition]
+cast_intpartn_inj [lemma, in Combi.Combi.partition]
+cast_intpartnKV [lemma, in Combi.Combi.partition]
+cast_intpartnK [lemma, in Combi.Combi.partition]
+cast_intpartn_id [lemma, in Combi.Combi.partition]
+cast_intpartnE [lemma, in Combi.Combi.partition]
+cast_intpartn [definition, in Combi.Combi.partition]
+cast_enum [lemma, in Combi.LRrule.stdplact]
+cast_set_inj [lemma, in Combi.SSRcomplements.ordcast]
+cast_set [definition, in Combi.SSRcomplements.ordcast]
+cast_map_cond [lemma, in Combi.SSRcomplements.ordcast]
+cast_lshift [lemma, in Combi.SymGroup.towerSn]
+cast_rshift [lemma, in Combi.SymGroup.towerSn]
+cast_cons [definition, in Combi.LRrule.Greene]
+cast_cycle_typeSN [lemma, in Combi.SymGroup.cycletype]
+cast_IirrS2 [lemma, in Combi.SymGroup.reprSn]
+Cast.d1 [variable, in Combi.MPoly.sympoly]
+Cast.d2 [variable, in Combi.MPoly.sympoly]
+Cast.eq_d [variable, in Combi.MPoly.sympoly]
+Cast.la [variable, in Combi.MPoly.sympoly]
+Cast.n0 [variable, in Combi.MPoly.sympoly]
+Cast.R [variable, in Combi.MPoly.sympoly]
+Cast.T [variable, in Combi.LRrule.Greene]
+Catalan [definition, in Combi.Combi.Dyckword]
+Catalan_binS [lemma, in Combi.Combi.bintree]
+Catalan_bin0 [lemma, in Combi.Combi.bintree]
+Catalan_bin_leqE [lemma, in Combi.Combi.bintree]
+Catalan_bin [definition, in Combi.Combi.bintree]
+Catalan_bin_leq [definition, in Combi.Combi.bintree]
+Catalan_binE [lemma, in Combi.Combi.Dyckword]
+CategoricalSystems [section, in Combi.MPoly.sympoly]
+CategoricalSystems.R [variable, in Combi.MPoly.sympoly]
+catlang [definition, in Combi.LRrule.freeSchur]
+catlangM [lemma, in Combi.LRrule.freeSchur]
+catleft_rotations [lemma, in Combi.Combi.bintree]
+cat_leftA [lemma, in Combi.Combi.bintree]
+cat_left_Node [lemma, in Combi.Combi.bintree]
+cat_leftt0 [lemma, in Combi.Combi.bintree]
+cat_left0t [lemma, in Combi.Combi.bintree]
+cat_left [definition, in Combi.Combi.bintree]
+cat_tuple_inj [lemma, in Combi.LRrule.freeSchur]
+cat_Dyck [definition, in Combi.Combi.Dyckword]
+cat3_equiv_cut3 [lemma, in Combi.Combi.vectNK]
+Cauchy [library]
+CauchyKernel [section, in Combi.MPoly.Cauchy]
+CauchyKernelField [section, in Combi.MPoly.Cauchy]
+CauchyKernelField.R [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.Big [section, in Combi.MPoly.Cauchy]
+CauchyKernel.Big.idx [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.Big.op [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.Big.R [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam [section, in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam.d [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam.famYinv_fun [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.d [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.m0 [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.n0 [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.R [variable, in Combi.MPoly.Cauchy]
+CauchyKernel.vecmx_index [variable, in Combi.MPoly.Cauchy]
+_ (XY) [notation, in Combi.MPoly.Cauchy]
+_ (X) [notation, in Combi.MPoly.Cauchy]
+_ (Y) [notation, in Combi.MPoly.Cauchy]
+_ *:M _ [notation, in Combi.MPoly.Cauchy]
+Cauchy_co_hpXY__canonical__Algebra_Additive [definition, in Combi.MPoly.Cauchy]
+Cauchy_co_hp__canonical__GRing_Linear [definition, in Combi.MPoly.Cauchy]
+Cauchy_co_hp__canonical__Algebra_Additive [definition, in Combi.MPoly.Cauchy]
+Cauchy_homsymp_zhomsymp [lemma, in Combi.MPoly.Cauchy]
+Cauchy_kernel_coeff_homog [lemma, in Combi.MPoly.Cauchy]
+Cauchy_kernel_coeff_symmetric [lemma, in Combi.MPoly.Cauchy]
+Cauchy_kernel_symmetric [lemma, in Combi.MPoly.Cauchy]
+Cauchy_homsyms_homsyms [lemma, in Combi.MPoly.Cauchy]
+Cauchy_homsymm_homsymh [lemma, in Combi.MPoly.Cauchy]
+Cauchy_symm_symh [lemma, in Combi.MPoly.Cauchy]
+Cauchy_kernel_dhomog [lemma, in Combi.MPoly.Cauchy]
+Cauchy_kernel [definition, in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_LRMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_Linear [definition, in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_RMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__Algebra_Additive [definition, in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_LRMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_Linear [definition, in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_RMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__Algebra_Additive [definition, in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_LRMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_Linear [definition, in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_RMorphism [definition, in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__Algebra_Additive [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzSemiAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzSemiAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzLalgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzLSemiAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzLalgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzLSemiAlgebra [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_Lmodule [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_LSemiModule [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzRing [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzSemiRing [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzRing [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzSemiRing [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_Zmodule [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_Nmodule [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddSemigroup [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddUMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseZmodule [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_ChoiceBaseAddUMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseAddUMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_ChoiceBaseAddMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseAddMagma [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__choice_Choice [definition, in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__eqtype_Equality [definition, in Combi.MPoly.Cauchy]
+Ccpo [library]
+cdval [projection, in Combi.SymGroup.presentSn]
+cent1_permE [lemma, in Combi.SymGroup.permcent]
+cent1_stab_iporbit_porbitgrpS [lemma, in Combi.SymGroup.permcent]
+cent1_act_on_iporbits [lemma, in Combi.SymGroup.permcent]
+cent1_act_on_porbits [lemma, in Combi.SymGroup.permcent]
+cent1_act_porbit [lemma, in Combi.SymGroup.permcent]
+cfdot [abbreviation, in Combi.SymGroup.Frobenius_char]
+cfdotr_ncfuniCT [lemma, in Combi.SymGroup.towerSn]
+cfdot_Ind_ncfuniCT [lemma, in Combi.SymGroup.towerSn]
+cfdot_Ind_cfuniCT [lemma, in Combi.SymGroup.towerSn]
+cfdot_classfun_part [lemma, in Combi.SymGroup.towerSn]
+cfdot_is_dot [lemma, in Combi.SymGroup.Frobenius_char]
+cfdot_is_hermitian [lemma, in Combi.SymGroup.Frobenius_char]
+cfdot_is_bilinear [lemma, in Combi.SymGroup.Frobenius_char]
+cfextprod [definition, in Combi.SymGroup.towerSn]
+cfextprodBl [lemma, in Combi.SymGroup.towerSn]
+cfextprodBr [lemma, in Combi.SymGroup.towerSn]
+CFExtProdDefs [section, in Combi.SymGroup.towerSn]
+CFExtProdDefs.aT [variable, in Combi.SymGroup.towerSn]
+CFExtProdDefs.G [variable, in Combi.SymGroup.towerSn]
+CFExtProdDefs.gT [variable, in Combi.SymGroup.towerSn]
+CFExtProdDefs.H [variable, in Combi.SymGroup.towerSn]
+cfextprodDl [lemma, in Combi.SymGroup.towerSn]
+cfextprodDr [lemma, in Combi.SymGroup.towerSn]
+cfextprodMnl [lemma, in Combi.SymGroup.towerSn]
+cfextprodMnr [lemma, in Combi.SymGroup.towerSn]
+cfextprodNl [lemma, in Combi.SymGroup.towerSn]
+cfextprodNr [lemma, in Combi.SymGroup.towerSn]
+CFExtProdTheory [section, in Combi.SymGroup.towerSn]
+CFExtProdTheory.aT [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.G [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.gT [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.H [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd [section, in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.n1 [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.n2 [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.rG [variable, in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.rH [variable, in Combi.SymGroup.towerSn]
+cfextprodZl [lemma, in Combi.SymGroup.towerSn]
+cfextprodZr [lemma, in Combi.SymGroup.towerSn]
+cfextprod_cfuni [lemma, in Combi.SymGroup.towerSn]
+cfextprod_char [lemma, in Combi.SymGroup.towerSn]
+cfextprod_suml [lemma, in Combi.SymGroup.towerSn]
+cfextprod_sumr [lemma, in Combi.SymGroup.towerSn]
+cfextprod_is_bilinear [lemma, in Combi.SymGroup.towerSn]
+cfextprod_subproof [lemma, in Combi.SymGroup.towerSn]
+cfextprod0l [lemma, in Combi.SymGroup.towerSn]
+cfextprod0r [lemma, in Combi.SymGroup.towerSn]
+cfRepr_extprod [lemma, in Combi.SymGroup.towerSn]
+cfRepr_sign2 [lemma, in Combi.SymGroup.reprSn]
+cfRepr_signed [lemma, in Combi.SymGroup.reprSn]
+cfRepr_sign [lemma, in Combi.SymGroup.reprSn]
+cfRepr_trivE [lemma, in Combi.SymGroup.reprSn]
+cfRepr_triv [lemma, in Combi.SymGroup.reprSn]
+cfRepr1_lin_char [lemma, in Combi.SymGroup.reprSn]
+cfuniCT [definition, in Combi.SymGroup.cycletype]
+cfuniCTE [lemma, in Combi.SymGroup.cycletype]
+cfuniCTnE [lemma, in Combi.SymGroup.cycletype]
+cfuni_Res [lemma, in Combi.SymGroup.towerSn]
+cfuni_tinj [lemma, in Combi.SymGroup.towerSn]
+ChangeBaseMonomial [section, in Combi.MPoly.sympoly]
+ChangeBaseMonomial.n [variable, in Combi.MPoly.sympoly]
+ChangeBaseMonomial.R [variable, in Combi.MPoly.sympoly]
+ChangeBasis [section, in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum [section, in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum.nvar0 [variable, in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum.R [variable, in Combi.MPoly.sympoly]
+\Pi _ [notation, in Combi.MPoly.sympoly]
+ChangeBasis.HandE [section, in Combi.MPoly.sympoly]
+ChangeBasis.HandE.E [variable, in Combi.MPoly.sympoly]
+ChangeBasis.HandE.E0 [variable, in Combi.MPoly.sympoly]
+ChangeBasis.HandE.H [variable, in Combi.MPoly.sympoly]
+ChangeBasis.HandE.Hanti [variable, in Combi.MPoly.sympoly]
+ChangeBasis.HandE.H0 [variable, in Combi.MPoly.sympoly]
+ChangeBasis.n0 [variable, in Combi.MPoly.sympoly]
+ChangeBasis.R [variable, in Combi.MPoly.sympoly]
+'Xn_ _ [notation, in Combi.MPoly.sympoly]
+'Xn [notation, in Combi.MPoly.sympoly]
+'XXn_ _ [notation, in Combi.MPoly.sympoly]
+ChangeField [section, in Combi.MPoly.homogsym]
+ChangeField.d [variable, in Combi.MPoly.homogsym]
+ChangeField.mor [variable, in Combi.MPoly.homogsym]
+ChangeField.n0 [variable, in Combi.MPoly.homogsym]
+ChangeField.R [variable, in Combi.MPoly.homogsym]
+ChangeField.S [variable, in Combi.MPoly.homogsym]
+ChangeNVar [section, in Combi.MPoly.homogsym]
+ChangeNVar [section, in Combi.MPoly.sympoly]
+ChangeNVar.d [variable, in Combi.MPoly.homogsym]
+ChangeNVar.d [variable, in Combi.MPoly.sympoly]
+ChangeNVar.Hd [variable, in Combi.MPoly.homogsym]
+ChangeNVar.Hd [variable, in Combi.MPoly.sympoly]
+ChangeNVar.m0 [variable, in Combi.MPoly.homogsym]
+ChangeNVar.m0 [variable, in Combi.MPoly.sympoly]
+ChangeNVar.n0 [variable, in Combi.MPoly.homogsym]
+ChangeNVar.n0 [variable, in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen [section, in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen.Gen [variable, in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen.Hcnvargen [variable, in Combi.MPoly.sympoly]
+ChangeNVar.R [variable, in Combi.MPoly.homogsym]
+ChangeNVar.R [variable, in Combi.MPoly.sympoly]
+changeTdropP [lemma, in Combi.LRrule.freeSchur]
+changeTtakeP [lemma, in Combi.LRrule.freeSchur]
+changeUdropP [lemma, in Combi.LRrule.freeSchur]
+changeUT [definition, in Combi.LRrule.freeSchur]
+changeUtakeP [lemma, in Combi.LRrule.freeSchur]
+changeUTK [lemma, in Combi.LRrule.freeSchur]
+CharDotProduct [section, in Combi.SymGroup.Frobenius_char]
+CharDotProduct.G [variable, in Combi.SymGroup.Frobenius_char]
+CharDotProduct.gT [variable, in Combi.SymGroup.Frobenius_char]
+charfun [definition, in Combi.HookFormula.hook]
+charfun_simplr [lemma, in Combi.HookFormula.hook]
+charfun_simpll [lemma, in Combi.HookFormula.hook]
+charSG0 [lemma, in Combi.SymGroup.reprSn]
+charSG1 [lemma, in Combi.SymGroup.reprSn]
+char_mpoly [lemma, in Combi.MPoly.antisym]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.std]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.std]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.std]
+choice_Choice__to__eqtype_hasDecEq [definition, in Combi.MPoly.homogsym]
+choice_Choice__to__choice_hasChoice [definition, in Combi.MPoly.homogsym]
+choice_Countable__to__choice_Choice_isCountable__78 [definition, in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq__76 [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice__74 [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable__21 [definition, in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq__19 [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice__17 [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable__18 [definition, in Combi.Combi.bintree]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.bintree]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.bintree]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.bintree]
+choice_Choice__to__choice_hasChoice [definition, in Combi.Combi.bintree]
+choice_Choice__to__eqtype_hasDecEq [definition, in Combi.LRrule.Schensted]
+choice_Choice__to__choice_hasChoice__14 [definition, in Combi.LRrule.Schensted]
+choice_Choice__to__choice_hasChoice [definition, in Combi.LRrule.Schensted]
+choice_Countable__to__choice_Choice_isCountable__25 [definition, in Combi.Combi.Dyckword]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.Dyckword]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.Dyckword]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.Dyckword]
+choice_Choice__to__choice_hasChoice [definition, in Combi.Combi.Dyckword]
+choice_Countable__to__choice_Choice_isCountable__20 [definition, in Combi.Combi.stdtab]
+choice_Countable__to__eqtype_hasDecEq__18 [definition, in Combi.Combi.stdtab]
+choice_Countable__to__choice_hasChoice__16 [definition, in Combi.Combi.stdtab]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.stdtab]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.stdtab]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.stdtab]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.tableau]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.tableau]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.tableau]
+choice_Countable__to__choice_Choice_isCountable__18 [definition, in Combi.Combi.ordtree]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.ordtree]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.ordtree]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.ordtree]
+choice_Choice__to__choice_hasChoice [definition, in Combi.Combi.ordtree]
+choice_Countable__to__choice_Choice_isCountable__24 [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__eqtype_hasDecEq__22 [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_hasChoice__20 [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.permuted]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.permuted]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.permuted]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.SymGroup.presentSn]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.SymGroup.presentSn]
+choice_Countable__to__choice_hasChoice [definition, in Combi.SymGroup.presentSn]
+choice_Choice__to__eqtype_hasDecEq [definition, in Combi.MPoly.sympoly]
+choice_Choice__to__choice_hasChoice [definition, in Combi.MPoly.sympoly]
+choice_Countable__to__choice_Choice_isCountable__28 [definition, in Combi.Combi.composition]
+choice_Countable__to__eqtype_hasDecEq__26 [definition, in Combi.Combi.composition]
+choice_Countable__to__choice_hasChoice__24 [definition, in Combi.Combi.composition]
+choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.composition]
+choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.composition]
+choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.composition]
+choose_corner [definition, in Combi.HookFormula.hook]
+choose_one_countE [lemma, in Combi.LRrule.implem]
+class [definition, in ALEA.Misc]
+classCT [definition, in Combi.SymGroup.cycletype]
+classCTP [lemma, in Combi.SymGroup.cycletype]
+classCT_inj [lemma, in Combi.SymGroup.cycletype]
+classes_of_permP [lemma, in Combi.SymGroup.cycletype]
+classfun_cfdot__canonical__sesquilinear_Dot [definition, in Combi.SymGroup.Frobenius_char]
+classfun_cfdot__canonical__sesquilinear_Hermitian [definition, in Combi.SymGroup.Frobenius_char]
+classfun_cfdot__canonical__sesquilinear_Bilinear [definition, in Combi.SymGroup.Frobenius_char]
+classGroup [section, in Combi.SymGroup.towerSn]
+classGroup.aT [variable, in Combi.SymGroup.towerSn]
+classGroup.G [variable, in Combi.SymGroup.towerSn]
+classGroup.gT [variable, in Combi.SymGroup.towerSn]
+classGroup.H [variable, in Combi.SymGroup.towerSn]
+classX [abbreviation, in Combi.SymGroup.towerSn]
+classXE [lemma, in Combi.SymGroup.towerSn]
+classXI [lemma, in Combi.SymGroup.towerSn]
+class_disj [lemma, in Combi.SymGroup.towerSn]
+class_double_neg [lemma, in ALEA.Misc]
+class_exc [lemma, in ALEA.Misc]
+class_and [lemma, in ALEA.Misc]
+class_orc [lemma, in ALEA.Misc]
+class_false [lemma, in ALEA.Misc]
+class_neg [lemma, in ALEA.Misc]
+class_braid1 [lemma, in Combi.SymGroup.presentSn]
+Close [constructor, in Combi.Combi.Dyckword]
+CMbij [definition, in Combi.SymGroup.cycletype]
+cnval [projection, in Combi.Combi.composition]
+cnvarhomsym [definition, in Combi.MPoly.homogsym]
+cnvarhomsyme [lemma, in Combi.MPoly.homogsym]
+cnvarhomsymh [lemma, in Combi.MPoly.homogsym]
+cnvarhomsymm [lemma, in Combi.MPoly.homogsym]
+cnvarhomsymp [lemma, in Combi.MPoly.homogsym]
+cnvarhomsyms [lemma, in Combi.MPoly.homogsym]
+cnvarhomsym_is_linear [lemma, in Combi.MPoly.homogsym]
+cnvarhomsym_subproof [lemma, in Combi.MPoly.homogsym]
+cnvarsym [definition, in Combi.MPoly.sympoly]
+cnvarsym_geq_trans [lemma, in Combi.MPoly.sympoly]
+cnvarsym_leq_trans [lemma, in Combi.MPoly.sympoly]
+cnvarsym_id [lemma, in Combi.MPoly.sympoly]
+cnvarsym_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+cnvarsym_is_linear [lemma, in Combi.MPoly.sympoly]
+cnvarsym_subproof [lemma, in Combi.MPoly.sympoly]
+cnvar_symm [lemma, in Combi.MPoly.sympoly]
+cnvar_syms [lemma, in Combi.MPoly.sympoly]
+cnvar_prodsymp [lemma, in Combi.MPoly.sympoly]
+cnvar_prodsymh [lemma, in Combi.MPoly.sympoly]
+cnvar_prodsyme [lemma, in Combi.MPoly.sympoly]
+cnvar_prodgen [lemma, in Combi.MPoly.sympoly]
+cnvar_symp [lemma, in Combi.MPoly.sympoly]
+cnvar_symh [lemma, in Combi.MPoly.sympoly]
+cnvar_syme [lemma, in Combi.MPoly.sympoly]
+cnvar_leq_symeE [lemma, in Combi.MPoly.sympoly]
+cocode [definition, in Combi.SymGroup.presentSn]
+cocodeE [lemma, in Combi.SymGroup.presentSn]
+cocodeP [lemma, in Combi.SymGroup.presentSn]
+cocode_straightenE [lemma, in Combi.SymGroup.presentSn]
+cocode_rec_cat [lemma, in Combi.SymGroup.presentSn]
+cocode_rec [definition, in Combi.SymGroup.presentSn]
+cocode2P [lemma, in Combi.SymGroup.presentSn]
+code [definition, in Combi.SymGroup.presentSn]
+Codes [section, in Combi.SymGroup.presentSn]
+codesz [record, in Combi.SymGroup.presentSn]
+codeszP [lemma, in Combi.SymGroup.presentSn]
+Codes.FinType [section, in Combi.SymGroup.presentSn]
+Codes.FinType.n [variable, in Combi.SymGroup.presentSn]
+code_ltn_size [lemma, in Combi.SymGroup.presentSn]
+coeff [projection, in ALEA.Qmeasure]
+coeffXdiff [lemma, in Combi.MPoly.antisym]
+coeff_prodXdiff [lemma, in Combi.MPoly.antisym]
+coeff_symh_to_symp [lemma, in Combi.MPoly.sympoly]
+coeff_prodgen_cast [lemma, in Combi.MPoly.sympoly]
+coeff_prodgen [definition, in Combi.MPoly.sympoly]
+coeff_prodgen_seq [definition, in Combi.MPoly.sympoly]
+colcomp [definition, in Combi.Combi.composition]
+colcompn [definition, in Combi.Combi.composition]
+colcompnP [lemma, in Combi.Combi.composition]
+colpartn [definition, in Combi.Combi.partition]
+colpartnE [lemma, in Combi.Combi.partition]
+colpartnP [lemma, in Combi.Combi.partition]
+colpartn_subproof [lemma, in Combi.Combi.partition]
+combclass [library]
+Combi [section, in Combi.SymGroup.presentSn]
+commtuple_morph [lemma, in Combi.LRrule.freeSchur]
+CommutativeImage [section, in Combi.LRrule.freeSchur]
+CommutativeImage.n [variable, in Combi.LRrule.freeSchur]
+CommutativeImage.R [variable, in Combi.LRrule.freeSchur]
+commute_cyclic [lemma, in Combi.SymGroup.permcent]
+commword [definition, in Combi.LRrule.freeSchur]
+commword_morph [lemma, in Combi.LRrule.freeSchur]
+comp [definition, in ALEA.Ccpo]
+compn [definition, in ALEA.Misc]
+CompOfn [section, in Combi.Combi.composition]
+CompOfn.n [variable, in Combi.Combi.composition]
+composition [library]
+composition_intcompn__canonical__fintype_SubFinite [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__fintype_Finite [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__choice_SubCountable [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__choice_Countable [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__choice_SubChoice [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__choice_Choice [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_SubEquality [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_SubType [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__choice_SubCountable [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__choice_Countable [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__choice_SubChoice [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__choice_Choice [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_SubEquality [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_SubType [definition, in Combi.Combi.composition]
+compS [lemma, in ALEA.Misc]
+CompSpec_rect [lemma, in ALEA.Misc]
+comp_symbe [lemma, in Combi.MPoly.homogsym]
+comp_monotonic2 [instance, in ALEA.Ccpo]
+comp_monotonic_left [lemma, in ALEA.Ccpo]
+comp_monotonic_right [lemma, in ALEA.Ccpo]
+comp_simpl [lemma, in ALEA.Ccpo]
+comp0 [lemma, in ALEA.Misc]
+comp0 [lemma, in Combi.Combi.composition]
+congr [library]
+congrrule [definition, in Combi.Basic.congr]
+congrruleP [lemma, in Combi.Basic.congr]
+congrrule_sym [lemma, in Combi.Basic.congr]
+congrrule_invar [lemma, in Combi.Basic.congr]
+congrrule_is_congr [lemma, in Combi.Basic.congr]
+CongruenceClosure [section, in Combi.Basic.congr]
+CongruenceClosure.Alph [variable, in Combi.Basic.congr]
+CongruenceClosure.Hinvar_congr [variable, in Combi.Basic.congr]
+CongruenceClosure.Hsym [variable, in Combi.Basic.congr]
+CongruenceClosure.inv [variable, in Combi.Basic.congr]
+CongruenceClosure.rule [variable, in Combi.Basic.congr]
+CongruenceFacts [section, in Combi.Basic.congr]
+CongruenceFacts.Alph [variable, in Combi.Basic.congr]
+CongruenceFacts.Hcongr [variable, in Combi.Basic.congr]
+CongruenceFacts.Hequiv [variable, in Combi.Basic.congr]
+CongruenceFacts.r [variable, in Combi.Basic.congr]
+congruence_rule [definition, in Combi.Basic.congr]
+congruence_rel [definition, in Combi.Basic.congr]
+congr_RS [lemma, in Combi.LRrule.plactic]
+congr_bump [lemma, in Combi.LRrule.plactic]
+congr_row_2 [lemma, in Combi.LRrule.plactic]
+congr_row_1 [lemma, in Combi.LRrule.plactic]
+congr_cat [lemma, in Combi.Basic.congr]
+congr_catr [lemma, in Combi.Basic.congr]
+congr_catl [lemma, in Combi.Basic.congr]
+congr_rcons [lemma, in Combi.Basic.congr]
+congr_cons [lemma, in Combi.Basic.congr]
+conj [abbreviation, in Combi.HookFormula.hook]
+Conj [section, in Combi.LRrule.freeSchur]
+conjbij [definition, in Combi.SymGroup.cycletype]
+conjbijK [lemma, in Combi.SymGroup.cycletype]
+conjbijP [lemma, in Combi.SymGroup.cycletype]
+conjg_porbits_homog [lemma, in Combi.SymGroup.cycletype]
+conjg_porbits_stab [lemma, in Combi.SymGroup.cycletype]
+conjg_cycle [lemma, in Combi.SymGroup.cycletype]
+ConjTab [section, in Combi.Combi.stdtab]
+ConjTab.disp [variable, in Combi.Combi.stdtab]
+ConjTab.T [variable, in Combi.Combi.stdtab]
+conj_porbitgrp [lemma, in Combi.SymGroup.permcent]
+conj_hookpartn [lemma, in Combi.Combi.partition]
+conj_colpartn [lemma, in Combi.Combi.partition]
+conj_rowpartn [lemma, in Combi.Combi.partition]
+conj_intpartnK [lemma, in Combi.Combi.partition]
+conj_intpartn [definition, in Combi.Combi.partition]
+conj_intpartnP [lemma, in Combi.Combi.partition]
+conj_intpartK [lemma, in Combi.Combi.partition]
+conj_intpart [definition, in Combi.Combi.partition]
+conj_leqE [lemma, in Combi.Combi.partition]
+conj_ltnE [lemma, in Combi.Combi.partition]
+conj_part_incr_nth [lemma, in Combi.Combi.partition]
+conj_partK [lemma, in Combi.Combi.partition]
+conj_part_ind [lemma, in Combi.Combi.partition]
+conj_nseq [lemma, in Combi.Combi.partition]
+conj_part [definition, in Combi.Combi.partition]
+conj_stdtabsh_bij [lemma, in Combi.Combi.stdtab]
+conj_stdtabsh [definition, in Combi.Combi.stdtab]
+conj_stdtabshP [lemma, in Combi.Combi.stdtab]
+conj_stdtabn [definition, in Combi.Combi.stdtab]
+conj_stdtabnP [lemma, in Combi.Combi.stdtab]
+conj_tab_expl2 [definition, in Combi.Combi.stdtab]
+conj_tab_expl1 [definition, in Combi.Combi.stdtab]
+conj_tabK [lemma, in Combi.Combi.stdtab]
+conj_tab_shapeK [lemma, in Combi.Combi.stdtab]
+conj_tab [definition, in Combi.Combi.stdtab]
+conj_permP [lemma, in Combi.SymGroup.cycletype]
+Conj.d1 [variable, in Combi.LRrule.freeSchur]
+Conj.d2 [variable, in Combi.LRrule.freeSchur]
+ConnectCompl [section, in Combi.Combi.skewpart]
+ConnectCompl.e [variable, in Combi.Combi.skewpart]
+ConnectCompl.T [variable, in Combi.Combi.skewpart]
+Connected4 [section, in Combi.Combi.skewpart]
+Connected4.inner [variable, in Combi.Combi.skewpart]
+Connected4.outer [variable, in Combi.Combi.skewpart]
+connect_from_sym [lemma, in Combi.Combi.skewpart]
+connect_rev [lemma, in Combi.Combi.skewpart]
+conn4_sym [lemma, in Combi.Combi.skewpart]
+conn4_skew [definition, in Combi.Combi.skewpart]
+cons_in_map_cons [lemma, in Combi.SSRcomplements.tools]
+cons_head_behead [lemma, in Combi.SSRcomplements.tools]
+cons_TamariP [lemma, in Combi.Combi.bintree]
+continuous [record, in ALEA.Ccpo]
+continuous [inductive, in ALEA.Ccpo]
+continuous_comp [lemma, in ALEA.Ccpo]
+continuous_sym [lemma, in ALEA.Ccpo]
+continuous_continuous2 [lemma, in ALEA.Ccpo]
+continuous_eq_compat [lemma, in ALEA.Ccpo]
+continuous2 [record, in ALEA.Ccpo]
+continuous2 [inductive, in ALEA.Ccpo]
+continuous2_app2 [lemma, in ALEA.Ccpo]
+continuous2_comp2 [lemma, in ALEA.Ccpo]
+continuous2_comp [lemma, in ALEA.Ccpo]
+continuous2_sym [lemma, in ALEA.Ccpo]
+continuous2_right [lemma, in ALEA.Ccpo]
+continuous2_left [lemma, in ALEA.Ccpo]
+continuous2_continuous [lemma, in ALEA.Ccpo]
+continuous2_eq_compat [lemma, in ALEA.Ccpo]
+continuous2_app [lemma, in ALEA.Ccpo]
+continuous2_intro [projection, in ALEA.Ccpo]
+continuous2_intro [constructor, in ALEA.Ccpo]
+cont_app_simpl [lemma, in ALEA.Ccpo]
+cont_app [definition, in ALEA.Ccpo]
+cont_app_monotonic [instance, in ALEA.Ccpo]
+cont_intro [projection, in ALEA.Ccpo]
+cont_intro [constructor, in ALEA.Ccpo]
+cont0 [instance, in ALEA.Ccpo]
+cont2 [definition, in ALEA.Ccpo]
+cont2_continuous [instance, in ALEA.Ccpo]
+coord [abbreviation, in Combi.MPoly.homogsym]
+Coord [section, in Combi.MPoly.homogsym]
+coord_zsymspsp [lemma, in Combi.MPoly.Cauchy]
+coord_zsympsps [lemma, in Combi.MPoly.Cauchy]
+coord_symbp [lemma, in Combi.MPoly.homogsym]
+coord_symbs [lemma, in Combi.MPoly.homogsym]
+coord_symbh [lemma, in Combi.MPoly.homogsym]
+coord_symbe [lemma, in Combi.MPoly.homogsym]
+coord_symbm [lemma, in Combi.MPoly.homogsym]
+coord_map_homsym [lemma, in Combi.MPoly.homogsym]
+Coord.d [variable, in Combi.MPoly.homogsym]
+Coord.Hd [variable, in Combi.MPoly.homogsym]
+Coord.n0 [variable, in Combi.MPoly.homogsym]
+Coord.R [variable, in Combi.MPoly.homogsym]
+corner_hook_length1 [lemma, in Combi.HookFormula.hook]
+corner_leg_length0 [lemma, in Combi.HookFormula.hook]
+corner_arm_length0 [lemma, in Combi.HookFormula.hook]
+corner_box_conj_part [lemma, in Combi.HookFormula.hook]
+corner_box_in_part [lemma, in Combi.HookFormula.hook]
+corner_box [definition, in Combi.HookFormula.hook]
+Corollary4 [lemma, in Combi.HookFormula.hook]
+Corollary4_eq [lemma, in Combi.HookFormula.hook]
+count_mem_iota [lemma, in Combi.SSRcomplements.tools]
+count_rcons [lemma, in Combi.SSRcomplements.tools]
+count_set_of_card [lemma, in Combi.Combi.partition]
+count_mem_vect_n_k_eq_1 [lemma, in Combi.Combi.vectNK]
+count_RS [lemma, in Combi.LRrule.Schensted]
+count_instab [lemma, in Combi.LRrule.Schensted]
+count_mem_height0 [lemma, in Combi.Combi.Dyckword]
+count_gt_dominate [lemma, in Combi.Combi.tableau]
+count_mem_LRyamtab_list [lemma, in Combi.LRrule.implem]
+count_unionP [lemma, in Combi.Basic.combclass]
+covers [definition, in Combi.Basic.ordtype]
+coversEV [lemma, in Combi.Basic.ordtype]
+CoversFinPOrder [section, in Combi.Basic.ordtype]
+CoversFinPOrder.disp [variable, in Combi.Basic.ordtype]
+CoversFinPOrder.T [variable, in Combi.Basic.ordtype]
+coversP [lemma, in Combi.Basic.ordtype]
+CoverSurgery [section, in Combi.LRrule.Greene_inv]
+CoverSurgery.N [variable, in Combi.LRrule.Greene_inv]
+CoverSurgery.P [variable, in Combi.LRrule.Greene_inv]
+CoverSurgery.Q [variable, in Combi.LRrule.Greene_inv]
+CoverSurgery.S [variable, in Combi.LRrule.Greene_inv]
+covers_Tamari [lemma, in Combi.Combi.bintree]
+covers_permP [lemma, in Combi.SymGroup.weak_order]
+covers_rind [lemma, in Combi.Basic.ordtype]
+covers_path [lemma, in Combi.Basic.ordtype]
+covers_connect [lemma, in Combi.Basic.ordtype]
+covers_ind [lemma, in Combi.Basic.ordtype]
+covers_dual [lemma, in Combi.Basic.ordtype]
+cover_iporbits [lemma, in Combi.SymGroup.permcent]
+cover_cast [lemma, in Combi.SSRcomplements.ordcast]
+cover_tabcols [lemma, in Combi.LRrule.Greene]
+cover_tabcols_rec [lemma, in Combi.LRrule.Greene]
+cover_nil [lemma, in Combi.LRrule.Greene]
+cover_setpart [lemma, in Combi.Combi.setpartition]
+co_hprXYE [lemma, in Combi.MPoly.Cauchy]
+co_hpYE [lemma, in Combi.MPoly.Cauchy]
+co_hpXY_is_zmod_morphism [lemma, in Combi.MPoly.Cauchy]
+co_hp_hp [lemma, in Combi.MPoly.Cauchy]
+co_hp_is_scalar [lemma, in Combi.MPoly.Cauchy]
+co_hp_is_zmod_morphism [lemma, in Combi.MPoly.Cauchy]
+co_hpXY [definition, in Combi.MPoly.Cauchy]
+co_hp [definition, in Combi.MPoly.Cauchy]
+cpo [record, in ALEA.Ccpo]
+cpo_ord_equiv [definition, in ALEA.Ccpo]
+cshift [definition, in ALEA.Ccpo]
+cshift_simpl [lemma, in ALEA.Ccpo]
+cshift_continuous2 [instance, in ALEA.Ccpo]
+ct [abbreviation, in Combi.SymGroup.towerSn]
+ct [abbreviation, in Combi.SymGroup.towerSn]
+ct [abbreviation, in Combi.SymGroup.towerSn]
+cte [definition, in ALEA.Ccpo]
+CTpartn [definition, in Combi.SymGroup.cycletype]
+CTpartnK [lemma, in Combi.SymGroup.cycletype]
+CTpartn_colpartn [lemma, in Combi.SymGroup.cycletype]
+cutcover [lemma, in Combi.LRrule.Greene]
+CutK [section, in Combi.Combi.vectNK]
+CutK.T [variable, in Combi.Combi.vectNK]
+cut_k_flatten [lemma, in Combi.Combi.vectNK]
+cut_k [definition, in Combi.Combi.vectNK]
+cut3 [definition, in Combi.Combi.vectNK]
+Cut3 [section, in Combi.Combi.vectNK]
+Cut3.match3 [variable, in Combi.Combi.vectNK]
+Cut3.T [variable, in Combi.Combi.vectNK]
+cval [projection, in Combi.Combi.composition]
+CycleDecSpec [constructor, in Combi.SymGroup.cycles]
+cycleij_in [lemma, in Combi.SymGroup.presentSn]
+cycleij_inS [lemma, in Combi.SymGroup.presentSn]
+cycleij_gt [lemma, in Combi.SymGroup.presentSn]
+cycleij_lt [lemma, in Combi.SymGroup.presentSn]
+cycleij_j [lemma, in Combi.SymGroup.presentSn]
+cycles [library]
+CycleType [section, in Combi.SymGroup.cycletype]
+cycletype [library]
+CycleTypeConj [section, in Combi.SymGroup.cycletype]
+CycleTypeConj.T [variable, in Combi.SymGroup.cycletype]
+CycleType.CFunIndicator [section, in Combi.SymGroup.cycletype]
+CycleType.CFunIndicator.ct [variable, in Combi.SymGroup.cycletype]
+'1_[ _ ] (ring_scope) [notation, in Combi.SymGroup.cycletype]
+CycleType.Classes [section, in Combi.SymGroup.cycletype]
+CycleType.Classes.ct [variable, in Combi.SymGroup.cycletype]
+CycleType.Permofcycletype [section, in Combi.SymGroup.cycletype]
+CycleType.T [variable, in Combi.SymGroup.cycletype]
+CycleType.TPerm [section, in Combi.SymGroup.cycletype]
+cycle_type_tinjC [lemma, in Combi.SymGroup.towerSn]
+cycle_type_tinj [lemma, in Combi.SymGroup.towerSn]
+cycle_typeSn_permCT [lemma, in Combi.SymGroup.cycletype]
+cycle_typeSn1 [lemma, in Combi.SymGroup.cycletype]
+cycle_typeSn [definition, in Combi.SymGroup.cycletype]
+cycle_type_tpermP [lemma, in Combi.SymGroup.cycletype]
+cycle_type_tperm [lemma, in Combi.SymGroup.cycletype]
+cycle_type_eq [lemma, in Combi.SymGroup.cycletype]
+cycle_dec_conjg [lemma, in Combi.SymGroup.cycletype]
+cycle_type_cyclic [lemma, in Combi.SymGroup.cycletype]
+cycle_type_conjg [lemma, in Combi.SymGroup.cycletype]
+cycle_typeV [lemma, in Combi.SymGroup.cycletype]
+cycle_type1 [lemma, in Combi.SymGroup.cycletype]
+cycle_type [definition, in Combi.SymGroup.cycletype]
+cycle_type_subproof [lemma, in Combi.SymGroup.cycletype]
+cycle_decP [lemma, in Combi.SymGroup.cycles]
+cycle_dec_spec [inductive, in Combi.SymGroup.cycles]
+cycle_decE [lemma, in Combi.SymGroup.cycles]
+cycle_dec [definition, in Combi.SymGroup.cycles]
+cycle_cyclic [lemma, in Combi.SymGroup.cycles]
+cycle_type_eltr [lemma, in Combi.SymGroup.reprSn]
+cyclic [definition, in Combi.SymGroup.cycles]
+cyclicP [lemma, in Combi.SymGroup.cycles]
+cyclic_conjg [lemma, in Combi.SymGroup.cycletype]
+cyclic_dec [lemma, in Combi.SymGroup.cycles]
+cymap [definition, in Combi.SymGroup.cycletype]
+cymapcan [definition, in Combi.SymGroup.cycletype]
+cymapcan_perm [lemma, in Combi.SymGroup.cycletype]
+cymapcan_aux [lemma, in Combi.SymGroup.cycletype]
+cymapE [lemma, in Combi.SymGroup.cycletype]
+cymapK [lemma, in Combi.SymGroup.cycletype]
+cymapP [lemma, in Combi.SymGroup.cycletype]
+cymap_comp [lemma, in Combi.SymGroup.cycletype]
+cymap_id [lemma, in Combi.SymGroup.cycletype]
+

D

+Datatypes_nat__canonical__ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhTBLattice [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinOrder [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+Datatypes_prod__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+Datatypes_option__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+Datatypes_unit__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+Dbot [projection, in ALEA.Ccpo]
+decomp_cf_triv [lemma, in Combi.SymGroup.towerSn]
+decr_nth_intpartE [lemma, in Combi.Combi.partition]
+decr_nth_intpart [definition, in Combi.Combi.partition]
+decr_nthK [lemma, in Combi.Combi.partition]
+decr_nth [definition, in Combi.Combi.partition]
+decr_yam [definition, in Combi.Combi.Yamanouchi]
+dec_exists_lt [lemma, in ALEA.Misc]
+dec_sig_lt [lemma, in ALEA.Misc]
+Defs [section, in Combi.Combi.partition]
+Defs [section, in Combi.LRrule.shuffle]
+Defs [section, in Combi.Combi.Dyckword]
+Defs [section, in Combi.LRrule.plactic]
+Defs [section, in Combi.Combi.setpartition]
+Defs [section, in Combi.Combi.composition]
+DefsFiber [section, in Combi.SymGroup.cycletype]
+DefsFiber.eqct [variable, in Combi.SymGroup.cycletype]
+DefsFiber.s [variable, in Combi.SymGroup.cycletype]
+DefsFiber.t [variable, in Combi.SymGroup.cycletype]
+DefsFiber.U [variable, in Combi.SymGroup.cycletype]
+DefsFiber.V [variable, in Combi.SymGroup.cycletype]
+DefsKostkaMon [section, in Combi.MPoly.Schur_altdef]
+DefsKostkaMon.d [variable, in Combi.MPoly.Schur_altdef]
+DefsKostkaMon.la [variable, in Combi.MPoly.Schur_altdef]
+DefsKostkaMon.n [variable, in Combi.MPoly.Schur_altdef]
+Defs.Alph [variable, in Combi.LRrule.shuffle]
+Defs.Alph [variable, in Combi.LRrule.plactic]
+Defs.disp [variable, in Combi.LRrule.plactic]
+Defs.S [variable, in Combi.Combi.setpartition]
+Defs.T [variable, in Combi.Combi.setpartition]
+Defs.word [variable, in Combi.LRrule.shuffle]
+Defs.word [variable, in Combi.LRrule.plactic]
+_ / _ (Combi_scope) [notation, in Combi.Combi.partition]
+DefTrivSign [section, in Combi.SymGroup.reprSn]
+DefTrivSign.d [variable, in Combi.SymGroup.reprSn]
+DefTrivSign.n [variable, in Combi.SymGroup.reprSn]
+DefType [section, in Combi.MPoly.homogsym]
+DefType [section, in Combi.MPoly.sympoly]
+DefType.d [variable, in Combi.MPoly.homogsym]
+DefType.Hvar [variable, in Combi.MPoly.homogsym]
+DefType.n [variable, in Combi.MPoly.homogsym]
+DefType.n [variable, in Combi.MPoly.sympoly]
+DefType.R [variable, in Combi.MPoly.homogsym]
+DefType.R [variable, in Combi.MPoly.sympoly]
+Delta [abbreviation, in Combi.MPoly.antisym]
+Delta [definition, in Combi.SymGroup.presentSn]
+DeltaP [lemma, in Combi.SymGroup.presentSn]
+del_rem_corner [lemma, in Combi.Combi.partition]
+Depend [section, in Combi.Basic.congr]
+Depend.Hsym [variable, in Combi.Basic.congr]
+Depend.inv [variable, in Combi.Basic.congr]
+Depend.rule [variable, in Combi.Basic.congr]
+Depend.T [variable, in Combi.Basic.congr]
+depth_tree_eq2P [lemma, in Combi.Combi.ordtree]
+depth_tree_eq1 [lemma, in Combi.Combi.ordtree]
+depth_ordtree_pos [lemma, in Combi.Combi.ordtree]
+depth_ordtreeE [lemma, in Combi.Combi.ordtree]
+depth_ordtree [definition, in Combi.Combi.ordtree]
+DescSet [section, in Combi.Combi.composition]
+descset [definition, in Combi.Combi.composition]
+descsetK [lemma, in Combi.Combi.composition]
+descset_bij [lemma, in Combi.Combi.composition]
+descset_inj [lemma, in Combi.Combi.composition]
+DescSet.n [variable, in Combi.Combi.composition]
+det_unitrig [lemma, in Combi.Basic.unitriginv]
+dhomog_of_sym_is_linear [lemma, in Combi.MPoly.homogsym]
+dhomog_of_homogsym [definition, in Combi.MPoly.homogsym]
+diag [definition, in ALEA.Ccpo]
+diag_shift [lemma, in ALEA.Ccpo]
+diag_le_compat [lemma, in ALEA.Ccpo]
+diffX_neq0 [lemma, in Combi.MPoly.antisym]
+diff_shapeK [lemma, in Combi.Combi.partition]
+diff_shape_pad0 [lemma, in Combi.Combi.partition]
+diff_shape_eq [lemma, in Combi.Combi.partition]
+diff_shape [definition, in Combi.Combi.partition]
+diff_nth_sumn_take [lemma, in Combi.Combi.composition]
+dim_homsym [lemma, in Combi.MPoly.homogsym]
+dim_cfReprSG [lemma, in Combi.SymGroup.Frobenius_char]
+dim_irrSG [lemma, in Combi.SymGroup.Frobenius_char]
+disjoint_psupport_dprodE [lemma, in Combi.SymGroup.permcent]
+disjoint_imset [lemma, in Combi.SSRcomplements.tools]
+disjoint_cover [lemma, in Combi.LRrule.Greene_inv]
+disjoint_inj_rec [lemma, in Combi.LRrule.Greene]
+disjoint_inj [lemma, in Combi.LRrule.Greene]
+disjoint_psupports_conjg [lemma, in Combi.SymGroup.cycletype]
+disjoint_psupports_porbits [lemma, in Combi.SymGroup.cycles]
+disjoint_psupports_cycles [lemma, in Combi.SymGroup.cycles]
+disjoint_psupports_of_decomp [lemma, in Combi.SymGroup.cycles]
+disjoint_cycle_dec [lemma, in Combi.SymGroup.cycles]
+disjoint_perm_dec [lemma, in Combi.SymGroup.cycles]
+disjoint_psupport_subset [lemma, in Combi.SymGroup.cycles]
+disjoint_psupports [definition, in Combi.SymGroup.cycles]
+disj_perm_of_setpart [lemma, in Combi.SymGroup.cycletype]
+disp [definition, in Combi.LRrule.extract]
+distr [record, in ALEA.Qmeasure]
+div_central_binomial [lemma, in Combi.Combi.Dyckword]
+dominant [definition, in Combi.MPoly.antisym]
+dominant_eq [lemma, in Combi.MPoly.antisym]
+dominant_mpart [lemma, in Combi.MPoly.sympoly]
+Dominate [section, in Combi.Combi.skewtab]
+dominate [definition, in Combi.Combi.tableau]
+Dominate [section, in Combi.Combi.tableau]
+dominateK_inspos [lemma, in Combi.LRrule.Schensted]
+dominateP [lemma, in Combi.Combi.tableau]
+dominate_inspos [lemma, in Combi.LRrule.Schensted]
+dominate_tl [lemma, in Combi.Combi.tableau]
+dominate_head [lemma, in Combi.Combi.tableau]
+dominate_cut [lemma, in Combi.Combi.tableau]
+dominate_take [lemma, in Combi.Combi.tableau]
+dominate_rcons [lemma, in Combi.Combi.tableau]
+dominate_rev_trans [definition, in Combi.Combi.tableau]
+dominate_trans [lemma, in Combi.Combi.tableau]
+dominate_recE [lemma, in Combi.Combi.tableau]
+dominate_rec [definition, in Combi.Combi.tableau]
+Dominate.disp [variable, in Combi.Combi.skewtab]
+Dominate.disp [variable, in Combi.Combi.tableau]
+Dominate.T [variable, in Combi.Combi.skewtab]
+Dominate.T [variable, in Combi.Combi.tableau]
+double_app [definition, in ALEA.Ccpo]
+double_lub_shift [lemma, in ALEA.Ccpo]
+double_lub_diag [lemma, in ALEA.Ccpo]
+drop_enumI [lemma, in Combi.SSRcomplements.tools]
+drop_nilE [lemma, in Combi.SSRcomplements.tools]
+Dual [section, in Combi.Basic.ordtype]
+Duality [section, in Combi.LRrule.Greene_inv]
+Duality.Alph [variable, in Combi.LRrule.Greene_inv]
+Duality.disp [variable, in Combi.LRrule.Greene_inv]
+Duality.k [variable, in Combi.LRrule.Greene_inv]
+Duality.w [variable, in Combi.LRrule.Greene_inv]
+Duality.word [variable, in Combi.LRrule.Greene_inv]
+DualRule [section, in Combi.LRrule.plactic]
+DualRule.Alph [variable, in Combi.LRrule.plactic]
+DualRule.disp [variable, in Combi.LRrule.plactic]
+DualRule.word [variable, in Combi.LRrule.plactic]
+Dual.hb_instance_111.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_111.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_111.hb_instance_111 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_100.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_100.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_100.hb_instance_100 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_91.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_91.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_91.hb_instance_91 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_80.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_80.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_80.hb_instance_80 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_69.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_69.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_69.hb_instance_69 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_60.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_60.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_60.hb_instance_60 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_53.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_53.d [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_53.hb_instance_53 [section, in Combi.Basic.ordtype]
+Dual.hb_instance_48.T [variable, in Combi.Basic.ordtype]
+Dual.hb_instance_48.hb_instance_48 [section, in Combi.Basic.ordtype]
+dvdn_zcard_fact [lemma, in Combi.SymGroup.permcent]
+dvdn_prodfact [lemma, in Combi.Combi.multinomial]
+dvdn_card_permuted [lemma, in Combi.Combi.permuted]
+dyck [definition, in Combi.Combi.Dyckword]
+Dyck [record, in Combi.Combi.Dyckword]
+dyckE [lemma, in Combi.Combi.Dyckword]
+DyckFactor [section, in Combi.Combi.Dyckword]
+DyckP [lemma, in Combi.Combi.Dyckword]
+DyckSetInd [section, in Combi.Combi.Dyckword]
+DyckSetInd.P [variable, in Combi.Combi.Dyckword]
+DyckSetInd.Pcons [variable, in Combi.Combi.Dyckword]
+DyckSetInd.Pnil [variable, in Combi.Combi.Dyckword]
+DyckToBal [section, in Combi.Combi.Dyckword]
+DyckToBal.HDyck [variable, in Combi.Combi.Dyckword]
+DyckToBal.Hrt [variable, in Combi.Combi.Dyckword]
+DyckToBal.rt [variable, in Combi.Combi.Dyckword]
+DyckToBal.w [variable, in Combi.Combi.Dyckword]
+DyckType [section, in Combi.Combi.Dyckword]
+dyckword [projection, in Combi.Combi.Dyckword]
+Dyckword [library]
+DyckWordRotationBijection [section, in Combi.Combi.Dyckword]
+DyckWordRotationBijection.n [variable, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_SubCountable [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_Countable [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_SubChoice [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_Choice [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_SubEquality [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_Equality [definition, in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_SubType [definition, in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__fintype_Finite [definition, in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__choice_Countable [definition, in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__choice_Choice [definition, in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__eqtype_Equality [definition, in Combi.Combi.Dyckword]
+Dyck_of_balK [lemma, in Combi.Combi.Dyckword]
+Dyck_of_Dyck_hsz [lemma, in Combi.Combi.Dyckword]
+Dyck_of_balP [lemma, in Combi.Combi.Dyckword]
+Dyck_of_bal [definition, in Combi.Combi.Dyckword]
+Dyck_hsz [definition, in Combi.Combi.Dyckword]
+Dyck_of_bintreeK [lemma, in Combi.Combi.Dyckword]
+Dyck_of_bintree [definition, in Combi.Combi.Dyckword]
+Dyck_size_even [lemma, in Combi.Combi.Dyckword]
+Dyck_ind [lemma, in Combi.Combi.Dyckword]
+Dyck_cut_ex [lemma, in Combi.Combi.Dyckword]
+Dyck_word_OwCw [lemma, in Combi.Combi.Dyckword]
+Dyck_word_flatten [lemma, in Combi.Combi.Dyckword]
+Dyck_word_cat [lemma, in Combi.Combi.Dyckword]
+Dyck_word_OwC [lemma, in Combi.Combi.Dyckword]
+Dyck_wordP [lemma, in Combi.Combi.Dyckword]
+Dyck_prefixP [lemma, in Combi.Combi.Dyckword]
+Dyck_word [definition, in Combi.Combi.Dyckword]
+Dyck_prefix [definition, in Combi.Combi.Dyckword]
+D0 [projection, in ALEA.Ccpo]
+

E

+E [abbreviation, in Combi.MPoly.homogsym]
+E [abbreviation, in Combi.MPoly.sympoly]
+E [abbreviation, in Combi.MPoly.sympoly]
+E [abbreviation, in Combi.MPoly.sympoly]
+ElemTransp [section, in Combi.SymGroup.presentSn]
+ElemTransp.n0 [variable, in Combi.SymGroup.presentSn]
+'s_[ _ ] [notation, in Combi.SymGroup.presentSn]
+'s_ _ [notation, in Combi.SymGroup.presentSn]
+elt [projection, in Combi.Combi.fibered_set]
+eltr [definition, in Combi.SymGroup.presentSn]
+EltrConj [section, in Combi.SymGroup.reprSn]
+EltrConj.n [variable, in Combi.SymGroup.reprSn]
+eltrD [definition, in Combi.SymGroup.presentSn]
+eltrD_ord [lemma, in Combi.SymGroup.presentSn]
+eltrK [lemma, in Combi.SymGroup.presentSn]
+eltrL [definition, in Combi.SymGroup.presentSn]
+eltrL_ord [lemma, in Combi.SymGroup.presentSn]
+eltrp [definition, in Combi.MPoly.antisym]
+EltrP [section, in Combi.MPoly.antisym]
+eltrpK [lemma, in Combi.MPoly.antisym]
+EltrP.i [variable, in Combi.MPoly.antisym]
+EltrP.n [variable, in Combi.MPoly.antisym]
+eltrR [definition, in Combi.SymGroup.presentSn]
+eltrR_ord [lemma, in Combi.SymGroup.presentSn]
+eltrV [lemma, in Combi.SymGroup.presentSn]
+eltr_conj [lemma, in Combi.SymGroup.reprSn]
+eltr_ind [lemma, in Combi.SymGroup.presentSn]
+eltr_genSn [lemma, in Combi.SymGroup.presentSn]
+eltr_exchange [lemma, in Combi.SymGroup.presentSn]
+eltr_comm [lemma, in Combi.SymGroup.presentSn]
+eltr_braid [lemma, in Combi.SymGroup.presentSn]
+eltr2 [lemma, in Combi.SymGroup.presentSn]
+Empty [section, in Combi.Combi.setpartition]
+empty_intpartP [lemma, in Combi.Combi.partition]
+empty_intpart [definition, in Combi.Combi.partition]
+Empty.T [variable, in Combi.Combi.setpartition]
+ends_at_rem_cornerE [lemma, in Combi.HookFormula.hook]
+ends_at [definition, in Combi.HookFormula.hook]
+EnumFintype [section, in Combi.Basic.combclass]
+EnumFintype.P [variable, in Combi.Basic.combclass]
+EnumFintype.subenum [variable, in Combi.Basic.combclass]
+EnumFintype.subenumP [variable, in Combi.Basic.combclass]
+EnumFintype.subenum_countE [variable, in Combi.Basic.combclass]
+EnumFintype.T [variable, in Combi.Basic.combclass]
+EnumFintype.TP [variable, in Combi.Basic.combclass]
+enumIMN [lemma, in Combi.LRrule.Greene]
+enumIsize_to_word [lemma, in Combi.LRrule.Greene]
+enum_stdwordn_uniq [lemma, in Combi.Combi.std]
+enum_stdwordnE [lemma, in Combi.Combi.std]
+enum_stdwordn [definition, in Combi.Combi.std]
+enum_box_in_uniq [lemma, in Combi.Combi.partition]
+enum_box_in [abbreviation, in Combi.Combi.partition]
+enum_box_skewE [lemma, in Combi.Combi.partition]
+enum_box_skew_uniq [lemma, in Combi.Combi.partition]
+enum_box_skew [definition, in Combi.Combi.partition]
+enum_intpartnE [lemma, in Combi.Combi.partition]
+enum_partnP [lemma, in Combi.Combi.partition]
+enum_partn_countE [lemma, in Combi.Combi.partition]
+enum_partn_allP [lemma, in Combi.Combi.partition]
+enum_partnsE [lemma, in Combi.Combi.partition]
+enum_partns_countE [lemma, in Combi.Combi.partition]
+enum_partns_allP [lemma, in Combi.Combi.partition]
+enum_partnskE [lemma, in Combi.Combi.partition]
+enum_partnsk_countE [lemma, in Combi.Combi.partition]
+enum_partnsk_allP [lemma, in Combi.Combi.partition]
+enum_partn [definition, in Combi.Combi.partition]
+enum_partns [definition, in Combi.Combi.partition]
+enum_partnsk [definition, in Combi.Combi.partition]
+enum_bintreesz_countE [lemma, in Combi.Combi.bintree]
+enum_bintreesz_uniq [lemma, in Combi.Combi.bintree]
+enum_bintreeszP [lemma, in Combi.Combi.bintree]
+enum_bintreeszE [lemma, in Combi.Combi.bintree]
+enum_bintreesz0 [lemma, in Combi.Combi.bintree]
+enum_bintreesz_leqE [lemma, in Combi.Combi.bintree]
+enum_bintreesz_leq_leqE [lemma, in Combi.Combi.bintree]
+enum_bintreesz [definition, in Combi.Combi.bintree]
+enum_bintreesz_leq [definition, in Combi.Combi.bintree]
+enum_cast_ord [lemma, in Combi.SSRcomplements.ordcast]
+enum_traceP [lemma, in Combi.HookFormula.hook]
+enum_trace_uniq [lemma, in Combi.HookFormula.hook]
+enum_trace [definition, in Combi.HookFormula.hook]
+enum_stdtabn [definition, in Combi.Combi.stdtab]
+enum_stdtabshE [lemma, in Combi.Combi.stdtab]
+enum_stdtabsh [definition, in Combi.Combi.stdtab]
+enum_subseqsE [lemma, in Combi.Combi.subseq]
+enum_ord_sorted [lemma, in Combi.SSRcomplements.sorted]
+enum_ord_sorted_ltn [lemma, in Combi.SSRcomplements.sorted]
+enum_ordtreesz_countE [lemma, in Combi.Combi.ordtree]
+enum_ordtreesz_uniq [lemma, in Combi.Combi.ordtree]
+enum_ordtreeszP [lemma, in Combi.Combi.ordtree]
+enum_ordtreesz [definition, in Combi.Combi.ordtree]
+enum_yamnE [lemma, in Combi.Combi.Yamanouchi]
+enum_yamevalE [lemma, in Combi.Combi.Yamanouchi]
+enum_yameval_countE [lemma, in Combi.Combi.Yamanouchi]
+enum_yamevalP [lemma, in Combi.Combi.Yamanouchi]
+enum_yameval [definition, in Combi.Combi.Yamanouchi]
+enum_yamevaln [definition, in Combi.Combi.Yamanouchi]
+enum_setpart_set1 [lemma, in Combi.Combi.setpartition]
+enum_setpart_set0 [lemma, in Combi.Combi.setpartition]
+enum_codeszE [lemma, in Combi.SymGroup.presentSn]
+enum_codesz_countE [lemma, in Combi.SymGroup.presentSn]
+enum_codeszP [lemma, in Combi.SymGroup.presentSn]
+enum_codesz [definition, in Combi.SymGroup.presentSn]
+enum_unionE [lemma, in Combi.Basic.combclass]
+enum_union [definition, in Combi.Basic.combclass]
+enum_sub_undupE [lemma, in Combi.Basic.combclass]
+enum_subE [lemma, in Combi.Basic.combclass]
+enum_intcompnE [lemma, in Combi.Combi.composition]
+enum_compnP [lemma, in Combi.Combi.composition]
+enum_compn_countE [lemma, in Combi.Combi.composition]
+enum_compn_allP [lemma, in Combi.Combi.composition]
+enum_compnE [lemma, in Combi.Combi.composition]
+enum_compn_any [lemma, in Combi.Combi.composition]
+enum_compn_rec_any [lemma, in Combi.Combi.composition]
+enum_compn [definition, in Combi.Combi.composition]
+enum_compn_rec [definition, in Combi.Combi.composition]
+enum0 [lemma, in Combi.SSRcomplements.tools]
+eqeval [definition, in Combi.MPoly.Schur_altdef]
+eqevalP [lemma, in Combi.MPoly.Schur_altdef]
+EqInvAltDef [section, in Combi.Combi.std]
+EqInvAltDef.disp1 [variable, in Combi.Combi.std]
+EqInvAltDef.disp2 [variable, in Combi.Combi.std]
+EqInvAltDef.disp3 [variable, in Combi.Combi.std]
+EqInvAltDef.S [variable, in Combi.Combi.std]
+EqInvAltDef.T [variable, in Combi.Combi.std]
+EqInvAltDef.U [variable, in Combi.Combi.std]
+EqInvDef [section, in Combi.Combi.std]
+EqInvDef.disp1 [variable, in Combi.Combi.std]
+EqInvDef.disp2 [variable, in Combi.Combi.std]
+EqInvDef.disp3 [variable, in Combi.Combi.std]
+EqInvDef.S [variable, in Combi.Combi.std]
+EqInvDef.T [variable, in Combi.Combi.std]
+EqInvDef.U [variable, in Combi.Combi.std]
+EqInvPosRemBig [section, in Combi.Combi.std]
+EqInvPosRemBig.disp1 [variable, in Combi.Combi.std]
+EqInvPosRemBig.disp2 [variable, in Combi.Combi.std]
+EqInvPosRemBig.S [variable, in Combi.Combi.std]
+EqInvPosRemBig.T [variable, in Combi.Combi.std]
+EqInvSkewTab [section, in Combi.Combi.skewtab]
+eqtype_Equality__to__eqtype_hasDecEq [definition, in Combi.LRrule.Schensted]
+eqtype_Equality__to__eqtype_hasDecEq [definition, in Combi.Combi.Dyckword]
+equivalence_partitionE [lemma, in Combi.Combi.setpartition]
+equiv_rtrans [lemma, in Combi.Basic.congr]
+equi_fbbij [lemma, in Combi.Combi.fibered_set]
+eq_inv_transp [lemma, in Combi.Combi.std]
+eq_inv_catr [lemma, in Combi.Combi.std]
+eq_inv_catl [lemma, in Combi.Combi.std]
+eq_inv_std [lemma, in Combi.Combi.std]
+eq_inv_rembig [lemma, in Combi.Combi.std]
+eq_inv_posbig [lemma, in Combi.Combi.std]
+eq_inv_allgtnX [lemma, in Combi.Combi.std]
+eq_inv_allgt_imply [lemma, in Combi.Combi.std]
+eq_inv_rconsK [lemma, in Combi.Combi.std]
+eq_inv_consK [lemma, in Combi.Combi.std]
+eq_inv_inversionP [lemma, in Combi.Combi.std]
+eq_invP [lemma, in Combi.Combi.std]
+eq_inv_size [lemma, in Combi.Combi.std]
+eq_inv_trans [lemma, in Combi.Combi.std]
+eq_inv_sym [lemma, in Combi.Combi.std]
+eq_inv_nil [lemma, in Combi.Combi.std]
+eq_inv_refl [lemma, in Combi.Combi.std]
+eq_inv [definition, in Combi.Combi.std]
+eq_from_nth_notin [lemma, in Combi.SSRcomplements.tools]
+eq_bintreeP [lemma, in Combi.Combi.bintree]
+eq_bintree [definition, in Combi.Combi.bintree]
+eq_inv_is_skew_tableau_reshape [lemma, in Combi.Combi.skewtab]
+eq_inv_is_skew_tableau_reshape_size [lemma, in Combi.Combi.skewtab]
+eq_inv_skew_dominate [lemma, in Combi.Combi.skewtab]
+eq_Greene_rel [lemma, in Combi.LRrule.Greene]
+eq_Greene_rel_t [lemma, in Combi.LRrule.Greene]
+eq_from_shape_get_tab [lemma, in Combi.Combi.stdtab]
+eq_inv_is_row [lemma, in Combi.Combi.stdtab]
+eq_ord_equiv [lemma, in ALEA.Ccpo]
+eq_ord [definition, in ALEA.Ccpo]
+eq_rel [definition, in ALEA.Ccpo]
+eq_in_porbit [lemma, in Combi.SymGroup.cycles]
+eq_ordtreeP [lemma, in Combi.Combi.ordtree]
+eq_ordtree [definition, in Combi.Combi.ordtree]
+eq_forest [definition, in Combi.Combi.ordtree]
+eq_interv_part [lemma, in Combi.Combi.skewpart]
+eq_seqE [lemma, in Combi.Combi.permuted]
+eq_nat2_dec [definition, in ALEA.Misc]
+eq_mnm1 [lemma, in Combi.MPoly.sympoly]
+Erdos_Szekeres [lemma, in Combi.Erdos_Szekeres.Erdos_Szekeres]
+Erdos_Szekeres [library]
+er_eqE [lemma, in Combi.MPoly.homogsym]
+Esym [lemma, in Combi.MPoly.homogsym]
+esympolyf_eval_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+esympolyf_eval_is_linear [lemma, in Combi.MPoly.sympoly]
+eval [definition, in Combi.MPoly.Schur_altdef]
+evalE [lemma, in Combi.MPoly.Schur_altdef]
+evalseq [definition, in Combi.Combi.Yamanouchi]
+evalseq_hyper_yam [lemma, in Combi.Combi.Yamanouchi]
+evalseq_decr_yam [lemma, in Combi.Combi.Yamanouchi]
+evalseq_eq_size [lemma, in Combi.Combi.Yamanouchi]
+evalseq_cons [lemma, in Combi.Combi.Yamanouchi]
+evalseq_countE [lemma, in Combi.Combi.Yamanouchi]
+evalseq_count [definition, in Combi.Combi.Yamanouchi]
+evalseq0 [lemma, in Combi.Combi.Yamanouchi]
+evalXY [definition, in Combi.MPoly.Cauchy]
+evalXY_homog [lemma, in Combi.MPoly.Cauchy]
+evalXY_XE [lemma, in Combi.MPoly.Cauchy]
+evalXY_is_linear [lemma, in Combi.MPoly.Cauchy]
+evalXY_is_monoid_morphism [lemma, in Combi.MPoly.Cauchy]
+evalXY_is_zmod_morphism [lemma, in Combi.MPoly.Cauchy]
+eval_ext_tab [lemma, in Combi.MPoly.Schur_altdef]
+eval_res_tab [lemma, in Combi.MPoly.Schur_altdef]
+eval_yameval [lemma, in Combi.Combi.Yamanouchi]
+eval_eq [projection, in Combi.LRrule.implem]
+eval_part [projection, in Combi.LRrule.implem]
+Examples [section, in Combi.Combi.std]
+Examples.u [variable, in Combi.Combi.std]
+Examples.v [variable, in Combi.Combi.std]
+Example1 [module, in Combi.Basic.combclass]
+Example1.all_isOne [lemma, in Combi.Basic.combclass]
+Example1.card_isOne [lemma, in Combi.Basic.combclass]
+Example1.choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Basic.combclass]
+Example1.choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Basic.combclass]
+Example1.choice_Countable__to__choice_hasChoice [definition, in Combi.Basic.combclass]
+Example1.enum_isOne [lemma, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__fintype_SubFinite [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__fintype_Finite [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_SubCountable [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_Countable [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_SubChoice [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_Choice [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_SubEquality [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_Equality [definition, in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_SubType [definition, in Combi.Basic.combclass]
+Example1.fintype_Finite__to__fintype_isFinite [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_15 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_10 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_9 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_8 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_7 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_3 [definition, in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_1 [definition, in Combi.Basic.combclass]
+Example1.isOne [record, in Combi.Basic.combclass]
+Example1.isOne_count_1 [lemma, in Combi.Basic.combclass]
+Example1.is_one [definition, in Combi.Basic.combclass]
+Example1.one [projection, in Combi.Basic.combclass]
+Example2 [module, in Combi.Basic.combclass]
+Example2.all_isoneE [lemma, in Combi.Basic.combclass]
+Example2.card_isOne [lemma, in Combi.Basic.combclass]
+Example2.choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Basic.combclass]
+Example2.choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Basic.combclass]
+Example2.choice_Countable__to__choice_hasChoice [definition, in Combi.Basic.combclass]
+Example2.enum_isOne [lemma, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__fintype_SubFinite [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__fintype_Finite [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_SubCountable [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_Countable [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_SubChoice [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_Choice [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_SubEquality [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_Equality [definition, in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_SubType [definition, in Combi.Basic.combclass]
+Example2.fintype_Finite__to__fintype_isFinite [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_30 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_25 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_24 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_23 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_22 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_18 [definition, in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_16 [definition, in Combi.Basic.combclass]
+Example2.isOne [record, in Combi.Basic.combclass]
+Example2.isOne_uniq [lemma, in Combi.Basic.combclass]
+Example2.is_one [definition, in Combi.Basic.combclass]
+Example2.one [projection, in Combi.Basic.combclass]
+Example3 [module, in Combi.Basic.combclass]
+Example3.all_isOne [lemma, in Combi.Basic.combclass]
+Example3.card_isOne [lemma, in Combi.Basic.combclass]
+Example3.choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Basic.combclass]
+Example3.choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Basic.combclass]
+Example3.choice_Countable__to__choice_hasChoice [definition, in Combi.Basic.combclass]
+Example3.enum_isOne [lemma, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__fintype_SubFinite [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__fintype_Finite [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_SubCountable [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_Countable [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_SubChoice [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_Choice [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_SubEquality [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_Equality [definition, in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_SubType [definition, in Combi.Basic.combclass]
+Example3.fintype_Finite__to__fintype_isFinite [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_45 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_40 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_39 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_38 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_37 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_33 [definition, in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_31 [definition, in Combi.Basic.combclass]
+Example3.isOne [record, in Combi.Basic.combclass]
+Example3.isOne_in [lemma, in Combi.Basic.combclass]
+Example3.is_one [definition, in Combi.Basic.combclass]
+Example3.one [projection, in Combi.Basic.combclass]
+exc [definition, in ALEA.Misc]
+excluded_middle [lemma, in ALEA.Misc]
+exc_intro_class [lemma, in ALEA.Misc]
+exc_intro [lemma, in ALEA.Misc]
+exists_minh [lemma, in Combi.Combi.Dyckword]
+exist_size_Sch [lemma, in Combi.LRrule.Schensted]
+expand_prod_add1_seq [lemma, in Combi.HookFormula.hook]
+expg_tinj_rshift [lemma, in Combi.SymGroup.towerSn]
+expg_tinj_lshift [lemma, in Combi.SymGroup.towerSn]
+expg_prod_of_disjoint [lemma, in Combi.SymGroup.cycles]
+expri2 [lemma, in Combi.MPoly.sympoly]
+expUmpartE [lemma, in Combi.MPoly.sympoly]
+expUmpartNE [lemma, in Combi.MPoly.sympoly]
+exp1sumnDsize [lemma, in Combi.MPoly.sympoly]
+extlsplit [lemma, in Combi.LRrule.Greene]
+extprod_mx_repr [definition, in Combi.SymGroup.towerSn]
+extprod_mx_repr_subproof [lemma, in Combi.SymGroup.towerSn]
+extract [definition, in Combi.LRrule.Greene]
+extract [library]
+extractIE [lemma, in Combi.LRrule.Greene]
+extractmaskE [lemma, in Combi.LRrule.Greene]
+extractpred [definition, in Combi.LRrule.Greene]
+extractS [lemma, in Combi.LRrule.Greene]
+extract_rev_set [lemma, in Combi.LRrule.Greene_inv]
+extract_cut [lemma, in Combi.LRrule.Greene_inv]
+extract_tabcols_rec [lemma, in Combi.LRrule.Greene]
+extract_tabrows_rec [lemma, in Combi.LRrule.Greene]
+extract_tabrows_0 [lemma, in Combi.LRrule.Greene]
+extract0 [lemma, in Combi.LRrule.Greene]
+extract1 [lemma, in Combi.LRrule.Greene]
+extract2 [lemma, in Combi.LRrule.Greene]
+extrsplit [lemma, in Combi.LRrule.Greene]
+extsubsI [lemma, in Combi.LRrule.Greene]
+extsubsIm [lemma, in Combi.LRrule.Greene]
+extsubsm [lemma, in Combi.LRrule.Greene]
+ext_tab_inj [lemma, in Combi.MPoly.Schur_altdef]
+ext_tabK [lemma, in Combi.MPoly.Schur_altdef]
+ext_tab [definition, in Combi.MPoly.Schur_altdef]
+ext_tab_subproof [lemma, in Combi.MPoly.Schur_altdef]
+ext_tab_fun [definition, in Combi.MPoly.Schur_altdef]
+ex_set_setpart_shape [lemma, in Combi.Combi.partition]
+ex_setpart_shape [lemma, in Combi.Combi.partition]
+ex_subset_card [lemma, in Combi.Combi.partition]
+ex_dropeq [lemma, in Combi.Combi.skewpart]
+

F

+factor_Dyck_seq [lemma, in Combi.Combi.Dyckword]
+factor_Dyck [lemma, in Combi.Combi.Dyckword]
+famY [definition, in Combi.MPoly.Cauchy]
+famYinv [definition, in Combi.MPoly.Cauchy]
+famYinv_subproof [lemma, in Combi.MPoly.Cauchy]
+famY_bij [lemma, in Combi.MPoly.Cauchy]
+famY_subproof [lemma, in Combi.MPoly.Cauchy]
+FastImplem [section, in Combi.MPoly.MurnaghanNakayama]
+FastImplem.ComRing [section, in Combi.MPoly.MurnaghanNakayama]
+FastImplem.ComRing.R [variable, in Combi.MPoly.MurnaghanNakayama]
+FastImplem.n0 [variable, in Combi.MPoly.MurnaghanNakayama]
+fbbij [definition, in Combi.Combi.fibered_set]
+fbbijK [lemma, in Combi.Combi.fibered_set]
+fbbijP [lemma, in Combi.Combi.fibered_set]
+fbbij_in_fiber [lemma, in Combi.Combi.fibered_set]
+fbfun [projection, in Combi.Combi.fibered_set]
+fbset [projection, in Combi.Combi.fibered_set]
+fbset_fbbij [lemma, in Combi.Combi.fibered_set]
+Fchar [definition, in Combi.SymGroup.Frobenius_char]
+FcharE [lemma, in Combi.SymGroup.Frobenius_char]
+FcharK [lemma, in Combi.SymGroup.Frobenius_char]
+FcharNvar [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_irrSGE [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_ind_morph [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_inv_isometry [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_isometry [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_sign [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_inv_homsymp [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_triv [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_invK [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_inv_is_linear [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_ncfuniCT [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_is_linear [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_invE [lemma, in Combi.SymGroup.Frobenius_char]
+Fchar_inv [definition, in Combi.SymGroup.Frobenius_char]
+fcomp [definition, in ALEA.Ccpo]
+fcomp_simpl [lemma, in ALEA.Ccpo]
+fcomp2 [definition, in ALEA.Ccpo]
+fcomp2_simpl [lemma, in ALEA.Ccpo]
+fcont [record, in ALEA.Ccpo]
+fcontinuous [projection, in ALEA.Ccpo]
+Fcontm [definition, in ALEA.Ccpo]
+fcontm [projection, in ALEA.Ccpo]
+fcontm_fcont_comp_simpl [lemma, in ALEA.Ccpo]
+Fcontm_continuous [instance, in ALEA.Ccpo]
+fcontm_monotonic [instance, in ALEA.Ccpo]
+fcont_compn_com [lemma, in ALEA.Ccpo]
+fcont_compn_Sn_simpl [lemma, in ALEA.Ccpo]
+fcont_compn [definition, in ALEA.Ccpo]
+fcont_SEQ_simpl [lemma, in ALEA.Ccpo]
+fcont_SEQ [definition, in ALEA.Ccpo]
+fcont_continuous2 [instance, in ALEA.Ccpo]
+fcont_continuous [lemma, in ALEA.Ccpo]
+fcont_eq_compat2 [lemma, in ALEA.Ccpo]
+fcont_le_compat2 [lemma, in ALEA.Ccpo]
+fcont_COMP_simpl [lemma, in ALEA.Ccpo]
+fcont_COMP [definition, in ALEA.Ccpo]
+fcont_Comp_continuous2 [instance, in ALEA.Ccpo]
+fcont_Comp_simpl [lemma, in ALEA.Ccpo]
+fcont_Comp [definition, in ALEA.Ccpo]
+fcont_comp_le_compat [lemma, in ALEA.Ccpo]
+fcont_comp_simpl [lemma, in ALEA.Ccpo]
+fcont_comp [definition, in ALEA.Ccpo]
+fcont_comp_continuous [instance, in ALEA.Ccpo]
+fcont_lub_simpl [lemma, in ALEA.Ccpo]
+fcont_app_continuous [lemma, in ALEA.Ccpo]
+fcont_mshift [definition, in ALEA.Ccpo]
+fcont_ishift [definition, in ALEA.Ccpo]
+fcont_app_simpl [lemma, in ALEA.Ccpo]
+fcont_app [definition, in ALEA.Ccpo]
+fcont_cpo [instance, in ALEA.Ccpo]
+fcont_lub [definition, in ALEA.Ccpo]
+fcont_lub_continuous [instance, in ALEA.Ccpo]
+fcont_eq [lemma, in ALEA.Ccpo]
+fcont_le [lemma, in ALEA.Ccpo]
+fcont_eq_elim [lemma, in ALEA.Ccpo]
+fcont_eq_intro [lemma, in ALEA.Ccpo]
+fcont_le_elim [lemma, in ALEA.Ccpo]
+fcont_le_intro [lemma, in ALEA.Ccpo]
+fcont_ord [instance, in ALEA.Ccpo]
+fcont_fun [definition, in ALEA.Ccpo]
+fcont0 [definition, in ALEA.Ccpo]
+fcont2_comp_simpl [lemma, in ALEA.Ccpo]
+fcont2_comp [definition, in ALEA.Ccpo]
+fcont2_COMP [definition, in ALEA.Ccpo]
+fcpo [instance, in ALEA.Ccpo]
+fcpo_lub_simpl [lemma, in ALEA.Ccpo]
+feq [definition, in ALEA.Misc]
+feq_trans [lemma, in ALEA.Misc]
+feq_sym [lemma, in ALEA.Misc]
+feq_refl [lemma, in ALEA.Misc]
+fiber [definition, in Combi.Combi.fibered_set]
+fibered_set [record, in Combi.Combi.fibered_set]
+fibered_set [library]
+fiber_slporbitE [lemma, in Combi.SymGroup.cycletype]
+Fif [definition, in ALEA.Ccpo]
+fif [definition, in ALEA.Ccpo]
+fif_continuous2 [lemma, in ALEA.Ccpo]
+Fif_continuous2 [instance, in ALEA.Ccpo]
+fif_continuous_right [lemma, in ALEA.Ccpo]
+fif_continuous_left2 [lemma, in ALEA.Ccpo]
+fif_continuous_left [lemma, in ALEA.Ccpo]
+Fif_continuous_left [lemma, in ALEA.Ccpo]
+Fif_continuous_right [lemma, in ALEA.Ccpo]
+Fif_simpl [lemma, in ALEA.Ccpo]
+fif_mon2 [instance, in ALEA.Ccpo]
+filtergtn_LRsupport [lemma, in Combi.LRrule.therule]
+FilterLeqGeq [section, in Combi.Combi.skewtab]
+FilterLeqGeq.disp [variable, in Combi.Combi.skewtab]
+FilterLeqGeq.T [variable, in Combi.Combi.skewtab]
+filterleq_LRsupport [lemma, in Combi.LRrule.therule]
+filter_gt_RS [lemma, in Combi.LRrule.shuffle]
+filter_le_first_row0 [lemma, in Combi.Combi.skewtab]
+filter_gt_first_row0 [lemma, in Combi.Combi.skewtab]
+filter_le_tab [definition, in Combi.Combi.skewtab]
+filter_le_dominate [lemma, in Combi.Combi.skewtab]
+filter_ext_tab [lemma, in Combi.MPoly.Schur_altdef]
+filter_to_word [lemma, in Combi.Combi.tableau]
+filter_gt_tab [definition, in Combi.Combi.tableau]
+filter_gt_dominate [lemma, in Combi.Combi.tableau]
+filter_le_row [lemma, in Combi.Combi.tableau]
+filter_gt_row [lemma, in Combi.Combi.tableau]
+filter_le_to_word [lemma, in Combi.LRrule.therule]
+filter_gt_to_word [lemma, in Combi.LRrule.therule]
+filter_gt_shiftn [lemma, in Combi.LRrule.therule]
+filter_le_shiftn [lemma, in Combi.LRrule.therule]
+fin [record, in ALEA.Qmeasure]
+FindCorner [section, in Combi.HookFormula.hook]
+FindCorner.EndsAt [section, in Combi.HookFormula.hook]
+FindCorner.EndsAt.Alpha [variable, in Combi.HookFormula.hook]
+FindCorner.EndsAt.Beta [variable, in Combi.HookFormula.hook]
+FindCorner.EndsAt.Hcorn [variable, in Combi.HookFormula.hook]
+FindCorner.Formula [section, in Combi.HookFormula.hook]
+FindCorner.Formula.alpha [variable, in Combi.HookFormula.hook]
+FindCorner.Formula.R [variable, in Combi.HookFormula.hook]
+FindCorner.Formula.T [variable, in Combi.HookFormula.hook]
+FindCorner.p [variable, in Combi.HookFormula.hook]
+FindCorner.Theorem2 [section, in Combi.HookFormula.hook]
+FindCorner.Theorem2.Alpha [variable, in Combi.HookFormula.hook]
+FindCorner.Theorem2.Beta [variable, in Combi.HookFormula.hook]
+FindCorner.Theorem2.Hcorn [variable, in Combi.HookFormula.hook]
+FindCorner.Theorem2.Hpartc' [variable, in Combi.HookFormula.hook]
+FindCorner.Theorem2.p' [variable, in Combi.HookFormula.hook]
+find_append_nth [lemma, in Combi.Combi.stdtab]
+FinerCard [section, in Combi.Combi.setpartition]
+FinerCard.finPQ [variable, in Combi.Combi.setpartition]
+FinerCard.P [variable, in Combi.Combi.setpartition]
+FinerCard.Q [variable, in Combi.Combi.setpartition]
+FinerCard.S [variable, in Combi.Combi.setpartition]
+FinerCard.T [variable, in Combi.Combi.setpartition]
+Finite [definition, in ALEA.Qmeasure]
+FiniteDistributions [section, in ALEA.Qmeasure]
+FiniteDistributions.A [variable, in ALEA.Qmeasure]
+FiniteDistributions.p [variable, in ALEA.Qmeasure]
+Finite_in_seq [lemma, in ALEA.Qmeasure]
+Finite_eq_out [lemma, in ALEA.Qmeasure]
+Finite_eq_in [lemma, in ALEA.Qmeasure]
+Finite_simpl [lemma, in ALEA.Qmeasure]
+finite_stable_sub [lemma, in ALEA.Qmeasure]
+finite_simpl [lemma, in ALEA.Qmeasure]
+finite_mon [instance, in ALEA.Qmeasure]
+finite_stdtabn [lemma, in Combi.Combi.stdtab]
+finite_stdtabsh [lemma, in Combi.Combi.stdtab]
+finite_tabsh [lemma, in Combi.Combi.tableau]
+finite_unionP [lemma, in Combi.Basic.combclass]
+finite_sub_undupP [lemma, in Combi.Basic.combclass]
+finite_subP [lemma, in Combi.Basic.combclass]
+finord_wf_down [lemma, in Combi.Basic.ordtype]
+finord_wf [lemma, in Combi.Basic.ordtype]
+FinSet [section, in Combi.SSRcomplements.tools]
+finset_set_of__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+FinSet.T [variable, in Combi.SSRcomplements.tools]
+FinType [section, in Combi.Combi.subseq]
+FinType [section, in Combi.Combi.tableau]
+FinType [section, in Combi.Combi.ordtree]
+FinType [section, in Combi.Combi.permuted]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.std]
+fintype_Finite__to__fintype_isFinite__87 [definition, in Combi.Combi.partition]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.partition]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.bintree]
+fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.bintree]
+fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.bintree]
+fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.bintree]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.Dyckword]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.subseq]
+fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.subseq]
+fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.subseq]
+fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.subseq]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.ordtree]
+fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.ordtree]
+fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.ordtree]
+fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.ordtree]
+fintype_Finite__to__fintype_isFinite__33 [definition, in Combi.Combi.Yamanouchi]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.Yamanouchi]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.setpartition]
+fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.setpartition]
+fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.setpartition]
+fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.setpartition]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.permuted]
+fintype_ordinal__canonical__Order_FinTBTotal [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBTotal [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BTotal [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBDistrLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBDistrLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BDistrLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderTBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBJoinSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BJoinSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinBMeetSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBMeetSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BMeetSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinBPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTJoinSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TTotal [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TDistrLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubTLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubTLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubTLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderTLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TLattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TJoinSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TMeetSemilattice [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TPOrder [definition, in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.SymGroup.presentSn]
+fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.composition]
+FinType.d [variable, in Combi.Combi.tableau]
+FinType.disp [variable, in Combi.Combi.tableau]
+FinType.hb_instance_22.n [variable, in Combi.Combi.ordtree]
+FinType.hb_instance_22.hb_instance_22 [section, in Combi.Combi.ordtree]
+FinType.n [variable, in Combi.Combi.permuted]
+FinType.sh [variable, in Combi.Combi.tableau]
+FinType.T [variable, in Combi.Combi.subseq]
+FinType.T [variable, in Combi.Combi.tableau]
+FinType.T [variable, in Combi.Combi.permuted]
+FinType.tabsh_enum [variable, in Combi.Combi.tableau]
+FinType.w [variable, in Combi.Combi.subseq]
+FinType.w [variable, in Combi.Combi.permuted]
+FIXP [definition, in ALEA.Ccpo]
+Fixp [definition, in ALEA.Ccpo]
+fixp [definition, in ALEA.Ccpo]
+fixp_le_lub [lemma, in ALEA.Ccpo]
+fixp_ind_rel [lemma, in ALEA.Ccpo]
+fixp_ind [lemma, in ALEA.Ccpo]
+fixp_double [lemma, in ALEA.Ccpo]
+FIXP_compn [lemma, in ALEA.Ccpo]
+FIXP_comp [lemma, in ALEA.Ccpo]
+FIXP_comp_com [lemma, in ALEA.Ccpo]
+FIXP_inv [lemma, in ALEA.Ccpo]
+FIXP_eq [lemma, in ALEA.Ccpo]
+FIXP_eq_compat [lemma, in ALEA.Ccpo]
+FIXP_le_compat [lemma, in ALEA.Ccpo]
+FIXP_simpl [lemma, in ALEA.Ccpo]
+Fixp_cont_continuous [instance, in ALEA.Ccpo]
+Fixp_cont_simpl [lemma, in ALEA.Ccpo]
+Fixp_cont [definition, in ALEA.Ccpo]
+fixp_continuous_eq [lemma, in ALEA.Ccpo]
+fixp_continuous [lemma, in ALEA.Ccpo]
+Fixp_simpl [lemma, in ALEA.Ccpo]
+fixp_monotonic [instance, in ALEA.Ccpo]
+fixp_le_compat [lemma, in ALEA.Ccpo]
+fixp_cte [definition, in ALEA.Ccpo]
+fixp_inv [lemma, in ALEA.Ccpo]
+fixp_eq [lemma, in ALEA.Ccpo]
+fixp_le [lemma, in ALEA.Ccpo]
+flatten_equiv_cut_k [lemma, in Combi.Combi.vectNK]
+Flip [definition, in ALEA.Qmeasure]
+flip [definition, in ALEA.Qmeasure]
+flip [definition, in Combi.Combi.bintree]
+flipK [lemma, in Combi.Combi.bintree]
+flipsz [definition, in Combi.Combi.bintree]
+flipszK [lemma, in Combi.Combi.bintree]
+flipsz_leftcomb [lemma, in Combi.Combi.bintree]
+flipsz_rightcomb [lemma, in Combi.Combi.bintree]
+flipsz_subproof [lemma, in Combi.Combi.bintree]
+Flip_simpl [lemma, in ALEA.Qmeasure]
+flip_false [lemma, in ALEA.Qmeasure]
+flip_true [lemma, in ALEA.Qmeasure]
+flip_prob [lemma, in ALEA.Qmeasure]
+flip_stable_sub [lemma, in ALEA.Qmeasure]
+flip_mon [instance, in ALEA.Qmeasure]
+flip_leftcomb [lemma, in Combi.Combi.bintree]
+flip_rightcomb [lemma, in Combi.Combi.bintree]
+fmon [record, in ALEA.Ccpo]
+fmono [instance, in ALEA.Ccpo]
+fmonotonic [projection, in ALEA.Ccpo]
+fmonotonic2 [instance, in ALEA.Ccpo]
+fmont [projection, in ALEA.Ccpo]
+fmon_lub_simpl [lemma, in ALEA.Ccpo]
+fmon_cpo [instance, in ALEA.Ccpo]
+fmon_cte_comp [lemma, in ALEA.Ccpo]
+fmon_le_compat2 [lemma, in ALEA.Ccpo]
+fmon_diag_simpl [lemma, in ALEA.Ccpo]
+fmon_eq [lemma, in ALEA.Ccpo]
+fmon_le [lemma, in ALEA.Ccpo]
+fmon2_mon [instance, in ALEA.Ccpo]
+fnatO_elim [lemma, in ALEA.Ccpo]
+fnatO_intro [definition, in ALEA.Ccpo]
+foldr_join_Dyck_inj [lemma, in Combi.Combi.Dyckword]
+foldr_maxn [lemma, in Combi.Combi.Yamanouchi]
+ford [instance, in ALEA.Ccpo]
+ford_eq_intro [lemma, in ALEA.Ccpo]
+ford_eq_elim [lemma, in ALEA.Ccpo]
+ford_le_intro [lemma, in ALEA.Ccpo]
+ford_le_elim [lemma, in ALEA.Ccpo]
+forest [abbreviation, in Combi.Combi.ordtree]
+forest_to_bintreeK [lemma, in Combi.Combi.ordtree]
+forest_to_bintree [definition, in Combi.Combi.ordtree]
+Formula1 [lemma, in Combi.HookFormula.hook]
+fprob [definition, in ALEA.Qmeasure]
+freeSchur [definition, in Combi.LRrule.freeSchur]
+FreeSchur [section, in Combi.LRrule.freeSchur]
+freeSchur [library]
+freeSchurP [lemma, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs [section, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.w [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.HTriple [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.Q [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij [section, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.Hdrop [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.Htake [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.w [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.T [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.disp [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop [section, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.Hsh2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.Hsh1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.T2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.U2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.T1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.U1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT [section, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport [section, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.d1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.d2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.P1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.P2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.Degree [section, in Combi.LRrule.freeSchur]
+FreeSchur.Degree.d [variable, in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule [section, in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.d1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.d2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.Q1 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.Q2 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.n0 [variable, in Combi.LRrule.freeSchur]
+FreeSchur.R [variable, in Combi.LRrule.freeSchur]
+free_LR_rule_alternate [lemma, in Combi.LRrule.freeSchur]
+free_LR_rule [lemma, in Combi.LRrule.freeSchur]
+Frobenius_char [lemma, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_coord [lemma, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_homsymdot [lemma, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar_inv__canonical__GRing_Linear [definition, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar_inv__canonical__Algebra_Additive [definition, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar__canonical__GRing_Linear [definition, in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar__canonical__Algebra_Additive [definition, in Combi.SymGroup.Frobenius_char]
+Frobenius_ident_rat [lemma, in Combi.HookFormula.Frobenius_ident]
+Frobenius_ident [lemma, in Combi.HookFormula.Frobenius_ident]
+Frobenius_char [library]
+Frobenius_ident [library]
+from_vct0 [lemma, in Combi.Combi.bintree]
+from_vct_cat [lemma, in Combi.Combi.bintree]
+from_vctK [lemma, in Combi.Combi.bintree]
+from_vct_cat_leftE [lemma, in Combi.Combi.bintree]
+from_vct_accE [lemma, in Combi.Combi.bintree]
+from_vct_fuelE [lemma, in Combi.Combi.bintree]
+from_vct_acc_nil [lemma, in Combi.Combi.bintree]
+from_vct [definition, in Combi.Combi.bintree]
+from_vct_acc [definition, in Combi.Combi.bintree]
+from_vct_rec [definition, in Combi.Combi.bintree]
+from_left_cat [lemma, in Combi.Combi.bintree]
+from_leftK [lemma, in Combi.Combi.bintree]
+from_left [definition, in Combi.Combi.bintree]
+from_revdualK [lemma, in Combi.LRrule.plactic]
+from_revdual [definition, in Combi.LRrule.plactic]
+from_descsetK [lemma, in Combi.Combi.composition]
+from_descset [definition, in Combi.Combi.composition]
+from_descset_spec [lemma, in Combi.Combi.composition]
+fs [projection, in Combi.SymGroup.cycletype]
+fstable [instance, in ALEA.Ccpo]
+fstable2 [instance, in ALEA.Ccpo]
+fs_porbitP [lemma, in Combi.SymGroup.cycletype]
+fs_homog [projection, in Combi.SymGroup.cycletype]
+fs_stab [projection, in Combi.SymGroup.cycletype]
+full_bound [lemma, in Combi.Basic.congr]
+fun_ext [definition, in ALEA.Ccpo]
+fun2 [definition, in ALEA.Ccpo]
+fun2_mon2 [lemma, in ALEA.Ccpo]
+fun2_monotonic [instance, in ALEA.Ccpo]
+

G

+G [abbreviation, in Combi.MPoly.sympoly]
+geL [abbreviation, in Combi.LRrule.plactic]
+genclass [definition, in Combi.Basic.congr]
+genclassE [lemma, in Combi.Basic.congr]
+genclass_hom [definition, in Combi.Basic.congr]
+genclass_multhom [definition, in Combi.Basic.congr]
+GenCongr [constructor, in Combi.Basic.congr]
+gencongr [definition, in Combi.Basic.congr]
+gencongrP [lemma, in Combi.Basic.congr]
+gencongr_hom [definition, in Combi.Basic.congr]
+gencongr_multhom [definition, in Combi.Basic.congr]
+gencongr_generic_ind [lemma, in Combi.Basic.congr]
+gencongr_unique [lemma, in Combi.Basic.congr]
+gencongr_imply [lemma, in Combi.Basic.congr]
+gencongr_invar [lemma, in Combi.Basic.congr]
+gencongr_ind [lemma, in Combi.Basic.congr]
+gencongr_min [lemma, in Combi.Basic.congr]
+gencongr_is_congr [lemma, in Combi.Basic.congr]
+gencongr_equiv [lemma, in Combi.Basic.congr]
+Generated_EquivCongruence [inductive, in Combi.Basic.congr]
+Generators [section, in Combi.MPoly.sympoly]
+Generators.gen [variable, in Combi.MPoly.sympoly]
+Generators.gen_homog [variable, in Combi.MPoly.sympoly]
+Generators.n [variable, in Combi.MPoly.sympoly]
+Generators.R [variable, in Combi.MPoly.sympoly]
+genfun_length [lemma, in Combi.SymGroup.presentSn]
+get_conj_tab [lemma, in Combi.Combi.stdtab]
+get_tab_append_nth [lemma, in Combi.Combi.stdtab]
+get_map_tab [lemma, in Combi.Combi.tableau]
+get_tab_default [lemma, in Combi.Combi.tableau]
+get_tab [definition, in Combi.Combi.tableau]
+Greene [library]
+GreeneCat [section, in Combi.LRrule.Greene]
+GreeneCat.Alph [variable, in Combi.LRrule.Greene]
+GreeneCat.HR [variable, in Combi.LRrule.Greene]
+GreeneCat.lsplit [variable, in Combi.LRrule.Greene]
+GreeneCat.M [variable, in Combi.LRrule.Greene]
+GreeneCat.N [variable, in Combi.LRrule.Greene]
+GreeneCat.R [variable, in Combi.LRrule.Greene]
+GreeneCat.rsplit [variable, in Combi.LRrule.Greene]
+GreeneCat.V [variable, in Combi.LRrule.Greene]
+GreeneCat.W [variable, in Combi.LRrule.Greene]
+GreeneDef [section, in Combi.LRrule.Greene]
+GreeneDef.Alph [variable, in Combi.LRrule.Greene]
+GreeneDef.HR [variable, in Combi.LRrule.Greene]
+GreeneDef.N [variable, in Combi.LRrule.Greene]
+GreeneDef.R [variable, in Combi.LRrule.Greene]
+GreeneDef.wt [variable, in Combi.LRrule.Greene]
+_ .-supp (form_scope) [notation, in Combi.LRrule.Greene]
+GreeneInj [section, in Combi.LRrule.Greene]
+GreeneInj.R1 [variable, in Combi.LRrule.Greene]
+GreeneInj.R2 [variable, in Combi.LRrule.Greene]
+GreeneInj.T1 [variable, in Combi.LRrule.Greene]
+GreeneInj.T2 [variable, in Combi.LRrule.Greene]
+GreeneInvariants [section, in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual [section, in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual.Alph [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual.disp [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual.word [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule [section, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.Alph [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.disp [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.k [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.u [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.v1 [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.v2 [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.w [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.word [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariants.Alph [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariants.disp [variable, in Combi.LRrule.Greene_inv]
+GreeneInvariants.word [variable, in Combi.LRrule.Greene_inv]
+GreenEqShape [section, in Combi.LRrule.Greene_inv]
+GreenEqShape.d1 [variable, in Combi.LRrule.Greene_inv]
+GreenEqShape.d2 [variable, in Combi.LRrule.Greene_inv]
+GreenEqShape.S [variable, in Combi.LRrule.Greene_inv]
+GreenEqShape.T [variable, in Combi.LRrule.Greene_inv]
+GreeneRec [section, in Combi.LRrule.Greene]
+GreeneRec.Alph [variable, in Combi.LRrule.Greene]
+GreeneRec.cast_set_tab [variable, in Combi.LRrule.Greene]
+GreeneRec.disp [variable, in Combi.LRrule.Greene]
+GreeneRec.Induction [section, in Combi.LRrule.Greene]
+GreeneRec.Induction.t [variable, in Combi.LRrule.Greene]
+GreeneRec.Induction.t0 [variable, in Combi.LRrule.Greene]
+GreeneRec.sym_cast [variable, in Combi.LRrule.Greene]
+GreeneSeq [section, in Combi.LRrule.Greene]
+GreeneSeq.Alph [variable, in Combi.LRrule.Greene]
+GreeneSeq.HnegR [variable, in Combi.LRrule.Greene]
+GreeneSeq.HR [variable, in Combi.LRrule.Greene]
+GreeneSeq.negR [variable, in Combi.LRrule.Greene]
+GreeneSeq.R [variable, in Combi.LRrule.Greene]
+GreeneTab [section, in Combi.LRrule.Greene]
+GreeneTab.Alph [variable, in Combi.LRrule.Greene]
+GreeneTab.disp [variable, in Combi.LRrule.Greene]
+Greene_rel_one [lemma, in Combi.Erdos_Szekeres.Erdos_Szekeres]
+Greene_invseq [lemma, in Combi.LRrule.stdplact]
+Greene_std [lemma, in Combi.LRrule.stdplact]
+Greene_col_eq_shape_RS [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_eq_shape_RS [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_RS [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_invar_plactic [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_RS [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_invar_plactic [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact2i [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact2 [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact1i [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact1 [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact2i [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact2 [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact1i [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact1 [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact2 [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact1i [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact1 [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact2i [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact2i [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact1 [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact2 [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact1i [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_dual [lemma, in Combi.LRrule.Greene_inv]
+Greene_row_dual [lemma, in Combi.LRrule.Greene_inv]
+Greene_col_tab_eq_shape [lemma, in Combi.LRrule.Greene]
+Greene_row_tab_eq_shape [lemma, in Combi.LRrule.Greene]
+Greene_col_tab [lemma, in Combi.LRrule.Greene]
+Greene_row_tab [lemma, in Combi.LRrule.Greene]
+Greene_col [definition, in Combi.LRrule.Greene]
+Greene_row [definition, in Combi.LRrule.Greene]
+Greene_rel_rev [lemma, in Combi.LRrule.Greene]
+Greene_rel_uniq [lemma, in Combi.LRrule.Greene]
+Greene_rel_seq [lemma, in Combi.LRrule.Greene]
+Greene_rel_cat [lemma, in Combi.LRrule.Greene]
+Greene_rel [definition, in Combi.LRrule.Greene]
+Greene_rel_t_cat [lemma, in Combi.LRrule.Greene]
+Greene_rel_t_uniq [lemma, in Combi.LRrule.Greene]
+Greene_rel_t_cast [lemma, in Combi.LRrule.Greene]
+Greene_rel_t_0 [lemma, in Combi.LRrule.Greene]
+Greene_rel_t_sup [lemma, in Combi.LRrule.Greene]
+Greene_rel_t_inf [lemma, in Combi.LRrule.Greene]
+Greene_rel_t [definition, in Combi.LRrule.Greene]
+Greene_inv [library]
+grel_neig4_sym [lemma, in Combi.Combi.skewpart]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__76 [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__70 [definition, in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__65 [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__59 [definition, in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__53 [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__48 [definition, in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__42 [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable [definition, in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.Cauchy]
+GRing_Zmodule_isLmodule__to__GRing_Nmodule_isLSemiModule [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__GRing_PzSemiRing_isNonZero [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__GRing_Nmodule_isPzSemiRing [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_AddMagma_isAddSemigroup [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseAddMagma_isAddMagma [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseAddUMagma_isAddUMagma [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseZmoduleNmodule_isZmodule [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasAdd [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasZero [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasOpp [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__eqtype_hasDecEq [definition, in Combi.MPoly.Cauchy]
+GRing_NzRing__to__choice_hasChoice [definition, in Combi.MPoly.Cauchy]
+GRing_isLinear__to__Algebra_isNmodMorphism__77 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__75 [definition, in Combi.MPoly.homogsym]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__65 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__63 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__58 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__56 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__45 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__43 [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseAddMagma_isAddMagma [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_isSubBaseAddUMagma [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasAdd [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_AddMagma_isAddSemigroup [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasZero [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseZmoduleNmodule_isZmodule [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasOpp [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseAddUMagma_isAddUMagma [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__GRing_isSubLSemiModule [definition, in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__GRing_Nmodule_isLSemiModule [definition, in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__Algebra_isAddClosed [definition, in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__Algebra_isOppClosed [definition, in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__GRing_isScaleClosed [definition, in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__GRing_isScaleClosed [definition, in Combi.MPoly.antisym]
+GRing_isZmodClosed__to__Algebra_isOppClosed [definition, in Combi.MPoly.antisym]
+GRing_isZmodClosed__to__Algebra_isAddClosed [definition, in Combi.MPoly.antisym]
+GRing_isLinear__to__Algebra_isNmodMorphism__17 [definition, in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__GRing_isScalable__15 [definition, in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__Algebra_isNmodMorphism [definition, in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__GRing_isScalable [definition, in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__Algebra_isNmodMorphism__108 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__106 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__99 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__97 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__90 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__88 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__81 [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__79 [definition, in Combi.MPoly.sympoly]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.sympoly]
+GRing_SubComUnitRing_isSubIntegralDomain__to__GRing_ComUnitRing_isIntegral [definition, in Combi.MPoly.sympoly]
+GRing_SubNzRing_isSubUnitRing__to__GRing_NzRing_hasMulInverse [definition, in Combi.MPoly.sympoly]
+GRing_SubSemiRing_isSubComSemiRing__to__GRing_SemiRing_hasCommutativeMul [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_AddMagma_isAddSemigroup [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasZero [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseZmoduleNmodule_isZmodule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasOpp [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseAddUMagma_isAddUMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_isSubPzSemiRing [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_Nmodule_isPzSemiRing [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseAddMagma_isAddMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_isSubLSemiModule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_Nmodule_isLSemiModule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_LSemiModule_isLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_isSubBaseAddUMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_LSemiAlgebra_isSemiAlgebra [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasAdd [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_PzSemiRing_isNonZero [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism [definition, in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasAdd [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_AddMagma_isAddSemigroup [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasZero [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseZmoduleNmodule_isZmodule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasOpp [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseAddUMagma_isAddUMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_isSubPzSemiRing [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_Nmodule_isPzSemiRing [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseAddMagma_isAddMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_isSubLSemiModule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_Nmodule_isLSemiModule [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_LSemiModule_isLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_isSubBaseAddUMagma [definition, in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_PzSemiRing_isNonZero [definition, in Combi.MPoly.sympoly]
+gtL [abbreviation, in Combi.LRrule.plactic]
+gtn_braidC [lemma, in Combi.SymGroup.presentSn]
+gt_trans [lemma, in Combi.LRrule.Greene]
+

H

+hasincr [definition, in Combi.MPoly.Schur_altdef]
+hasincr0 [lemma, in Combi.MPoly.Schur_altdef]
+has_no_square [definition, in Combi.Combi.skewpart]
+HBeta [lemma, in Combi.HookFormula.hook]
+HBeta' [lemma, in Combi.HookFormula.hook]
+Hbound [projection, in Combi.Basic.congr]
+HB_unnamed_mixin_77 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_74 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_72 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_68 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_66 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_63 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_61 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_57 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_55 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_54 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_51 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_50 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_46 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_44 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_43 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_40 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_39 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_36 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_34 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_33 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_31 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_29 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_27 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_26 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_24 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_23 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_22 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_21 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_20 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_19 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_18 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_17 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_16 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_15 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_14 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_13 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_factory_1 [definition, in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_15 [definition, in Combi.Combi.std]
+HB_unnamed_factory_10 [definition, in Combi.Combi.std]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.std]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.std]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.std]
+HB_unnamed_factory_3 [definition, in Combi.Combi.std]
+HB_unnamed_factory_1 [definition, in Combi.Combi.std]
+HB_unnamed_factory_86 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_84 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_83 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_80 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_79 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_78 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_73 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_71 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_70 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_68 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_67 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_66 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_61 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_60 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_59 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_54 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_53 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_51 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_50 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_48 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_47 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_46 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_41 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_40 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_39 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_36 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_35 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_34 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_33 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_32 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_31 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_30 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_29 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_28 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_27 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_26 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_15 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_14 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_13 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_12 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_8 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_7 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_6 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_3 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_factory_1 [definition, in Combi.MPoly.homogsym]
+HB_unnamed_mixin_88 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_82 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_81 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_80 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_79 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_72 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_70 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_35 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_32 [definition, in Combi.Combi.partition]
+hb_instance_31.d [variable, in Combi.Combi.partition]
+hb_instance_31.hb_instance_31 [section, in Combi.Combi.partition]
+HB_unnamed_mixin_30 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_25 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_24 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_23 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_22 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_15 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_13 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_12 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_10 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_3 [definition, in Combi.Combi.partition]
+HB_unnamed_factory_1 [definition, in Combi.Combi.partition]
+HB_unnamed_mixin_28 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_23 [definition, in Combi.Combi.bintree]
+hb_instance_22.n [variable, in Combi.Combi.bintree]
+hb_instance_22.hb_instance_22 [section, in Combi.Combi.bintree]
+HB_unnamed_mixin_21 [definition, in Combi.Combi.bintree]
+HB_unnamed_mixin_20 [definition, in Combi.Combi.bintree]
+HB_unnamed_mixin_19 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_14 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_12 [definition, in Combi.Combi.bintree]
+HB_unnamed_mixin_11 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_7 [definition, in Combi.Combi.bintree]
+HB_unnamed_mixin_6 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_3 [definition, in Combi.Combi.bintree]
+HB_unnamed_factory_1 [definition, in Combi.Combi.bintree]
+HB_unnamed_mixin_3 [definition, in Combi.SymGroup.towerSn]
+HB_unnamed_factory_1 [definition, in Combi.SymGroup.towerSn]
+HB_unnamed_mixin_10 [definition, in Combi.MPoly.antisym]
+HB_unnamed_factory_6 [definition, in Combi.MPoly.antisym]
+HB_unnamed_mixin_5 [definition, in Combi.MPoly.antisym]
+HB_unnamed_mixin_4 [definition, in Combi.MPoly.antisym]
+HB_unnamed_factory_1 [definition, in Combi.MPoly.antisym]
+hb_strip_rowE [lemma, in Combi.Combi.skewtab]
+hb_strip_shape_res_tab [lemma, in Combi.MPoly.Schur_altdef]
+HB_unnamed_mixin_17 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_mixin_16 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_factory_12 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_factory_10 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_mixin_9 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_factory_6 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_mixin_5 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_factory_3 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_factory_1 [definition, in Combi.LRrule.Schensted]
+HB_unnamed_mixin_19 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_18 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_13 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_12 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_11 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_8 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_6 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_4 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_3 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_1 [definition, in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_28 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_27 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_26 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_21 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_19 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_18 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_13 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_12 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_8 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_4 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_mixin_3 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_1 [definition, in Combi.Combi.Dyckword]
+HB_unnamed_factory_24 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_23 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_22 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_21 [definition, in Combi.Combi.stdtab]
+HB_unnamed_factory_14 [definition, in Combi.Combi.stdtab]
+HB_unnamed_factory_12 [definition, in Combi.Combi.stdtab]
+HB_unnamed_factory_10 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.stdtab]
+HB_unnamed_factory_3 [definition, in Combi.Combi.stdtab]
+HB_unnamed_factory_1 [definition, in Combi.Combi.stdtab]
+HB_unnamed_mixin_11 [definition, in Combi.Combi.subseq]
+HB_unnamed_mixin_10 [definition, in Combi.Combi.subseq]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.subseq]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.subseq]
+HB_unnamed_factory_3 [definition, in Combi.Combi.subseq]
+HB_unnamed_factory_1 [definition, in Combi.Combi.subseq]
+HB_unnamed_factory_10 [definition, in Combi.Combi.tableau]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.tableau]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.tableau]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.tableau]
+HB_unnamed_factory_3 [definition, in Combi.Combi.tableau]
+HB_unnamed_factory_1 [definition, in Combi.Combi.tableau]
+HB_unnamed_mixin_28 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_23 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_21 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_20 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_19 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_14 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_12 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_11 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_7 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_6 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_3 [definition, in Combi.Combi.ordtree]
+HB_unnamed_factory_1 [definition, in Combi.Combi.ordtree]
+HB_unnamed_mixin_34 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_28 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_27 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_26 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_25 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_18 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_16 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_15 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_10 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_3 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_1 [definition, in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_14 [definition, in Combi.Combi.setpartition]
+HB_unnamed_factory_12 [definition, in Combi.Combi.setpartition]
+HB_unnamed_mixin_11 [definition, in Combi.Combi.setpartition]
+HB_unnamed_mixin_10 [definition, in Combi.Combi.setpartition]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.setpartition]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.setpartition]
+HB_unnamed_factory_3 [definition, in Combi.Combi.setpartition]
+HB_unnamed_factory_1 [definition, in Combi.Combi.setpartition]
+hb_strip_conjE [lemma, in Combi.Combi.skewpart]
+hb_strip_conj [lemma, in Combi.Combi.skewpart]
+hb_stripP [lemma, in Combi.Combi.skewpart]
+hb_strip_size [lemma, in Combi.Combi.skewpart]
+hb_strip_included [lemma, in Combi.Combi.skewpart]
+hb_strip [definition, in Combi.Combi.skewpart]
+HB_unnamed_mixin_15 [definition, in Combi.Combi.permuted]
+HB_unnamed_factory_10 [definition, in Combi.Combi.permuted]
+HB_unnamed_mixin_9 [definition, in Combi.Combi.permuted]
+HB_unnamed_mixin_8 [definition, in Combi.Combi.permuted]
+HB_unnamed_mixin_7 [definition, in Combi.Combi.permuted]
+HB_unnamed_factory_3 [definition, in Combi.Combi.permuted]
+HB_unnamed_factory_1 [definition, in Combi.Combi.permuted]
+HB_unnamed_factory_112 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_101 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_92 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_81 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_70 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_61 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_54 [definition, in Combi.Basic.ordtype]
+HB_unnamed_mixin_52 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_49 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_44 [definition, in Combi.Basic.ordtype]
+hb_instance_43.n [variable, in Combi.Basic.ordtype]
+hb_instance_43.hb_instance_43 [section, in Combi.Basic.ordtype]
+HB_unnamed_factory_39 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_35 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_33 [definition, in Combi.Basic.ordtype]
+hb_instance_32.T' [variable, in Combi.Basic.ordtype]
+hb_instance_32.T [variable, in Combi.Basic.ordtype]
+hb_instance_32.hb_instance_32 [section, in Combi.Basic.ordtype]
+HB_unnamed_mixin_31 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_28 [definition, in Combi.Basic.ordtype]
+hb_instance_27.T [variable, in Combi.Basic.ordtype]
+hb_instance_27.hb_instance_27 [section, in Combi.Basic.ordtype]
+HB_unnamed_mixin_26 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_23 [definition, in Combi.Basic.ordtype]
+hb_instance_22.T [variable, in Combi.Basic.ordtype]
+hb_instance_22.hb_instance_22 [section, in Combi.Basic.ordtype]
+HB_unnamed_mixin_21 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_18 [definition, in Combi.Basic.ordtype]
+hb_instance_17.n [variable, in Combi.Basic.ordtype]
+hb_instance_17.hb_instance_17 [section, in Combi.Basic.ordtype]
+HB_unnamed_mixin_16 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_13 [definition, in Combi.Basic.ordtype]
+HB_unnamed_mixin_12 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_9 [definition, in Combi.Basic.ordtype]
+HB_unnamed_mixin_8 [definition, in Combi.Basic.ordtype]
+HB_unnamed_factory_6 [definition, in Combi.Basic.ordtype]
+HB_unnamed_mixin_15 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_factory_10 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_9 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_8 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_7 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_factory_3 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_factory_1 [definition, in Combi.SymGroup.presentSn]
+HB_unnamed_factory_111 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_110 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_109 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_104 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_102 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_101 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_100 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_95 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_93 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_92 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_91 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_86 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_84 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_83 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_82 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_77 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_75 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_73 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_72 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_70 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_69 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_67 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_66 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_64 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_63 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_61 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_60 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_44 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_42 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_41 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_40 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_37 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_36 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_35 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_34 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_33 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_32 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_31 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_30 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_29 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_28 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_27 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_26 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_25 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_24 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_23 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_8 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_7 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_6 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_3 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_factory_1 [definition, in Combi.MPoly.sympoly]
+HB_unnamed_mixin_37 [definition, in Combi.Combi.composition]
+HB_unnamed_factory_32 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_31 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_30 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_29 [definition, in Combi.Combi.composition]
+HB_unnamed_factory_22 [definition, in Combi.Combi.composition]
+HB_unnamed_factory_20 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_19 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_18 [definition, in Combi.Combi.composition]
+HB_unnamed_mixin_17 [definition, in Combi.Combi.composition]
+HB_unnamed_factory_13 [definition, in Combi.Combi.composition]
+HB_unnamed_factory_11 [definition, in Combi.Combi.composition]
+Hcrn [lemma, in Combi.HookFormula.hook]
+head_pad [lemma, in Combi.Combi.partition]
+head_tableau_non_nil [lemma, in Combi.LRrule.Schensted]
+head_leq_invbumped [lemma, in Combi.LRrule.Schensted]
+head_lt_invins [lemma, in Combi.LRrule.Schensted]
+head_instab [lemma, in Combi.LRrule.Schensted]
+head_ins_lt_bumped [lemma, in Combi.LRrule.Schensted]
+head_leq_last_sorted [lemma, in Combi.SSRcomplements.sorted]
+head_filter_gt_tab [lemma, in Combi.Combi.tableau]
+head_leq_last_row [definition, in Combi.Combi.tableau]
+head_revcode [lemma, in Combi.SymGroup.presentSn]
+head_row_skew_yam [lemma, in Combi.LRrule.implem]
+height [definition, in Combi.Combi.Dyckword]
+height_take_leq [lemma, in Combi.Combi.Dyckword]
+height_nseq [lemma, in Combi.Combi.Dyckword]
+height_rev [lemma, in Combi.Combi.Dyckword]
+height_drop [lemma, in Combi.Combi.Dyckword]
+height_simpl [definition, in Combi.Combi.Dyckword]
+height_cat [lemma, in Combi.Combi.Dyckword]
+height_rcons [lemma, in Combi.Combi.Dyckword]
+height_cons [lemma, in Combi.Combi.Dyckword]
+height_nil [lemma, in Combi.Combi.Dyckword]
+Hinvar_all [projection, in Combi.Basic.congr]
+Hinvar_refl [projection, in Combi.Basic.congr]
+Hinvst [lemma, in Combi.LRrule.stdplact]
+Hinvts [lemma, in Combi.LRrule.stdplact]
+Hla [definition, in Combi.MPoly.Schur_altdef]
+Hlamu [definition, in Combi.MPoly.Schur_altdef]
+HLF [abbreviation, in Combi.HookFormula.hook]
+homlang [abbreviation, in Combi.LRrule.freeSchur]
+homlang [definition, in Combi.LRrule.freeSchur]
+homogsym [record, in Combi.MPoly.homogsym]
+homogsym [library]
+HomogSymLModType [section, in Combi.MPoly.homogsym]
+HomogSymLModType.d [variable, in Combi.MPoly.homogsym]
+HomogSymLModType.n [variable, in Combi.MPoly.homogsym]
+HomogSymLModType.R [variable, in Combi.MPoly.homogsym]
+HomogSymProd [section, in Combi.MPoly.homogsym]
+HomogSymProd.c [variable, in Combi.MPoly.homogsym]
+HomogSymProd.d [variable, in Combi.MPoly.homogsym]
+HomogSymProd.n [variable, in Combi.MPoly.homogsym]
+HomogSymProd.R [variable, in Combi.MPoly.homogsym]
+_ *h _ [notation, in Combi.MPoly.homogsym]
+homogsym_homsymdot__canonical__sesquilinear_Dot [definition, in Combi.MPoly.homogsym]
+homogsym_homsymdot__canonical__sesquilinear_Hermitian [definition, in Combi.MPoly.homogsym]
+homogsym_homsymdot__canonical__sesquilinear_Bilinear [definition, in Combi.MPoly.homogsym]
+homogsym_cnvarhomsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_cnvarhomsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_map_homsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_map_homsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_omegahomsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_omegahomsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_in_homsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_in_homsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_homsymprod__canonical__sesquilinear_Bilinear [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__vector_Vector [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__vector_SemiVector [definition, in Combi.MPoly.homogsym]
+homogsym_vecaxiom [lemma, in Combi.MPoly.homogsym]
+homogsym_dhomog_of_homogsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_dhomog_of_homogsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_homsym__canonical__GRing_Linear [definition, in Combi.MPoly.homogsym]
+homogsym_homsym__canonical__Algebra_Additive [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_SubLmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_Lmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_SubLSemiModule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_LSemiModule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubZmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubNmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_Zmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_Nmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddSemigroup [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubAddUMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddUMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubBaseAddUMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseZmodule [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_ChoiceBaseAddUMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseAddUMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_ChoiceBaseAddMagma [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseAddMagma [definition, in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__GRing_SubmodClosed [definition, in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_ZmodClosed [definition, in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_AddClosed [definition, in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_OppClosed [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__choice_SubChoice [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__choice_Choice [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_SubEquality [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_Equality [definition, in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_SubType [definition, in Combi.MPoly.homogsym]
+homog_X_mPo_gen [lemma, in Combi.MPoly.sympoly]
+homog_symmE [lemma, in Combi.MPoly.sympoly]
+homsym [projection, in Combi.MPoly.homogsym]
+homsymdot [definition, in Combi.MPoly.homogsym]
+homsymdotBl [lemma, in Combi.MPoly.homogsym]
+homsymdotBr [lemma, in Combi.MPoly.homogsym]
+homsymdotC [lemma, in Combi.MPoly.homogsym]
+homsymdotDl [lemma, in Combi.MPoly.homogsym]
+homsymdotDr [lemma, in Combi.MPoly.homogsym]
+homsymdotE [lemma, in Combi.MPoly.homogsym]
+homsymdotMnl [lemma, in Combi.MPoly.homogsym]
+homsymdotMnr [lemma, in Combi.MPoly.homogsym]
+homsymdotNl [lemma, in Combi.MPoly.homogsym]
+homsymdotNr [lemma, in Combi.MPoly.homogsym]
+homsymdotpp [lemma, in Combi.MPoly.homogsym]
+homsymdotss [lemma, in Combi.MPoly.Cauchy]
+homsymdotZl [lemma, in Combi.MPoly.homogsym]
+homsymdotZr [lemma, in Combi.MPoly.homogsym]
+homsymdot_omegasf [lemma, in Combi.MPoly.homogsym]
+homsymdot_is_dot [lemma, in Combi.MPoly.homogsym]
+homsymdot_sumr [lemma, in Combi.MPoly.homogsym]
+homsymdot_suml [lemma, in Combi.MPoly.homogsym]
+homsymdot_is_hermitian [lemma, in Combi.MPoly.homogsym]
+homsymdot_is_bilinear [lemma, in Combi.MPoly.homogsym]
+homsymdot0l [lemma, in Combi.MPoly.homogsym]
+homsymdot0r [lemma, in Combi.MPoly.homogsym]
+homsyme [definition, in Combi.MPoly.homogsym]
+homsymE [lemma, in Combi.MPoly.homogsym]
+homsyme_character [lemma, in Combi.SymGroup.Frobenius_char]
+HomSymField [section, in Combi.MPoly.homogsym]
+HomSymField.d [variable, in Combi.MPoly.homogsym]
+HomSymField.Hd [variable, in Combi.MPoly.homogsym]
+HomSymField.n0 [variable, in Combi.MPoly.homogsym]
+HomSymField.R [variable, in Combi.MPoly.homogsym]
+homsymh [definition, in Combi.MPoly.homogsym]
+homsymh_character [lemma, in Combi.SymGroup.Frobenius_char]
+homsymm [definition, in Combi.MPoly.homogsym]
+homsymmE [lemma, in Combi.MPoly.homogsym]
+homsymp [definition, in Combi.MPoly.homogsym]
+homsymprod [definition, in Combi.MPoly.homogsym]
+homsymprodBl [lemma, in Combi.MPoly.homogsym]
+homsymprodBr [lemma, in Combi.MPoly.homogsym]
+homsymprodDl [lemma, in Combi.MPoly.homogsym]
+homsymprodDr [lemma, in Combi.MPoly.homogsym]
+HomSymProdGen [section, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons [section, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.d [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.Hla [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.Hlla [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.la [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.l0 [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Merge [section, in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.d1 [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.d2 [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.la [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.mu [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.n0 [variable, in Combi.MPoly.homogsym]
+HomSymProdGen.R [variable, in Combi.MPoly.homogsym]
+homsymprodMnl [lemma, in Combi.MPoly.homogsym]
+homsymprodMnr [lemma, in Combi.MPoly.homogsym]
+homsymprodNl [lemma, in Combi.MPoly.homogsym]
+homsymprodNr [lemma, in Combi.MPoly.homogsym]
+homsymprodZl [lemma, in Combi.MPoly.homogsym]
+homsymprodZr [lemma, in Combi.MPoly.homogsym]
+homsymprod_h1p [lemma, in Combi.MPoly.homogsym]
+homsymprod_h1e [lemma, in Combi.MPoly.homogsym]
+homsymprod_h1h [lemma, in Combi.MPoly.homogsym]
+homsymprod_hp [lemma, in Combi.MPoly.homogsym]
+homsymprod_he [lemma, in Combi.MPoly.homogsym]
+homsymprod_hh [lemma, in Combi.MPoly.homogsym]
+homsymprod_suml [lemma, in Combi.MPoly.homogsym]
+homsymprod_sumr [lemma, in Combi.MPoly.homogsym]
+homsymprod_is_bilinear [lemma, in Combi.MPoly.homogsym]
+homsymprod_subproof [lemma, in Combi.MPoly.homogsym]
+homsymprod0l [lemma, in Combi.MPoly.homogsym]
+homsymprod0r [lemma, in Combi.MPoly.homogsym]
+homsymp_orthogonal [lemma, in Combi.MPoly.homogsym]
+homsyms [definition, in Combi.MPoly.homogsym]
+homsyms_orthonormal [lemma, in Combi.MPoly.Cauchy]
+homsyms_homsympM [lemma, in Combi.MPoly.MurnaghanNakayama]
+homsym_is_dhomog [lemma, in Combi.MPoly.homogsym]
+homsym_is_linear [lemma, in Combi.MPoly.homogsym]
+homsym_inj [lemma, in Combi.MPoly.homogsym]
+hook [library]
+HookLengthFormula [lemma, in Combi.HookFormula.hook]
+HookLengthFormula_rat [lemma, in Combi.HookFormula.hook]
+hookpart [definition, in Combi.Combi.partition]
+hookpartn [definition, in Combi.Combi.partition]
+hookpartnE [lemma, in Combi.Combi.partition]
+hookpartnP [lemma, in Combi.Combi.partition]
+hookpartnPE [lemma, in Combi.Combi.partition]
+hookpartn_row [lemma, in Combi.Combi.partition]
+hookpartn_col [lemma, in Combi.Combi.partition]
+hookpartn_subproof [lemma, in Combi.Combi.partition]
+hook_length_prod_div [lemma, in Combi.HookFormula.hook]
+hook_length_prod_nat [lemma, in Combi.HookFormula.hook]
+hook_length_prod_non0 [lemma, in Combi.HookFormula.hook]
+hook_length_last_rectangle [lemma, in Combi.HookFormula.hook]
+hook_boxes_empty [lemma, in Combi.HookFormula.hook]
+hook_boxes [definition, in Combi.HookFormula.hook]
+hook_box [definition, in Combi.HookFormula.hook]
+hook_box_indices [definition, in Combi.HookFormula.hook]
+hook_length_pred [lemma, in Combi.HookFormula.hook]
+hook_length_incr_nth [lemma, in Combi.HookFormula.hook]
+hook_length_incr_nth_col [lemma, in Combi.HookFormula.hook]
+hook_length_incr_nth_row [lemma, in Combi.HookFormula.hook]
+hook_length_corner_box [lemma, in Combi.HookFormula.hook]
+hook_length1_corner [lemma, in Combi.HookFormula.hook]
+hook_length_ltr [lemma, in Combi.HookFormula.hook]
+hook_length_ltl [lemma, in Combi.HookFormula.hook]
+hook_length_conj_part [lemma, in Combi.HookFormula.hook]
+hook_length_geq1 [lemma, in Combi.HookFormula.hook]
+hook_length_prod [definition, in Combi.HookFormula.hook]
+hook_length [definition, in Combi.HookFormula.hook]
+Hp [lemma, in Combi.HookFormula.hook]
+Hpart' [lemma, in Combi.HookFormula.hook]
+HS [abbreviation, in Combi.SymGroup.Frobenius_char]
+HS [abbreviation, in Combi.SymGroup.Frobenius_char]
+HSC [abbreviation, in Combi.MPoly.Cauchy]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.homogsym]
+HSF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+HSF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+HSFR [abbreviation, in Combi.MPoly.homogsym]
+HSFS [abbreviation, in Combi.MPoly.homogsym]
+Hszrcons [lemma, in Combi.MPoly.Schur_altdef]
+hyper_stdtabn [definition, in Combi.LRrule.freeSchur]
+hyper_stdtabnP [lemma, in Combi.LRrule.freeSchur]
+hyper_stdtabP [lemma, in Combi.LRrule.freeSchur]
+hyper_stdtab [definition, in Combi.LRrule.freeSchur]
+hyper_stdtabsh [definition, in Combi.Combi.stdtab]
+hyper_stdtabsh_subproof [lemma, in Combi.Combi.stdtab]
+hyper_yameval [definition, in Combi.Combi.Yamanouchi]
+hyper_yam_of_eval [lemma, in Combi.Combi.Yamanouchi]
+hyper_yamP [lemma, in Combi.Combi.Yamanouchi]
+hyper_yam [definition, in Combi.Combi.Yamanouchi]
+hyper_yam_rev [definition, in Combi.Combi.Yamanouchi]
+

I

+Id [definition, in ALEA.Ccpo]
+id [definition, in ALEA.Ccpo]
+Identity [section, in Combi.HookFormula.Frobenius_ident]
+Identity.n [variable, in Combi.HookFormula.Frobenius_ident]
+IdomainSchurSym [section, in Combi.MPoly.Schur_altdef]
+IdomainSchurSym.n0 [variable, in Combi.MPoly.Schur_altdef]
+IdomainSchurSym.R [variable, in Combi.MPoly.Schur_altdef]
+'a_ _ [notation, in Combi.MPoly.Schur_altdef]
+'s_ _ [notation, in Combi.MPoly.Schur_altdef]
+Id_simpl [lemma, in ALEA.Ccpo]
+Id_mon [instance, in ALEA.Ccpo]
+ieqi1F [lemma, in Combi.SymGroup.presentSn]
+ieqi2F [lemma, in Combi.SymGroup.presentSn]
+ifte [definition, in ALEA.Misc]
+if_else_not [lemma, in ALEA.Misc]
+if_then_not [lemma, in ALEA.Misc]
+if_else [lemma, in ALEA.Misc]
+if_then [lemma, in ALEA.Misc]
+if_beq_nat_nat_eq_dec [lemma, in ALEA.Misc]
+Ik [abbreviation, in Combi.LRrule.Greene]
+image_map_finer [lemma, in Combi.Combi.setpartition]
+Imon [definition, in ALEA.Ccpo]
+imon [definition, in ALEA.Ccpo]
+Imonotonic [lemma, in ALEA.Ccpo]
+Imonotonic2 [lemma, in ALEA.Ccpo]
+Imon_simpl [lemma, in ALEA.Ccpo]
+imon_simpl [lemma, in ALEA.Ccpo]
+imon2 [definition, in ALEA.Ccpo]
+Imon2 [definition, in ALEA.Ccpo]
+imon2_simpl [lemma, in ALEA.Ccpo]
+Imon2_simpl [lemma, in ALEA.Ccpo]
+implem [library]
+imsetD [lemma, in Combi.SSRcomplements.tools]
+ImsetInj [section, in Combi.SSRcomplements.tools]
+ImsetInj.f [variable, in Combi.SSRcomplements.tools]
+ImsetInj.f_inj [variable, in Combi.SSRcomplements.tools]
+ImsetInj.T [variable, in Combi.SSRcomplements.tools]
+ImsetInj.T1 [variable, in Combi.SSRcomplements.tools]
+ImsetInj.T2 [variable, in Combi.SSRcomplements.tools]
+imset_trivIset [lemma, in Combi.SSRcomplements.tools]
+imset_inj [lemma, in Combi.SSRcomplements.tools]
+imset_transversal_preim [lemma, in Combi.Combi.Dyckword]
+imset_classCT [lemma, in Combi.SymGroup.cycletype]
+incl [lemma, in Combi.Combi.skewpart]
+incl [projection, in Combi.LRrule.implem]
+included [definition, in Combi.Combi.partition]
+includedP [lemma, in Combi.Combi.partition]
+included_pad0 [lemma, in Combi.Combi.partition]
+included_conj_partE [lemma, in Combi.Combi.partition]
+included_conj_part [lemma, in Combi.Combi.partition]
+included_anti [lemma, in Combi.Combi.partition]
+included_sumnE [lemma, in Combi.Combi.partition]
+included_incr_nth_inner [lemma, in Combi.Combi.partition]
+included_decr_nth [lemma, in Combi.Combi.partition]
+included_incr_nth [lemma, in Combi.Combi.partition]
+included_trans [lemma, in Combi.Combi.partition]
+included_refl [lemma, in Combi.Combi.partition]
+included_behead [lemma, in Combi.Combi.partition]
+included_shape_filter_gt [lemma, in Combi.Combi.skewtab]
+included_shape_filter_gt_tab [lemma, in Combi.LRrule.therule]
+included_add_ribbon [lemma, in Combi.Combi.skewpart]
+IncrMap [section, in Combi.LRrule.plactic]
+IncrMap [section, in Combi.Combi.tableau]
+IncrMap.disp1 [variable, in Combi.LRrule.plactic]
+IncrMap.disp1 [variable, in Combi.Combi.tableau]
+IncrMap.disp2 [variable, in Combi.LRrule.plactic]
+IncrMap.disp2 [variable, in Combi.Combi.tableau]
+IncrMap.F [variable, in Combi.LRrule.plactic]
+IncrMap.F [variable, in Combi.Combi.tableau]
+IncrMap.Hincr [variable, in Combi.LRrule.plactic]
+IncrMap.T1 [variable, in Combi.LRrule.plactic]
+IncrMap.T1 [variable, in Combi.Combi.tableau]
+IncrMap.T2 [variable, in Combi.LRrule.plactic]
+IncrMap.T2 [variable, in Combi.Combi.tableau]
+IncrMap.u [variable, in Combi.LRrule.plactic]
+IncrMap.v [variable, in Combi.LRrule.plactic]
+incr_first_n_nthC [lemma, in Combi.Combi.partition]
+incr_first_n [definition, in Combi.Combi.partition]
+incr_nthK [lemma, in Combi.Combi.partition]
+incr_nth_injl [lemma, in Combi.Combi.stdtab]
+incr_equiv [lemma, in Combi.SSRcomplements.sorted]
+incr_tab [lemma, in Combi.Combi.tableau]
+incr_nth_size [lemma, in Combi.Combi.Yamanouchi]
+index_invstd [lemma, in Combi.LRrule.shuffle]
+index_sfilterleq [lemma, in Combi.LRrule.shuffle]
+index_leq_filter [lemma, in Combi.LRrule.shuffle]
+indporbit [definition, in Combi.SymGroup.cycletype]
+indporbitP [lemma, in Combi.SymGroup.cycletype]
+indporbit_cymap [lemma, in Combi.SymGroup.cycletype]
+indtree [definition, in Combi.Combi.ordtree]
+indtreeforest [definition, in Combi.Combi.ordtree]
+Induction [section, in Combi.SymGroup.towerSn]
+Induction.m [variable, in Combi.SymGroup.towerSn]
+Induction.n [variable, in Combi.SymGroup.towerSn]
+'z_ _ [notation, in Combi.SymGroup.towerSn]
+'1z_[ _ ] [notation, in Combi.SymGroup.towerSn]
+inh [definition, in Combi.Basic.ordtype]
+Inhabited [abbreviation, in Combi.Basic.ordtype]
+Inhabited [module, in Combi.Basic.ordtype]
+InhabitedElpiOperations [module, in Combi.Basic.ordtype]
+Inhabited.axioms_ [record, in Combi.Basic.ordtype]
+Inhabited.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+Inhabited.class [projection, in Combi.Basic.ordtype]
+Inhabited.clone [abbreviation, in Combi.Basic.ordtype]
+Inhabited.copy [abbreviation, in Combi.Basic.ordtype]
+Inhabited.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+Inhabited.Exports [module, in Combi.Basic.ordtype]
+Inhabited.Exports.inhType [abbreviation, in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited__to__choice_Choice [definition, in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+Inhabited.on [abbreviation, in Combi.Basic.ordtype]
+Inhabited.on_ [abbreviation, in Combi.Basic.ordtype]
+Inhabited.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+Inhabited.pack_ [definition, in Combi.Basic.ordtype]
+Inhabited.phant_on_ [definition, in Combi.Basic.ordtype]
+Inhabited.phant_clone [definition, in Combi.Basic.ordtype]
+Inhabited.sort [projection, in Combi.Basic.ordtype]
+Inhabited.type [record, in Combi.Basic.ordtype]
+InhFinite [abbreviation, in Combi.Basic.ordtype]
+InhFinite [module, in Combi.Basic.ordtype]
+InhFiniteElpiOperations [module, in Combi.Basic.ordtype]
+InhFinite.axioms_ [record, in Combi.Basic.ordtype]
+InhFinite.choice_Choice_isCountable_mixin [projection, in Combi.Basic.ordtype]
+InhFinite.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhFinite.class [projection, in Combi.Basic.ordtype]
+InhFinite.clone [abbreviation, in Combi.Basic.ordtype]
+InhFinite.copy [abbreviation, in Combi.Basic.ordtype]
+InhFinite.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhFinite.Exports [module, in Combi.Basic.ordtype]
+InhFinite.Exports.inhFinType [abbreviation, in Combi.Basic.ordtype]
+InhFinite.Exports.join_ordtype_InhFinite_between_fintype_Finite_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.join_ordtype_InhFinite_between_choice_Countable_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__fintype_Finite [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__fintype_Finite_class [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__choice_Countable [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__choice_Countable_class [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhFinite.fintype_isFinite_mixin [projection, in Combi.Basic.ordtype]
+InhFinite.on [abbreviation, in Combi.Basic.ordtype]
+InhFinite.on_ [abbreviation, in Combi.Basic.ordtype]
+InhFinite.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhFinite.pack_ [definition, in Combi.Basic.ordtype]
+InhFinite.phant_on_ [definition, in Combi.Basic.ordtype]
+InhFinite.phant_clone [definition, in Combi.Basic.ordtype]
+InhFinite.sort [projection, in Combi.Basic.ordtype]
+InhFinite.type [record, in Combi.Basic.ordtype]
+InhFinLattice [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice [module, in Combi.Basic.ordtype]
+InhFinLatticeElpiOperations [module, in Combi.Basic.ordtype]
+InhFinLattice.axioms_ [record, in Combi.Basic.ordtype]
+InhFinLattice.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.choice_Choice_isCountable_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.class [projection, in Combi.Basic.ordtype]
+InhFinLattice.clone [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice.copy [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.Exports [module, in Combi.Basic.ordtype]
+InhFinLattice.Exports.inhFinLatticeType [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_Lattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinPOrder_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinPreorder_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_Lattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_fintype_Finite_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_choice_Countable_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinLattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinJoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinMeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhFinPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinPreorder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinPreorder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhFinite_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__fintype_Finite [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__fintype_Finite_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__choice_Countable [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__choice_Countable_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhLattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_Lattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_Lattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_JoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_MeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhFinLattice.fintype_isFinite_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.on [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice.on_ [abbreviation, in Combi.Basic.ordtype]
+InhFinLattice.Order_POrder_isJoinSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.Order_POrder_isMeetSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhFinLattice.pack_ [definition, in Combi.Basic.ordtype]
+InhFinLattice.phant_on_ [definition, in Combi.Basic.ordtype]
+InhFinLattice.phant_clone [definition, in Combi.Basic.ordtype]
+InhFinLattice.sort [projection, in Combi.Basic.ordtype]
+InhFinLattice.type [record, in Combi.Basic.ordtype]
+InhFinOrder [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder [module, in Combi.Basic.ordtype]
+InhFinOrderElpiOperations [module, in Combi.Basic.ordtype]
+InhFinOrder.axioms_ [record, in Combi.Basic.ordtype]
+InhFinOrder.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.choice_Choice_isCountable_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.class [projection, in Combi.Basic.ordtype]
+InhFinOrder.clone [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder.copy [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Exports [module, in Combi.Basic.ordtype]
+InhFinOrder.Exports.inhFinOrderType [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinLattice_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinLattice_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinLattice_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinJoinSemilattice_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinMeetSemilattice_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinPOrder_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinPOrder_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinPOrder_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinPreorder_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinite_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinite_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_fintype_Finite_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_choice_Countable_and_ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinTotal [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinTotal_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinDistrLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinDistrLattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinLattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinLattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinJoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinMeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinPreorder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinPreorder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinite_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__fintype_Finite [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__fintype_Finite_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__choice_Countable [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__choice_Countable_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhOrder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Total [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Total_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_DistrLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_DistrLattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhLattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Lattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Lattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_JoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_MeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhFinOrder.fintype_isFinite_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.on [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder.on_ [abbreviation, in Combi.Basic.ordtype]
+InhFinOrder.Order_DistrLattice_isTotal_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Order_Lattice_isDistributive_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Order_POrder_isJoinSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Order_POrder_isMeetSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhFinOrder.pack_ [definition, in Combi.Basic.ordtype]
+InhFinOrder.phant_on_ [definition, in Combi.Basic.ordtype]
+InhFinOrder.phant_clone [definition, in Combi.Basic.ordtype]
+InhFinOrder.sort [projection, in Combi.Basic.ordtype]
+InhFinOrder.type [record, in Combi.Basic.ordtype]
+InhFinPOrder [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder [module, in Combi.Basic.ordtype]
+InhFinPOrderElpiOperations [module, in Combi.Basic.ordtype]
+InhFinPOrder.axioms_ [record, in Combi.Basic.ordtype]
+InhFinPOrder.choice_Choice_isCountable_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.class [projection, in Combi.Basic.ordtype]
+InhFinPOrder.clone [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder.copy [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.Exports [module, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.inhFinPOrderType [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_Order_POrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_Order_Preorder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_fintype_Finite_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_choice_Countable_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_FinPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_FinPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_FinPreorder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_FinPreorder_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_InhFinite_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__fintype_Finite [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__fintype_Finite_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__choice_Countable [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__choice_Countable_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhFinPOrder.fintype_isFinite_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.on [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder.on_ [abbreviation, in Combi.Basic.ordtype]
+InhFinPOrder.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhFinPOrder.pack_ [definition, in Combi.Basic.ordtype]
+InhFinPOrder.phant_on_ [definition, in Combi.Basic.ordtype]
+InhFinPOrder.phant_clone [definition, in Combi.Basic.ordtype]
+InhFinPOrder.sort [projection, in Combi.Basic.ordtype]
+InhFinPOrder.type [record, in Combi.Basic.ordtype]
+InhLattice [abbreviation, in Combi.Basic.ordtype]
+InhLattice [module, in Combi.Basic.ordtype]
+InhLatticeElpiOperations [module, in Combi.Basic.ordtype]
+InhLattice.axioms_ [record, in Combi.Basic.ordtype]
+InhLattice.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.class [projection, in Combi.Basic.ordtype]
+InhLattice.clone [abbreviation, in Combi.Basic.ordtype]
+InhLattice.copy [abbreviation, in Combi.Basic.ordtype]
+InhLattice.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.Exports [module, in Combi.Basic.ordtype]
+InhLattice.Exports.inhLatticeType [abbreviation, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_Lattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_Lattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_Lattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_Lattice_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_JoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_MeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhLattice.on [abbreviation, in Combi.Basic.ordtype]
+InhLattice.on_ [abbreviation, in Combi.Basic.ordtype]
+InhLattice.Order_POrder_isJoinSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.Order_POrder_isMeetSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhLattice.pack_ [definition, in Combi.Basic.ordtype]
+InhLattice.phant_on_ [definition, in Combi.Basic.ordtype]
+InhLattice.phant_clone [definition, in Combi.Basic.ordtype]
+InhLattice.sort [projection, in Combi.Basic.ordtype]
+InhLattice.type [record, in Combi.Basic.ordtype]
+InHomSym [section, in Combi.MPoly.homogsym]
+InHomSym.d [variable, in Combi.MPoly.homogsym]
+InHomSym.n0 [variable, in Combi.MPoly.homogsym]
+InHomSym.R [variable, in Combi.MPoly.homogsym]
+'pi_ _ [notation, in Combi.MPoly.homogsym]
+InhOrder [abbreviation, in Combi.Basic.ordtype]
+InhOrder [module, in Combi.Basic.ordtype]
+InhOrderElpiOperations [module, in Combi.Basic.ordtype]
+InhOrder.axioms_ [record, in Combi.Basic.ordtype]
+InhOrder.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.class [projection, in Combi.Basic.ordtype]
+InhOrder.clone [abbreviation, in Combi.Basic.ordtype]
+InhOrder.copy [abbreviation, in Combi.Basic.ordtype]
+InhOrder.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Exports [module, in Combi.Basic.ordtype]
+InhOrder.Exports.inhOrderType [abbreviation, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_InhLattice_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_InhPOrder_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_Inhabited_and_Order_Total [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Total [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Total_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_DistrLattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_DistrLattice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_InhLattice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Lattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Lattice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_JoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_MeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhOrder.on [abbreviation, in Combi.Basic.ordtype]
+InhOrder.on_ [abbreviation, in Combi.Basic.ordtype]
+InhOrder.Order_DistrLattice_isTotal_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Order_Lattice_isDistributive_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Order_POrder_isJoinSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Order_POrder_isMeetSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhOrder.pack_ [definition, in Combi.Basic.ordtype]
+InhOrder.phant_on_ [definition, in Combi.Basic.ordtype]
+InhOrder.phant_clone [definition, in Combi.Basic.ordtype]
+InhOrder.sort [projection, in Combi.Basic.ordtype]
+InhOrder.type [record, in Combi.Basic.ordtype]
+InhPOrder [abbreviation, in Combi.Basic.ordtype]
+InhPOrder [module, in Combi.Basic.ordtype]
+InhPOrderElpiOperations [module, in Combi.Basic.ordtype]
+InhPOrder.axioms_ [record, in Combi.Basic.ordtype]
+InhPOrder.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhPOrder.class [projection, in Combi.Basic.ordtype]
+InhPOrder.clone [abbreviation, in Combi.Basic.ordtype]
+InhPOrder.copy [abbreviation, in Combi.Basic.ordtype]
+InhPOrder.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhPOrder.Exports [module, in Combi.Basic.ordtype]
+InhPOrder.Exports.inhPOrderType [abbreviation, in Combi.Basic.ordtype]
+InhPOrder.Exports.join_ordtype_InhPOrder_between_ordtype_Inhabited_and_Order_POrder [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.join_ordtype_InhPOrder_between_ordtype_Inhabited_and_Order_Preorder [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhPOrder.on [abbreviation, in Combi.Basic.ordtype]
+InhPOrder.on_ [abbreviation, in Combi.Basic.ordtype]
+InhPOrder.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhPOrder.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhPOrder.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhPOrder.pack_ [definition, in Combi.Basic.ordtype]
+InhPOrder.phant_on_ [definition, in Combi.Basic.ordtype]
+InhPOrder.phant_clone [definition, in Combi.Basic.ordtype]
+InhPOrder.sort [projection, in Combi.Basic.ordtype]
+InhPOrder.type [record, in Combi.Basic.ordtype]
+InhTBLattice [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice [module, in Combi.Basic.ordtype]
+InhTBLatticeElpiOperations [module, in Combi.Basic.ordtype]
+InhTBLattice.axioms_ [record, in Combi.Basic.ordtype]
+InhTBLattice.choice_hasChoice_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.class [projection, in Combi.Basic.ordtype]
+InhTBLattice.clone [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice.copy [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice.eqtype_hasDecEq_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Exports [module, in Combi.Basic.ordtype]
+InhTBLattice.Exports.inhTBLatticeType [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBLattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BLattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBJoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BJoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBMeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BMeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBPOrder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BPOrder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBPreorder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BPreorder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_InhLattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_InhPOrder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_Inhabited_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TLattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TLattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_Lattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_Lattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TJoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TJoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_JoinSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_JoinSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TMeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TMeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_MeetSemilattice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_MeetSemilattice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TPOrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TPOrder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_POrder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_POrder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TPreorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TPreorder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_Preorder [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_Preorder_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__choice_Choice [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__choice_Choice_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__eqtype_Equality [definition, in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__eqtype_Equality_class [definition, in Combi.Basic.ordtype]
+InhTBLattice.on [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice.on_ [abbreviation, in Combi.Basic.ordtype]
+InhTBLattice.Order_POrder_isJoinSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Order_POrder_isMeetSemilattice_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Order_Preorder_isDuallyPOrder_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Order_hasBottom_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Order_hasTop_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.Order_isDuallyPreorder_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.ordtype_isInhabited_mixin [projection, in Combi.Basic.ordtype]
+InhTBLattice.pack_ [definition, in Combi.Basic.ordtype]
+InhTBLattice.phant_on_ [definition, in Combi.Basic.ordtype]
+InhTBLattice.phant_clone [definition, in Combi.Basic.ordtype]
+InhTBLattice.sort [projection, in Combi.Basic.ordtype]
+InhTBLattice.type [record, in Combi.Basic.ordtype]
+inh_chooseE [lemma, in Combi.Basic.ordtype]
+inh_xchooseE [lemma, in Combi.Basic.ordtype]
+inh_ex [definition, in Combi.Basic.ordtype]
+inj_strict_mon [lemma, in ALEA.Ccpo]
+inner_part [projection, in Combi.LRrule.implem]
+inordi [lemma, in Combi.SymGroup.presentSn]
+inordi_neq_i1 [lemma, in Combi.SymGroup.presentSn]
+inordi1 [lemma, in Combi.SymGroup.presentSn]
+inord_predS [lemma, in Combi.SymGroup.presentSn]
+inord1i [lemma, in Combi.SymGroup.presentSn]
+inporbits [definition, in Combi.SymGroup.permcent]
+inporbits_im [lemma, in Combi.SymGroup.permcent]
+inporbits1 [lemma, in Combi.SymGroup.permcent]
+inputSpec [record, in Combi.LRrule.implem]
+ins [abbreviation, in Combi.LRrule.Schensted]
+ins [definition, in Combi.LRrule.Schensted]
+inscode [definition, in Combi.SymGroup.presentSn]
+inscodeP [lemma, in Combi.SymGroup.presentSn]
+insE [lemma, in Combi.LRrule.Schensted]
+insmin [definition, in Combi.LRrule.Schensted]
+inspos [abbreviation, in Combi.LRrule.Schensted]
+inspos [definition, in Combi.LRrule.Schensted]
+insposE [lemma, in Combi.LRrule.Schensted]
+inspos_leq_exP [lemma, in Combi.LRrule.Schensted]
+inspos_lt_size_ins [lemma, in Combi.LRrule.Schensted]
+inspos_leq_size [lemma, in Combi.LRrule.Schensted]
+inspos_rcons [lemma, in Combi.LRrule.plactic]
+inspred [definition, in Combi.LRrule.Schensted]
+inspredN_lt_inspos [lemma, in Combi.LRrule.Schensted]
+inspred_mininspred [lemma, in Combi.LRrule.Schensted]
+inspred_inspos [lemma, in Combi.LRrule.Schensted]
+inspred_any_bump [lemma, in Combi.LRrule.Schensted]
+insRow [abbreviation, in Combi.LRrule.Schensted]
+insrow [definition, in Combi.LRrule.Schensted]
+insrowE [lemma, in Combi.LRrule.Schensted]
+insrow_head_lt [lemma, in Combi.LRrule.Schensted]
+instab [definition, in Combi.LRrule.Schensted]
+instabnrow [definition, in Combi.LRrule.Schensted]
+instabnrowE [lemma, in Combi.LRrule.Schensted]
+instabnrowinvK [lemma, in Combi.LRrule.Schensted]
+instab_non_nil [lemma, in Combi.LRrule.Schensted]
+insub_wordcdK [lemma, in Combi.SymGroup.presentSn]
+ins_bumprowE [lemma, in Combi.LRrule.Schensted]
+ins_non_nil [lemma, in Combi.LRrule.Schensted]
+ins_leq [lemma, in Combi.LRrule.Schensted]
+ins_head_lt [lemma, in Combi.LRrule.Schensted]
+intcomp [record, in Combi.Combi.composition]
+intcompn [record, in Combi.Combi.composition]
+intcompnP [lemma, in Combi.Combi.composition]
+intcompn_behead [definition, in Combi.MPoly.sympoly]
+intcompn_behead_sub_proof [lemma, in Combi.MPoly.sympoly]
+intcompn_cons [definition, in Combi.MPoly.sympoly]
+intcompn_cons_sub_proof [lemma, in Combi.MPoly.sympoly]
+intcompn_castE [lemma, in Combi.Combi.composition]
+intcompn_cast [definition, in Combi.Combi.composition]
+intcompn_sumn [lemma, in Combi.Combi.composition]
+intcompn0 [lemma, in Combi.Combi.composition]
+intcompn1 [lemma, in Combi.Combi.composition]
+intcompn2 [lemma, in Combi.Combi.composition]
+intcompP [lemma, in Combi.Combi.composition]
+intcomp_of_intcompn [definition, in Combi.Combi.composition]
+intpart [record, in Combi.Combi.partition]
+intpartn [record, in Combi.Combi.partition]
+IntPartN [section, in Combi.Combi.skewpart]
+IntpartnCons [section, in Combi.Combi.partition]
+IntpartnCons.d [variable, in Combi.Combi.partition]
+IntpartnCons.Hla [variable, in Combi.Combi.partition]
+IntpartnCons.Hlla [variable, in Combi.Combi.partition]
+IntpartnCons.la [variable, in Combi.Combi.partition]
+IntpartnCons.l0 [variable, in Combi.Combi.partition]
+IntPartNDom [module, in Combi.Combi.partition]
+IntPartNDom.botEintpartndom [lemma, in Combi.Combi.partition]
+IntPartNDom.double_minn [lemma, in Combi.Combi.partition]
+IntPartNDom.Exports [module, in Combi.Combi.partition]
+IntPartNDom.Exports.botEintpartndom [definition, in Combi.Combi.partition]
+IntPartNDom.Exports.join_intpartnE [definition, in Combi.Combi.partition]
+IntPartNDom.Exports.leEpartdom [definition, in Combi.Combi.partition]
+IntPartNDom.Exports.partdom_conj_intpartn [definition, in Combi.Combi.partition]
+IntPartNDom.Exports.sumn_take_pardom_meet [definition, in Combi.Combi.partition]
+IntPartNDom.Exports.topEintpartndom [definition, in Combi.Combi.partition]
+'PDom_ _ [notation, in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.partition]
+IntPartNDom.from_parttupleK [lemma, in Combi.Combi.partition]
+IntPartNDom.from_parttupleP [lemma, in Combi.Combi.partition]
+IntPartNDom.from_parttuple [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_142 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_140 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_139 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_138 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_135 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_134 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_133 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_130 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_129 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_125 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_124 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_123 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_122 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_121 [definition, in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_116 [definition, in Combi.Combi.partition]
+IntPartNDom.intpartndom [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom [section, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhTBLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhLattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_Lattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_MeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_JoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_POrder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_Preorder [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinite [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_Inhabited [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__fintype_Finite [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__choice_Countable [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__choice_Choice [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+IntPartNDom.IntPartNDom.d [variable, in Combi.Combi.partition]
+_ ^# [notation, in Combi.Combi.partition]
+'PDom [notation, in Combi.Combi.partition]
+IntPartNDom.IntPartNTopBottom [section, in Combi.Combi.partition]
+IntPartNDom.IntPartNTopBottom.d [variable, in Combi.Combi.partition]
+'PDom [notation, in Combi.Combi.partition]
+IntPartNDom.is_parttupleP [lemma, in Combi.Combi.partition]
+IntPartNDom.is_parttuple [definition, in Combi.Combi.partition]
+IntPartNDom.join_intpartnE [lemma, in Combi.Combi.partition]
+IntPartNDom.join_intpartnP [lemma, in Combi.Combi.partition]
+IntPartNDom.join_intpartn [definition, in Combi.Combi.partition]
+IntPartNDom.leEpartdom [lemma, in Combi.Combi.partition]
+IntPartNDom.le_meet_intpartn [lemma, in Combi.Combi.partition]
+IntPartNDom.meet_intpartnP [lemma, in Combi.Combi.partition]
+IntPartNDom.meet_intpartnC [lemma, in Combi.Combi.partition]
+IntPartNDom.meet_intpartn [definition, in Combi.Combi.partition]
+IntPartNDom.nth_parttuple_minn [lemma, in Combi.Combi.partition]
+IntPartNDom.nth_parttuple [lemma, in Combi.Combi.partition]
+IntPartNDom.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNDom.Order_Le_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.Combi.partition]
+IntPartNDom.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.partition]
+IntPartNDom.ordtype_Inhabited__to__ordtype_isInhabited [definition, in Combi.Combi.partition]
+IntPartNDom.partdom_colpartn [lemma, in Combi.Combi.partition]
+IntPartNDom.partdom_rowpartn [lemma, in Combi.Combi.partition]
+IntPartNDom.partdom_conj_intpartn [lemma, in Combi.Combi.partition]
+IntPartNDom.partdom_display [lemma, in Combi.Combi.partition]
+IntPartNDom.partdom_antisym [lemma, in Combi.Combi.partition]
+IntPartNDom.parttuple [definition, in Combi.Combi.partition]
+IntPartNDom.parttupleK [lemma, in Combi.Combi.partition]
+IntPartNDom.parttupleP [lemma, in Combi.Combi.partition]
+IntPartNDom.parttuplePK [lemma, in Combi.Combi.partition]
+IntPartNDom.parttuple_minnP [lemma, in Combi.Combi.partition]
+IntPartNDom.parttuple_minnC [lemma, in Combi.Combi.partition]
+IntPartNDom.parttuple_minn [definition, in Combi.Combi.partition]
+IntPartNDom.part_fromtuple [definition, in Combi.Combi.partition]
+IntPartNDom.sumn_take_pardom_meet [lemma, in Combi.Combi.partition]
+IntPartNDom.sumn_take_part_fromtuple [lemma, in Combi.Combi.partition]
+IntPartNDom.sum_diff_tuple [lemma, in Combi.Combi.partition]
+IntPartNDom.sum_diff [lemma, in Combi.Combi.partition]
+IntPartNDom.take_intpartn_over [lemma, in Combi.Combi.partition]
+IntPartNDom.topEintpartndom [lemma, in Combi.Combi.partition]
+IntPartNLexi [module, in Combi.Combi.partition]
+IntPartNLexi.botEintpartnlexi [lemma, in Combi.Combi.partition]
+IntPartNLexi.colpartn_bot [lemma, in Combi.Combi.partition]
+IntPartNLexi.eqtype_SubType__to__eqtype_isSub [definition, in Combi.Combi.partition]
+IntPartNLexi.Exports [module, in Combi.Combi.partition]
+IntPartNLexi.Exports.botEintpartnlexi [definition, in Combi.Combi.partition]
+IntPartNLexi.Exports.leEintpartnlexi [definition, in Combi.Combi.partition]
+IntPartNLexi.Exports.ltEintpartnlexi [definition, in Combi.Combi.partition]
+IntPartNLexi.Exports.topEintpartnlexi [definition, in Combi.Combi.partition]
+'PLexi_ _ [notation, in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_69 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_67 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_65 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_63 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_62 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_61 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_60 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_59 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_58 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_57 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_48 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_47 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_46 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_45 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_44 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_39 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_38 [definition, in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_36 [definition, in Combi.Combi.partition]
+IntPartNLexi.intpartnlexi [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi [section, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinite [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhTBLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_Inhabited [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBDistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBDistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TDistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BDistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Total [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinDistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_DistrLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinLattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Lattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_JoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_MeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_POrder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Preorder [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__fintype_SubFinite [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__fintype_Finite [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_SubCountable [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_Countable [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_SubChoice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_Choice [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_SubEquality [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_SubType [definition, in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi.d [variable, in Combi.Combi.partition]
+'PLexi [notation, in Combi.Combi.partition]
+IntPartNLexi.leEintpartnlexi [lemma, in Combi.Combi.partition]
+IntPartNLexi.ltEintpartnlexi [lemma, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_DistrLattice_isTotal [definition, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_Lattice_isDistributive [definition, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_isDuallyPreorder [definition, in Combi.Combi.partition]
+IntPartNLexi.ordtype_isInhabitedType__to__ordtype_isInhabited [definition, in Combi.Combi.partition]
+IntPartNLexi.rowpartn_top [lemma, in Combi.Combi.partition]
+IntPartNLexi.topEintpartnlexi [lemma, in Combi.Combi.partition]
+intpartnP [lemma, in Combi.Combi.partition]
+intpartnsk_nb [definition, in Combi.Combi.partition]
+intpartns_nb [definition, in Combi.Combi.partition]
+intpartn_of_mon [definition, in Combi.MPoly.homogsym]
+intpartn_cons [lemma, in Combi.Combi.partition]
+intpartn_nb [definition, in Combi.Combi.partition]
+intpartn_nth0 [lemma, in Combi.Combi.partition]
+intpartn_count_leq2E [lemma, in Combi.Combi.partition]
+intpartn_leq [lemma, in Combi.Combi.partition]
+intpartn_leq_head [lemma, in Combi.Combi.partition]
+intpartn_sorted [lemma, in Combi.Combi.partition]
+IntPartN.la [variable, in Combi.Combi.skewpart]
+IntPartN.m [variable, in Combi.Combi.skewpart]
+IntPartN.nbox [variable, in Combi.Combi.skewpart]
+IntPartN.val_id [variable, in Combi.Combi.skewpart]
+'Pr [notation, in Combi.Combi.skewpart]
+intpartn0 [lemma, in Combi.Combi.partition]
+intpartn1 [lemma, in Combi.Combi.partition]
+intpartn2 [lemma, in Combi.Combi.partition]
+intpartn3 [lemma, in Combi.Combi.partition]
+intpartP [lemma, in Combi.Combi.partition]
+intpart_of_monP [lemma, in Combi.MPoly.homogsym]
+intpart_of_mon [definition, in Combi.MPoly.homogsym]
+intpart_rem_corner_ind [lemma, in Combi.Combi.partition]
+intpart_of_intpartn [definition, in Combi.Combi.partition]
+intpart_sumn_take_inj [lemma, in Combi.Combi.partition]
+intpart_eqP [lemma, in Combi.Combi.partition]
+intpart_sorted [lemma, in Combi.Combi.partition]
+invar [projection, in Combi.Basic.congr]
+InvarContHom [section, in Combi.Basic.congr]
+InvarContHom.Alph [variable, in Combi.Basic.congr]
+InvarContHom.Hcongr [variable, in Combi.Basic.congr]
+InvarContHom.Hhom [variable, in Combi.Basic.congr]
+InvarContHom.rule [variable, in Combi.Basic.congr]
+InvarContHom.szinvar [variable, in Combi.Basic.congr]
+InvarContMultHom [section, in Combi.Basic.congr]
+InvarContMultHom.Alph [variable, in Combi.Basic.congr]
+InvarContMultHom.Hcongr [variable, in Combi.Basic.congr]
+InvarContMultHom.Hmulthom [variable, in Combi.Basic.congr]
+InvarContMultHom.Hsym [variable, in Combi.Basic.congr]
+InvarContMultHom.rule [variable, in Combi.Basic.congr]
+invariant_context [record, in Combi.Basic.congr]
+invar_rewrite_path [lemma, in Combi.Basic.congr]
+invar_step [lemma, in Combi.Basic.congr]
+invar_undupE [lemma, in Combi.Basic.congr]
+invbump [definition, in Combi.LRrule.Schensted]
+invbumped [definition, in Combi.LRrule.Schensted]
+invbumprow [definition, in Combi.LRrule.Schensted]
+invbumprowK [lemma, in Combi.LRrule.Schensted]
+invbump_dom [lemma, in Combi.LRrule.Schensted]
+invbump_geq_head [lemma, in Combi.LRrule.Schensted]
+invcont_size [definition, in Combi.Basic.congr]
+invcont_perm [definition, in Combi.Basic.congr]
+invcont_congr [definition, in Combi.Basic.congr]
+invins [definition, in Combi.LRrule.Schensted]
+invinstabnrow [definition, in Combi.LRrule.Schensted]
+invinstabnrowK [lemma, in Combi.LRrule.Schensted]
+invseq [definition, in Combi.Combi.std]
+InvSeq [section, in Combi.Combi.std]
+invseqE [lemma, in Combi.Combi.std]
+invseqK [lemma, in Combi.LRrule.stdplact]
+invseqRSE [lemma, in Combi.LRrule.stdplact]
+invseqRSPQE [lemma, in Combi.LRrule.stdplact]
+invseq_invstd [lemma, in Combi.Combi.std]
+invseq_nthE [lemma, in Combi.Combi.std]
+invseq_is_std [lemma, in Combi.Combi.std]
+invseq_sym [lemma, in Combi.Combi.std]
+invset [definition, in Combi.SymGroup.presentSn]
+InvSet [section, in Combi.SymGroup.presentSn]
+invsetK [lemma, in Combi.SymGroup.presentSn]
+invsetP [lemma, in Combi.SymGroup.presentSn]
+invset_eltrR [lemma, in Combi.SymGroup.presentSn]
+invset_eltrL [lemma, in Combi.SymGroup.presentSn]
+invset_inj [lemma, in Combi.SymGroup.presentSn]
+invset_maxpermMl [lemma, in Combi.SymGroup.presentSn]
+invset_maxpermMr [lemma, in Combi.SymGroup.presentSn]
+invset_maxperm [lemma, in Combi.SymGroup.presentSn]
+invset_permV [lemma, in Combi.SymGroup.presentSn]
+invset_Delta [lemma, in Combi.SymGroup.presentSn]
+InvSet.n [variable, in Combi.SymGroup.presentSn]
+invstd [definition, in Combi.Combi.std]
+invstdK [lemma, in Combi.Combi.std]
+invstdRSE [lemma, in Combi.LRrule.stdplact]
+invstd_inj [lemma, in Combi.Combi.std]
+invstd_is_std [lemma, in Combi.Combi.std]
+invstd_cat_in_shsh [lemma, in Combi.LRrule.shuffle]
+invstd_catleq [lemma, in Combi.LRrule.shuffle]
+invstd_catgtn [lemma, in Combi.LRrule.shuffle]
+inv_weight_pos [lemma, in ALEA.Qmeasure]
+in_seq_sum [lemma, in ALEA.Qmeasure]
+in_std_ltn_size [lemma, in Combi.Combi.std]
+in_homsym_comp_symbe [lemma, in Combi.MPoly.homogsym]
+in_homsymE [lemma, in Combi.MPoly.homogsym]
+in_homsym_is_linear [lemma, in Combi.MPoly.homogsym]
+in_homsym [definition, in Combi.MPoly.homogsym]
+in_conj_part [lemma, in Combi.Combi.partition]
+in_conj_part_impl [lemma, in Combi.Combi.partition]
+in_part_is_part [lemma, in Combi.Combi.partition]
+in_part_le [lemma, in Combi.Combi.partition]
+in_part_non0 [lemma, in Combi.Combi.partition]
+in_shape_size [lemma, in Combi.Combi.partition]
+in_skew_nil [lemma, in Combi.Combi.partition]
+in_shape_nil [lemma, in Combi.Combi.partition]
+in_skew_in [lemma, in Combi.Combi.partition]
+in_skew_out [lemma, in Combi.Combi.partition]
+in_skewE [lemma, in Combi.Combi.partition]
+in_skew [definition, in Combi.Combi.partition]
+in_shape [definition, in Combi.Combi.partition]
+in_vect_n_k [lemma, in Combi.Combi.vectNK]
+in_hook_boxesP [lemma, in Combi.HookFormula.hook]
+in_hook_shape [lemma, in Combi.HookFormula.hook]
+in_hook [definition, in Combi.HookFormula.hook]
+in_porbit_setP [lemma, in Combi.SymGroup.cycles]
+in_psupport [lemma, in Combi.SymGroup.cycles]
+in_shape_tab [lemma, in Combi.Combi.tableau]
+in_shape_tab_size [lemma, in Combi.Combi.tableau]
+in_maxL [lemma, in Combi.Basic.ordtype]
+Iord [definition, in ALEA.Ccpo]
+Iord_app [lemma, in ALEA.Ccpo]
+iotagtnk [lemma, in Combi.LRrule.Greene]
+iota_geq [lemma, in Combi.SSRcomplements.tools]
+iota_ltn [lemma, in Combi.SSRcomplements.tools]
+iota_hookE [lemma, in Combi.HookFormula.hook]
+iota_cut_i [lemma, in Combi.SymGroup.presentSn]
+irev_w [lemma, in Combi.LRrule.Greene_inv]
+irrSG [definition, in Combi.SymGroup.Frobenius_char]
+irrSGP [lemma, in Combi.SymGroup.Frobenius_char]
+irrSG_char_int [lemma, in Combi.SymGroup.Frobenius_char]
+irrSG_irr [lemma, in Combi.SymGroup.Frobenius_char]
+irrSG_orthonormal [lemma, in Combi.SymGroup.Frobenius_char]
+irr_S2 [lemma, in Combi.SymGroup.reprSn]
+isantisymP [lemma, in Combi.MPoly.antisym]
+isantisym_alt [lemma, in Combi.MPoly.antisym]
+isantisym_mlead_rho [lemma, in Combi.MPoly.antisym]
+isantisym_mlead_iota [lemma, in Combi.MPoly.antisym]
+isantisym_msupp_uniq [lemma, in Combi.MPoly.antisym]
+isantisym_eltrP [lemma, in Combi.MPoly.antisym]
+isantisym_msupp [lemma, in Combi.MPoly.antisym]
+isantisym_tpermP [lemma, in Combi.MPoly.antisym]
+isglb [definition, in ALEA.Ccpo]
+isglb_decr_lift [lemma, in ALEA.Ccpo]
+isglb_decr_ext [lemma, in ALEA.Ccpo]
+isglb_eq_compat_right [lemma, in ALEA.Ccpo]
+isglb_eq_compat_left [lemma, in ALEA.Ccpo]
+isglb_eq_compat [lemma, in ALEA.Ccpo]
+isglb_le [lemma, in ALEA.Ccpo]
+ishift [definition, in ALEA.Ccpo]
+ishift_continuous [instance, in ALEA.Ccpo]
+ishift_le_compat [lemma, in ALEA.Ccpo]
+ishift_simpl [lemma, in ALEA.Ccpo]
+ishift_mon [instance, in ALEA.Ccpo]
+ishomogsym1 [definition, in Combi.MPoly.homogsym]
+isInhabited [abbreviation, in Combi.Basic.ordtype]
+isInhabited [module, in Combi.Basic.ordtype]
+isInhabitedType [abbreviation, in Combi.Basic.ordtype]
+isInhabitedType [module, in Combi.Basic.ordtype]
+isInhabitedType.axioms [abbreviation, in Combi.Basic.ordtype]
+isInhabitedType.axioms_ [record, in Combi.Basic.ordtype]
+isInhabitedType.Build [abbreviation, in Combi.Basic.ordtype]
+isInhabitedType.Exports [module, in Combi.Basic.ordtype]
+isInhabitedType.isInhabitedType.isInhabitedType [section, in Combi.Basic.ordtype]
+isInhabitedType.isInhabitedType.T [variable, in Combi.Basic.ordtype]
+isInhabitedType.phant_axioms [definition, in Combi.Basic.ordtype]
+isInhabitedType.phant_Build [definition, in Combi.Basic.ordtype]
+isInhabitedType.x [projection, in Combi.Basic.ordtype]
+isInhabited.axioms [abbreviation, in Combi.Basic.ordtype]
+isInhabited.axioms_ [record, in Combi.Basic.ordtype]
+isInhabited.Build [abbreviation, in Combi.Basic.ordtype]
+isInhabited.Exports [module, in Combi.Basic.ordtype]
+isInhabited.identity_builder [definition, in Combi.Basic.ordtype]
+isInhabited.inh_ex [projection, in Combi.Basic.ordtype]
+isInhabited.isInhabited.isInhabited [section, in Combi.Basic.ordtype]
+isInhabited.isInhabited.T [variable, in Combi.Basic.ordtype]
+isInhabited.phant_axioms [definition, in Combi.Basic.ordtype]
+isInhabited.phant_Build [definition, in Combi.Basic.ordtype]
+IsInvset [constructor, in Combi.SymGroup.presentSn]
+islub [record, in ALEA.Ccpo]
+islub_lub [lemma, in ALEA.Ccpo]
+islub_mlub [lemma, in ALEA.Ccpo]
+islub_fun_intro [lemma, in ALEA.Ccpo]
+islub_unique [lemma, in ALEA.Ccpo]
+islub_unique_eq [lemma, in ALEA.Ccpo]
+islub_decr [lemma, in ALEA.Ccpo]
+islub_exch [lemma, in ALEA.Ccpo]
+islub_incr_lift [lemma, in ALEA.Ccpo]
+islub_incr_ext [lemma, in ALEA.Ccpo]
+islub_eq_compat_right [lemma, in ALEA.Ccpo]
+islub_eq_compat_left [lemma, in ALEA.Ccpo]
+islub_eq_compat [lemma, in ALEA.Ccpo]
+islub_le [projection, in ALEA.Ccpo]
+IsoBottom [abbreviation, in Combi.Combi.composition]
+IsoBottom [module, in Combi.Combi.composition]
+IsoBottom.axioms [abbreviation, in Combi.Combi.composition]
+IsoBottom.axioms_ [record, in Combi.Combi.composition]
+IsoBottom.Build [abbreviation, in Combi.Combi.composition]
+IsoBottom.disp' [projection, in Combi.Combi.composition]
+IsoBottom.Exports [module, in Combi.Combi.composition]
+IsoBottom.f [projection, in Combi.Combi.composition]
+IsoBottom.f_mono [projection, in Combi.Combi.composition]
+IsoBottom.f_can [projection, in Combi.Combi.composition]
+IsoBottom.f' [projection, in Combi.Combi.composition]
+IsoBottom.f'_can [projection, in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__Order_POrder [definition, in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__Order_Preorder [definition, in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__choice_Choice [definition, in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+IsoBottom.IsoBottom.disp [variable, in Combi.Combi.composition]
+IsoBottom.IsoBottom.IsoBottom [section, in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_Order_Preorder_isDuallyPOrder [variable, in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_Order_isDuallyPreorder [variable, in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_eqtype_hasDecEq [variable, in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_choice_hasChoice [variable, in Combi.Combi.composition]
+IsoBottom.IsoBottom.T [variable, in Combi.Combi.composition]
+IsoBottom.phant_axioms [definition, in Combi.Combi.composition]
+IsoBottom.phant_Build [definition, in Combi.Combi.composition]
+IsoBottom.T' [projection, in Combi.Combi.composition]
+isom_tinj [lemma, in Combi.SymGroup.towerSn]
+IsoTop [abbreviation, in Combi.Combi.composition]
+IsoTop [module, in Combi.Combi.composition]
+IsoTop.axioms [abbreviation, in Combi.Combi.composition]
+IsoTop.axioms_ [record, in Combi.Combi.composition]
+IsoTop.Build [abbreviation, in Combi.Combi.composition]
+IsoTop.disp' [projection, in Combi.Combi.composition]
+IsoTop.Exports [module, in Combi.Combi.composition]
+IsoTop.f [projection, in Combi.Combi.composition]
+IsoTop.f_mono [projection, in Combi.Combi.composition]
+IsoTop.f_can [projection, in Combi.Combi.composition]
+IsoTop.f' [projection, in Combi.Combi.composition]
+IsoTop.f'_can [projection, in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__Order_POrder [definition, in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__Order_Preorder [definition, in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__choice_Choice [definition, in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+IsoTop.IsoTop.disp [variable, in Combi.Combi.composition]
+IsoTop.IsoTop.IsoTop [section, in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_Order_Preorder_isDuallyPOrder [variable, in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_Order_isDuallyPreorder [variable, in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_eqtype_hasDecEq [variable, in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_choice_hasChoice [variable, in Combi.Combi.composition]
+IsoTop.IsoTop.T [variable, in Combi.Combi.composition]
+IsoTop.phant_axioms [definition, in Combi.Combi.composition]
+IsoTop.phant_Build [definition, in Combi.Combi.composition]
+IsoTop.T' [projection, in Combi.Combi.composition]
+isperm_of_porbit [lemma, in Combi.SymGroup.cycletype]
+issym_eltrP [lemma, in Combi.MPoly.antisym]
+issym_tpermP [lemma, in Combi.MPoly.antisym]
+is_std_of_n [definition, in Combi.Combi.std]
+is_std_wordpermP [lemma, in Combi.Combi.std]
+is_stdP [lemma, in Combi.Combi.std]
+is_std [definition, in Combi.Combi.std]
+is_homsym_submod_closed [lemma, in Combi.MPoly.homogsym]
+is_homsym [abbreviation, in Combi.MPoly.homogsym]
+is_homsym [definition, in Combi.MPoly.homogsym]
+is_part_decr_nth_part [lemma, in Combi.Combi.partition]
+is_part_of_nsk [definition, in Combi.Combi.partition]
+is_part_of_ns [definition, in Combi.Combi.partition]
+is_part_of_n [definition, in Combi.Combi.partition]
+is_add_corner_conj_part [lemma, in Combi.Combi.partition]
+is_part_conj [lemma, in Combi.Combi.partition]
+is_part_incr_first_n [lemma, in Combi.Combi.partition]
+is_part_nseq1 [lemma, in Combi.Combi.partition]
+is_part_decr_nth [lemma, in Combi.Combi.partition]
+is_rem_cornerP [lemma, in Combi.Combi.partition]
+is_part_incr_nth [lemma, in Combi.Combi.partition]
+is_part_incr_nth_size [lemma, in Combi.Combi.partition]
+is_add_corner [definition, in Combi.Combi.partition]
+is_rem_corner [definition, in Combi.Combi.partition]
+is_part_rem_trail0 [lemma, in Combi.Combi.partition]
+is_part_catr [lemma, in Combi.Combi.partition]
+is_part_catl [lemma, in Combi.Combi.partition]
+is_part_rconsK [lemma, in Combi.Combi.partition]
+is_part_subseq [lemma, in Combi.Combi.partition]
+is_part_behead [lemma, in Combi.Combi.partition]
+is_part_consK [lemma, in Combi.Combi.partition]
+is_part_sortedE [lemma, in Combi.Combi.partition]
+is_part_ijP [lemma, in Combi.Combi.partition]
+is_partP [lemma, in Combi.Combi.partition]
+is_part [definition, in Combi.Combi.partition]
+is_Tamari [definition, in Combi.Combi.bintree]
+is_col_dual [lemma, in Combi.LRrule.Greene_inv]
+is_row_dual [lemma, in Combi.LRrule.Greene_inv]
+is_dominant_nth_partm [lemma, in Combi.MPoly.antisym]
+is_dominant_partm [lemma, in Combi.MPoly.antisym]
+is_stdtab_of_n_LRtriple [lemma, in Combi.LRrule.shuffle]
+is_tableau_std [lemma, in Combi.Combi.skewtab]
+is_tableau_reshape_std [lemma, in Combi.Combi.skewtab]
+is_skew_tableau_reshape_std [lemma, in Combi.Combi.skewtab]
+is_skew_tableau_skew_reshape_pad0 [lemma, in Combi.Combi.skewtab]
+is_skew_tableau_filter_le [lemma, in Combi.Combi.skewtab]
+is_skew_tableau_filter_le_tmp [lemma, in Combi.Combi.skewtab]
+is_skew_tableau_pad0 [lemma, in Combi.Combi.skewtab]
+is_skew_tableau0 [lemma, in Combi.Combi.skewtab]
+is_skew_tableauP [lemma, in Combi.Combi.skewtab]
+is_skew_tableau [definition, in Combi.Combi.skewtab]
+is_part_skew_yam [lemma, in Combi.Combi.skewtab]
+is_skew_yamE [lemma, in Combi.Combi.skewtab]
+is_skew_yam [definition, in Combi.Combi.skewtab]
+is_stdtab_RStabmap2 [lemma, in Combi.LRrule.Schensted]
+is_RStabpair [definition, in Combi.LRrule.Schensted]
+is_tableau_instabnrowinv1 [lemma, in Combi.LRrule.Schensted]
+is_RSpair [definition, in Combi.LRrule.Schensted]
+is_yam_RSmap2 [lemma, in Combi.LRrule.Schensted]
+is_tableau_RSmap1 [lemma, in Combi.LRrule.Schensted]
+is_rem_corner_instabnrow [lemma, in Combi.LRrule.Schensted]
+is_row_invins [lemma, in Combi.LRrule.Schensted]
+is_tableau_RS [lemma, in Combi.LRrule.Schensted]
+is_tableau_instab [lemma, in Combi.LRrule.Schensted]
+is_row_Sch [lemma, in Combi.LRrule.Schensted]
+is_row_ins [lemma, in Combi.LRrule.Schensted]
+is_trace_corner_nil [lemma, in Combi.HookFormula.hook]
+is_trace_in_shape [lemma, in Combi.HookFormula.hook]
+is_trace_in_in_shape [lemma, in Combi.HookFormula.hook]
+is_trace_lastl [lemma, in Combi.HookFormula.hook]
+is_trace_lastr [lemma, in Combi.HookFormula.hook]
+is_trace_tlr [lemma, in Combi.HookFormula.hook]
+is_trace_tll [lemma, in Combi.HookFormula.hook]
+is_trace [definition, in Combi.HookFormula.hook]
+is_dyckword [projection, in Combi.Combi.Dyckword]
+is_yam_plactic [lemma, in Combi.LRrule.Yam_plact]
+is_part_incr_nth1E [lemma, in Combi.LRrule.Yam_plact]
+is_part_incr_nthE [lemma, in Combi.LRrule.Yam_plact]
+is_stdtab_conj [lemma, in Combi.Combi.stdtab]
+is_part_shape_deg [lemma, in Combi.Combi.stdtab]
+is_stdtab_of_n [definition, in Combi.Combi.stdtab]
+is_stdtab_of_shape [definition, in Combi.Combi.stdtab]
+is_row_stdE [lemma, in Combi.Combi.stdtab]
+is_stdtab_remn [lemma, in Combi.Combi.stdtab]
+is_tab_append_nth_size [lemma, in Combi.Combi.stdtab]
+is_tab_append_nth_size_alternative_proof [lemma, in Combi.Combi.stdtab]
+is_stdtab [definition, in Combi.Combi.stdtab]
+is_tab_of_shape [definition, in Combi.Combi.tableau]
+is_tableau_filter_gt [lemma, in Combi.Combi.tableau]
+is_tableau_getP [lemma, in Combi.Combi.tableau]
+is_tableau_sorted_dominate [lemma, in Combi.Combi.tableau]
+is_part_sht [lemma, in Combi.Combi.tableau]
+is_tableau_catr [lemma, in Combi.Combi.tableau]
+is_tableau_catl [lemma, in Combi.Combi.tableau]
+is_tableau_rconsK [lemma, in Combi.Combi.tableau]
+is_tableauP [lemma, in Combi.Combi.tableau]
+is_tableau [definition, in Combi.Combi.tableau]
+is_row_set_nth [lemma, in Combi.Combi.tableau]
+is_row [abbreviation, in Combi.Combi.tableau]
+is_row_cat2 [definition, in Combi.Combi.tableau]
+is_row_drop [definition, in Combi.Combi.tableau]
+is_row_take [definition, in Combi.Combi.tableau]
+is_row_last [definition, in Combi.Combi.tableau]
+is_row_rconsK [definition, in Combi.Combi.tableau]
+is_row_rcons [definition, in Combi.Combi.tableau]
+is_row_consK [definition, in Combi.Combi.tableau]
+is_row_cons [definition, in Combi.Combi.tableau]
+is_rowP [definition, in Combi.Combi.tableau]
+is_row1P [definition, in Combi.Combi.tableau]
+is_row [abbreviation, in Combi.Combi.tableau]
+is_row_yamrow [lemma, in Combi.LRrule.therule]
+is_skew_reshape_tableauP [lemma, in Combi.LRrule.therule]
+is_skew_reshape_tableau [definition, in Combi.LRrule.therule]
+is_skew_tableau_map_shiftn [lemma, in Combi.LRrule.therule]
+is_yam_of_size [definition, in Combi.Combi.Yamanouchi]
+is_yam_cat_any [lemma, in Combi.Combi.Yamanouchi]
+is_add_corner_yam [lemma, in Combi.Combi.Yamanouchi]
+is_rem_corner_yam [lemma, in Combi.Combi.Yamanouchi]
+is_yam_decr [lemma, in Combi.Combi.Yamanouchi]
+is_yam_catr [lemma, in Combi.Combi.Yamanouchi]
+is_yam_tl [lemma, in Combi.Combi.Yamanouchi]
+is_part_eval_yam [lemma, in Combi.Combi.Yamanouchi]
+is_yam_ijP [lemma, in Combi.Combi.Yamanouchi]
+is_yamP [lemma, in Combi.Combi.Yamanouchi]
+is_yam_of_eval [definition, in Combi.Combi.Yamanouchi]
+is_yam [definition, in Combi.Combi.Yamanouchi]
+is_finer_card [lemma, in Combi.Combi.setpartition]
+is_invset_tclosureU [lemma, in Combi.SymGroup.weak_order]
+is_part_of_add_ribbon [lemma, in Combi.Combi.skewpart]
+is_part_add_ribbon [lemma, in Combi.Combi.skewpart]
+is_part_add_ribbon_on [lemma, in Combi.Combi.skewpart]
+is_code_straighten [lemma, in Combi.SymGroup.presentSn]
+is_invset_Delta [lemma, in Combi.SymGroup.presentSn]
+is_invset [inductive, in Combi.SymGroup.presentSn]
+is_code_of_size [definition, in Combi.SymGroup.presentSn]
+is_code_rconsK [lemma, in Combi.SymGroup.presentSn]
+is_code_rcons [lemma, in Combi.SymGroup.presentSn]
+is_codeP [lemma, in Combi.SymGroup.presentSn]
+is_part_pad0 [lemma, in Combi.LRrule.implem]
+is_comp_of_n [definition, in Combi.Combi.composition]
+is_comp_cat [lemma, in Combi.Combi.composition]
+is_comp_rcons [lemma, in Combi.Combi.composition]
+is_comp_cons [lemma, in Combi.Combi.composition]
+is_comp1 [lemma, in Combi.Combi.composition]
+is_compP [lemma, in Combi.Combi.composition]
+is_comp [definition, in Combi.Combi.composition]
+Iter [definition, in ALEA.Ccpo]
+iter [definition, in ALEA.Ccpo]
+iterO_simpl [lemma, in ALEA.Ccpo]
+iterS_simpl [lemma, in ALEA.Ccpo]
+IterS_simpl [lemma, in ALEA.Ccpo]
+iter_continuous_eq [lemma, in ALEA.Ccpo]
+iter_continuous [lemma, in ALEA.Ccpo]
+iter_monotonic [instance, in ALEA.Ccpo]
+iter_mon [instance, in ALEA.Ccpo]
+iter_incr [lemma, in ALEA.Ccpo]
+iter_ [definition, in ALEA.Ccpo]
+i1eqiF [lemma, in Combi.SymGroup.presentSn]
+i2eqiF [lemma, in Combi.SymGroup.presentSn]
+

J

+joingU1 [lemma, in Combi.SymGroup.presentSn]
+join_tab_skew [lemma, in Combi.Combi.skewtab]
+join_tab_filter [lemma, in Combi.Combi.skewtab]
+join_tab [definition, in Combi.Combi.skewtab]
+join_Dyck_inj [lemma, in Combi.Combi.Dyckword]
+join_Dyck [definition, in Combi.Combi.Dyckword]
+join_stdtab_in_shuffle [lemma, in Combi.LRrule.therule]
+join_stdtab [lemma, in Combi.LRrule.therule]
+

K

+Kostka [definition, in Combi.MPoly.Schur_altdef]
+Kostka [section, in Combi.MPoly.Schur_altdef]
+KostkaEq [section, in Combi.MPoly.Schur_altdef]
+KostkaEq.d [variable, in Combi.MPoly.Schur_altdef]
+KostkaEq.la [variable, in Combi.MPoly.Schur_altdef]
+KostkaInv [definition, in Combi.MPoly.Schur_altdef]
+KostkaInv_unitrig [lemma, in Combi.MPoly.Schur_altdef]
+KostkaMon [definition, in Combi.MPoly.Schur_altdef]
+KostkaMon_partdom [lemma, in Combi.MPoly.Schur_altdef]
+KostkaMon_sumeval [lemma, in Combi.MPoly.Schur_altdef]
+KostkaRec [section, in Combi.MPoly.Schur_altdef]
+KostkaStd [lemma, in Combi.MPoly.Schur_altdef]
+KostkaTab [definition, in Combi.MPoly.Schur_altdef]
+Kostka_unitrig [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_recE [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_rec_size0 [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_expl3 [definition, in Combi.MPoly.Schur_altdef]
+Kostka_expl2 [definition, in Combi.MPoly.Schur_altdef]
+Kostka_expl1 [definition, in Combi.MPoly.Schur_altdef]
+Kostka_rec [definition, in Combi.MPoly.Schur_altdef]
+Kostka_ind [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_diag [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_partdom [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_size0 [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_sumnE [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_any [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_mnmwiden [lemma, in Combi.MPoly.Schur_altdef]
+Kostka_Coeff [lemma, in Combi.MPoly.Schur_altdef]
+Kostka.d [variable, in Combi.MPoly.Schur_altdef]
+'K ( _ , _ ) (nat_scope) [notation, in Combi.MPoly.Schur_altdef]
+'K ( _ , _ ) (ring_scope) [notation, in Combi.MPoly.Schur_altdef]
+Kostka0 [lemma, in Combi.MPoly.Schur_altdef]
+ksupp [definition, in Combi.LRrule.Greene]
+ksuppCol_inj_plact2i [lemma, in Combi.LRrule.Greene_inv]
+ksuppCol_inj_plact1 [lemma, in Combi.LRrule.Greene_inv]
+KsuppInj [section, in Combi.LRrule.stdplact]
+KsuppInj.Hinv [variable, in Combi.LRrule.stdplact]
+KsuppInj.s [variable, in Combi.LRrule.stdplact]
+KsuppInj.t [variable, in Combi.LRrule.stdplact]
+ksuppRow_inj_plact2 [lemma, in Combi.LRrule.Greene_inv]
+ksuppRow_inj_plact1i [lemma, in Combi.LRrule.Greene_inv]
+ksupp_inj_invseq [lemma, in Combi.LRrule.stdplact]
+ksupp_inj_stdI [lemma, in Combi.LRrule.stdplact]
+ksupp_inj_std [lemma, in Combi.LRrule.stdplact]
+ksupp_gt_tabcolsk [lemma, in Combi.LRrule.Greene]
+ksupp_leqX_tabrowsk [lemma, in Combi.LRrule.Greene]
+ksupp_inj_rev [lemma, in Combi.LRrule.Greene]
+ksupp_inj [definition, in Combi.LRrule.Greene]
+ksupp_cast [lemma, in Combi.LRrule.Greene]
+ksupp0 [lemma, in Combi.LRrule.Greene]
+

L

+langQ [definition, in Combi.LRrule.shuffle]
+langQE [lemma, in Combi.LRrule.shuffle]
+last_behead_rcons [lemma, in Combi.SSRcomplements.tools]
+last_incr_nth_non0 [lemma, in Combi.Combi.partition]
+last_rot_pfminh [lemma, in Combi.Combi.Dyckword]
+last_big_append_nth [lemma, in Combi.Combi.stdtab]
+last_bigP [lemma, in Combi.Combi.stdtab]
+last_big [definition, in Combi.Combi.stdtab]
+last_ins_lt [lemma, in Combi.LRrule.plactic]
+last_yam [lemma, in Combi.Combi.Yamanouchi]
+law1 [lemma, in ALEA.Qmeasure]
+law2 [lemma, in ALEA.Qmeasure]
+law3 [lemma, in ALEA.Qmeasure]
+lcast_com [lemma, in Combi.LRrule.Greene]
+LeafP [lemma, in Combi.Combi.bintree]
+leftcomb [definition, in Combi.Combi.bintree]
+leftcombsz [definition, in Combi.Combi.bintree]
+leftcomb_rotations [lemma, in Combi.Combi.bintree]
+left_branchK [lemma, in Combi.Combi.bintree]
+left_branch [definition, in Combi.Combi.bintree]
+leg_length_corner_box [lemma, in Combi.HookFormula.hook]
+leg_length_lel [lemma, in Combi.HookFormula.hook]
+leg_length_ltr [lemma, in Combi.HookFormula.hook]
+leg_length [definition, in Combi.HookFormula.hook]
+leL [abbreviation, in Combi.LRrule.plactic]
+leL_geLdualE [lemma, in Combi.LRrule.plactic]
+Lemma3 [lemma, in Combi.HookFormula.hook]
+Length [section, in Combi.SymGroup.presentSn]
+length [definition, in Combi.SymGroup.presentSn]
+lengthKL [lemma, in Combi.SymGroup.presentSn]
+lengthKR [lemma, in Combi.SymGroup.presentSn]
+lengthM [lemma, in Combi.SymGroup.presentSn]
+lengthME [lemma, in Combi.SymGroup.presentSn]
+lengthV [lemma, in Combi.SymGroup.presentSn]
+length_eq1 [lemma, in Combi.SymGroup.presentSn]
+length_eq0 [lemma, in Combi.SymGroup.presentSn]
+length_permcd [lemma, in Combi.SymGroup.presentSn]
+length_prods [lemma, in Combi.SymGroup.presentSn]
+length_eltr [lemma, in Combi.SymGroup.presentSn]
+length_descR [lemma, in Combi.SymGroup.presentSn]
+length_sub1R [lemma, in Combi.SymGroup.presentSn]
+length_add1R [lemma, in Combi.SymGroup.presentSn]
+length_descL [lemma, in Combi.SymGroup.presentSn]
+length_sub1L [lemma, in Combi.SymGroup.presentSn]
+length_add1L [lemma, in Combi.SymGroup.presentSn]
+length_maxpermMl [lemma, in Combi.SymGroup.presentSn]
+length_maxpermMr [lemma, in Combi.SymGroup.presentSn]
+length_maxpermE [lemma, in Combi.SymGroup.presentSn]
+length_maxperm [lemma, in Combi.SymGroup.presentSn]
+length_max [lemma, in Combi.SymGroup.presentSn]
+Length.n0 [variable, in Combi.SymGroup.presentSn]
+Length.PartCode [section, in Combi.SymGroup.presentSn]
+Length.PartCode.is_partcode [variable, in Combi.SymGroup.presentSn]
+Length.PartCode.word_of_partcocode [variable, in Combi.SymGroup.presentSn]
+'s_[ _ ] [notation, in Combi.SymGroup.presentSn]
+'s_ _ [notation, in Combi.SymGroup.presentSn]
+length1 [lemma, in Combi.SymGroup.presentSn]
+LEPermTheory [section, in Combi.SymGroup.weak_order]
+LEPermTheory.n0 [variable, in Combi.SymGroup.weak_order]
+leperm_invset [lemma, in Combi.SymGroup.weak_order]
+leperm_succ [lemma, in Combi.SymGroup.weak_order]
+leperm_factorP [lemma, in Combi.SymGroup.weak_order]
+leperm_maxperm [lemma, in Combi.SymGroup.weak_order]
+leperm_maxpermMl [lemma, in Combi.SymGroup.weak_order]
+leperm1p [lemma, in Combi.SymGroup.weak_order]
+LeqGeqOrder [module, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.anti_geq [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_trans [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_total [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_refl [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.gtn_irr [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.gtn_trans [definition, in Combi.SSRcomplements.sorted]
+LeqGeqOrder.ltn_irr [definition, in Combi.SSRcomplements.sorted]
+leq_sumn_in [lemma, in Combi.SSRcomplements.tools]
+leq_addE [lemma, in Combi.SSRcomplements.tools]
+leq_head_sumn [lemma, in Combi.Combi.partition]
+leq_Greene [lemma, in Combi.LRrule.Greene]
+less [lemma, in Combi.Combi.skewpart]
+lessz [lemma, in Combi.Combi.skewpart]
+let_indep_distr [lemma, in ALEA.Qmeasure]
+let_indep [lemma, in ALEA.Qmeasure]
+le_intpartndomlexi [lemma, in Combi.Combi.partition]
+le_mlub [lemma, in ALEA.Ccpo]
+le_lub [projection, in ALEA.Ccpo]
+le_isglb [lemma, in ALEA.Ccpo]
+le_islub [projection, in ALEA.Ccpo]
+le_Ole [lemma, in ALEA.Ccpo]
+LinRepr [section, in Combi.SymGroup.reprSn]
+LinRepr.G [variable, in Combi.SymGroup.reprSn]
+LinRepr.gT [variable, in Combi.SymGroup.reprSn]
+linvseq [definition, in Combi.Combi.std]
+linvseqK [lemma, in Combi.LRrule.stdplact]
+linvseqP [lemma, in Combi.Combi.std]
+linvseq_sizeP [lemma, in Combi.Combi.std]
+linvseq_subset_iota [lemma, in Combi.Combi.std]
+linvseq_ltn_szt [lemma, in Combi.Combi.std]
+lin_char_Sn [lemma, in Combi.SymGroup.reprSn]
+lin_char_reprP [lemma, in Combi.SymGroup.reprSn]
+LR [section, in Combi.LRrule.therule]
+LR [section, in Combi.LRrule.implem]
+LRcoeff [definition, in Combi.LRrule.implem]
+LRcoeffE [lemma, in Combi.LRrule.implem]
+LRcoeffP [lemma, in Combi.LRrule.implem]
+LRcoeff_computeP [lemma, in Combi.LRrule.therule]
+LRrule_langQ_alternate [lemma, in Combi.LRrule.shuffle]
+LRrule_langQ [lemma, in Combi.LRrule.shuffle]
+LRrule_Pieri.R [variable, in Combi.MPoly.sympoly]
+LRrule_Pieri.n0 [variable, in Combi.MPoly.sympoly]
+LRrule_Pieri [section, in Combi.MPoly.sympoly]
+lrshift_recF [lemma, in Combi.LRrule.Greene]
+LRsupport [definition, in Combi.LRrule.freeSchur]
+LRsupport_conj [lemma, in Combi.LRrule.freeSchur]
+LRtab_coeff_conj [lemma, in Combi.LRrule.freeSchur]
+LRtab_coeff_shapeE [lemma, in Combi.LRrule.freeSchur]
+LRtab_coeffP [lemma, in Combi.LRrule.freeSchur]
+LRtab_coeff [definition, in Combi.LRrule.freeSchur]
+LRtab_set [definition, in Combi.LRrule.freeSchur]
+LRtab_set_included [lemma, in Combi.LRrule.therule]
+LRtab_coeffP [lemma, in Combi.LRrule.implem]
+LRTriple [constructor, in Combi.LRrule.shuffle]
+LRtriple [inductive, in Combi.LRrule.shuffle]
+LRTriple [section, in Combi.LRrule.shuffle]
+LRtripleP [lemma, in Combi.LRrule.shuffle]
+LRtriple_conj [lemma, in Combi.LRrule.shuffle]
+LRtriple_cat_equiv [lemma, in Combi.LRrule.shuffle]
+LRtriple_cat_langQ [lemma, in Combi.LRrule.shuffle]
+LRtriple_fastE [lemma, in Combi.LRrule.shuffle]
+LRtriple_sind [definition, in Combi.LRrule.shuffle]
+LRtriple_ind [definition, in Combi.LRrule.shuffle]
+LRTriple.Alph [variable, in Combi.LRrule.shuffle]
+LRTriple.disp [variable, in Combi.LRrule.shuffle]
+LRTriple.word [variable, in Combi.LRrule.shuffle]
+LRyamtabP [lemma, in Combi.LRrule.implem]
+LRyamtab_spec_recip [lemma, in Combi.LRrule.implem]
+LRyamtab_all [lemma, in Combi.LRrule.implem]
+LRyamtab_eval [lemma, in Combi.LRrule.implem]
+LRyamtab_skew_tableau [lemma, in Combi.LRrule.implem]
+LRyamtab_shape [lemma, in Combi.LRrule.implem]
+LRyamtab_included [lemma, in Combi.LRrule.implem]
+LRyamtab_yam [lemma, in Combi.LRrule.implem]
+LRyamtab_list [definition, in Combi.LRrule.implem]
+LRyamtab_list_countE [lemma, in Combi.LRrule.implem]
+LRyamtab_list_skew_tableau0 [lemma, in Combi.LRrule.implem]
+LRyamtab_list_shape0 [lemma, in Combi.LRrule.implem]
+LRyamtab_list_size [lemma, in Combi.LRrule.implem]
+LRyamtab_list_pad0 [lemma, in Combi.LRrule.implem]
+LRyamtab_list_included [lemma, in Combi.LRrule.implem]
+LRyamtab_list_recP [lemma, in Combi.LRrule.implem]
+LRyamtab_count_rec [definition, in Combi.LRrule.implem]
+LRyamtab_list_rec [definition, in Combi.LRrule.implem]
+LRyam_coeff_colpartn [lemma, in Combi.LRrule.therule]
+LRyam_coeff_rowpart [lemma, in Combi.LRrule.therule]
+LRyam_coeffP [lemma, in Combi.LRrule.therule]
+LRyam_compute [definition, in Combi.LRrule.therule]
+LRyam_enum [definition, in Combi.LRrule.therule]
+LRyam_coeffE [lemma, in Combi.LRrule.therule]
+LRyam_coeff [definition, in Combi.LRrule.therule]
+LRyam_set [definition, in Combi.LRrule.therule]
+LRyam_spec_recip [lemma, in Combi.LRrule.implem]
+LRyam_list [definition, in Combi.LRrule.implem]
+LR_rule_irrSG [lemma, in Combi.SymGroup.Frobenius_char]
+LR_rule_tab [lemma, in Combi.LRrule.freeSchur]
+LR.d1 [variable, in Combi.LRrule.therule]
+LR.d1 [variable, in Combi.LRrule.implem]
+LR.d2 [variable, in Combi.LRrule.therule]
+LR.d2 [variable, in Combi.LRrule.implem]
+LR.n0 [variable, in Combi.LRrule.implem]
+LR.Pieri [section, in Combi.LRrule.therule]
+LR.Pieri.n0 [variable, in Combi.LRrule.therule]
+LR.Pieri.R [variable, in Combi.LRrule.therule]
+LR.P1 [variable, in Combi.LRrule.implem]
+LR.P2 [variable, in Combi.LRrule.implem]
+LR.R [variable, in Combi.LRrule.implem]
+LR.TheRule [section, in Combi.LRrule.therule]
+LR.TheRule.n0 [variable, in Combi.LRrule.therule]
+LR.TheRule.OneCoeff [section, in Combi.LRrule.therule]
+LR.TheRule.OneCoeff.Hincl [variable, in Combi.LRrule.therule]
+LR.TheRule.OneCoeff.P [variable, in Combi.LRrule.therule]
+LR.TheRule.P1 [variable, in Combi.LRrule.therule]
+LR.TheRule.P2 [variable, in Combi.LRrule.therule]
+LR.TheRule.R [variable, in Combi.LRrule.therule]
+lsh [abbreviation, in Combi.LRrule.Greene]
+lshift_in_rshift_recF [lemma, in Combi.LRrule.Greene]
+lshift_recP [lemma, in Combi.LRrule.Greene]
+lsh_rec [definition, in Combi.LRrule.Greene]
+lsplit_rec_tab [lemma, in Combi.LRrule.Greene]
+lsplit_rec [definition, in Combi.LRrule.Greene]
+ltcovers [lemma, in Combi.Basic.ordtype]
+ltL [abbreviation, in Combi.LRrule.plactic]
+ltL_gtLdualE [lemma, in Combi.LRrule.plactic]
+ltnPred [lemma, in Combi.HookFormula.hook]
+ltn_braidC [lemma, in Combi.SymGroup.presentSn]
+ltperm_invset [lemma, in Combi.SymGroup.weak_order]
+ltperm_length [lemma, in Combi.SymGroup.weak_order]
+lt_intpartndomlexi [lemma, in Combi.Combi.partition]
+lt_bumped [lemma, in Combi.LRrule.Schensted]
+lt_inspos_nth [lemma, in Combi.LRrule.Schensted]
+Lub [definition, in ALEA.Ccpo]
+lub [projection, in ALEA.Ccpo]
+lub_le_fixp [lemma, in ALEA.Ccpo]
+lub_continuous [instance, in ALEA.Ccpo]
+lub_cont2_app2_eq [lemma, in ALEA.Ccpo]
+lub_app2_eq [lemma, in ALEA.Ccpo]
+lub_comp_eq [lemma, in ALEA.Ccpo]
+lub_app2_le [lemma, in ALEA.Ccpo]
+lub_comp_le [lemma, in ALEA.Ccpo]
+lub_fcpo_mon [lemma, in ALEA.Ccpo]
+lub_mon_fcpo [lemma, in ALEA.Ccpo]
+lub_ishift [lemma, in ALEA.Ccpo]
+lub_shift_mon [instance, in ALEA.Ccpo]
+lub_fun [definition, in ALEA.Ccpo]
+lub_Olt [lemma, in ALEA.Ccpo]
+lub_seq_eq [lemma, in ALEA.Ccpo]
+lub_eq_lift [lemma, in ALEA.Ccpo]
+lub_le_lift [lemma, in ALEA.Ccpo]
+lub_lift_left [lemma, in ALEA.Ccpo]
+lub_lift_right [lemma, in ALEA.Ccpo]
+lub_cte [lemma, in ALEA.Ccpo]
+lub_mon [instance, in ALEA.Ccpo]
+lub_le [projection, in ALEA.Ccpo]
+

M

+m [abbreviation, in Combi.MPoly.Cauchy]
+M [definition, in ALEA.Qmeasure]
+m [abbreviation, in Combi.MPoly.homogsym]
+m [abbreviation, in Combi.MPoly.sympoly]
+map_homsymbs [lemma, in Combi.MPoly.homogsym]
+map_homsymbp [lemma, in Combi.MPoly.homogsym]
+map_homsymbh [lemma, in Combi.MPoly.homogsym]
+map_homsymbe [lemma, in Combi.MPoly.homogsym]
+map_homsymbm [lemma, in Combi.MPoly.homogsym]
+map_homsyms [lemma, in Combi.MPoly.homogsym]
+map_homsymp [lemma, in Combi.MPoly.homogsym]
+map_homsymh [lemma, in Combi.MPoly.homogsym]
+map_homsyme [lemma, in Combi.MPoly.homogsym]
+map_homsymm [lemma, in Combi.MPoly.homogsym]
+map_homsym_is_scalable [lemma, in Combi.MPoly.homogsym]
+map_homsym_is_zmod_morphism [lemma, in Combi.MPoly.homogsym]
+map_homsym [definition, in Combi.MPoly.homogsym]
+map_sympoly_d_homog [lemma, in Combi.MPoly.homogsym]
+map_filter_comp [lemma, in Combi.SSRcomplements.tools]
+map_mpolyX [lemma, in Combi.MPoly.antisym]
+map_finer_pblock [lemma, in Combi.Combi.setpartition]
+map_finer_in [lemma, in Combi.Combi.setpartition]
+map_finer_subset [lemma, in Combi.Combi.setpartition]
+map_finer [definition, in Combi.Combi.setpartition]
+map_syms [lemma, in Combi.MPoly.sympoly]
+map_symp_prod [lemma, in Combi.MPoly.sympoly]
+map_symp [lemma, in Combi.MPoly.sympoly]
+map_symh_prod [lemma, in Combi.MPoly.sympoly]
+map_symh [lemma, in Combi.MPoly.sympoly]
+map_syme_prod [lemma, in Combi.MPoly.sympoly]
+map_syme [lemma, in Combi.MPoly.sympoly]
+map_symm [lemma, in Combi.MPoly.sympoly]
+map_sympoly_is_scalable [lemma, in Combi.MPoly.sympoly]
+map_sympoly_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+map_sympoly_is_zmod_morphism [lemma, in Combi.MPoly.sympoly]
+map_sympoly [definition, in Combi.MPoly.sympoly]
+map_mpoly_issym [lemma, in Combi.MPoly.sympoly]
+mask_injP [lemma, in Combi.Combi.subseq]
+mask1E [lemma, in Combi.Combi.subseq]
+Mat [definition, in Combi.Basic.unitriginv]
+maxL [definition, in Combi.Basic.ordtype]
+maxLb [lemma, in Combi.Basic.ordtype]
+maxLP [lemma, in Combi.Basic.ordtype]
+maxLPt [lemma, in Combi.Basic.ordtype]
+maxL_iota_n [lemma, in Combi.Basic.ordtype]
+maxL_iota [lemma, in Combi.Basic.ordtype]
+maxL_LbR [lemma, in Combi.Basic.ordtype]
+maxL_perm [lemma, in Combi.Basic.ordtype]
+maxL_cat [lemma, in Combi.Basic.ordtype]
+maxperm [definition, in Combi.SymGroup.presentSn]
+MaxPerm [section, in Combi.SymGroup.presentSn]
+maxpermK [lemma, in Combi.SymGroup.presentSn]
+maxpermV [lemma, in Combi.SymGroup.presentSn]
+MaxPerm.n [variable, in Combi.SymGroup.presentSn]
+MaxSeq [section, in Combi.Basic.ordtype]
+MaxSeq.disp [variable, in Combi.Basic.ordtype]
+MaxSeq.T [variable, in Combi.Basic.ordtype]
+maxXL [lemma, in Combi.Basic.ordtype]
+mcoeffXU [lemma, in Combi.MPoly.sympoly]
+mcoeff_symbs [lemma, in Combi.MPoly.homogsym]
+mcoeff_alt [lemma, in Combi.MPoly.antisym]
+mcoeff_arbound [lemma, in Combi.MPoly.antisym]
+mcoeff_alt_SchurE [lemma, in Combi.MPoly.Schur_altdef]
+mcoeff_symm [lemma, in Combi.MPoly.sympoly]
+mcoeff_symm_pol [lemma, in Combi.MPoly.sympoly]
+mcoeff_symh [lemma, in Combi.MPoly.sympoly]
+mcoeff_symh_pol [lemma, in Combi.MPoly.sympoly]
+mcoeff_symh_pol_bound [lemma, in Combi.MPoly.sympoly]
+mdeg_tnth_monsY [lemma, in Combi.MPoly.Cauchy]
+mdeg_monX [lemma, in Combi.MPoly.Cauchy]
+mdeg_rho [lemma, in Combi.MPoly.antisym]
+mdeg_mpart [lemma, in Combi.MPoly.antisym]
+MeasureProp [section, in ALEA.Qmeasure]
+MeasureProp.A [variable, in ALEA.Qmeasure]
+MeasureProp.m [variable, in ALEA.Qmeasure]
+mem_std [lemma, in Combi.Combi.std]
+mem_drop_enumI [lemma, in Combi.SSRcomplements.tools]
+mem_take_enumI [lemma, in Combi.SSRcomplements.tools]
+mem_takeP [lemma, in Combi.SSRcomplements.tools]
+mem_enum_box_in [lemma, in Combi.Combi.partition]
+mem_enum_box_skew [lemma, in Combi.Combi.partition]
+mem_enum_bintreesz [lemma, in Combi.Combi.bintree]
+mem_cast [lemma, in Combi.SSRcomplements.ordcast]
+mem_shsh [lemma, in Combi.LRrule.shuffle]
+mem_sfilterleqK [lemma, in Combi.LRrule.shuffle]
+mem_shuffle [lemma, in Combi.LRrule.shuffle]
+mem_shuffle_pred [lemma, in Combi.LRrule.shuffle]
+mem_shuffle_predU [lemma, in Combi.LRrule.shuffle]
+mem_shape_vect_n_k [lemma, in Combi.Combi.vectNK]
+mem_enum_seqE [lemma, in Combi.LRrule.Greene]
+mem_RSclass [lemma, in Combi.LRrule.Schensted]
+mem_prefixesP [lemma, in Combi.Combi.Dyckword]
+mem_preim_partition [lemma, in Combi.Combi.Dyckword]
+mem_to_word [lemma, in Combi.Combi.tableau]
+mem_enum_ordtreesz [lemma, in Combi.Combi.ordtree]
+mem_pblock_setpart [lemma, in Combi.Combi.setpartition]
+mem_setpart_pblock [lemma, in Combi.Combi.setpartition]
+mem_eq_pblock [lemma, in Combi.Combi.setpartition]
+mem_pblockC [lemma, in Combi.Combi.setpartition]
+mem_enum_permuted [lemma, in Combi.Combi.permuted]
+mem_fiber [lemma, in Combi.Combi.fibered_set]
+mem_Delta [lemma, in Combi.SymGroup.presentSn]
+mem_partsum_gt [lemma, in Combi.Combi.composition]
+mem_partsum_non0 [lemma, in Combi.Combi.composition]
+merge_cons [lemma, in Combi.Combi.partition]
+merge_sortedE [lemma, in Combi.Combi.partition]
+merge_is_part [lemma, in Combi.Combi.partition]
+mesymlm_rbound [lemma, in Combi.MPoly.antisym]
+mesym_SchurE [lemma, in Combi.MPoly.Schur_mpoly]
+MF [definition, in ALEA.Qmeasure]
+mfinite [definition, in ALEA.Qmeasure]
+MFO [instance, in ALEA.Qmeasure]
+mfun2 [definition, in ALEA.Ccpo]
+mfun2_mon [instance, in ALEA.Ccpo]
+mfun2_simpl [lemma, in ALEA.Ccpo]
+mindropeq [definition, in Combi.Combi.skewpart]
+MinDropEq [section, in Combi.Combi.skewpart]
+mindropeqC [lemma, in Combi.Combi.skewpart]
+mindropeq_non0 [lemma, in Combi.Combi.skewpart]
+mindropeq_nthP [lemma, in Combi.Combi.skewpart]
+mindropeq_cons_neq [lemma, in Combi.Combi.skewpart]
+mindropeq_cons_eq [lemma, in Combi.Combi.skewpart]
+mindropeq_nil [lemma, in Combi.Combi.skewpart]
+mindropeq_leq [lemma, in Combi.Combi.skewpart]
+mindropeq_eq [lemma, in Combi.Combi.skewpart]
+MinDropEq.T [variable, in Combi.Combi.skewpart]
+mindropeq0 [lemma, in Combi.Combi.skewpart]
+minh [definition, in Combi.Combi.Dyckword]
+minhE [lemma, in Combi.Combi.Dyckword]
+minhP [lemma, in Combi.Combi.Dyckword]
+minh_rrw [lemma, in Combi.Combi.Dyckword]
+minh_neg [lemma, in Combi.Combi.Dyckword]
+mininspred [definition, in Combi.LRrule.Schensted]
+minSS [lemma, in Combi.SSRcomplements.tools]
+Minv [definition, in Combi.Basic.unitriginv]
+Minvl [lemma, in Combi.Basic.unitriginv]
+Minvr [lemma, in Combi.Basic.unitriginv]
+Minv_lincombr [lemma, in Combi.Basic.unitriginv]
+Minv_lincombl [lemma, in Combi.Basic.unitriginv]
+Minv_unitrig [lemma, in Combi.Basic.unitriginv]
+Minv_uni [lemma, in Combi.Basic.unitriginv]
+Minv_trig [lemma, in Combi.Basic.unitriginv]
+Misc [library]
+mk_isglb [lemma, in ALEA.Ccpo]
+mlead_antisym_sorted [lemma, in Combi.MPoly.antisym]
+MLet [definition, in ALEA.Qmeasure]
+Mlet [definition, in ALEA.Qmeasure]
+Mlet_assoc [lemma, in ALEA.Qmeasure]
+Mlet_ext [lemma, in ALEA.Qmeasure]
+Mlet_unit [lemma, in ALEA.Qmeasure]
+Mlet_eq_compat [lemma, in ALEA.Qmeasure]
+MLet_simpl [lemma, in ALEA.Qmeasure]
+Mlet_mon2 [instance, in ALEA.Qmeasure]
+Mlet_le_compat [lemma, in ALEA.Qmeasure]
+Mlet_simpl_eq [lemma, in ALEA.Qmeasure]
+Mlet_simpl [lemma, in ALEA.Qmeasure]
+mlub_lift_left [lemma, in ALEA.Ccpo]
+mlub_lift_right [lemma, in ALEA.Ccpo]
+mlub_le [lemma, in ALEA.Ccpo]
+mlub_eq_compat [lemma, in ALEA.Ccpo]
+mlub_le_compat [lemma, in ALEA.Ccpo]
+mnm_perm [lemma, in Combi.MPoly.antisym]
+mnm_n0E [lemma, in Combi.MPoly.sympoly]
+MNRule [section, in Combi.MPoly.MurnaghanNakayama]
+MNRule.n0 [variable, in Combi.MPoly.MurnaghanNakayama]
+MNRule.R [variable, in Combi.MPoly.MurnaghanNakayama]
+MN_coeffE [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_homogP [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_recP [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_consE [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_notincl [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_szE [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_fast [definition, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec [definition, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_homogP [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeffP [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeffP_int [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_consE [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff0 [lemma, in Combi.MPoly.MurnaghanNakayama]
+MN_coeff [definition, in Combi.MPoly.MurnaghanNakayama]
+MO [instance, in ALEA.Qmeasure]
+MonDistrib [section, in ALEA.Qmeasure]
+MonDistrib.A [variable, in ALEA.Qmeasure]
+MonDistrib.B [variable, in ALEA.Qmeasure]
+MonomPart [section, in Combi.MPoly.antisym]
+MonomPart.n [variable, in Combi.MPoly.antisym]
+_ # _ [notation, in Combi.MPoly.antisym]
+monotonic [record, in ALEA.Ccpo]
+monotonic [inductive, in ALEA.Ccpo]
+monotonic_sym [lemma, in ALEA.Ccpo]
+monotonic_lub_comp [instance, in ALEA.Ccpo]
+monotonic_comp_mon [instance, in ALEA.Ccpo]
+monotonic_comp [instance, in ALEA.Ccpo]
+monotonic_stable [instance, in ALEA.Ccpo]
+monotonic_intro [lemma, in ALEA.Ccpo]
+monotonic_def [projection, in ALEA.Ccpo]
+monotonic_def [constructor, in ALEA.Ccpo]
+monotonic2 [record, in ALEA.Ccpo]
+monotonic2 [inductive, in ALEA.Ccpo]
+monotonic2_sym [lemma, in ALEA.Ccpo]
+monotonic2_stable2 [instance, in ALEA.Ccpo]
+monotonic2_intro [projection, in ALEA.Ccpo]
+monotonic2_intro [constructor, in ALEA.Ccpo]
+monsY [definition, in Combi.MPoly.Cauchy]
+monsYK [lemma, in Combi.MPoly.Cauchy]
+monsY_bij [lemma, in Combi.MPoly.Cauchy]
+mons2mE [lemma, in Combi.MPoly.Schur_mpoly]
+monX [definition, in Combi.MPoly.Cauchy]
+mon_fun_lub [instance, in ALEA.Ccpo]
+mon_ord_equiv_simpl [lemma, in ALEA.Ccpo]
+mon_ord_equiv [definition, in ALEA.Ccpo]
+mon_app2_mon [instance, in ALEA.Ccpo]
+mon_app2 [instance, in ALEA.Ccpo]
+mon_diag [instance, in ALEA.Ccpo]
+mon_cte [instance, in ALEA.Ccpo]
+mon_id [instance, in ALEA.Ccpo]
+mon_seq_lift_right [instance, in ALEA.Ccpo]
+mon_seq_lift_left [instance, in ALEA.Ccpo]
+mon_fun_mon [instance, in ALEA.Ccpo]
+mon_comp [definition, in ALEA.Ccpo]
+mon_fun_eq [lemma, in ALEA.Ccpo]
+mon_fun_subst [definition, in ALEA.Ccpo]
+mon_fun_eq_monotonic [lemma, in ALEA.Ccpo]
+mon_eq_compat [lemma, in ALEA.Ccpo]
+mon_le_compat [lemma, in ALEA.Ccpo]
+mon_simpl [lemma, in ALEA.Ccpo]
+mon2 [definition, in ALEA.Ccpo]
+mon2_fun2 [lemma, in ALEA.Ccpo]
+mon2_le_compat [lemma, in ALEA.Ccpo]
+mon2_simpl [lemma, in ALEA.Ccpo]
+mon2_elim2 [lemma, in ALEA.Ccpo]
+mon2_elim1 [lemma, in ALEA.Ccpo]
+mon2_intro [instance, in ALEA.Ccpo]
+morph_of_tinj [definition, in Combi.SymGroup.towerSn]
+morph_eltr [lemma, in Combi.SymGroup.presentSn]
+mpart [definition, in Combi.MPoly.antisym]
+mpartE [lemma, in Combi.MPoly.antisym]
+mpartK [lemma, in Combi.MPoly.antisym]
+mpartS [lemma, in Combi.MPoly.Schur_altdef]
+mpart_partm_perm [lemma, in Combi.MPoly.antisym]
+mpart_is_dominant [lemma, in Combi.MPoly.antisym]
+mpart0 [lemma, in Combi.MPoly.antisym]
+MPoESymHomog [section, in Combi.MPoly.sympoly]
+MPoESymHomog.n [variable, in Combi.MPoly.sympoly]
+MPoESymHomog.R [variable, in Combi.MPoly.sympoly]
+MPolySym [section, in Combi.MPoly.antisym]
+MPolySym.n [variable, in Combi.MPoly.antisym]
+MPolySym.R [variable, in Combi.MPoly.antisym]
+_ # _ [notation, in Combi.MPoly.antisym]
+mseq_cte [definition, in ALEA.Ccpo]
+mseq_lift_right_left [lemma, in ALEA.Ccpo]
+mseq_lift_right_le_compat [lemma, in ALEA.Ccpo]
+mseq_lift_right_simpl [lemma, in ALEA.Ccpo]
+mseq_lift_right [definition, in ALEA.Ccpo]
+mseq_lift_left_le_compat [lemma, in ALEA.Ccpo]
+mseq_lift_left_simpl [lemma, in ALEA.Ccpo]
+mseq_lift_left [definition, in ALEA.Ccpo]
+mshift [definition, in ALEA.Ccpo]
+mshift_continuous [instance, in ALEA.Ccpo]
+mshift_continuous2 [lemma, in ALEA.Ccpo]
+mshift_le_compat [lemma, in ALEA.Ccpo]
+mshift_simpl [lemma, in ALEA.Ccpo]
+mshift2_eq [lemma, in ALEA.Ccpo]
+Mstable_linear [lemma, in ALEA.Qmeasure]
+Mstable_mull [lemma, in ALEA.Qmeasure]
+Mstable_divi [lemma, in ALEA.Qmeasure]
+Mstable_muli [lemma, in ALEA.Qmeasure]
+Mstable_addi [lemma, in ALEA.Qmeasure]
+Mstable_divn [lemma, in ALEA.Qmeasure]
+Mstable_subn [lemma, in ALEA.Qmeasure]
+Mstable_addn [lemma, in ALEA.Qmeasure]
+Mstable_add [lemma, in ALEA.Qmeasure]
+Mstable_opp [lemma, in ALEA.Qmeasure]
+Mstable_eq [lemma, in ALEA.Qmeasure]
+Mstable0 [lemma, in ALEA.Qmeasure]
+mstar [definition, in ALEA.Qmeasure]
+msuppX1 [lemma, in Combi.MPoly.antisym]
+msym_pihomog [lemma, in Combi.MPoly.homogsym]
+msym_map_mpoly [lemma, in Combi.MPoly.antisym]
+msym_fundamental_symh_un [lemma, in Combi.MPoly.sympoly]
+mu [projection, in ALEA.Qmeasure]
+MultAlternSymp [section, in Combi.MPoly.MurnaghanNakayama]
+MultAlternSymp.n0 [variable, in Combi.MPoly.MurnaghanNakayama]
+MultAlternSymp.R [variable, in Combi.MPoly.MurnaghanNakayama]
+'a_ _ [notation, in Combi.MPoly.MurnaghanNakayama]
+MultinomCompl [section, in Combi.MPoly.sympoly]
+MultinomCompl.n [variable, in Combi.MPoly.sympoly]
+MultinomCompl.R [variable, in Combi.MPoly.sympoly]
+multinomial [definition, in Combi.Combi.multinomial]
+multinomial [library]
+multinomialE [lemma, in Combi.Combi.multinomial]
+multinomial_filter_neq0 [lemma, in Combi.Combi.multinomial]
+multinomial_cat [lemma, in Combi.Combi.multinomial]
+multinomial_factd [lemma, in Combi.Combi.multinomial]
+multinomial_nseq1 [lemma, in Combi.Combi.multinomial]
+multinomial_nseq [lemma, in Combi.Combi.multinomial]
+multinomial_fact [lemma, in Combi.Combi.multinomial]
+multinomial_rec [definition, in Combi.Combi.multinomial]
+multinomial0 [lemma, in Combi.Combi.multinomial]
+multinomial1 [lemma, in Combi.Combi.multinomial]
+multinomial2 [lemma, in Combi.Combi.multinomial]
+MultSymsSymp [section, in Combi.MPoly.MurnaghanNakayama]
+MultSymsSympIDomain [section, in Combi.MPoly.MurnaghanNakayama]
+MultSymsSympIDomain.n0 [variable, in Combi.MPoly.MurnaghanNakayama]
+MultSymsSymp.n0 [variable, in Combi.MPoly.MurnaghanNakayama]
+MultSymsSymp.R [variable, in Combi.MPoly.MurnaghanNakayama]
+mult_altern_pmap [lemma, in Combi.MPoly.MurnaghanNakayama]
+mult_altern_oapp [lemma, in Combi.MPoly.MurnaghanNakayama]
+mult_altern_symp_pol [lemma, in Combi.MPoly.MurnaghanNakayama]
+mult_syme_U [lemma, in Combi.MPoly.sympoly]
+mult_symh_powersum [lemma, in Combi.MPoly.sympoly]
+mult_symh_U [lemma, in Combi.MPoly.sympoly]
+mul_ek_pk [lemma, in Combi.MPoly.sympoly]
+mul_ek_p1 [lemma, in Combi.MPoly.sympoly]
+Munit [definition, in ALEA.Qmeasure]
+munit [definition, in ALEA.Qmeasure]
+Munit_eq_compat [lemma, in ALEA.Qmeasure]
+Munit_simpl_eq [lemma, in ALEA.Qmeasure]
+Munit_simpl [lemma, in ALEA.Qmeasure]
+MurnaghanNakayama [library]
+Murnaghan_NakayamaCT [lemma, in Combi.SymGroup.Frobenius_char]
+Murnaghan_Nakayama_char [lemma, in Combi.SymGroup.Frobenius_char]
+mu_pos_cond [lemma, in ALEA.Qmeasure]
+mu_bool_cond [lemma, in ALEA.Qmeasure]
+mu_in_seq [lemma, in ALEA.Qmeasure]
+mu_stable_sum [lemma, in ALEA.Qmeasure]
+mu_random_sum [lemma, in ALEA.Qmeasure]
+mu_uniform_sum [lemma, in ALEA.Qmeasure]
+mu_bool_negb1 [lemma, in ALEA.Qmeasure]
+mu_bool_negb [lemma, in ALEA.Qmeasure]
+mu_bool_negb0 [lemma, in ALEA.Qmeasure]
+mu_bool_impl1 [lemma, in ALEA.Qmeasure]
+mu_bool_impl [lemma, in ALEA.Qmeasure]
+mu_bool_0le [lemma, in ALEA.Qmeasure]
+mu_bool_le1 [lemma, in ALEA.Qmeasure]
+mu_eq_compat [lemma, in ALEA.Qmeasure]
+mu_le_compat [lemma, in ALEA.Qmeasure]
+mu_stable_inv_inv [lemma, in ALEA.Qmeasure]
+mu_stable_mulr [lemma, in ALEA.Qmeasure]
+mu_cte [lemma, in ALEA.Qmeasure]
+mu_stable_le1 [lemma, in ALEA.Qmeasure]
+mu_stable_pos [lemma, in ALEA.Qmeasure]
+mu_add_zero [lemma, in ALEA.Qmeasure]
+mu_stable_mull [lemma, in ALEA.Qmeasure]
+mu_stable_add [lemma, in ALEA.Qmeasure]
+mu_stable_inv [lemma, in ALEA.Qmeasure]
+mu_one_eq [lemma, in ALEA.Qmeasure]
+mu_zero_eq [lemma, in ALEA.Qmeasure]
+mu_zero [lemma, in ALEA.Qmeasure]
+mu_stable_eq [lemma, in ALEA.Qmeasure]
+mu_monotonic [lemma, in ALEA.Qmeasure]
+mu_prob [projection, in ALEA.Qmeasure]
+mu_stable_sub [projection, in ALEA.Qmeasure]
+mu_walk_to_corner_is_trace [lemma, in Combi.HookFormula.hook]
+mwmwgt_homogP [lemma, in Combi.MPoly.sympoly]
+mxvec_indexK [lemma, in Combi.MPoly.Cauchy]
+mxvec_index [abbreviation, in Combi.MPoly.Cauchy]
+

N

+n [abbreviation, in Combi.MPoly.Cauchy]
+n [abbreviation, in Combi.MPoly.Cauchy]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.homogsym]
+n [abbreviation, in Combi.MPoly.Schur_mpoly]
+n [abbreviation, in Combi.MPoly.Schur_mpoly]
+n [abbreviation, in Combi.Basic.unitriginv]
+n [abbreviation, in Combi.MPoly.Schur_altdef]
+n [abbreviation, in Combi.MPoly.Schur_altdef]
+n [abbreviation, in Combi.MPoly.Schur_altdef]
+n [abbreviation, in Combi.MPoly.Schur_altdef]
+n [abbreviation, in Combi.LRrule.freeSchur]
+n [abbreviation, in Combi.LRrule.therule]
+n [abbreviation, in Combi.LRrule.therule]
+n [abbreviation, in Combi.SymGroup.weak_order]
+n [abbreviation, in Combi.SymGroup.weak_order]
+n [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+n [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+n [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+n [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+n [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+n [abbreviation, in Combi.SymGroup.presentSn]
+n [abbreviation, in Combi.SymGroup.presentSn]
+n [abbreviation, in Combi.SymGroup.presentSn]
+n [abbreviation, in Combi.SymGroup.presentSn]
+n [abbreviation, in Combi.LRrule.implem]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+n [abbreviation, in Combi.MPoly.sympoly]
+natO [instance, in ALEA.Ccpo]
+nat_monotonic_inv [lemma, in ALEA.Ccpo]
+nat_monotonic [lemma, in ALEA.Ccpo]
+nat_repr [definition, in Combi.SymGroup.reprSn]
+nat_mx_repr [lemma, in Combi.SymGroup.reprSn]
+nat_mx [definition, in Combi.SymGroup.reprSn]
+nat_compare_specT [lemma, in ALEA.Misc]
+nbump_bumprowE [lemma, in Combi.LRrule.Schensted]
+nbump_ins_rconsE [lemma, in Combi.LRrule.Schensted]
+nbump_size_ins [lemma, in Combi.LRrule.Schensted]
+nbump_inspos_eq_size [lemma, in Combi.LRrule.Schensted]
+nbump_insposE [lemma, in Combi.LRrule.Schensted]
+nbump_mininspredE [lemma, in Combi.LRrule.Schensted]
+nbump_bumprow_rconsE [lemma, in Combi.LRrule.plactic]
+ncfuniCT [definition, in Combi.SymGroup.towerSn]
+ncfuniCT_Ind [lemma, in Combi.SymGroup.towerSn]
+ncfuniCT_gen [lemma, in Combi.SymGroup.towerSn]
+neig4 [abbreviation, in Combi.Combi.skewpart]
+neig4 [abbreviation, in Combi.Combi.skewpart]
+neig4 [definition, in Combi.Combi.skewpart]
+neig4box [definition, in Combi.Combi.skewpart]
+neig4boxE [lemma, in Combi.Combi.skewpart]
+neig4box_sym [lemma, in Combi.Combi.skewpart]
+neig4_sym [lemma, in Combi.Combi.skewpart]
+Neq_lt_0 [lemma, in ALEA.Misc]
+neq0zcard [lemma, in Combi.SymGroup.permcent]
+neq0zcoeff [lemma, in Combi.SymGroup.towerSn]
+Newton_syme [lemma, in Combi.MPoly.sympoly]
+Newton_syme1 [lemma, in Combi.MPoly.sympoly]
+Newton_symh1 [lemma, in Combi.MPoly.sympoly]
+Newton_symh [lemma, in Combi.MPoly.sympoly]
+nil_Dyck [definition, in Combi.Combi.Dyckword]
+NirrSn [lemma, in Combi.SymGroup.reprSn]
+NirrS2 [lemma, in Combi.SymGroup.reprSn]
+Nlt0_le1 [lemma, in ALEA.Misc]
+nohasincr_setdiff [lemma, in Combi.MPoly.Schur_altdef]
+NonEmpty [section, in Combi.LRrule.Schensted]
+NonEmpty.Bijection [section, in Combi.LRrule.Schensted]
+NonEmpty.Bump [section, in Combi.LRrule.Schensted]
+NonEmpty.Bump.HRow [variable, in Combi.LRrule.Schensted]
+NonEmpty.Bump.l [variable, in Combi.LRrule.Schensted]
+NonEmpty.Bump.Row [variable, in Combi.LRrule.Schensted]
+NonEmpty.Classes [section, in Combi.LRrule.Schensted]
+NonEmpty.disp [variable, in Combi.LRrule.Schensted]
+NonEmpty.Dominate [section, in Combi.LRrule.Schensted]
+NonEmpty.Insert [section, in Combi.LRrule.Schensted]
+NonEmpty.Insert.HRow [variable, in Combi.LRrule.Schensted]
+NonEmpty.Insert.l [variable, in Combi.LRrule.Schensted]
+NonEmpty.Insert.Row [variable, in Combi.LRrule.Schensted]
+NonEmpty.Inverse [section, in Combi.LRrule.Schensted]
+NonEmpty.InverseBump [section, in Combi.LRrule.Schensted]
+NonEmpty.Schensted [section, in Combi.LRrule.Schensted]
+NonEmpty.Statistics [section, in Combi.LRrule.Schensted]
+NonEmpty.T [variable, in Combi.LRrule.Schensted]
+NonEmpty.Tableaux [section, in Combi.LRrule.Schensted]
+non_decr_equiv [lemma, in Combi.SSRcomplements.sorted]
+NoSetContainingBoth [module, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case [section, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.a [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.Alph [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.b [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.disp [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.HnoBoth [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.k [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.P [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.Px [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.R [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.u [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.v [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.word [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.x [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.y [variable, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.extract_swap_set [lemma, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Hcast [lemma, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.ksupp_Q [lemma, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.posa [abbreviation, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.posb [abbreviation, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Q [definition, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.size_cover_Q [lemma, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swapX [abbreviation, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_set_inj [lemma, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_set [definition, in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_setX [abbreviation, in Combi.LRrule.Greene_inv]
+notbump [lemma, in Combi.LRrule.Schensted]
+notin0_part [lemma, in Combi.Combi.partition]
+not_hasincr_part [lemma, in Combi.MPoly.Schur_altdef]
+not_and_elim_right [lemma, in ALEA.Misc]
+not_and_elim_left [lemma, in ALEA.Misc]
+Npos [lemma, in ALEA.Misc]
+Nsucc_pred_pos [lemma, in ALEA.Misc]
+NThAddRibbon [section, in Combi.Combi.skewpart]
+NThAddRibbon.lesmin [variable, in Combi.Combi.skewpart]
+NThAddRibbon.less [variable, in Combi.Combi.skewpart]
+NThAddRibbon.lessz [variable, in Combi.Combi.skewpart]
+NThAddRibbon.rem [variable, in Combi.Combi.skewpart]
+NThAddRibbon.sh [variable, in Combi.Combi.skewpart]
+NThAddRibbon.start [variable, in Combi.Combi.skewpart]
+NThAddRibbon.stop [variable, in Combi.Combi.skewpart]
+NThAddRibbon.sztd [variable, in Combi.Combi.skewpart]
+nth_sizeu2 [lemma, in Combi.Combi.std]
+nth_sizeu1 [lemma, in Combi.Combi.std]
+nth_sizeu [lemma, in Combi.Combi.std]
+nth_transp [lemma, in Combi.Combi.std]
+nth_set_nth_any [lemma, in Combi.SSRcomplements.tools]
+nth_set_nth_expand [lemma, in Combi.SSRcomplements.tools]
+nth_diff_shape [lemma, in Combi.Combi.partition]
+nth_pad [lemma, in Combi.Combi.partition]
+nth_conjE [lemma, in Combi.Combi.partition]
+nth_incr_first_n [lemma, in Combi.Combi.partition]
+nth_decr_nth_neq [lemma, in Combi.Combi.partition]
+nth_decr_nth [lemma, in Combi.Combi.partition]
+nth_rem_trail0 [lemma, in Combi.Combi.partition]
+nth_part_non0 [lemma, in Combi.Combi.partition]
+nth_std_pos [lemma, in Combi.LRrule.stdplact]
+nth_vctmin [lemma, in Combi.Combi.bintree]
+nth_add_setdiff [lemma, in Combi.MPoly.Schur_altdef]
+nth_lt_inspos [lemma, in Combi.LRrule.Schensted]
+nth_inspos_ins [lemma, in Combi.LRrule.Schensted]
+nth_pfminh [lemma, in Combi.Combi.Dyckword]
+nth_evalseq [lemma, in Combi.Combi.Yamanouchi]
+nth_add_ribbon_stop_lt [lemma, in Combi.Combi.skewpart]
+nth_add_ribbon_in [lemma, in Combi.Combi.skewpart]
+nth_add_ribbon_start [lemma, in Combi.Combi.skewpart]
+nth_add_ribbon_lt_start [lemma, in Combi.Combi.skewpart]
+nth_inspos [lemma, in Combi.Basic.ordtype]
+nth_rembig [lemma, in Combi.Basic.ordtype]
+nth_lt_posbig [lemma, in Combi.Basic.ordtype]
+nth_posbig [lemma, in Combi.Basic.ordtype]
+nvar [abbreviation, in Combi.SymGroup.Frobenius_char]
+NVar [section, in Combi.SymGroup.Frobenius_char]
+nvar [abbreviation, in Combi.MPoly.sympoly]
+NVar.Character [section, in Combi.SymGroup.Frobenius_char]
+NVar.Character.Hn [variable, in Combi.SymGroup.Frobenius_char]
+NVar.Character.n [variable, in Combi.SymGroup.Frobenius_char]
+NVar.Defs [section, in Combi.SymGroup.Frobenius_char]
+NVar.Defs.Hn [variable, in Combi.SymGroup.Frobenius_char]
+NVar.Defs.n [variable, in Combi.SymGroup.Frobenius_char]
+NVar.nvar0 [variable, in Combi.SymGroup.Frobenius_char]
+N2Nat_inj_pos [lemma, in ALEA.Misc]
+N2Nat_inj_le [lemma, in ALEA.Misc]
+N2Nat_inj_lt [lemma, in ALEA.Misc]
+N2Nat_le_mono [lemma, in ALEA.Misc]
+N2Nat_lt_mono [lemma, in ALEA.Misc]
+

O

+odd_cycle_type [lemma, in Combi.SymGroup.cycletype]
+odd_size_permE [lemma, in Combi.SymGroup.presentSn]
+odd_eltr [lemma, in Combi.SymGroup.presentSn]
+Odistr [instance, in ALEA.Qmeasure]
+Oeq [projection, in ALEA.Ccpo]
+Oeq_trans [lemma, in ALEA.Ccpo]
+Oeq_le_sym [lemma, in ALEA.Ccpo]
+Oeq_le [lemma, in ALEA.Ccpo]
+Oeq_sym [lemma, in ALEA.Ccpo]
+Oeq_refl_eq [lemma, in ALEA.Ccpo]
+Oeq_refl [lemma, in ALEA.Ccpo]
+OfSize [section, in Combi.Combi.ordtree]
+OfSize.n [variable, in Combi.Combi.ordtree]
+Oge [definition, in ALEA.Ccpo]
+Ole [projection, in ALEA.Ccpo]
+Ole_not_lt [lemma, in ALEA.Ccpo]
+Ole_lt_trans [lemma, in ALEA.Ccpo]
+Ole_notle_lt [lemma, in ALEA.Ccpo]
+Ole_diff_lt [lemma, in ALEA.Ccpo]
+Ole_eq_left [lemma, in ALEA.Ccpo]
+Ole_eq_right [lemma, in ALEA.Ccpo]
+Ole_eq_compat [lemma, in ALEA.Ccpo]
+Ole_antisym [lemma, in ALEA.Ccpo]
+Ole_refl [lemma, in ALEA.Ccpo]
+Ole_trans [lemma, in ALEA.Ccpo]
+Ole_refl_eq_inv [lemma, in ALEA.Ccpo]
+Ole_refl_eq [lemma, in ALEA.Ccpo]
+Olt [definition, in ALEA.Ccpo]
+Olt_le_trans [lemma, in ALEA.Ccpo]
+Olt_antirefl [lemma, in ALEA.Ccpo]
+Olt_trans [lemma, in ALEA.Ccpo]
+Olt_notle [lemma, in ALEA.Ccpo]
+Olt_le [lemma, in ALEA.Ccpo]
+Olt_neq_rev [lemma, in ALEA.Ccpo]
+Olt_neq [lemma, in ALEA.Ccpo]
+Olt_eq_compat [lemma, in ALEA.Ccpo]
+Omega [section, in Combi.MPoly.sympoly]
+omegahomsym [definition, in Combi.MPoly.homogsym]
+OmegaHomSym [section, in Combi.MPoly.homogsym]
+omegahomsym_rmorph [lemma, in Combi.MPoly.homogsym]
+omegahomsym_is_linear [lemma, in Combi.MPoly.homogsym]
+omegahomsym_subproof [lemma, in Combi.MPoly.homogsym]
+OmegaHomSym.d [variable, in Combi.MPoly.homogsym]
+OmegaHomSym.n0 [variable, in Combi.MPoly.homogsym]
+OmegaHomSym.R [variable, in Combi.MPoly.homogsym]
+OmegaProd [section, in Combi.MPoly.homogsym]
+OmegaProd.n0 [variable, in Combi.MPoly.homogsym]
+OmegaProd.R [variable, in Combi.MPoly.homogsym]
+omegasf [definition, in Combi.MPoly.sympoly]
+omegasfK [lemma, in Combi.MPoly.sympoly]
+omegasf_syms [lemma, in Combi.MPoly.sympoly]
+omegasf_prodsymp [lemma, in Combi.MPoly.sympoly]
+omegasf_prodsymh [lemma, in Combi.MPoly.sympoly]
+omegasf_prodsyme [lemma, in Combi.MPoly.sympoly]
+omegasf_compsymh [lemma, in Combi.MPoly.sympoly]
+omegasf_sympolyf_eval [lemma, in Combi.MPoly.sympoly]
+omegasf_homogE [lemma, in Combi.MPoly.sympoly]
+omegasf_homog [lemma, in Combi.MPoly.sympoly]
+omegasf_symp [lemma, in Combi.MPoly.sympoly]
+omegasf_symh [lemma, in Combi.MPoly.sympoly]
+omegasf_syme [lemma, in Combi.MPoly.sympoly]
+omegasf_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+omegasf_is_linear [lemma, in Combi.MPoly.sympoly]
+omegasf_is_symmetric [lemma, in Combi.MPoly.sympoly]
+omega_homsymp [lemma, in Combi.MPoly.homogsym]
+omega_homsyms [lemma, in Combi.MPoly.homogsym]
+omega_homsyme [lemma, in Combi.MPoly.homogsym]
+omega_homsymh [lemma, in Combi.MPoly.homogsym]
+omega_Fchar [lemma, in Combi.SymGroup.Frobenius_char]
+omega_Fchar_inv [lemma, in Combi.SymGroup.Frobenius_char]
+Omega.d [variable, in Combi.MPoly.sympoly]
+Omega.n0 [variable, in Combi.MPoly.sympoly]
+Omega.R [variable, in Combi.MPoly.sympoly]
+one_letter_included [lemma, in Combi.LRrule.implem]
+one_letter_choicesP [lemma, in Combi.LRrule.implem]
+one_letter_choices [definition, in Combi.LRrule.implem]
+Open [constructor, in Combi.Combi.Dyckword]
+OperDistr [section, in ALEA.Qmeasure]
+OperDistr.A [variable, in ALEA.Qmeasure]
+OperDistr.B [variable, in ALEA.Qmeasure]
+OperDistr.C [variable, in ALEA.Qmeasure]
+OperDistr.MuBool [section, in ALEA.Qmeasure]
+OperDistr.MuBool.m [variable, in ALEA.Qmeasure]
+orc [definition, in ALEA.Misc]
+orc_intro [lemma, in ALEA.Misc]
+orc_right [lemma, in ALEA.Misc]
+orc_left [lemma, in ALEA.Misc]
+ord [record, in ALEA.Ccpo]
+ordcast [library]
+Order [record, in ALEA.Ccpo]
+OrderEqRefl [instance, in ALEA.Ccpo]
+OrderEqSym [instance, in ALEA.Ccpo]
+OrderEqTrans [instance, in ALEA.Ccpo]
+OrderEquiv [instance, in ALEA.Ccpo]
+order_rel [projection, in ALEA.Ccpo]
+order_eq [projection, in ALEA.Ccpo]
+order_cyclic [lemma, in Combi.SymGroup.cycles]
+Order_dual__canonical__ordtype_InhFinOrder [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinLattice [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinPOrder [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinite [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhOrder [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhTBLattice [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhLattice [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhPOrder [definition, in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_Inhabited [definition, in Combi.Basic.ordtype]
+OrdNode [constructor, in Combi.Combi.ordtree]
+OrdNode_inj [lemma, in Combi.Combi.ordtree]
+OrdSetoid [lemma, in ALEA.Ccpo]
+OrdTableau [section, in Combi.Combi.tableau]
+OrdTableau.d [variable, in Combi.Combi.tableau]
+OrdTableau.Hszs [variable, in Combi.Combi.tableau]
+OrdTableau.n [variable, in Combi.Combi.tableau]
+OrdTableau.sh [variable, in Combi.Combi.tableau]
+ordtree [inductive, in Combi.Combi.ordtree]
+ordtree [library]
+ordtreesz [record, in Combi.Combi.ordtree]
+ordtreeszP [lemma, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__fintype_SubFinite [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__fintype_Finite [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_SubCountable [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_Countable [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_SubChoice [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_Choice [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_SubEquality [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_Equality [definition, in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_SubType [definition, in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__choice_Countable [definition, in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__choice_Choice [definition, in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__eqtype_Equality [definition, in Combi.Combi.ordtree]
+ordtype [library]
+ordtype_isInhabitedType__to__ordtype_isInhabited__34 [definition, in Combi.Combi.partition]
+ordtype_isInhabitedType__to__ordtype_isInhabited [definition, in Combi.Combi.partition]
+ordtype_isInhabitedType__to__ordtype_isInhabited [definition, in Combi.Combi.setpartition]
+ordtype_InhFinOrder__to__Order_DistrLattice_isTotal [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_Lattice_isDistributive [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_POrder_isJoinSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_POrder_isMeetSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__fintype_isFinite [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__choice_Choice_isCountable [definition, in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__fintype_isFinite [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__choice_Choice_isCountable [definition, in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__fintype_isFinite [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__choice_Choice_isCountable [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_DistrLattice_isTotal [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_Lattice_isDistributive [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_POrder_isJoinSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_POrder_isMeetSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhOrder__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_hasBottom [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_hasTop [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhLattice__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__Order_isDuallyPreorder [definition, in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__51 [definition, in Combi.Basic.ordtype]
+ordtype_Inhabited__to__eqtype_hasDecEq [definition, in Combi.Basic.ordtype]
+ordtype_Inhabited__to__choice_hasChoice [definition, in Combi.Basic.ordtype]
+ordtype_Inhabited__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__30 [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__25 [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__20 [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__15 [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__11 [definition, in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited [definition, in Combi.Basic.ordtype]
+ord_to_bintreeK [lemma, in Combi.Combi.ordtree]
+ord_to_bintree [definition, in Combi.Combi.ordtree]
+ord0_in_map_liftF [lemma, in Combi.LRrule.Greene]
+outer_shape_pad0 [lemma, in Combi.Combi.partition]
+outer_shapeK [lemma, in Combi.Combi.partition]
+outer_shape [definition, in Combi.Combi.partition]
+outer_part [projection, in Combi.LRrule.implem]
+OutEval [section, in Combi.LRrule.implem]
+OutEval.outev [variable, in Combi.LRrule.implem]
+OutEval.tsumn [variable, in Combi.LRrule.implem]
+outputSpec [record, in Combi.LRrule.implem]
+outputSpecP [lemma, in Combi.LRrule.implem]
+outputSpec_count_mem [lemma, in Combi.LRrule.implem]
+out_perm_prod [lemma, in Combi.SymGroup.cycles]
+

P

+P [abbreviation, in Combi.MPoly.Schur_altdef]
+PackedSpec [section, in Combi.LRrule.implem]
+PackedSpec.eval [variable, in Combi.LRrule.implem]
+PackedSpec.inner [variable, in Combi.LRrule.implem]
+PackedSpec.outer [variable, in Combi.LRrule.implem]
+pact [abbreviation, in Combi.Combi.permuted]
+pad [definition, in Combi.Combi.partition]
+Pair [abbreviation, in Combi.LRrule.Schensted]
+partdom [definition, in Combi.Combi.partition]
+partdomP [lemma, in Combi.Combi.partition]
+partdomsh [definition, in Combi.Combi.partition]
+partdomshP [lemma, in Combi.Combi.partition]
+partdomsh_merge [lemma, in Combi.Combi.partition]
+partdomsh_merge1 [lemma, in Combi.Combi.partition]
+partdomsh_cons2E [lemma, in Combi.Combi.partition]
+partdomsh_cons2 [lemma, in Combi.Combi.partition]
+partdomsh_add [lemma, in Combi.Combi.partition]
+partdom_union_intpart [lemma, in Combi.Combi.partition]
+partdom_union_intpartr [lemma, in Combi.Combi.partition]
+partdom_union_intpartl [lemma, in Combi.Combi.partition]
+partdom_consK [lemma, in Combi.Combi.partition]
+partdom_anti [lemma, in Combi.Combi.partition]
+partdom_trans [lemma, in Combi.Combi.partition]
+partdom_refl [lemma, in Combi.Combi.partition]
+partdom_nil [lemma, in Combi.Combi.partition]
+partition [library]
+partition_box_skew__canonical__fintype_SubFinite [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__fintype_Finite [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__choice_SubCountable [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__choice_Countable [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__choice_SubChoice [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__choice_Choice [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_SubEquality [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_SubType [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__ordtype_InhFinite [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__ordtype_Inhabited [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__fintype_SubFinite [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__fintype_Finite [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__choice_SubCountable [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__choice_Countable [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__choice_SubChoice [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__choice_Choice [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_SubEquality [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_SubType [definition, in Combi.Combi.partition]
+partition_intpart__canonical__ordtype_Inhabited [definition, in Combi.Combi.partition]
+partition_intpart__canonical__choice_SubCountable [definition, in Combi.Combi.partition]
+partition_intpart__canonical__choice_Countable [definition, in Combi.Combi.partition]
+partition_intpart__canonical__choice_SubChoice [definition, in Combi.Combi.partition]
+partition_intpart__canonical__choice_Choice [definition, in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_SubEquality [definition, in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_SubType [definition, in Combi.Combi.partition]
+partition_psupport [lemma, in Combi.SymGroup.cycles]
+partition_porbits [lemma, in Combi.SymGroup.cycles]
+partm [definition, in Combi.MPoly.antisym]
+partmE [lemma, in Combi.MPoly.antisym]
+partmK [lemma, in Combi.MPoly.antisym]
+partmP [lemma, in Combi.MPoly.antisym]
+partm_permK [lemma, in Combi.MPoly.antisym]
+partnCT [definition, in Combi.SymGroup.cycletype]
+partnCTE [lemma, in Combi.SymGroup.cycletype]
+partnCTK [lemma, in Combi.SymGroup.cycletype]
+partnCT_congr [lemma, in Combi.SymGroup.cycletype]
+partn_of_compn [definition, in Combi.Combi.composition]
+PartOfn [section, in Combi.Combi.partition]
+PartOfn.n [variable, in Combi.Combi.partition]
+'P [notation, in Combi.Combi.partition]
+partsums [definition, in Combi.Combi.composition]
+partsums_cons [lemma, in Combi.Combi.composition]
+partsums_cat [lemma, in Combi.Combi.composition]
+part_rem_corner_ind [lemma, in Combi.Combi.partition]
+part_nseq1P [lemma, in Combi.Combi.partition]
+part_includedP [lemma, in Combi.Combi.partition]
+part_sumn_rectangle [lemma, in Combi.Combi.partition]
+part_eqP [lemma, in Combi.Combi.partition]
+part_leq_head [lemma, in Combi.Combi.partition]
+part_head_non0 [lemma, in Combi.Combi.partition]
+part_head0F [lemma, in Combi.Combi.partition]
+part_yam_of_stdtab [lemma, in Combi.Combi.stdtab]
+part_rcons_ind [lemma, in Combi.Combi.Yamanouchi]
+part_set1_eq [lemma, in Combi.Combi.setpartition]
+part_ordinal1 [lemma, in Combi.Combi.setpartition]
+part_ordinal0 [lemma, in Combi.Combi.setpartition]
+part_sumn_count [lemma, in Combi.MPoly.sympoly]
+part_of_comp_subproof [lemma, in Combi.Combi.composition]
+part_is_comp [lemma, in Combi.Combi.composition]
+part0 [lemma, in Combi.Combi.partition]
+path_braidred_catl [lemma, in Combi.SymGroup.presentSn]
+pblock_trivsetpart [lemma, in Combi.Combi.setpartition]
+pblock_setpart1 [lemma, in Combi.Combi.setpartition]
+pblock_notin [lemma, in Combi.Combi.setpartition]
+pblock_in [lemma, in Combi.Combi.setpartition]
+pchar0_algC [lemma, in Combi.MPoly.Cauchy]
+pchar0_rat [lemma, in Combi.MPoly.Cauchy]
+pchar0_algC [lemma, in Combi.MPoly.homogsym]
+pchar0_algC [lemma, in Combi.SymGroup.Frobenius_char]
+pchar0_rat [lemma, in Combi.SymGroup.Frobenius_char]
+permcent [library]
+PermComp [section, in Combi.SSRcomplements.permcomp]
+permcomp [library]
+PermComp.T [variable, in Combi.SSRcomplements.permcomp]
+permCT [definition, in Combi.SymGroup.cycletype]
+permCTP [lemma, in Combi.SymGroup.cycletype]
+permCT_colpartn [lemma, in Combi.SymGroup.cycletype]
+permCT_colpartn_card [lemma, in Combi.SymGroup.cycletype]
+permCT_exists [lemma, in Combi.SymGroup.cycletype]
+permcycles [definition, in Combi.SymGroup.permcent]
+PermCycles [section, in Combi.SymGroup.permcent]
+PermCycles [section, in Combi.SymGroup.cycles]
+permcyclesC [lemma, in Combi.SymGroup.permcent]
+permcyclesK [lemma, in Combi.SymGroup.permcent]
+permcyclesM [lemma, in Combi.SymGroup.permcent]
+permcyclesP [lemma, in Combi.SymGroup.permcent]
+permcycles_inj [lemma, in Combi.SymGroup.permcent]
+permcycles_morphism [definition, in Combi.SymGroup.permcent]
+PermCycles.CM [section, in Combi.SymGroup.permcent]
+PermCycles.CM.s [variable, in Combi.SymGroup.permcent]
+PermCycles.T [variable, in Combi.SymGroup.permcent]
+PermCycles.T [variable, in Combi.SymGroup.cycles]
+'CC ( _ ) (group_scope) [notation, in Combi.SymGroup.permcent]
+PermEq [section, in Combi.Combi.std]
+PermEq.Alph [variable, in Combi.Combi.std]
+PermEq.disp [variable, in Combi.Combi.std]
+permKP [lemma, in Combi.SSRcomplements.permcomp]
+PermLattice [module, in Combi.SymGroup.weak_order]
+PermLattice.Exports [module, in Combi.SymGroup.weak_order]
+PermLattice.Exports.bottom_perm [lemma, in Combi.SymGroup.weak_order]
+PermLattice.Exports.invset_join [lemma, in Combi.SymGroup.weak_order]
+PermLattice.Exports.n [abbreviation, in Combi.SymGroup.weak_order]
+PermLattice.Exports.PermLattice [section, in Combi.SymGroup.weak_order]
+PermLattice.Exports.PermLattice.n0 [variable, in Combi.SymGroup.weak_order]
+PermLattice.Exports.perm_join_meetE [lemma, in Combi.SymGroup.weak_order]
+PermLattice.Exports.top_perm [lemma, in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_factory_22 [definition, in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_factory_20 [definition, in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_mixin_19 [definition, in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_mixin_18 [definition, in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_factory_15 [definition, in Combi.SymGroup.weak_order]
+PermLattice.infperm [definition, in Combi.SymGroup.weak_order]
+PermLattice.infperm_is_meet [lemma, in Combi.SymGroup.weak_order]
+PermLattice.invset_supperm [lemma, in Combi.SymGroup.weak_order]
+PermLattice.n [abbreviation, in Combi.SymGroup.weak_order]
+PermLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.PermLattice [section, in Combi.SymGroup.weak_order]
+PermLattice.PermLattice.n0 [variable, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBLattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBLattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TLattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BLattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BPOrder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BPreorder [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinLattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinMeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_Lattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_MeetSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinJoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_JoinSemilattice [definition, in Combi.SymGroup.weak_order]
+PermLattice.supperm [definition, in Combi.SymGroup.weak_order]
+PermLattice.suppermC [lemma, in Combi.SymGroup.weak_order]
+PermLattice.suppermP [lemma, in Combi.SymGroup.weak_order]
+PermLattice.suppermPl [lemma, in Combi.SymGroup.weak_order]
+PermLattice.suppermPr [lemma, in Combi.SymGroup.weak_order]
+PermLattice.supperm_is_join [lemma, in Combi.SymGroup.weak_order]
+PermOfInvSetEltr [section, in Combi.SymGroup.presentSn]
+PermOfInvSetEltr.n0 [variable, in Combi.SymGroup.presentSn]
+'s_[ _ ] [notation, in Combi.SymGroup.presentSn]
+'s_ _ [notation, in Combi.SymGroup.presentSn]
+permuted [record, in Combi.Combi.permuted]
+Permuted [section, in Combi.Combi.permuted]
+permuted [library]
+permutedact [definition, in Combi.Combi.permuted]
+permutedact_is_action [lemma, in Combi.Combi.permuted]
+permutedact_subproof [lemma, in Combi.Combi.permuted]
+permutedP [lemma, in Combi.Combi.permuted]
+permuted_action_trans [lemma, in Combi.Combi.permuted]
+permuted_action [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__fintype_SubFinite [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__fintype_Finite [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_SubCountable [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_Countable [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_SubChoice [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_Choice [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_SubEquality [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_Equality [definition, in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_SubType [definition, in Combi.Combi.permuted]
+permuted_seq [definition, in Combi.Combi.permuted]
+permuted_tuple [definition, in Combi.Combi.permuted]
+Permuted.SizeN [section, in Combi.Combi.permuted]
+Permuted.SizeN.n [variable, in Combi.Combi.permuted]
+Permuted.T [variable, in Combi.Combi.permuted]
+perm_size_uniq [lemma, in Combi.Combi.std]
+perm_stdE [lemma, in Combi.Combi.std]
+perm_of_std [lemma, in Combi.Combi.std]
+perm_std [lemma, in Combi.Combi.std]
+perm_union_intpartn [lemma, in Combi.Combi.partition]
+perm_union_intpart [lemma, in Combi.Combi.partition]
+perm_enum_basis [lemma, in Combi.MPoly.Schur_mpoly]
+perm_smalln [lemma, in Combi.MPoly.antisym]
+perm_mpart_partm [lemma, in Combi.MPoly.antisym]
+perm_partm [lemma, in Combi.MPoly.antisym]
+perm_mpart [lemma, in Combi.MPoly.antisym]
+perm_sfilterleq [lemma, in Combi.LRrule.shuffle]
+perm_sfiltergtn [lemma, in Combi.LRrule.shuffle]
+perm_shiftn_std [lemma, in Combi.LRrule.shuffle]
+perm_shuffle [lemma, in Combi.LRrule.shuffle]
+perm_join_tab [lemma, in Combi.Combi.skewtab]
+perm_KostkaMon [lemma, in Combi.MPoly.Schur_altdef]
+perm_RS [lemma, in Combi.LRrule.Schensted]
+perm_tabword_of_tuple [lemma, in Combi.LRrule.freeSchur]
+perm_commword [lemma, in Combi.LRrule.freeSchur]
+perm_multinomial [lemma, in Combi.Combi.multinomial]
+perm_append_nth [lemma, in Combi.Combi.stdtab]
+perm_of_setpartE [lemma, in Combi.SymGroup.cycletype]
+perm_of_setpart [definition, in Combi.SymGroup.cycletype]
+perm_of_porbitE [lemma, in Combi.SymGroup.cycletype]
+perm_of_porbit [definition, in Combi.SymGroup.cycletype]
+perm_of_porbit_subproof [lemma, in Combi.SymGroup.cycletype]
+perm_decE [lemma, in Combi.SymGroup.cycles]
+perm_dec [definition, in Combi.SymGroup.cycles]
+perm_evalseq [lemma, in Combi.Combi.Yamanouchi]
+perm_display [lemma, in Combi.SymGroup.weak_order]
+perm_eq_permuted_tuple [lemma, in Combi.Combi.permuted]
+perm_invar_congr [lemma, in Combi.Basic.congr]
+perm_invar [lemma, in Combi.Basic.congr]
+perm_bound [lemma, in Combi.Basic.congr]
+perm_rembig [lemma, in Combi.Basic.ordtype]
+perm_allLtnE [lemma, in Combi.Basic.ordtype]
+perm_allLtn [lemma, in Combi.Basic.ordtype]
+perm_allLeqE [lemma, in Combi.Basic.ordtype]
+perm_allLeq [lemma, in Combi.Basic.ordtype]
+perm_on_prods_length [lemma, in Combi.SymGroup.presentSn]
+perm_on_prods_length_ord [lemma, in Combi.SymGroup.presentSn]
+perm_on_prods [lemma, in Combi.SymGroup.presentSn]
+perm_on_cocode_recP [lemma, in Combi.SymGroup.presentSn]
+perm_of_invsetK [lemma, in Combi.SymGroup.presentSn]
+perm_of_invset [definition, in Combi.SymGroup.presentSn]
+perm_of_relP [lemma, in Combi.SymGroup.presentSn]
+perm_partn [definition, in Combi.MPoly.sympoly]
+pfminh [definition, in Combi.Combi.Dyckword]
+pfminhE [lemma, in Combi.Combi.Dyckword]
+pfminhP [lemma, in Combi.Combi.Dyckword]
+pfminh_rrw [lemma, in Combi.Combi.Dyckword]
+pfminh_pos [lemma, in Combi.Combi.Dyckword]
+pfminh_min [lemma, in Combi.Combi.Dyckword]
+pfminh_size [lemma, in Combi.Combi.Dyckword]
+Pieri_colpartn [lemma, in Combi.LRrule.therule]
+Pieri_rowpartn [lemma, in Combi.LRrule.therule]
+pihomog_sym [lemma, in Combi.MPoly.homogsym]
+pihomog_mPo [lemma, in Combi.MPoly.sympoly]
+Pla [abbreviation, in Combi.MPoly.homogsym]
+plactcongr [definition, in Combi.LRrule.plactic]
+PlactDual [section, in Combi.LRrule.plactic]
+PlactDual.Alph [variable, in Combi.LRrule.plactic]
+PlactDual.disp [variable, in Combi.LRrule.plactic]
+PlactDual.word [variable, in Combi.LRrule.plactic]
+plactic [library]
+plactic_RS [lemma, in Combi.LRrule.Greene_inv]
+plactic_shapeRS [lemma, in Combi.LRrule.Greene_inv]
+plactic_shapeRS_row_proof [lemma, in Combi.LRrule.Greene_inv]
+plactic_filter_lt [lemma, in Combi.LRrule.plactic]
+plactic_filter_le [lemma, in Combi.LRrule.plactic]
+plactic_filter_gt [lemma, in Combi.LRrule.plactic]
+plactic_filter_ge [lemma, in Combi.LRrule.plactic]
+plactrule [definition, in Combi.LRrule.plactic]
+plactruleP [lemma, in Combi.LRrule.plactic]
+plactrule_homog [lemma, in Combi.LRrule.plactic]
+plactrule_sym [lemma, in Combi.LRrule.plactic]
+plact_changeUT [lemma, in Combi.LRrule.freeSchur]
+plact_changeUT_drop [lemma, in Combi.LRrule.freeSchur]
+plact_changeUT_take [lemma, in Combi.LRrule.freeSchur]
+plact_from_yam [lemma, in Combi.LRrule.Yam_plact]
+plact_map_in_incr [lemma, in Combi.LRrule.plactic]
+plact_from_dualE [lemma, in Combi.LRrule.plactic]
+plact_dualE [lemma, in Combi.LRrule.plactic]
+plact_from_revdual [lemma, in Combi.LRrule.plactic]
+plact_revdual [lemma, in Combi.LRrule.plactic]
+plact_uniq_revE [lemma, in Combi.LRrule.plactic]
+plact_uniq_rev [lemma, in Combi.LRrule.plactic]
+plact_col [lemma, in Combi.LRrule.plactic]
+plact_row [lemma, in Combi.LRrule.plactic]
+plact_homog [lemma, in Combi.LRrule.plactic]
+plact_cat [definition, in Combi.LRrule.plactic]
+plact_catr [definition, in Combi.LRrule.plactic]
+plact_catl [definition, in Combi.LRrule.plactic]
+plact_rcons [definition, in Combi.LRrule.plactic]
+plact_cons [definition, in Combi.LRrule.plactic]
+plact_is_congr [lemma, in Combi.LRrule.plactic]
+plact_trans [lemma, in Combi.LRrule.plactic]
+plact_ltrans [lemma, in Combi.LRrule.plactic]
+plact_sym [lemma, in Combi.LRrule.plactic]
+plact_refl [lemma, in Combi.LRrule.plactic]
+plact_equiv [lemma, in Combi.LRrule.plactic]
+plact1 [definition, in Combi.LRrule.plactic]
+plact1dual [lemma, in Combi.LRrule.plactic]
+plact1I [lemma, in Combi.LRrule.plactic]
+plact1i [definition, in Combi.LRrule.plactic]
+plact1idual [lemma, in Combi.LRrule.plactic]
+plact1iP [lemma, in Combi.LRrule.plactic]
+plact1i_homog [lemma, in Combi.LRrule.plactic]
+plact1P [lemma, in Combi.LRrule.plactic]
+plact1_ge [lemma, in Combi.LRrule.plactic]
+plact1_homog [lemma, in Combi.LRrule.plactic]
+plact2 [definition, in Combi.LRrule.plactic]
+plact2dual [lemma, in Combi.LRrule.plactic]
+plact2I [lemma, in Combi.LRrule.plactic]
+plact2i [definition, in Combi.LRrule.plactic]
+plact2idual [lemma, in Combi.LRrule.plactic]
+plact2iP [lemma, in Combi.LRrule.plactic]
+plact2i_homog [lemma, in Combi.LRrule.plactic]
+plact2P [lemma, in Combi.LRrule.plactic]
+plact2_ge [lemma, in Combi.LRrule.plactic]
+plact2_homog [lemma, in Combi.LRrule.plactic]
+Plla [abbreviation, in Combi.MPoly.homogsym]
+Pm [abbreviation, in Combi.MPoly.Schur_altdef]
+pnval [projection, in Combi.Combi.partition]
+points [projection, in ALEA.Qmeasure]
+pol [abbreviation, in Combi.MPoly.Cauchy]
+Pol [abbreviation, in Combi.MPoly.homogsym]
+polX [abbreviation, in Combi.MPoly.Cauchy]
+polXY [abbreviation, in Combi.MPoly.Cauchy]
+polXY [definition, in Combi.MPoly.Cauchy]
+polXY_scaleAr [lemma, in Combi.MPoly.Cauchy]
+polXY_scaleAl [lemma, in Combi.MPoly.Cauchy]
+polXY_scale [definition, in Combi.MPoly.Cauchy]
+polX_XY_is_linear [lemma, in Combi.MPoly.Cauchy]
+polX_XY_is_monoid_morphism [lemma, in Combi.MPoly.Cauchy]
+polX_XY_is_zmod_morphism [lemma, in Combi.MPoly.Cauchy]
+polX_XY [definition, in Combi.MPoly.Cauchy]
+polY [abbreviation, in Combi.MPoly.Cauchy]
+polylang [definition, in Combi.LRrule.freeSchur]
+polyXY_scale [lemma, in Combi.MPoly.Cauchy]
+polyX_inj [lemma, in Combi.MPoly.antisym]
+polY_XY_is_linear [lemma, in Combi.MPoly.Cauchy]
+polY_XY_is_monoid_morphism [lemma, in Combi.MPoly.Cauchy]
+polY_XY_is_zmod_morphism [lemma, in Combi.MPoly.Cauchy]
+polY_XY [definition, in Combi.MPoly.Cauchy]
+polZ [abbreviation, in Combi.MPoly.Cauchy]
+PorbitBijection [section, in Combi.SymGroup.cycletype]
+PorbitBijection.CM [variable, in Combi.SymGroup.cycletype]
+PorbitBijection.s [variable, in Combi.SymGroup.cycletype]
+PorbitBijection.t [variable, in Combi.SymGroup.cycletype]
+PorbitBijection.U [variable, in Combi.SymGroup.cycletype]
+PorbitBijection.V [variable, in Combi.SymGroup.cycletype]
+porbitgrpE [lemma, in Combi.SymGroup.permcent]
+porbitPb [lemma, in Combi.SymGroup.cycletype]
+porbits_tinj [lemma, in Combi.SymGroup.towerSn]
+porbits_tperm [lemma, in Combi.SymGroup.cycletype]
+porbits_perm_of_setpart [lemma, in Combi.SymGroup.cycletype]
+porbits_of_set [lemma, in Combi.SymGroup.cycletype]
+porbits_conjg [lemma, in Combi.SymGroup.cycletype]
+porbits_map [record, in Combi.SymGroup.cycletype]
+porbit_permcycles [lemma, in Combi.SymGroup.permcent]
+porbit_tinj_rshift [lemma, in Combi.SymGroup.towerSn]
+porbit_tinj_lshift [lemma, in Combi.SymGroup.towerSn]
+porbit_tpermR [lemma, in Combi.SymGroup.cycletype]
+porbit_tpermL [lemma, in Combi.SymGroup.cycletype]
+porbit_tpermD [lemma, in Combi.SymGroup.cycletype]
+porbit_set_of_set [lemma, in Combi.SymGroup.cycletype]
+porbit_conjg [lemma, in Combi.SymGroup.cycletype]
+porbit_cymap [lemma, in Combi.SymGroup.cycletype]
+porbit_cymapcan [lemma, in Combi.SymGroup.cycletype]
+porbit_set_of_disjoint [lemma, in Combi.SymGroup.cycles]
+porbit_set_restr [lemma, in Combi.SymGroup.cycles]
+porbit_restr_perm [lemma, in Combi.SymGroup.cycles]
+porbit_set_astabs [lemma, in Combi.SymGroup.cycles]
+porbit_set_eq0 [lemma, in Combi.SymGroup.cycles]
+porbit_set [definition, in Combi.SymGroup.cycles]
+porbit_mod [lemma, in Combi.SymGroup.cycles]
+porbit_fix [lemma, in Combi.SymGroup.cycles]
+pos [abbreviation, in Combi.LRrule.Schensted]
+posbig [definition, in Combi.Basic.ordtype]
+posbigE [lemma, in Combi.Basic.ordtype]
+posbig_invseq [lemma, in Combi.LRrule.stdplact]
+posbig_take_dropE [lemma, in Combi.Basic.ordtype]
+posbig_size [lemma, in Combi.Basic.ordtype]
+posbig_size_cons [lemma, in Combi.Basic.ordtype]
+pqpair [projection, in Combi.LRrule.Schensted]
+pqpair_inj [lemma, in Combi.LRrule.Schensted]
+predi [definition, in Combi.MPoly.antisym]
+predi_eltrpE [lemma, in Combi.MPoly.antisym]
+predi_eltrp [lemma, in Combi.MPoly.antisym]
+predLR_bij_LRsupport [lemma, in Combi.LRrule.freeSchur]
+pred_LRtriple_conj [lemma, in Combi.LRrule.shuffle]
+pred_LRtriple_fast_filter_gt [lemma, in Combi.LRrule.shuffle]
+pred_LRtriple_fast [definition, in Combi.LRrule.shuffle]
+pred_LRtriple [definition, in Combi.LRrule.shuffle]
+pred_LRtriple_fast_bijLRyam [lemma, in Combi.LRrule.therule]
+pred0_std [lemma, in Combi.LRrule.shuffle]
+prefixes [definition, in Combi.Combi.Dyckword]
+PreimPartition [section, in Combi.Combi.Dyckword]
+PreimPartition.D [variable, in Combi.Combi.Dyckword]
+PreimPartition.f [variable, in Combi.Combi.Dyckword]
+PreimPartition.rT [variable, in Combi.Combi.Dyckword]
+PreimPartition.T [variable, in Combi.Combi.Dyckword]
+preimset_trivIset [lemma, in Combi.SSRcomplements.tools]
+preim_Dyck_of_balE [lemma, in Combi.Combi.Dyckword]
+PresentationSn [section, in Combi.SymGroup.presentSn]
+PresentationSn.eltrG [variable, in Combi.SymGroup.presentSn]
+PresentationSn.gT [variable, in Combi.SymGroup.presentSn]
+PresentationSn.n [variable, in Combi.SymGroup.presentSn]
+'g_ _ [notation, in Combi.SymGroup.presentSn]
+presentation_S4 [lemma, in Combi.SymGroup.presentSn]
+presentation_S3 [lemma, in Combi.SymGroup.presentSn]
+presentation_S2 [lemma, in Combi.SymGroup.presentSn]
+presentation_Sn_eltr [lemma, in Combi.SymGroup.presentSn]
+presentSn [library]
+presentSn_codesz__canonical__fintype_SubFinite [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__fintype_Finite [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_SubCountable [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_Countable [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_SubChoice [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_Choice [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_SubEquality [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_Equality [definition, in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_SubType [definition, in Combi.SymGroup.presentSn]
+prob_choose_corner_ends_at [lemma, in Combi.HookFormula.hook]
+ProdGen [section, in Combi.MPoly.sympoly]
+ProdGen.co [variable, in Combi.MPoly.sympoly]
+ProdGen.Defs [section, in Combi.MPoly.sympoly]
+ProdGen.Defs.gen [variable, in Combi.MPoly.sympoly]
+ProdGen.Defs.gen_homog [variable, in Combi.MPoly.sympoly]
+'g_ _ [notation, in Combi.MPoly.sympoly]
+'g[ _ ] [notation, in Combi.MPoly.sympoly]
+ProdGen.gA [variable, in Combi.MPoly.sympoly]
+ProdGen.gB [variable, in Combi.MPoly.sympoly]
+ProdGen.n [variable, in Combi.MPoly.sympoly]
+ProdGen.R [variable, in Combi.MPoly.sympoly]
+'co[ _ ]_ _ [notation, in Combi.MPoly.sympoly]
+'co[ _ ] [notation, in Combi.MPoly.sympoly]
+'gA_ _ [notation, in Combi.MPoly.sympoly]
+'gA[ _ ] [notation, in Combi.MPoly.sympoly]
+'gB_ _ [notation, in Combi.MPoly.sympoly]
+'gB[ _ ] [notation, in Combi.MPoly.sympoly]
+prodsK [lemma, in Combi.SymGroup.presentSn]
+prodsV [lemma, in Combi.SymGroup.presentSn]
+prods_straighten [lemma, in Combi.SymGroup.presentSn]
+prods_reducesE [lemma, in Combi.SymGroup.presentSn]
+prods_wordcd_inj [lemma, in Combi.SymGroup.presentSn]
+prods_codesz_bij [lemma, in Combi.SymGroup.presentSn]
+prods_codesz [definition, in Combi.SymGroup.presentSn]
+prods_iota_ltmi [lemma, in Combi.SymGroup.presentSn]
+prods_iota_mi [lemma, in Combi.SymGroup.presentSn]
+prod_sympXY [lemma, in Combi.MPoly.Cauchy]
+prod_conjg [lemma, in Combi.SymGroup.towerSn]
+prod_hook_length_quot_row [lemma, in Combi.HookFormula.hook]
+prod_of_disjoint [lemma, in Combi.SymGroup.cycles]
+prod_homog [lemma, in Combi.MPoly.sympoly]
+prod_partsum [definition, in Combi.MPoly.sympoly]
+prod_symp_homog [definition, in Combi.MPoly.sympoly]
+prod_symp [definition, in Combi.MPoly.sympoly]
+prod_symh_homog [definition, in Combi.MPoly.sympoly]
+prod_symh [definition, in Combi.MPoly.sympoly]
+prod_syme_homog [definition, in Combi.MPoly.sympoly]
+prod_syme [definition, in Combi.MPoly.sympoly]
+prod_prodgen [lemma, in Combi.MPoly.sympoly]
+prod_gen_cast [lemma, in Combi.MPoly.sympoly]
+prod_gen_colpartn [lemma, in Combi.MPoly.sympoly]
+prod_genM [lemma, in Combi.MPoly.sympoly]
+prod_gen0 [lemma, in Combi.MPoly.sympoly]
+prod_gen_homog [lemma, in Combi.MPoly.sympoly]
+prod_gen [definition, in Combi.MPoly.sympoly]
+psupport [definition, in Combi.SymGroup.cycles]
+psupports_perm_of_porbit [lemma, in Combi.SymGroup.cycletype]
+psupport_perm_of_porbit [lemma, in Combi.SymGroup.cycletype]
+psupport_conjg [lemma, in Combi.SymGroup.cycletype]
+psupport_of_disjoint [lemma, in Combi.SymGroup.cycles]
+psupport_disjointC [lemma, in Combi.SymGroup.cycles]
+psupport_cycle_dec [lemma, in Combi.SymGroup.cycles]
+psupport_restr_perm [lemma, in Combi.SymGroup.cycles]
+psupport_restr_perm_incl [lemma, in Combi.SymGroup.cycles]
+psupport_card_porbit [lemma, in Combi.SymGroup.cycles]
+psupport_stable [lemma, in Combi.SymGroup.cycles]
+psupport_eq0 [lemma, in Combi.SymGroup.cycles]
+psupport_perm_on [lemma, in Combi.SymGroup.cycles]
+psupport_expg [lemma, in Combi.SymGroup.cycles]
+psupport1 [lemma, in Combi.SymGroup.cycles]
+pval [projection, in Combi.Combi.partition]
+pyampair [projection, in Combi.LRrule.Schensted]
+pyampair_inj [lemma, in Combi.LRrule.Schensted]
+

Q

+Qmeasure [library]
+QTableau [section, in Combi.LRrule.Schensted]
+QTableau.disp [variable, in Combi.LRrule.Schensted]
+QTableau.T [variable, in Combi.LRrule.Schensted]
+

R

+Random [definition, in ALEA.Qmeasure]
+Random_in_range [lemma, in ALEA.Qmeasure]
+Random_eq_out [lemma, in ALEA.Qmeasure]
+Random_eq_in [lemma, in ALEA.Qmeasure]
+Random_simpl [lemma, in ALEA.Qmeasure]
+ratO [instance, in ALEA.Qmeasure]
+rcast_com [lemma, in Combi.LRrule.Greene]
+rclass [definition, in Combi.Basic.congr]
+RCons [section, in Combi.Combi.subseq]
+rconsK [lemma, in Combi.SSRcomplements.tools]
+rcons_nilF [lemma, in Combi.SSRcomplements.tools]
+rcons_set_nth [lemma, in Combi.SSRcomplements.tools]
+rcons_bal_of_Dyck [lemma, in Combi.Combi.Dyckword]
+rcons_rcons [lemma, in Combi.LRrule.plactic]
+RCons.T [variable, in Combi.Combi.subseq]
+recforest [definition, in Combi.Combi.ordtree]
+rectree [definition, in Combi.Combi.ordtree]
+Recursion [section, in Combi.Combi.ordtree]
+Recursion.HPnil [variable, in Combi.Combi.ordtree]
+Recursion.IHforest [variable, in Combi.Combi.ordtree]
+Recursion.IHtree [variable, in Combi.Combi.ordtree]
+Recursion.P [variable, in Combi.Combi.ordtree]
+Recursion.PF [variable, in Combi.Combi.ordtree]
+rec_tree [lemma, in Combi.Combi.ordtree]
+reduced [abbreviation, in Combi.SymGroup.presentSn]
+reduced [abbreviation, in Combi.SymGroup.presentSn]
+Reduced [section, in Combi.SymGroup.presentSn]
+reducedM [lemma, in Combi.SymGroup.presentSn]
+reducedP [lemma, in Combi.SymGroup.presentSn]
+reduced_braid [lemma, in Combi.SymGroup.presentSn]
+reduced_wcord [lemma, in Combi.SymGroup.presentSn]
+reduced_rconsK [lemma, in Combi.SymGroup.presentSn]
+reduced_consK [lemma, in Combi.SymGroup.presentSn]
+reduced_catl [lemma, in Combi.SymGroup.presentSn]
+reduced_catr [lemma, in Combi.SymGroup.presentSn]
+reduced_sprod_code [lemma, in Combi.SymGroup.presentSn]
+reduced_revE [lemma, in Combi.SymGroup.presentSn]
+reduced_rev [lemma, in Combi.SymGroup.presentSn]
+reduced_iiF [lemma, in Combi.SymGroup.presentSn]
+reduced_nil [lemma, in Combi.SymGroup.presentSn]
+reduced_word [definition, in Combi.SymGroup.presentSn]
+Reduced.n [variable, in Combi.SymGroup.presentSn]
+_ =Br _ [notation, in Combi.SymGroup.presentSn]
+'s_[ _ ] [notation, in Combi.SymGroup.presentSn]
+'s_ _ [notation, in Combi.SymGroup.presentSn]
+reduceP [lemma, in Combi.SymGroup.presentSn]
+reduces [definition, in Combi.SymGroup.presentSn]
+reducesP [lemma, in Combi.SymGroup.presentSn]
+reduces_catl [lemma, in Combi.SymGroup.presentSn]
+RefinementOrder [module, in Combi.Combi.composition]
+RefinementOrder.botEcompnref [lemma, in Combi.Combi.composition]
+RefinementOrder.compnref_rev [lemma, in Combi.Combi.composition]
+RefinementOrder.compnref_display [lemma, in Combi.Combi.composition]
+RefinementOrder.composition_IsoTop__to__Order_hasTop [definition, in Combi.Combi.composition]
+RefinementOrder.composition_IsoBottom__to__Order_hasBottom [definition, in Combi.Combi.composition]
+RefinementOrder.descset_colcompn [lemma, in Combi.Combi.composition]
+RefinementOrder.descset_rowcompn [lemma, in Combi.Combi.composition]
+RefinementOrder.descset_join [lemma, in Combi.Combi.composition]
+RefinementOrder.descset_meet [lemma, in Combi.Combi.composition]
+RefinementOrder.descset_mono [lemma, in Combi.Combi.composition]
+RefinementOrder.eqtype_SubType__to__eqtype_isSub [definition, in Combi.Combi.composition]
+RefinementOrder.Exports [module, in Combi.Combi.composition]
+RefinementOrder.Exports.botEcompnref [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.compnref_rev [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.descset_join [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.descset_meet [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.descset_mono [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.intcompnref [abbreviation, in Combi.Combi.composition]
+RefinementOrder.Exports.leEcompnref [definition, in Combi.Combi.composition]
+RefinementOrder.Exports.topEcompnref [definition, in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__fintype_isFinite [definition, in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__choice_hasChoice [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_73 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_71 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_70 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_68 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_67 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_63 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_62 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_61 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_58 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_57 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_55 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_54 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_53 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_50 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_49 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_48 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_47 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_46 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_41 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_40 [definition, in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_38 [definition, in Combi.Combi.composition]
+RefinementOrder.leEcompnref [lemma, in Combi.Combi.composition]
+RefinementOrder.Order_IsoDistrLattice__to__Order_Lattice_isDistributive [definition, in Combi.Combi.composition]
+RefinementOrder.Order_IsoLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.Order_IsoLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.Order_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.Order_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.ordtype_isInhabitedType__to__ordtype_isInhabited [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder [section, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBDistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhTBLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBDistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TDistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BDistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinDistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_DistrLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinMeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhLattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_Lattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_MeetSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinJoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_JoinSemilattice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinite [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_Inhabited [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinPOrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinPreorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_POrder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_Preorder [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__fintype_SubFinite [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__fintype_Finite [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_SubCountable [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_Countable [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_SubChoice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_Choice [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_SubEquality [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_Equality [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_SubType [definition, in Combi.Combi.composition]
+RefinementOrder.RefinementOrder.n [variable, in Combi.Combi.composition]
+'CRef [notation, in Combi.Combi.composition]
+RefinementOrder.SetIn [abbreviation, in Combi.Combi.composition]
+RefinementOrder.topEcompnref [lemma, in Combi.Combi.composition]
+RefinementOrder.type [definition, in Combi.Combi.composition]
+RefinmentOrder [module, in Combi.Combi.setpartition]
+RefinmentOrder.Exports [module, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.Finer [section, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.Finer.S [variable, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.Finer.T [variable, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.is_finer_subpartP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.is_finer_pblockP [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.is_finerP [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.join_finer_eq [abbreviation, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.join_finerE [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.join_finer_eq_in_S [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.mem_meet_finerP [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.setpart_topE [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.Exports.setpart_bottomE [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_28 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_27 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_24 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_22 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_20 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_19 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_18 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_15 [definition, in Combi.Combi.setpartition]
+RefinmentOrder.is_finerP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_setpart_anti [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_trans [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_refl [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_pblockP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.is_finer [definition, in Combi.Combi.setpartition]
+RefinmentOrder.join_finerP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.join_finerC [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.join_finer [definition, in Combi.Combi.setpartition]
+RefinmentOrder.join_finer_equivalence [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.join_finer_eq [definition, in Combi.Combi.setpartition]
+RefinmentOrder.le_join_finer [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.le_meet_finer [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.MeetSpec [constructor, in Combi.Combi.setpartition]
+RefinmentOrder.meet_finerP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.meet_finerC [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.meet_spec [inductive, in Combi.Combi.setpartition]
+RefinmentOrder.meet_finer [definition, in Combi.Combi.setpartition]
+RefinmentOrder.meet_finer_subproof [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.mem_meet_finerP [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Order_Le_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder [section, in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder.S [variable, in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder.T [variable, in Combi.Combi.setpartition]
+RefinmentOrder.setpartfiner_display [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhTBLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhFinLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TLattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_Lattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TMeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_MeetSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TJoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_JoinSemilattice [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhFinPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinPreorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhPOrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_POrder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_Preorder [definition, in Combi.Combi.setpartition]
+RefinmentOrder.setpart_conn [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.setpart1_bottom [lemma, in Combi.Combi.setpartition]
+RefinmentOrder.trivsetpart_top [lemma, in Combi.Combi.setpartition]
+reflexive [projection, in ALEA.Ccpo]
+RelatSn [constructor, in Combi.SymGroup.presentSn]
+relat_Sn [inductive, in Combi.SymGroup.presentSn]
+rembig [definition, in Combi.Basic.ordtype]
+rembigE [lemma, in Combi.Basic.ordtype]
+rembigP [lemma, in Combi.Basic.ordtype]
+rembig_ins_std [lemma, in Combi.Combi.std]
+rembig_RS [lemma, in Combi.LRrule.plactic]
+rembig_RS_last_big [lemma, in Combi.LRrule.plactic]
+rembig_plactcongr [lemma, in Combi.LRrule.plactic]
+rembig_plact [lemma, in Combi.LRrule.plactic]
+rembig_plact2i [lemma, in Combi.LRrule.plactic]
+rembig_plact2 [lemma, in Combi.LRrule.plactic]
+rembig_plact1i [lemma, in Combi.LRrule.plactic]
+rembig_plact1 [lemma, in Combi.LRrule.plactic]
+rembig_iota [lemma, in Combi.Basic.ordtype]
+rembig_uniq [lemma, in Combi.Basic.ordtype]
+rembig_subseq [lemma, in Combi.Basic.ordtype]
+rembig_rev_uniq [lemma, in Combi.Basic.ordtype]
+rembig_eq_permR [lemma, in Combi.Basic.ordtype]
+rembig_eq_permL [lemma, in Combi.Basic.ordtype]
+rembig_cat [lemma, in Combi.Basic.ordtype]
+rembig_catL [lemma, in Combi.Basic.ordtype]
+rembig_catR [lemma, in Combi.Basic.ordtype]
+remn [definition, in Combi.Combi.stdtab]
+remnP [lemma, in Combi.Combi.stdtab]
+remn_rec [definition, in Combi.Combi.stdtab]
+RemoveBig [section, in Combi.LRrule.plactic]
+RemoveBig [section, in Combi.Basic.ordtype]
+RemoveBig.Alph [variable, in Combi.LRrule.plactic]
+RemoveBig.disp [variable, in Combi.LRrule.plactic]
+RemoveBig.disp [variable, in Combi.Basic.ordtype]
+RemoveBig.T [variable, in Combi.Basic.ordtype]
+RemoveBig.word [variable, in Combi.LRrule.plactic]
+RemoveBig.Z [variable, in Combi.Basic.ordtype]
+rem_corner_conj_part [lemma, in Combi.Combi.partition]
+rem_corner_incr_first_nE [lemma, in Combi.Combi.partition]
+rem_corner_incr_first_n [lemma, in Combi.Combi.partition]
+rem_corners_uniq [lemma, in Combi.Combi.partition]
+rem_corners [definition, in Combi.Combi.partition]
+rem_corner_incr_nth [lemma, in Combi.Combi.partition]
+rem_trail0 [definition, in Combi.Combi.partition]
+rem_irr [lemma, in Combi.SymGroup.Frobenius_char]
+rem_irr1 [lemma, in Combi.SymGroup.Frobenius_char]
+reprS [abbreviation, in Combi.SymGroup.reprSn]
+reprSn [library]
+repr_S2 [lemma, in Combi.SymGroup.reprSn]
+repr1 [lemma, in Combi.SymGroup.reprSn]
+repr1_S1 [lemma, in Combi.SymGroup.reprSn]
+repr1_S0 [lemma, in Combi.SymGroup.reprSn]
+res [abbreviation, in Combi.Combi.skewpart]
+reshape_index_walk_to [lemma, in Combi.HookFormula.hook]
+Restriction [section, in Combi.SymGroup.towerSn]
+Restriction.m [variable, in Combi.SymGroup.towerSn]
+Restriction.n [variable, in Combi.SymGroup.towerSn]
+RestrIntervBig [section, in Combi.LRrule.plactic]
+RestrIntervBig.Alph [variable, in Combi.LRrule.plactic]
+RestrIntervBig.disp [variable, in Combi.LRrule.plactic]
+RestrIntervBig.L [variable, in Combi.LRrule.plactic]
+RestrIntervBig.word [variable, in Combi.LRrule.plactic]
+RestrIntervSmall [section, in Combi.LRrule.plactic]
+RestrIntervSmall.Alph [variable, in Combi.LRrule.plactic]
+RestrIntervSmall.disp [variable, in Combi.LRrule.plactic]
+RestrIntervSmall.word [variable, in Combi.LRrule.plactic]
+restr_perm_porbits [lemma, in Combi.SymGroup.permcent]
+restr_perm_genC [lemma, in Combi.SymGroup.permcent]
+restr_perm_inj [lemma, in Combi.SymGroup.cycles]
+restr_perm_psupportE [lemma, in Combi.SymGroup.cycles]
+restr_perm_neq [lemma, in Combi.SymGroup.cycles]
+res_tab_inj [lemma, in Combi.MPoly.Schur_altdef]
+res_tabK [lemma, in Combi.MPoly.Schur_altdef]
+res_tabP [lemma, in Combi.MPoly.Schur_altdef]
+res_tab [definition, in Combi.MPoly.Schur_altdef]
+Rev [section, in Combi.LRrule.Greene]
+Rev [section, in Combi.LRrule.plactic]
+RevConj [section, in Combi.LRrule.Greene_inv]
+RevConj.Alph [variable, in Combi.LRrule.Greene_inv]
+RevConj.d [variable, in Combi.LRrule.Greene_inv]
+revdual [definition, in Combi.LRrule.plactic]
+revdualK [lemma, in Combi.LRrule.plactic]
+revset [definition, in Combi.LRrule.Greene]
+revsetK [lemma, in Combi.LRrule.Greene]
+rev_is_ksupp_col [lemma, in Combi.LRrule.Greene_inv]
+rev_is_ksupp_row [lemma, in Combi.LRrule.Greene_inv]
+rev_enum [lemma, in Combi.LRrule.Greene_inv]
+rev_ksuppKV [lemma, in Combi.LRrule.Greene_inv]
+rev_ksuppK [lemma, in Combi.LRrule.Greene_inv]
+rev_set_inj [lemma, in Combi.LRrule.Greene_inv]
+rev_ord_cast_inj [lemma, in Combi.LRrule.Greene_inv]
+rev_ksupp_inv [definition, in Combi.LRrule.Greene_inv]
+rev_ksupp [definition, in Combi.LRrule.Greene_inv]
+rev_set [definition, in Combi.LRrule.Greene_inv]
+rev_ord_cast [definition, in Combi.LRrule.Greene_inv]
+rev_intcompnK [lemma, in Combi.Combi.composition]
+rev_intcompn [definition, in Combi.Combi.composition]
+rev_intcompn_spec [lemma, in Combi.Combi.composition]
+Rev.Alph [variable, in Combi.LRrule.Greene]
+Rev.Alph [variable, in Combi.LRrule.plactic]
+Rev.disp [variable, in Combi.LRrule.plactic]
+Rev.word [variable, in Combi.LRrule.plactic]
+Rew [constructor, in Combi.Basic.congr]
+rewrite_path_min [lemma, in Combi.Basic.congr]
+rewrite_path_sym [lemma, in Combi.Basic.congr]
+rewrite_path_trans [lemma, in Combi.Basic.congr]
+rewrite_path [inductive, in Combi.Basic.congr]
+rho [abbreviation, in Combi.MPoly.antisym]
+rho [abbreviation, in Combi.MPoly.antisym]
+rho [definition, in Combi.MPoly.antisym]
+rho [abbreviation, in Combi.MPoly.Schur_altdef]
+rho [abbreviation, in Combi.MPoly.Schur_altdef]
+rho [abbreviation, in Combi.MPoly.Schur_altdef]
+rho [abbreviation, in Combi.MPoly.Schur_altdef]
+rho [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+rho_uniq [lemma, in Combi.MPoly.antisym]
+rho_iota [lemma, in Combi.MPoly.antisym]
+RHSL3 [definition, in Combi.HookFormula.hook]
+RHSL3_trace [definition, in Combi.HookFormula.hook]
+ribbon [definition, in Combi.Combi.skewpart]
+ribbonE [lemma, in Combi.Combi.skewpart]
+RibbonOn [section, in Combi.Combi.skewpart]
+RibbonOn.Hrib [variable, in Combi.Combi.skewpart]
+RibbonOn.inner [variable, in Combi.Combi.skewpart]
+RibbonOn.outer [variable, in Combi.Combi.skewpart]
+RibbonOn.partinn [variable, in Combi.Combi.skewpart]
+RibbonOn.partout [variable, in Combi.Combi.skewpart]
+RibbonOn.start [variable, in Combi.Combi.skewpart]
+RibbonOn.stop [variable, in Combi.Combi.skewpart]
+ribbonP [lemma, in Combi.Combi.skewpart]
+ribbontb_start [definition, in Combi.Combi.skewpart]
+ribbontb_start_subproof [lemma, in Combi.Combi.skewpart]
+ribbontb_stop_ltn [lemma, in Combi.Combi.skewpart]
+ribbon_textbookE [lemma, in Combi.Combi.skewpart]
+ribbon_textbook_on [lemma, in Combi.Combi.skewpart]
+ribbon_on_no_square [lemma, in Combi.Combi.skewpart]
+ribbon_on_conn4_skew [lemma, in Combi.Combi.skewpart]
+ribbon_on_conn4 [lemma, in Combi.Combi.skewpart]
+ribbon_on_conn4_box_ex [lemma, in Combi.Combi.skewpart]
+ribbon_on_box_ex [definition, in Combi.Combi.skewpart]
+ribbon_on_box_exP [lemma, in Combi.Combi.skewpart]
+ribbon_on_box_exE [lemma, in Combi.Combi.skewpart]
+ribbon_box_ex [definition, in Combi.Combi.skewpart]
+ribbon_textbook [definition, in Combi.Combi.skewpart]
+ribbon_addE [lemma, in Combi.Combi.skewpart]
+ribbon_on_addE [lemma, in Combi.Combi.skewpart]
+ribbon_startrem [lemma, in Combi.Combi.skewpart]
+ribbon_on_startrem [lemma, in Combi.Combi.skewpart]
+ribbon_on_startrem_acc [lemma, in Combi.Combi.skewpart]
+ribbon_on0_startrem [lemma, in Combi.Combi.skewpart]
+ribbon_sumn_diffE [lemma, in Combi.Combi.skewpart]
+ribbon_sumn_lt [lemma, in Combi.Combi.skewpart]
+ribbon_included [lemma, in Combi.Combi.skewpart]
+ribbon_from_included [lemma, in Combi.Combi.skewpart]
+ribbon_mindropeq [lemma, in Combi.Combi.skewpart]
+ribbon_from_mindropeq [lemma, in Combi.Combi.skewpart]
+ribbon_fromP [lemma, in Combi.Combi.skewpart]
+ribbon_on_inj [lemma, in Combi.Combi.skewpart]
+ribbon_on_sumn [lemma, in Combi.Combi.skewpart]
+ribbon_on_startE [lemma, in Combi.Combi.skewpart]
+ribbon_on_stopE [lemma, in Combi.Combi.skewpart]
+ribbon_on_mindropeq [lemma, in Combi.Combi.skewpart]
+ribbon_on_height [lemma, in Combi.Combi.skewpart]
+ribbon_on_stop_lt [lemma, in Combi.Combi.skewpart]
+ribbon_on_start_le [lemma, in Combi.Combi.skewpart]
+ribbon_on_included [lemma, in Combi.Combi.skewpart]
+ribbon_on_is_skew [lemma, in Combi.Combi.skewpart]
+ribbon_on_nth_leq [lemma, in Combi.Combi.skewpart]
+ribbon_on_start_stop [lemma, in Combi.Combi.skewpart]
+ribbon_onSS [lemma, in Combi.Combi.skewpart]
+ribbon_noneq [lemma, in Combi.Combi.skewpart]
+ribbon_from_noneq [lemma, in Combi.Combi.skewpart]
+ribbon_consK [lemma, in Combi.Combi.skewpart]
+ribbon_from_impl [lemma, in Combi.Combi.skewpart]
+ribbon_height [definition, in Combi.Combi.skewpart]
+ribbon_on [definition, in Combi.Combi.skewpart]
+ribbon_from [definition, in Combi.Combi.skewpart]
+rightcomb [definition, in Combi.Combi.bintree]
+rightcombsz [definition, in Combi.Combi.bintree]
+rightcomb_rotationsE [lemma, in Combi.Combi.bintree]
+rightcomb_rotations [lemma, in Combi.Combi.bintree]
+right_sizesK [lemma, in Combi.Combi.bintree]
+right_sizesP [lemma, in Combi.Combi.bintree]
+right_sizes_cat_left [lemma, in Combi.Combi.bintree]
+right_sizes_from_left [lemma, in Combi.Combi.bintree]
+right_sizes_left_comb [lemma, in Combi.Combi.bintree]
+right_sizes [definition, in Combi.Combi.bintree]
+RingSchurSym [section, in Combi.MPoly.Schur_altdef]
+RingSchurSym.n0 [variable, in Combi.MPoly.Schur_altdef]
+RingSchurSym.R [variable, in Combi.MPoly.Schur_altdef]
+'s_ _ [notation, in Combi.MPoly.Schur_altdef]
+rotationP [lemma, in Combi.Combi.bintree]
+rotations [definition, in Combi.Combi.bintree]
+rotations_right_sizesP [lemma, in Combi.Combi.bintree]
+rotations_add_bounded [lemma, in Combi.Combi.bintree]
+rotations_add [lemma, in Combi.Combi.bintree]
+rotations_add_head [lemma, in Combi.Combi.bintree]
+rotations_vctleq_impl [lemma, in Combi.Combi.bintree]
+rotations_neq [lemma, in Combi.Combi.bintree]
+rotations_flip [lemma, in Combi.Combi.bintree]
+rotations_flip_impl [lemma, in Combi.Combi.bintree]
+rotations_right_sub [lemma, in Combi.Combi.bintree]
+rotations_left_sub [lemma, in Combi.Combi.bintree]
+rot_is_Dyck [lemma, in Combi.Combi.Dyckword]
+rot_pfminhE [lemma, in Combi.Combi.Dyckword]
+rowcomp [definition, in Combi.Combi.composition]
+rowcompn [definition, in Combi.Combi.composition]
+rowcompnP [lemma, in Combi.Combi.composition]
+rowpartn [definition, in Combi.Combi.partition]
+rowpartnE [lemma, in Combi.Combi.partition]
+rowpartnSE [lemma, in Combi.Combi.partition]
+rowpartn_subproof [lemma, in Combi.Combi.partition]
+rowpartn0E [lemma, in Combi.Combi.partition]
+Rows [section, in Combi.Combi.tableau]
+RowsAndCols [section, in Combi.LRrule.plactic]
+RowsAndCols.Alph [variable, in Combi.LRrule.plactic]
+RowsAndCols.disp [variable, in Combi.LRrule.plactic]
+RowsAndCols.word [variable, in Combi.LRrule.plactic]
+Rows.disp [variable, in Combi.Combi.tableau]
+Rows.T [variable, in Combi.Combi.tableau]
+row_hb_strip [lemma, in Combi.Combi.skewtab]
+row_dominate [lemma, in Combi.Combi.tableau]
+row_lt_by_pos [lemma, in Combi.Combi.tableau]
+row_free1 [lemma, in Combi.SymGroup.reprSn]
+rrw_bal1 [lemma, in Combi.Combi.Dyckword]
+RS [definition, in Combi.LRrule.extract]
+RS [definition, in Combi.LRrule.Schensted]
+RSbij [definition, in Combi.LRrule.Schensted]
+RSbijinv [definition, in Combi.LRrule.Schensted]
+RSbijinvnat [definition, in Combi.LRrule.extract]
+RSbijnat [definition, in Combi.LRrule.extract]
+RSclass [definition, in Combi.LRrule.Schensted]
+RSclassE [lemma, in Combi.LRrule.Schensted]
+RSclassP [lemma, in Combi.LRrule.Schensted]
+RSclass_countE [lemma, in Combi.LRrule.Schensted]
+rsh [abbreviation, in Combi.LRrule.Greene]
+rshift_in_lshift_recF [lemma, in Combi.LRrule.Greene]
+rshift_recP [lemma, in Combi.LRrule.Greene]
+rsh_rec [definition, in Combi.LRrule.Greene]
+RSinvstdE [lemma, in Combi.LRrule.stdplact]
+RSmap [definition, in Combi.LRrule.Schensted]
+RSmapE [lemma, in Combi.LRrule.Schensted]
+RSmapinv [definition, in Combi.LRrule.Schensted]
+RSmapinv2 [definition, in Combi.LRrule.Schensted]
+RSmapinv2K [lemma, in Combi.LRrule.Schensted]
+RSmapK [lemma, in Combi.LRrule.Schensted]
+RSmap_std [lemma, in Combi.LRrule.stdplact]
+RSmap_spec [lemma, in Combi.LRrule.Schensted]
+RSmap_rev [definition, in Combi.LRrule.Schensted]
+rspair [record, in Combi.LRrule.Schensted]
+RSpairyamQ [lemma, in Combi.LRrule.freeSchur]
+RSperm [lemma, in Combi.LRrule.Schensted]
+rsplit_rec_tab [lemma, in Combi.LRrule.Greene]
+rsplit_rec [definition, in Combi.LRrule.Greene]
+RSstdE [lemma, in Combi.LRrule.Schensted]
+RStab [definition, in Combi.LRrule.Schensted]
+RStabE [lemma, in Combi.LRrule.freeSchur]
+RStabinv [definition, in Combi.LRrule.Schensted]
+RStabinvK [lemma, in Combi.LRrule.Schensted]
+RStabinvnat [definition, in Combi.LRrule.extract]
+RStabK [lemma, in Combi.LRrule.Schensted]
+RStabmap [definition, in Combi.LRrule.Schensted]
+RStabmapE [lemma, in Combi.LRrule.Schensted]
+RSTabmapstdE [lemma, in Combi.LRrule.stdplact]
+RStabmap_std [lemma, in Combi.LRrule.stdplact]
+RStabmap_spec [lemma, in Combi.LRrule.Schensted]
+RStabnat [definition, in Combi.LRrule.extract]
+rstabpair [record, in Combi.LRrule.Schensted]
+RSToPlactic [section, in Combi.LRrule.plactic]
+RSToPlactic.Alph [variable, in Combi.LRrule.plactic]
+RSToPlactic.disp [variable, in Combi.LRrule.plactic]
+RSToPlactic.word [variable, in Combi.LRrule.plactic]
+rsymrel [definition, in Combi.SymGroup.presentSn]
+rsymrel_total [lemma, in Combi.SymGroup.presentSn]
+rsymrel_trans [lemma, in Combi.SymGroup.presentSn]
+rsymrel_anti [lemma, in Combi.SymGroup.presentSn]
+rsymrel_refl [lemma, in Combi.SymGroup.presentSn]
+rsym_invsetP [lemma, in Combi.SymGroup.presentSn]
+rsym_invset_total [lemma, in Combi.SymGroup.presentSn]
+rsym_invset_trans [lemma, in Combi.SymGroup.presentSn]
+rsym_invset_anti [lemma, in Combi.SymGroup.presentSn]
+rsym_invset_refl [lemma, in Combi.SymGroup.presentSn]
+RS_rev_uniq [lemma, in Combi.LRrule.Greene_inv]
+RS_tabE [lemma, in Combi.LRrule.Greene_inv]
+RS_rev [definition, in Combi.LRrule.Schensted]
+RS_yam [lemma, in Combi.LRrule.Yam_plact]
+RS_yam_RS [lemma, in Combi.LRrule.Yam_plact]
+rtrans [definition, in Combi.Basic.congr]
+rtransP [lemma, in Combi.Basic.congr]
+rtrans_min [lemma, in Combi.Basic.congr]
+rtrans_ind [lemma, in Combi.Basic.congr]
+rule_gencongr [lemma, in Combi.Basic.congr]
+rule_congrrule [lemma, in Combi.Basic.congr]
+rule_rtrans [lemma, in Combi.Basic.congr]
+

S

+S [abbreviation, in Combi.MPoly.sympoly]
+same_connect_rev [lemma, in Combi.Combi.skewpart]
+Scalar [section, in Combi.MPoly.Cauchy]
+ScalarChange [section, in Combi.MPoly.antisym]
+ScalarChange [section, in Combi.MPoly.sympoly]
+ScalarChange.mor [variable, in Combi.MPoly.antisym]
+ScalarChange.mor [variable, in Combi.MPoly.sympoly]
+ScalarChange.n [variable, in Combi.MPoly.antisym]
+ScalarChange.n0 [variable, in Combi.MPoly.sympoly]
+ScalarChange.R [variable, in Combi.MPoly.antisym]
+ScalarChange.R [variable, in Combi.MPoly.sympoly]
+ScalarChange.S [variable, in Combi.MPoly.antisym]
+ScalarChange.S [variable, in Combi.MPoly.sympoly]
+ScalarProduct [section, in Combi.MPoly.homogsym]
+ScalarProduct.d [variable, in Combi.MPoly.homogsym]
+ScalarProduct.Hd [variable, in Combi.MPoly.homogsym]
+ScalarProduct.n0 [variable, in Combi.MPoly.homogsym]
+'[ _ | _ ] (ring_scope) [notation, in Combi.MPoly.homogsym]
+'[ _ | _ ] (ring_scope) [notation, in Combi.MPoly.homogsym]
+Scalar.d [variable, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_73.mu [variable, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_73.la [variable, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_73.hb_instance_73 [section, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_67.p [variable, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_67.hb_instance_67 [section, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_62.la [variable, in Combi.MPoly.Cauchy]
+Scalar.hb_instance_62.hb_instance_62 [section, in Combi.MPoly.Cauchy]
+Scalar.Hd [variable, in Combi.MPoly.Cauchy]
+Scalar.n0 [variable, in Combi.MPoly.Cauchy]
+_ (X) [notation, in Combi.MPoly.Cauchy]
+_ (Y) [notation, in Combi.MPoly.Cauchy]
+'hpC[ _ ] [notation, in Combi.MPoly.Cauchy]
+'hsC[ _ ] [notation, in Combi.MPoly.Cauchy]
+scale_polXYE [lemma, in Combi.MPoly.Cauchy]
+scale_polXYDl [lemma, in Combi.MPoly.Cauchy]
+scale_polXYDr [lemma, in Combi.MPoly.Cauchy]
+scale_polXY1m [lemma, in Combi.MPoly.Cauchy]
+scale_polXYA [lemma, in Combi.MPoly.Cauchy]
+Sch [definition, in Combi.LRrule.Schensted]
+Schensted [library]
+Schensted_rstabpair__canonical__choice_SubChoice [definition, in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__choice_Choice [definition, in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_SubEquality [definition, in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_Equality [definition, in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_SubType [definition, in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__choice_SubChoice [definition, in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__choice_Choice [definition, in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_SubEquality [definition, in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_Equality [definition, in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_SubType [definition, in Combi.LRrule.Schensted]
+Schur [definition, in Combi.MPoly.Schur_mpoly]
+Schur [section, in Combi.MPoly.Schur_mpoly]
+Schur [abbreviation, in Combi.LRrule.freeSchur]
+Schur [abbreviation, in Combi.LRrule.therule]
+Schur [abbreviation, in Combi.LRrule.therule]
+Schur [abbreviation, in Combi.LRrule.implem]
+Schur [section, in Combi.MPoly.sympoly]
+SchurAlternantDef [section, in Combi.MPoly.Schur_altdef]
+SchurAlternantDef.n0 [variable, in Combi.MPoly.Schur_altdef]
+SchurAlternantDef.R [variable, in Combi.MPoly.Schur_altdef]
+'a_ _ [notation, in Combi.MPoly.Schur_altdef]
+'s_[ _ ] [notation, in Combi.MPoly.Schur_altdef]
+SchurComRingType [section, in Combi.MPoly.Schur_mpoly]
+SchurComRingType.n0 [variable, in Combi.MPoly.Schur_mpoly]
+SchurComRingType.R [variable, in Combi.MPoly.Schur_mpoly]
+SchurE [lemma, in Combi.LRrule.freeSchur]
+Schur_rowpartn [lemma, in Combi.MPoly.Schur_mpoly]
+Schur_oversize [lemma, in Combi.MPoly.Schur_mpoly]
+Schur_tabsh_readingE [lemma, in Combi.MPoly.Schur_mpoly]
+Schur_homog [lemma, in Combi.MPoly.Schur_altdef]
+Schur_sym [lemma, in Combi.MPoly.Schur_altdef]
+Schur_sym_idomain [lemma, in Combi.MPoly.Schur_altdef]
+Schur_cast [lemma, in Combi.MPoly.Schur_altdef]
+Schur_freeSchurE [lemma, in Combi.LRrule.freeSchur]
+Schur_altdef [library]
+Schur_mpoly [library]
+Schur.n0 [variable, in Combi.MPoly.Schur_mpoly]
+Schur.n0 [variable, in Combi.MPoly.sympoly]
+Schur.R [variable, in Combi.MPoly.Schur_mpoly]
+Schur.R [variable, in Combi.MPoly.sympoly]
+'s[ _ ] [notation, in Combi.MPoly.sympoly]
+Schur0 [lemma, in Combi.MPoly.Schur_mpoly]
+Schur1 [lemma, in Combi.MPoly.Schur_mpoly]
+Schutzenberger_shuffle_plact [lemma, in Combi.LRrule.shuffle]
+Sch_max_size [lemma, in Combi.LRrule.Schensted]
+Sch_leq_last [lemma, in Combi.LRrule.Schensted]
+Sch_exists [lemma, in Combi.LRrule.Schensted]
+Sch_size [lemma, in Combi.LRrule.Schensted]
+Sch_rcons [lemma, in Combi.LRrule.Schensted]
+Sch_rev [definition, in Combi.LRrule.Schensted]
+Sch_plact [lemma, in Combi.LRrule.plactic]
+scover [abbreviation, in Combi.LRrule.Greene]
+SeqLemmas [section, in Combi.SSRcomplements.tools]
+SeqLemmas.T [variable, in Combi.SSRcomplements.tools]
+seq_masks_uniq [lemma, in Combi.Combi.subseq]
+seq_lift_right [definition, in ALEA.Ccpo]
+seq_lift_left [definition, in ALEA.Ccpo]
+seq_finType [definition, in Combi.Basic.combclass]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear__82 [definition, in Combi.MPoly.homogsym]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [definition, in Combi.MPoly.homogsym]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [definition, in Combi.SymGroup.towerSn]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [definition, in Combi.SymGroup.Frobenius_char]
+SetAct [section, in Combi.SSRcomplements.permcomp]
+setactC [lemma, in Combi.SSRcomplements.permcomp]
+setactI [lemma, in Combi.SSRcomplements.permcomp]
+setactU [lemma, in Combi.SSRcomplements.permcomp]
+setactU1 [lemma, in Combi.SSRcomplements.permcomp]
+SetAct.aT [variable, in Combi.SSRcomplements.permcomp]
+SetAct.D [variable, in Combi.SSRcomplements.permcomp]
+SetAct.rT [variable, in Combi.SSRcomplements.permcomp]
+SetAct.to [variable, in Combi.SSRcomplements.permcomp]
+setact0 [lemma, in Combi.SSRcomplements.permcomp]
+setact1 [lemma, in Combi.SSRcomplements.permcomp]
+SetContainingBothLeft [module, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case [section, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.a [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Alph [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.b [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn [section, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.Hposb [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.HT [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.T [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BNotIn [section, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BNotIn.HbNin [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.c [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.disp [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Hposa [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Hposc [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.HRabc [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.HS [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.k [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.P [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Px [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.R [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.S [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.u [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.v [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.word [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.x [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.x' [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.y [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.coverS1T1 [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.cover_bin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.disjointS1T1 [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.enumUltV [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.exists_Qy [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.exists_Q_noboth [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_bT [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_Tb [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_cS [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_Sa [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_T1E [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_S1E [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_swap_setSE [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_SE [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hba [projection, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hbax [projection, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hbc [projection, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hxba [projection, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.hypRabc [record, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.inPQE [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ksupp_bin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ksupp_bnotin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posa [abbreviation, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posaS1_bin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posaT1_bin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posa_inTF [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posb [abbreviation, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posbinSF [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posb_inSF [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posc [definition, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posc_inTF [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posc_subproof [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbin [definition, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbin_noboth [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbnotin [definition, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbnotin_noboth [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RabcGtnX [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RabcLeqX [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis [section, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis.Alph [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis.disp [variable, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Rtrans [projection, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.size_cover_bnotin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_T1 [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_S1 [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_swap_set_S [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_swap_set [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ST_cover_disjoint [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.swap [abbreviation, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.swap_set [abbreviation, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.S1 [definition, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.S1_subsST [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.tnth_posc [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.trivIset_Qbin [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.TSneq [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.TS_disjoint [lemma, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.T1 [definition, in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.T1_subsST [lemma, in Combi.LRrule.Greene_inv]
+setdiff [definition, in Combi.MPoly.Schur_altdef]
+setpart [record, in Combi.Combi.setpartition]
+SetPartition [section, in Combi.SSRcomplements.tools]
+setpartition [library]
+SetPartitionShape [section, in Combi.Combi.partition]
+SetPartitionShape.T [variable, in Combi.Combi.partition]
+setpartition_setpart__canonical__ordtype_InhFinite [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__ordtype_Inhabited [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__fintype_SubFinite [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__fintype_Finite [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_SubCountable [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_Countable [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_SubChoice [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_Choice [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_SubEquality [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_Equality [definition, in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_SubType [definition, in Combi.Combi.setpartition]
+SetPartition.T [variable, in Combi.SSRcomplements.tools]
+setpartP [lemma, in Combi.Combi.setpartition]
+setpartval [projection, in Combi.Combi.setpartition]
+setpart_shape_imset [lemma, in Combi.Combi.partition]
+setpart_shape_union [lemma, in Combi.Combi.partition]
+setpart_shapeP [lemma, in Combi.Combi.partition]
+setpart_shape [definition, in Combi.Combi.partition]
+setpart_set1_eq_set1 [lemma, in Combi.Combi.setpartition]
+setpart_set1 [definition, in Combi.Combi.setpartition]
+setpart_set0_eq_set0 [lemma, in Combi.Combi.setpartition]
+setpart_set0 [definition, in Combi.Combi.setpartition]
+setpart_eq [lemma, in Combi.Combi.setpartition]
+setpart_subset [lemma, in Combi.Combi.setpartition]
+setpart_non0 [lemma, in Combi.Combi.setpartition]
+setpart1 [definition, in Combi.Combi.setpartition]
+setpart1_subproof [lemma, in Combi.Combi.setpartition]
+setU1E [lemma, in Combi.SSRcomplements.tools]
+set_nth_non_nil [lemma, in Combi.SSRcomplements.tools]
+set_nth_rcons [lemma, in Combi.SSRcomplements.tools]
+set_head_default [lemma, in Combi.SSRcomplements.tools]
+set_nil [lemma, in Combi.LRrule.Greene]
+set_nth_LxR [lemma, in Combi.LRrule.plactic]
+set1_disjoint [lemma, in Combi.SSRcomplements.tools]
+SF [abbreviation, in Combi.MPoly.homogsym]
+SF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+SF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+SF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+SF [abbreviation, in Combi.MPoly.MurnaghanNakayama]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+SF [abbreviation, in Combi.MPoly.sympoly]
+sfiltergtn [definition, in Combi.LRrule.shuffle]
+sfiltergtn_invstd [lemma, in Combi.LRrule.shuffle]
+sfiltergtn_is_std [lemma, in Combi.LRrule.shuffle]
+sfilterleq [definition, in Combi.LRrule.shuffle]
+sfilterleqE [lemma, in Combi.LRrule.shuffle]
+sfilterleqK [lemma, in Combi.LRrule.shuffle]
+sfilterleq_invstd [lemma, in Combi.LRrule.shuffle]
+sfilterleq_is_std [lemma, in Combi.LRrule.shuffle]
+sfilterleq_LRsupport_skew [lemma, in Combi.LRrule.therule]
+sfilterleq_LRsupportP [lemma, in Combi.LRrule.therule]
+shaped_hyper_stdtabnP [lemma, in Combi.LRrule.freeSchur]
+shape_invseq [lemma, in Combi.LRrule.stdplact]
+shape_RS_std [lemma, in Combi.LRrule.stdplact]
+shape_RS_revdual [lemma, in Combi.LRrule.Greene_inv]
+shape_join_tab_skew_reshape [lemma, in Combi.Combi.skewtab]
+shape_join_tab [lemma, in Combi.Combi.skewtab]
+shape_inner_filter_le [lemma, in Combi.Combi.skewtab]
+shape_skew_reshape [lemma, in Combi.Combi.skewtab]
+shape_tabcols [lemma, in Combi.LRrule.Greene]
+shape_res_tab [definition, in Combi.MPoly.Schur_altdef]
+shape_res_tab_subproof [lemma, in Combi.MPoly.Schur_altdef]
+shape_RStabmapE [lemma, in Combi.LRrule.Schensted]
+shape_instabnrowinv1 [lemma, in Combi.LRrule.Schensted]
+shape_RSmap_eq [lemma, in Combi.LRrule.Schensted]
+shape_instabnrow [lemma, in Combi.LRrule.Schensted]
+shape_dropRS [lemma, in Combi.LRrule.freeSchur]
+shape_takeRS [lemma, in Combi.LRrule.freeSchur]
+shape_bij_LRsupport [lemma, in Combi.LRrule.freeSchur]
+shape_hyper_stdtabnP [lemma, in Combi.LRrule.freeSchur]
+shape_RS_yam [lemma, in Combi.LRrule.Yam_plact]
+shape_yamtab [lemma, in Combi.LRrule.Yam_plact]
+shape_conj_tab [lemma, in Combi.Combi.stdtab]
+shape_deg_stdtabn_of_sh [lemma, in Combi.Combi.stdtab]
+shape_deg [definition, in Combi.Combi.stdtab]
+shape_stdtabsh [lemma, in Combi.Combi.stdtab]
+shape_yam_of_stdtab [lemma, in Combi.Combi.stdtab]
+shape_stdtab_of_yam [lemma, in Combi.Combi.stdtab]
+shape_append_nth [lemma, in Combi.Combi.stdtab]
+shape_map_tab [lemma, in Combi.Combi.tableau]
+shape_tabsh [lemma, in Combi.Combi.tableau]
+shape_bijLR [lemma, in Combi.LRrule.therule]
+shape0 [lemma, in Combi.Combi.stdtab]
+shaps_eq [projection, in Combi.LRrule.implem]
+shcols [definition, in Combi.LRrule.Greene]
+shcol_cards [lemma, in Combi.LRrule.Greene]
+shift [definition, in ALEA.Ccpo]
+ShiftedShuffle [section, in Combi.LRrule.shuffle]
+shiftinv_pos_homo [lemma, in Combi.Basic.ordtype]
+shiftinv_posK [lemma, in Combi.Basic.ordtype]
+shiftinv_pos [definition, in Combi.Basic.ordtype]
+shiftn [definition, in Combi.LRrule.shuffle]
+shiftn_skew_dominate [lemma, in Combi.LRrule.therule]
+shiftset [definition, in Combi.LRrule.Greene]
+shiftuK [lemma, in Combi.LRrule.shuffle]
+shift_plactcongr [lemma, in Combi.LRrule.shuffle]
+shift_mon2 [instance, in ALEA.Ccpo]
+shift_fun_mon [instance, in ALEA.Ccpo]
+shift_le_compat [lemma, in ALEA.Ccpo]
+shift_simpl [lemma, in ALEA.Ccpo]
+shift_mon_fun [instance, in ALEA.Ccpo]
+shift_pos_mono [lemma, in Combi.Basic.ordtype]
+shift_posK [lemma, in Combi.Basic.ordtype]
+shift_pos [definition, in Combi.Basic.ordtype]
+shrows [definition, in Combi.LRrule.Greene]
+shrows_cards [lemma, in Combi.LRrule.Greene]
+shsh [definition, in Combi.LRrule.shuffle]
+shsh_sfilterleq [lemma, in Combi.LRrule.shuffle]
+shsh_sfiltergtn [lemma, in Combi.LRrule.shuffle]
+shuffle [definition, in Combi.LRrule.shuffle]
+shuffle [library]
+shuffleC [lemma, in Combi.LRrule.shuffle]
+shuffle_nil [lemma, in Combi.LRrule.shuffle]
+shuffle_from_rec [definition, in Combi.LRrule.shuffle]
+signed_repr [definition, in Combi.SymGroup.reprSn]
+signed_mx_repr [lemma, in Combi.SymGroup.reprSn]
+signed_mx [definition, in Combi.SymGroup.reprSn]
+sign_Chi2 [lemma, in Combi.SymGroup.reprSn]
+sign_char2 [lemma, in Combi.SymGroup.reprSn]
+sign_charP [lemma, in Combi.SymGroup.reprSn]
+sign_char [definition, in Combi.SymGroup.reprSn]
+sign_char_subproof [lemma, in Combi.SymGroup.reprSn]
+sign_irr [lemma, in Combi.SymGroup.reprSn]
+sign_repr [definition, in Combi.SymGroup.reprSn]
+sign_mx_repr [lemma, in Combi.SymGroup.reprSn]
+sign_mx [definition, in Combi.SymGroup.reprSn]
+SimpleCalculation [lemma, in Combi.HookFormula.hook]
+SimpleRecursion [section, in Combi.Combi.ordtree]
+SimpleRecursion.IHtree [variable, in Combi.Combi.ordtree]
+SimpleRecursion.P [variable, in Combi.Combi.ordtree]
+simplexp [definition, in Combi.MPoly.antisym]
+Singleton [section, in Combi.Combi.setpartition]
+Singleton.T [variable, in Combi.Combi.setpartition]
+Singleton.x [variable, in Combi.Combi.setpartition]
+Size [section, in Combi.Combi.bintree]
+sizeIk [lemma, in Combi.LRrule.Greene]
+SizeN [section, in Combi.Combi.bintree]
+SizeN.n [variable, in Combi.Combi.bintree]
+size_invstd [lemma, in Combi.Combi.std]
+size_invseq [lemma, in Combi.Combi.std]
+size_leq_invseq [lemma, in Combi.Combi.std]
+size_all_leq [lemma, in Combi.Combi.std]
+size_std [lemma, in Combi.Combi.std]
+size_std_rec [lemma, in Combi.Combi.std]
+size_sdtn [lemma, in Combi.Combi.std]
+size_enum_partn [lemma, in Combi.Combi.partition]
+size_enum_partns [lemma, in Combi.Combi.partition]
+size_enum_partnsk [lemma, in Combi.Combi.partition]
+size_hookpartn [lemma, in Combi.Combi.partition]
+size_rowpartn [lemma, in Combi.Combi.partition]
+size_colpartn [lemma, in Combi.Combi.partition]
+size_diff_shape [lemma, in Combi.Combi.partition]
+size_included [lemma, in Combi.Combi.partition]
+size_conj_part [lemma, in Combi.Combi.partition]
+size_incr_first_n [lemma, in Combi.Combi.partition]
+size_rem_trail0 [lemma, in Combi.Combi.partition]
+size_part [lemma, in Combi.Combi.partition]
+size_vctmin [lemma, in Combi.Combi.bintree]
+size_from_vct [lemma, in Combi.Combi.bintree]
+size_from_vct_acc [lemma, in Combi.Combi.bintree]
+size_right_sizes [lemma, in Combi.Combi.bintree]
+size_rotations [lemma, in Combi.Combi.bintree]
+size_flip [lemma, in Combi.Combi.bintree]
+size_leftcomb [lemma, in Combi.Combi.bintree]
+size_rightcomb [lemma, in Combi.Combi.bintree]
+size_left_branch [lemma, in Combi.Combi.bintree]
+size_cat_left [lemma, in Combi.Combi.bintree]
+size_from_left [lemma, in Combi.Combi.bintree]
+size_mem_enum_bintreeszP [lemma, in Combi.Combi.bintree]
+size_enum_bintreeszE [lemma, in Combi.Combi.bintree]
+size_enum_bintreesz [lemma, in Combi.Combi.bintree]
+size_Catalan_bin_leq [lemma, in Combi.Combi.bintree]
+size_tree_eq0 [lemma, in Combi.Combi.bintree]
+size_tree [definition, in Combi.Combi.bintree]
+size_cover_rev [lemma, in Combi.LRrule.Greene_inv]
+size_rev_ksupp [lemma, in Combi.LRrule.Greene_inv]
+size_partm [lemma, in Combi.MPoly.antisym]
+size_langQ [lemma, in Combi.LRrule.shuffle]
+size_sfilterleq_cat [lemma, in Combi.LRrule.shuffle]
+size_sfiltergtn_cat [lemma, in Combi.LRrule.shuffle]
+size_sfilterleq [lemma, in Combi.LRrule.shuffle]
+size_sfiltergtn [lemma, in Combi.LRrule.shuffle]
+size_shuffle [lemma, in Combi.LRrule.shuffle]
+size_cut_k [lemma, in Combi.Combi.vectNK]
+size_join_tab [lemma, in Combi.Combi.skewtab]
+size_skew_reshape [lemma, in Combi.Combi.skewtab]
+size_row_extract [lemma, in Combi.LRrule.Greene]
+size_cover_tabcolsk [lemma, in Combi.LRrule.Greene]
+size_cover_tabrows [lemma, in Combi.LRrule.Greene]
+size_tabcols_cons [lemma, in Combi.LRrule.Greene]
+size_to_word_cons [lemma, in Combi.LRrule.Greene]
+size_tabrows [lemma, in Combi.LRrule.Greene]
+size_shrows [lemma, in Combi.LRrule.Greene]
+size_tabcols [lemma, in Combi.LRrule.Greene]
+size_shcols [lemma, in Combi.LRrule.Greene]
+size_shcols_cons [lemma, in Combi.LRrule.Greene]
+size_cover_inj [lemma, in Combi.LRrule.Greene]
+size_extract [lemma, in Combi.LRrule.Greene]
+size_coverI [lemma, in Combi.LRrule.Greene]
+size_RS [lemma, in Combi.LRrule.Schensted]
+size_instab [lemma, in Combi.LRrule.Schensted]
+size_invins [lemma, in Combi.LRrule.Schensted]
+size_RSmap2 [lemma, in Combi.LRrule.Schensted]
+size_ndec_Sch [lemma, in Combi.LRrule.Schensted]
+size_ins_non_0 [lemma, in Combi.LRrule.Schensted]
+size_ins_sup [lemma, in Combi.LRrule.Schensted]
+size_ins_inf [lemma, in Combi.LRrule.Schensted]
+size_tracel [lemma, in Combi.HookFormula.hook]
+size_tracer [lemma, in Combi.HookFormula.hook]
+size_hook_box_indices [lemma, in Combi.HookFormula.hook]
+size_RSmapinv2_yam [lemma, in Combi.LRrule.freeSchur]
+size_RS_tuple [lemma, in Combi.LRrule.freeSchur]
+size_bal_of_Dyck [lemma, in Combi.Combi.Dyckword]
+size_Dyck_of_bal [lemma, in Combi.Combi.Dyckword]
+size_UDn [lemma, in Combi.Combi.Dyckword]
+size_UnDn [lemma, in Combi.Combi.Dyckword]
+size_bintree_of_Dyck [lemma, in Combi.Combi.Dyckword]
+size_Dyck_of_bintree [lemma, in Combi.Combi.Dyckword]
+size_foldr_join_Dyck [lemma, in Combi.Combi.Dyckword]
+size_count_braceE [lemma, in Combi.Combi.Dyckword]
+size_conj_tab [lemma, in Combi.Combi.stdtab]
+size_tab_stdtabn [lemma, in Combi.Combi.stdtab]
+size_yam_of_stdtab [lemma, in Combi.Combi.stdtab]
+size_notin_stdtab_of_yam [lemma, in Combi.Combi.stdtab]
+size_yam_of_stdtab_rec [lemma, in Combi.Combi.stdtab]
+size_tab_remn [lemma, in Combi.Combi.stdtab]
+size_stdtab_of_yam [lemma, in Combi.Combi.stdtab]
+size_append_nth [lemma, in Combi.Combi.stdtab]
+size_cycle_type [lemma, in Combi.SymGroup.cycletype]
+size_revdual [lemma, in Combi.LRrule.plactic]
+size_plact [lemma, in Combi.LRrule.plactic]
+size_tabsh [lemma, in Combi.Combi.tableau]
+size_to_word [lemma, in Combi.Combi.tableau]
+size_tab [definition, in Combi.Combi.tableau]
+size_zip2 [lemma, in Combi.LRrule.therule]
+size_leq_skew_reshape [lemma, in Combi.LRrule.therule]
+size_mem_enum_ordtreeszP [lemma, in Combi.Combi.ordtree]
+size_ord_to_bintree [lemma, in Combi.Combi.ordtree]
+size_bin_to_ordtree [lemma, in Combi.Combi.ordtree]
+size_tree_eq1 [lemma, in Combi.Combi.ordtree]
+size_ordtree_pos [lemma, in Combi.Combi.ordtree]
+size_ordtreeE [lemma, in Combi.Combi.ordtree]
+size_ordtree [definition, in Combi.Combi.ordtree]
+size_yamn [lemma, in Combi.Combi.Yamanouchi]
+size_yameval [lemma, in Combi.Combi.Yamanouchi]
+size_hyper_yam [lemma, in Combi.Combi.Yamanouchi]
+size_add_ribbon [lemma, in Combi.Combi.skewpart]
+size_count_mem_undup [lemma, in Combi.Combi.permuted]
+size_permuted_seq [lemma, in Combi.Combi.permuted]
+size_permuted_tuple [lemma, in Combi.Combi.permuted]
+size_invar_congr [lemma, in Combi.Basic.congr]
+size_invar_refl [lemma, in Combi.Basic.congr]
+size_invar [lemma, in Combi.Basic.congr]
+size_bound [lemma, in Combi.Basic.congr]
+size_rembig [lemma, in Combi.Basic.ordtype]
+size_straighten [lemma, in Combi.SymGroup.presentSn]
+size_inscode [lemma, in Combi.SymGroup.presentSn]
+size_braid [lemma, in Combi.SymGroup.presentSn]
+size_canword [lemma, in Combi.SymGroup.presentSn]
+size_cocode [lemma, in Combi.SymGroup.presentSn]
+size_cocode_rec [lemma, in Combi.SymGroup.presentSn]
+size_codesz [lemma, in Combi.SymGroup.presentSn]
+size_wordcd [lemma, in Combi.SymGroup.presentSn]
+size_LRyamtab_listE [lemma, in Combi.LRrule.implem]
+size_mpart_in_supp [lemma, in Combi.MPoly.sympoly]
+size_partsums [lemma, in Combi.Combi.composition]
+size_comp [lemma, in Combi.Combi.composition]
+Size.n [variable, in Combi.Combi.bintree]
+skew [projection, in Combi.LRrule.implem]
+skewpart [library]
+skewtab [library]
+skew_reshapeK [lemma, in Combi.Combi.skewtab]
+skew_reshape [definition, in Combi.Combi.skewtab]
+skew_dominate_cut [lemma, in Combi.Combi.skewtab]
+skew_dominate_consl [lemma, in Combi.Combi.skewtab]
+skew_dominate_no_overlap [lemma, in Combi.Combi.skewtab]
+skew_dominate_take [lemma, in Combi.Combi.skewtab]
+skew_dominate0 [lemma, in Combi.Combi.skewtab]
+skew_dominate [definition, in Combi.Combi.skewtab]
+skew_yam_included [lemma, in Combi.Combi.skewtab]
+skew_yam_catK [lemma, in Combi.Combi.skewtab]
+skew_yam_consK [lemma, in Combi.Combi.skewtab]
+skew_yam_catrK [lemma, in Combi.Combi.skewtab]
+skew_yam_cat [lemma, in Combi.Combi.skewtab]
+skew_nil_yamE [lemma, in Combi.Combi.skewtab]
+skew_yam_nil [lemma, in Combi.Combi.skewtab]
+slporbits [definition, in Combi.SymGroup.cycletype]
+Sn [section, in Combi.SymGroup.cycletype]
+SnXm [abbreviation, in Combi.SymGroup.towerSn]
+SnXm [abbreviation, in Combi.SymGroup.towerSn]
+Sn.n [variable, in Combi.SymGroup.cycletype]
+sorted [abbreviation, in Combi.SSRcomplements.sorted]
+Sorted [section, in Combi.SSRcomplements.sorted]
+sorted [library]
+sorted_geq_nth0E [lemma, in Combi.Combi.partition]
+sorted_geq_count_leq2E [lemma, in Combi.Combi.partition]
+sorted_std_extract [lemma, in Combi.LRrule.stdplact]
+sorted_gt_tabcols [lemma, in Combi.LRrule.Greene]
+sorted_leqX_tabrows [lemma, in Combi.LRrule.Greene]
+sorted_extract_cond [lemma, in Combi.LRrule.Greene]
+sorted_leq_last [lemma, in Combi.HookFormula.hook]
+sorted_in_leq_last [lemma, in Combi.HookFormula.hook]
+sorted_center [lemma, in Combi.LRrule.plactic]
+sorted_subseq_iota_rcons [lemma, in Combi.Combi.subseq]
+sorted_subseq_inP [lemma, in Combi.Combi.subseq]
+sorted_subseqP [lemma, in Combi.Combi.subseq]
+sorted_sumn_iotaE [lemma, in Combi.SSRcomplements.sorted]
+sorted_ltn_ind [lemma, in Combi.SSRcomplements.sorted]
+sorted_lt_by_pos [lemma, in Combi.SSRcomplements.sorted]
+sorted_last [lemma, in Combi.SSRcomplements.sorted]
+sorted_cons [lemma, in Combi.SSRcomplements.sorted]
+sorted_strictP [lemma, in Combi.SSRcomplements.sorted]
+sorted_rcons [lemma, in Combi.SSRcomplements.sorted]
+sorted_rconsK [lemma, in Combi.SSRcomplements.sorted]
+sorted_consK [lemma, in Combi.SSRcomplements.sorted]
+sorted_is_part [lemma, in Combi.LRrule.implem]
+sorted_ltn_partsums [lemma, in Combi.Combi.composition]
+Sorted.Hanti [variable, in Combi.SSRcomplements.sorted]
+Sorted.R [variable, in Combi.SSRcomplements.sorted]
+Sorted.Rrefl [variable, in Combi.SSRcomplements.sorted]
+Sorted.Rtrans [variable, in Combi.SSRcomplements.sorted]
+Sorted.T [variable, in Combi.SSRcomplements.sorted]
+Sorted.Z [variable, in Combi.SSRcomplements.sorted]
+_ <=R _ [notation, in Combi.SSRcomplements.sorted]
+sorted2P [lemma, in Combi.SSRcomplements.sorted]
+SP [abbreviation, in Combi.MPoly.sympoly]
+Spec [section, in Combi.Combi.std]
+Spec [section, in Combi.Combi.skewpart]
+Spec [section, in Combi.LRrule.implem]
+Spec.disp1 [variable, in Combi.Combi.std]
+Spec.disp2 [variable, in Combi.Combi.std]
+Spec.d1 [variable, in Combi.LRrule.implem]
+Spec.d2 [variable, in Combi.LRrule.implem]
+Spec.hgt [variable, in Combi.Combi.skewpart]
+Spec.Hincl [variable, in Combi.LRrule.implem]
+Spec.Hret [variable, in Combi.Combi.skewpart]
+Spec.nbox [variable, in Combi.Combi.skewpart]
+Spec.P [variable, in Combi.LRrule.implem]
+Spec.partsh [variable, in Combi.Combi.skewpart]
+Spec.pos [variable, in Combi.Combi.skewpart]
+Spec.P1 [variable, in Combi.LRrule.implem]
+Spec.P2 [variable, in Combi.LRrule.implem]
+Spec.res [variable, in Combi.Combi.skewpart]
+Spec.S [variable, in Combi.Combi.std]
+Spec.sh [variable, in Combi.Combi.skewpart]
+Spec.T [variable, in Combi.Combi.std]
+splitsetK [lemma, in Combi.LRrule.Greene]
+split_rec_cover [lemma, in Combi.LRrule.Greene]
+split_recabK [lemma, in Combi.LRrule.Greene]
+srel [definition, in Combi.SymGroup.presentSn]
+SRel [section, in Combi.SymGroup.presentSn]
+srelE [lemma, in Combi.SymGroup.presentSn]
+SRel.T [variable, in Combi.SymGroup.presentSn]
+ssrbool_has_quality__canonical__GRing_SubmodClosed [definition, in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_ZmodClosed [definition, in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_OppClosed [definition, in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_AddClosed [definition, in Combi.MPoly.antisym]
+SSRComplFinset [section, in Combi.SSRcomplements.tools]
+SSRComplFinset.aT [variable, in Combi.SSRcomplements.tools]
+SSRComplFinset.f [variable, in Combi.SSRcomplements.tools]
+SSRComplFinset.rT [variable, in Combi.SSRcomplements.tools]
+StabilityProperties [section, in ALEA.Qmeasure]
+StabilityProperties.A [variable, in ALEA.Qmeasure]
+StabilityProperties.m [variable, in ALEA.Qmeasure]
+StabilityProperties.Mstable_sub [variable, in ALEA.Qmeasure]
+stable [record, in ALEA.Ccpo]
+stable [inductive, in ALEA.Ccpo]
+stable_opp [definition, in ALEA.Qmeasure]
+stable_mull [definition, in ALEA.Qmeasure]
+stable_sub [definition, in ALEA.Qmeasure]
+stable_add [definition, in ALEA.Qmeasure]
+stable_intro [lemma, in ALEA.Ccpo]
+stable_def [projection, in ALEA.Ccpo]
+stable_def [constructor, in ALEA.Ccpo]
+stable2 [record, in ALEA.Ccpo]
+stable2 [inductive, in ALEA.Ccpo]
+stable2_intro [projection, in ALEA.Ccpo]
+stable2_intro [constructor, in ALEA.Ccpo]
+stab_iporbitsE [lemma, in Combi.SymGroup.permcent]
+stab_iporbitsE_prod [lemma, in Combi.SymGroup.permcent]
+stab_iporbits_map_inj [lemma, in Combi.SymGroup.permcent]
+stab_iporbits_map [definition, in Combi.SymGroup.permcent]
+stab_iporbits_porbitmap [definition, in Combi.SymGroup.permcent]
+stab_iporbits_homog [lemma, in Combi.SymGroup.permcent]
+stab_iporbits_stab [lemma, in Combi.SymGroup.permcent]
+stab_iporbits [definition, in Combi.SymGroup.permcent]
+stab_porbit [lemma, in Combi.SymGroup.permcent]
+stab_tuple_dprod [lemma, in Combi.Combi.permuted]
+stab_tuple_prod [lemma, in Combi.Combi.permuted]
+Standardisation [section, in Combi.Combi.std]
+Standardisation.Alph [variable, in Combi.Combi.std]
+Standardisation.disp [variable, in Combi.Combi.std]
+StandardWords [section, in Combi.Combi.std]
+start [abbreviation, in Combi.Combi.skewpart]
+startrem [definition, in Combi.Combi.skewpart]
+startremE [lemma, in Combi.Combi.skewpart]
+startrem_accP [lemma, in Combi.Combi.skewpart]
+startrem_accE [lemma, in Combi.Combi.skewpart]
+startrem_leq [lemma, in Combi.Combi.skewpart]
+startrem_leq_size [lemma, in Combi.Combi.skewpart]
+startrem_leq_pos [lemma, in Combi.Combi.skewpart]
+startrem_acc_geq [lemma, in Combi.Combi.skewpart]
+startrem0P [lemma, in Combi.Combi.skewpart]
+starts_at [definition, in Combi.HookFormula.hook]
+star_stable_sub [lemma, in ALEA.Qmeasure]
+star_le_compat [lemma, in ALEA.Qmeasure]
+star_monotonic [lemma, in ALEA.Qmeasure]
+star_stable_eq [lemma, in ALEA.Qmeasure]
+star_simpl [lemma, in ALEA.Qmeasure]
+std [definition, in Combi.Combi.std]
+std [library]
+StdCombClass [section, in Combi.Combi.std]
+StdCombClass.n [variable, in Combi.Combi.std]
+StdKostka [section, in Combi.MPoly.Schur_altdef]
+StdKostka.d [variable, in Combi.MPoly.Schur_altdef]
+StdKostka.la [variable, in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar [section, in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.Hd [variable, in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.n [variable, in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.td [variable, in Combi.MPoly.Schur_altdef]
+stdP [lemma, in Combi.Combi.std]
+stdplact [library]
+StdRS [section, in Combi.LRrule.stdplact]
+StdRS.Alph [variable, in Combi.LRrule.stdplact]
+StdRS.disp [variable, in Combi.LRrule.stdplact]
+StdSpec [constructor, in Combi.Combi.std]
+stdtab [library]
+StdtabCombClass [section, in Combi.Combi.stdtab]
+StdtabCombClass.n [variable, in Combi.Combi.stdtab]
+StdtabCombClass.stdtabn_enum [variable, in Combi.Combi.stdtab]
+stdtabn [record, in Combi.Combi.stdtab]
+StdtabnOfStdtabsh [section, in Combi.Combi.stdtab]
+StdtabnOfStdtabsh.n [variable, in Combi.Combi.stdtab]
+StdtabnOfStdtabsh.sh [variable, in Combi.Combi.stdtab]
+stdtabnP [lemma, in Combi.Combi.stdtab]
+stdtabnval [projection, in Combi.Combi.stdtab]
+stdtabn_of_sh [definition, in Combi.Combi.stdtab]
+stdtabn_of_sh_subproof [lemma, in Combi.Combi.stdtab]
+StdtabOfShape [section, in Combi.Combi.stdtab]
+StdtabOfShape.sh [variable, in Combi.Combi.stdtab]
+StdtabOfShape.stdtabsh_enum [variable, in Combi.Combi.stdtab]
+stdtabP [lemma, in Combi.Combi.stdtab]
+stdtabsh [record, in Combi.Combi.stdtab]
+stdtabshcast [definition, in Combi.Combi.stdtab]
+stdtabshP [lemma, in Combi.Combi.stdtab]
+stdtabshval [projection, in Combi.Combi.stdtab]
+stdtabsh_eval_to_word [lemma, in Combi.MPoly.Schur_altdef]
+stdtab_get_tabNE [lemma, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__fintype_SubFinite [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__fintype_Finite [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_SubCountable [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_Countable [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_SubChoice [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_Choice [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_SubEquality [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_Equality [definition, in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_SubType [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__fintype_SubFinite [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__fintype_Finite [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_SubCountable [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_Countable [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_SubChoice [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_Choice [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_SubEquality [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_Equality [definition, in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_SubType [definition, in Combi.Combi.stdtab]
+stdtab_of_yamK [lemma, in Combi.Combi.stdtab]
+stdtab_of_yam_inj [lemma, in Combi.Combi.stdtab]
+stdtab_of_yam_nil [lemma, in Combi.Combi.stdtab]
+stdtab_of_yamP [lemma, in Combi.Combi.stdtab]
+stdtab_of_yam [definition, in Combi.Combi.stdtab]
+StdTakeDrop [section, in Combi.Combi.std]
+StdTakeDrop.disp1 [variable, in Combi.Combi.std]
+StdTakeDrop.disp2 [variable, in Combi.Combi.std]
+StdTakeDrop.S [variable, in Combi.Combi.std]
+StdTakeDrop.T [variable, in Combi.Combi.std]
+stdwordn [record, in Combi.Combi.std]
+stdwordnP [lemma, in Combi.Combi.std]
+stdwordnval [projection, in Combi.Combi.std]
+std_cutabc [lemma, in Combi.Combi.std]
+std_transp [lemma, in Combi.Combi.std]
+std_drop_std [lemma, in Combi.Combi.std]
+std_take_std [lemma, in Combi.Combi.std]
+std_rconsK [lemma, in Combi.Combi.std]
+std_eq_invP [lemma, in Combi.Combi.std]
+std_specP [lemma, in Combi.Combi.std]
+std_spec_uniq [lemma, in Combi.Combi.std]
+std_spec [inductive, in Combi.Combi.std]
+std_eq_inv [lemma, in Combi.Combi.std]
+std_stdE [lemma, in Combi.Combi.std]
+std_std [lemma, in Combi.Combi.std]
+std_posbig [lemma, in Combi.Combi.std]
+std_rembig [lemma, in Combi.Combi.std]
+std_is_std [lemma, in Combi.Combi.std]
+std_rec [definition, in Combi.Combi.std]
+std_stdwordn__canonical__fintype_SubFinite [definition, in Combi.Combi.std]
+std_stdwordn__canonical__fintype_Finite [definition, in Combi.Combi.std]
+std_stdwordn__canonical__choice_SubCountable [definition, in Combi.Combi.std]
+std_stdwordn__canonical__choice_Countable [definition, in Combi.Combi.std]
+std_stdwordn__canonical__choice_SubChoice [definition, in Combi.Combi.std]
+std_stdwordn__canonical__choice_Choice [definition, in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_SubEquality [definition, in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_Equality [definition, in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_SubType [definition, in Combi.Combi.std]
+std_max [lemma, in Combi.Combi.std]
+std_uniq [lemma, in Combi.Combi.std]
+std_perm [lemma, in Combi.Combi.std]
+std_rcons_shiftinv [lemma, in Combi.LRrule.stdplact]
+std_plact [lemma, in Combi.LRrule.stdplact]
+std_plact2 [lemma, in Combi.LRrule.stdplact]
+std_plact1 [lemma, in Combi.LRrule.stdplact]
+std_shsh [lemma, in Combi.LRrule.shuffle]
+std_of_yam [lemma, in Combi.Combi.stdtab]
+step [definition, in Combi.Basic.congr]
+step_closed [lemma, in Combi.Basic.congr]
+step_mem [lemma, in Combi.Basic.congr]
+stop [abbreviation, in Combi.Combi.skewpart]
+stpn [abbreviation, in Combi.HookFormula.Frobenius_ident]
+straighten [definition, in Combi.SymGroup.presentSn]
+straighten_path_npos [lemma, in Combi.SymGroup.presentSn]
+straighten_rev [definition, in Combi.SymGroup.presentSn]
+subdescsetP [lemma, in Combi.Combi.composition]
+subdescset_partsumP [lemma, in Combi.Combi.composition]
+subenum_countE [lemma, in Combi.Basic.combclass]
+subseq [library]
+subseqrow [definition, in Combi.LRrule.Schensted]
+subseqrow_n [definition, in Combi.LRrule.Schensted]
+subseqs [record, in Combi.Combi.subseq]
+SubseqSorted [section, in Combi.Combi.subseq]
+SubseqSortedIn [section, in Combi.Combi.subseq]
+SubseqSortedIn.leT [variable, in Combi.Combi.subseq]
+SubseqSortedIn.T [variable, in Combi.Combi.subseq]
+SubseqSorted.leT [variable, in Combi.Combi.subseq]
+SubseqSorted.T [variable, in Combi.Combi.subseq]
+subseqsP [lemma, in Combi.Combi.subseq]
+subseqsval [projection, in Combi.Combi.subseq]
+subseqs_masks_uniq [lemma, in Combi.Combi.subseq]
+Subseqs_maskK [lemma, in Combi.Combi.subseq]
+subseq_take [lemma, in Combi.LRrule.Greene]
+subseq_subseqs__canonical__fintype_SubFinite [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__fintype_Finite [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_SubCountable [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_Countable [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_SubChoice [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_Choice [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_SubEquality [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_Equality [definition, in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_SubType [definition, in Combi.Combi.subseq]
+subseq_rcons_neq [lemma, in Combi.Combi.subseq]
+subseq_rcons_eq [lemma, in Combi.Combi.subseq]
+subseq_partsumE [lemma, in Combi.Combi.composition]
+subset_imsetK [lemma, in Combi.SSRcomplements.tools]
+subset_abc [lemma, in Combi.LRrule.plactic]
+subset_set1 [lemma, in Combi.Combi.setpartition]
+subset_s_trans_s [lemma, in Combi.Basic.congr]
+subset_undup_step [lemma, in Combi.Basic.congr]
+subset_step [lemma, in Combi.Basic.congr]
+SubtypesDisjointUnion [section, in Combi.Basic.combclass]
+SubtypesDisjointUnion.FI [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.Hpart [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.HPTi [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.P [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.Pi [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.PI [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.T [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.TI [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.TP [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.TPi [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.TPI [variable, in Combi.Basic.combclass]
+SubtypesDisjointUnion.union_enum [variable, in Combi.Basic.combclass]
+subType_unionE [lemma, in Combi.Basic.combclass]
+subType_seqP [lemma, in Combi.Basic.combclass]
+subType_seq [definition, in Combi.Basic.combclass]
+SubUndup [section, in Combi.Basic.combclass]
+SubUndup.P [variable, in Combi.Basic.combclass]
+SubUndup.subenum [variable, in Combi.Basic.combclass]
+SubUndup.subenumP [variable, in Combi.Basic.combclass]
+SubUndup.subenum_in [variable, in Combi.Basic.combclass]
+SubUndup.T [variable, in Combi.Basic.combclass]
+SubUndup.TP [variable, in Combi.Basic.combclass]
+sub_full [definition, in Combi.Combi.subseq]
+sub_nil [definition, in Combi.Combi.subseq]
+sub_enumE [lemma, in Combi.Basic.combclass]
+succ_neq0 [lemma, in ALEA.Qmeasure]
+sumndiff [lemma, in Combi.MPoly.Schur_altdef]
+sumnpSPE [lemma, in Combi.HookFormula.hook]
+sumn_sort [lemma, in Combi.SSRcomplements.tools]
+sumn_pred1_iota [lemma, in Combi.SSRcomplements.tools]
+sumn_nth_le [lemma, in Combi.SSRcomplements.tools]
+sumn_drop [lemma, in Combi.SSRcomplements.tools]
+sumn_take [lemma, in Combi.SSRcomplements.tools]
+sumn_mapE [lemma, in Combi.SSRcomplements.tools]
+sumn_map_condE [lemma, in Combi.SSRcomplements.tools]
+sumn_take_merge [lemma, in Combi.Combi.partition]
+sumn_union_part [lemma, in Combi.Combi.partition]
+sumn_intpartn [lemma, in Combi.Combi.partition]
+sumn_diff_shape [lemma, in Combi.Combi.partition]
+sumn_diff_shape_eq [lemma, in Combi.Combi.partition]
+sumn_included [lemma, in Combi.Combi.partition]
+sumn_take_inj [lemma, in Combi.Combi.partition]
+sumn_take_leq [lemma, in Combi.Combi.partition]
+sumn_conj_part [lemma, in Combi.Combi.partition]
+sumn_incr_first_n [lemma, in Combi.Combi.partition]
+sumn_decr_nth [lemma, in Combi.Combi.partition]
+sumn_incr_nth [lemma, in Combi.Combi.partition]
+sumn_rem_trail0 [lemma, in Combi.Combi.partition]
+sumn_right_sizes_gt [lemma, in Combi.Combi.bintree]
+sumn_partm [lemma, in Combi.MPoly.antisym]
+sumn_mpart [lemma, in Combi.MPoly.antisym]
+sumn_eval [lemma, in Combi.MPoly.Schur_altdef]
+sumn_shape_stdtabnE [lemma, in Combi.Combi.stdtab]
+sumn_diff_shape_intpartE [lemma, in Combi.LRrule.therule]
+sumn_add_ribbon [lemma, in Combi.Combi.skewpart]
+sumn_mapS [lemma, in Combi.Combi.skewpart]
+sumn_cocode [lemma, in Combi.SymGroup.presentSn]
+sumn_eq [projection, in Combi.LRrule.implem]
+sum_take [lemma, in Combi.SSRcomplements.tools]
+sum_minn [lemma, in Combi.SSRcomplements.tools]
+sum_conj [lemma, in Combi.Combi.partition]
+sum_count_mem [lemma, in Combi.Basic.combclass]
+sum_syme_symh [lemma, in Combi.MPoly.sympoly]
+sum_symh_syme [lemma, in Combi.MPoly.sympoly]
+sum_symmE [lemma, in Combi.MPoly.sympoly]
+Swap [module, in Combi.LRrule.Greene_inv]
+Swap.enum_cut [lemma, in Combi.LRrule.Greene_inv]
+Swap.pos0 [definition, in Combi.LRrule.Greene_inv]
+Swap.pos0_subproof [lemma, in Combi.LRrule.Greene_inv]
+Swap.pos01F [lemma, in Combi.LRrule.Greene_inv]
+Swap.pos1 [definition, in Combi.LRrule.Greene_inv]
+Swap.pos1_subproof [lemma, in Combi.LRrule.Greene_inv]
+Swap.size_cut_sizeu [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap [definition, in Combi.LRrule.Greene_inv]
+Swap.Swap [section, in Combi.LRrule.Greene_inv]
+Swap.swapL [lemma, in Combi.LRrule.Greene_inv]
+Swap.swapR [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_size_cover [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_cover [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_set_inj [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_set_invol [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_set [definition, in Combi.LRrule.Greene_inv]
+Swap.swap_inj [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap_invol [lemma, in Combi.LRrule.Greene_inv]
+Swap.Swap.Alph [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.disp [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.l0 [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.l1 [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.R [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.u [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.v [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.word [variable, in Combi.LRrule.Greene_inv]
+Swap.Swap.x [variable, in Combi.LRrule.Greene_inv]
+Swap.swap0 [lemma, in Combi.LRrule.Greene_inv]
+Swap.swap1 [lemma, in Combi.LRrule.Greene_inv]
+Swap.tnth_pos1 [lemma, in Combi.LRrule.Greene_inv]
+Swap.tnth_pos0 [lemma, in Combi.LRrule.Greene_inv]
+symbe [definition, in Combi.MPoly.homogsym]
+symbeE [lemma, in Combi.MPoly.homogsym]
+symbe_free [lemma, in Combi.MPoly.homogsym]
+symbe_basis [lemma, in Combi.MPoly.homogsym]
+symbh [definition, in Combi.MPoly.homogsym]
+symbhE [lemma, in Combi.MPoly.homogsym]
+symbh_free [lemma, in Combi.MPoly.homogsym]
+symbh_basis [lemma, in Combi.MPoly.homogsym]
+symbm [definition, in Combi.MPoly.homogsym]
+symbmE [lemma, in Combi.MPoly.homogsym]
+symbm_basis [lemma, in Combi.MPoly.homogsym]
+symbm_free [lemma, in Combi.MPoly.homogsym]
+symbp [definition, in Combi.MPoly.homogsym]
+symbpE [lemma, in Combi.MPoly.homogsym]
+symbp_free [lemma, in Combi.MPoly.homogsym]
+symbp_basis [lemma, in Combi.MPoly.homogsym]
+symbs [definition, in Combi.MPoly.homogsym]
+symbsE [lemma, in Combi.MPoly.homogsym]
+symbs_free [lemma, in Combi.MPoly.homogsym]
+symbs_basis [lemma, in Combi.MPoly.homogsym]
+symE [lemma, in Combi.MPoly.sympoly]
+syme [definition, in Combi.MPoly.sympoly]
+syme_syms_partdom [lemma, in Combi.MPoly.sympoly]
+syme_syms [lemma, in Combi.MPoly.sympoly]
+syme_syms_partdom_int [lemma, in Combi.MPoly.sympoly]
+syme_syms_int [lemma, in Combi.MPoly.sympoly]
+syme_to_symh [lemma, in Combi.MPoly.sympoly]
+syme_symhE [lemma, in Combi.MPoly.sympoly]
+syme_cast [lemma, in Combi.MPoly.sympoly]
+syme_to_symm [lemma, in Combi.MPoly.sympoly]
+syme_homog [lemma, in Combi.MPoly.sympoly]
+syme_geqnE [lemma, in Combi.MPoly.sympoly]
+syme_sym [lemma, in Combi.MPoly.sympoly]
+syme0 [lemma, in Combi.MPoly.sympoly]
+syme1 [lemma, in Combi.MPoly.sympoly]
+symh [definition, in Combi.MPoly.sympoly]
+SymheSyms [section, in Combi.MPoly.sympoly]
+SymheSymsInt [section, in Combi.MPoly.sympoly]
+SymheSymsInt.d [variable, in Combi.MPoly.sympoly]
+SymheSymsInt.n [variable, in Combi.MPoly.sympoly]
+_ ^~ [notation, in Combi.MPoly.sympoly]
+SymheSyms.d [variable, in Combi.MPoly.sympoly]
+SymheSyms.n [variable, in Combi.MPoly.sympoly]
+SymheSyms.R [variable, in Combi.MPoly.sympoly]
+_ ^~ [notation, in Combi.MPoly.sympoly]
+symHE_intpartn [lemma, in Combi.MPoly.sympoly]
+symHE_intcompn [lemma, in Combi.MPoly.sympoly]
+symHE_prod_intcomp [lemma, in Combi.MPoly.sympoly]
+symHE_rec [lemma, in Combi.MPoly.sympoly]
+symhe1E [lemma, in Combi.MPoly.sympoly]
+symh_basisE [lemma, in Combi.MPoly.Schur_mpoly]
+symh_to_symp [lemma, in Combi.MPoly.sympoly]
+symh_to_symp_intpartn [lemma, in Combi.MPoly.sympoly]
+symh_to_symp_prod_partsum [lemma, in Combi.MPoly.sympoly]
+symh_syms_partdom [lemma, in Combi.MPoly.sympoly]
+symh_syms [lemma, in Combi.MPoly.sympoly]
+symh_syms_partdom_int [lemma, in Combi.MPoly.sympoly]
+symh_syms_int [lemma, in Combi.MPoly.sympoly]
+symh_to_syme [lemma, in Combi.MPoly.sympoly]
+symh_symeE [lemma, in Combi.MPoly.sympoly]
+symh_cast [lemma, in Combi.MPoly.sympoly]
+symh_to_symm [lemma, in Combi.MPoly.sympoly]
+symh_homog [lemma, in Combi.MPoly.sympoly]
+symh_sym [lemma, in Combi.MPoly.sympoly]
+symh_pol_any [lemma, in Combi.MPoly.sympoly]
+symh_pol [definition, in Combi.MPoly.sympoly]
+symh_pol_bound [definition, in Combi.MPoly.sympoly]
+symh0 [lemma, in Combi.MPoly.sympoly]
+symm [definition, in Combi.MPoly.sympoly]
+symmX [lemma, in Combi.MPoly.Cauchy]
+symm_syms_partdom [lemma, in Combi.MPoly.sympoly]
+symm_syms [lemma, in Combi.MPoly.sympoly]
+symm_syms_partdom_int [lemma, in Combi.MPoly.sympoly]
+symm_syms_int [lemma, in Combi.MPoly.sympoly]
+symm_cast [lemma, in Combi.MPoly.sympoly]
+symm_unique0 [lemma, in Combi.MPoly.sympoly]
+symm_unique [lemma, in Combi.MPoly.sympoly]
+symm_homog [lemma, in Combi.MPoly.sympoly]
+symm_oversize [lemma, in Combi.MPoly.sympoly]
+symm_sym [lemma, in Combi.MPoly.sympoly]
+symm_pol [definition, in Combi.MPoly.sympoly]
+symm0 [lemma, in Combi.MPoly.sympoly]
+symp [definition, in Combi.MPoly.sympoly]
+sympe1E [lemma, in Combi.MPoly.sympoly]
+sympol [projection, in Combi.MPoly.sympoly]
+SymPolF [section, in Combi.MPoly.sympoly]
+SymPolF.m [variable, in Combi.MPoly.sympoly]
+SymPolF.R [variable, in Combi.MPoly.sympoly]
+sympolP [lemma, in Combi.MPoly.sympoly]
+sympoly [record, in Combi.MPoly.sympoly]
+sympoly [library]
+SymPolyComRingType [section, in Combi.MPoly.sympoly]
+SymPolyComRingType.n [variable, in Combi.MPoly.sympoly]
+SymPolyComRingType.R [variable, in Combi.MPoly.sympoly]
+sympolyf [definition, in Combi.MPoly.sympoly]
+sympolyfK [lemma, in Combi.MPoly.sympoly]
+sympolyfP [lemma, in Combi.MPoly.sympoly]
+sympolyf_evalX [lemma, in Combi.MPoly.sympoly]
+sympolyf_evalK [lemma, in Combi.MPoly.sympoly]
+sympolyf_evalE [lemma, in Combi.MPoly.sympoly]
+sympolyf_eval [definition, in Combi.MPoly.sympoly]
+sympolyf_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+sympolyf_is_linear [lemma, in Combi.MPoly.sympoly]
+SymPolyHomogKey [module, in Combi.MPoly.homogsym]
+SymPolyHomogKey.homogsym1_keyed [definition, in Combi.MPoly.homogsym]
+SymPolyHomogKey.homogsym1_key [lemma, in Combi.MPoly.homogsym]
+SymPolyIdomainType [section, in Combi.MPoly.sympoly]
+SymPolyIdomainType.n [variable, in Combi.MPoly.sympoly]
+SymPolyIdomainType.R [variable, in Combi.MPoly.sympoly]
+SymPolyRingType [section, in Combi.MPoly.sympoly]
+SymPolyRingType.n [variable, in Combi.MPoly.sympoly]
+SymPolyRingType.R [variable, in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubIntegralDomain [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_IntegralDomain [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComUnitAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComUnitRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComUnitRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubUnitRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_UnitAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_UnitRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComNzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComNzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComPzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComPzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_LRMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_RMorphism [definition, in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_Linear [definition, in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__Algebra_Additive [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzLalgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzLalgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzLalgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzLalgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubLmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_Lmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzLSemiAlgebra [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubLSemiModule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_LSemiModule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzSemiRing [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubZmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubNmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_Zmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_Nmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddSemigroup [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubAddUMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddUMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseZmodule [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubBaseAddUMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_ChoiceBaseAddUMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseAddUMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_ChoiceBaseAddMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseAddMagma [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__choice_SubChoice [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__choice_Choice [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_SubEquality [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_Equality [definition, in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_SubType [definition, in Combi.MPoly.sympoly]
+sympol_is_monoid_morphism [lemma, in Combi.MPoly.sympoly]
+sympol_is_linear [lemma, in Combi.MPoly.sympoly]
+sympol_inj [lemma, in Combi.MPoly.sympoly]
+sympXY [lemma, in Combi.MPoly.Cauchy]
+symp_cast [lemma, in Combi.MPoly.sympoly]
+symp_to_symm [lemma, in Combi.MPoly.sympoly]
+symp_homog [lemma, in Combi.MPoly.sympoly]
+symp_sym [lemma, in Combi.MPoly.sympoly]
+symp_pol [definition, in Combi.MPoly.sympoly]
+symp0 [lemma, in Combi.MPoly.sympoly]
+syms [definition, in Combi.MPoly.sympoly]
+SymsSymm [section, in Combi.MPoly.sympoly]
+SymsSymmInt [section, in Combi.MPoly.sympoly]
+SymsSymmInt.d [variable, in Combi.MPoly.sympoly]
+SymsSymmInt.n [variable, in Combi.MPoly.sympoly]
+SymsSymm.d [variable, in Combi.MPoly.sympoly]
+SymsSymm.n [variable, in Combi.MPoly.sympoly]
+SymsSymm.R [variable, in Combi.MPoly.sympoly]
+syms_prod_sympM [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_prod_sympM_int [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_sympM [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_pmap [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_oapp [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_oapp_int [lemma, in Combi.MPoly.MurnaghanNakayama]
+syms_syme_partdom [lemma, in Combi.MPoly.sympoly]
+syms_syme [lemma, in Combi.MPoly.sympoly]
+syms_symh_partdom [lemma, in Combi.MPoly.sympoly]
+syms_symh [lemma, in Combi.MPoly.sympoly]
+syms_syme_partdom_int [lemma, in Combi.MPoly.sympoly]
+syms_syme_int [lemma, in Combi.MPoly.sympoly]
+syms_symh_partdom_int [lemma, in Combi.MPoly.sympoly]
+syms_symh_int [lemma, in Combi.MPoly.sympoly]
+syms_symm_partdom [lemma, in Combi.MPoly.sympoly]
+syms_symm [lemma, in Combi.MPoly.sympoly]
+syms_symm_partdom_int [lemma, in Combi.MPoly.sympoly]
+syms_symm_int [lemma, in Combi.MPoly.sympoly]
+syms_symeM [lemma, in Combi.MPoly.sympoly]
+syms_symhM [lemma, in Combi.MPoly.sympoly]
+syms_symsM [lemma, in Combi.MPoly.sympoly]
+syms_cast [lemma, in Combi.MPoly.sympoly]
+syms_oversize [lemma, in Combi.MPoly.sympoly]
+syms_colpartn [lemma, in Combi.MPoly.sympoly]
+syms_rowpartn [lemma, in Combi.MPoly.sympoly]
+syms_homog [lemma, in Combi.MPoly.sympoly]
+syms0 [lemma, in Combi.MPoly.sympoly]
+syms1 [lemma, in Combi.MPoly.sympoly]
+sym_VanprodM [lemma, in Combi.MPoly.antisym]
+sym_antiE [lemma, in Combi.MPoly.antisym]
+sym_antisym_char_not2 [lemma, in Combi.MPoly.antisym]
+sym_anti [lemma, in Combi.MPoly.antisym]
+sym_smalln [lemma, in Combi.MPoly.antisym]
+sym_fundamental_symh [lemma, in Combi.MPoly.sympoly]
+sym_fundamental_symh_homog [lemma, in Combi.MPoly.sympoly]
+sym_fundamental_homog [lemma, in Combi.MPoly.sympoly]
+sym_symmE [lemma, in Combi.MPoly.sympoly]
+sz [abbreviation, in Combi.MPoly.Schur_altdef]
+sztd [lemma, in Combi.Combi.skewpart]
+

T

+T [abbreviation, in Combi.MPoly.Schur_altdef]
+tabcols [definition, in Combi.LRrule.Greene]
+tabcolsk [definition, in Combi.LRrule.Greene]
+tabcols_cons [lemma, in Combi.LRrule.Greene]
+tabcol_cut [lemma, in Combi.LRrule.Greene]
+Tableau [section, in Combi.Combi.tableau]
+tableau [library]
+TableauReading [section, in Combi.LRrule.freeSchur]
+TableauReading [section, in Combi.Combi.tableau]
+TableauReading.A [variable, in Combi.LRrule.freeSchur]
+TableauReading.A [variable, in Combi.Combi.tableau]
+TableauReading.disp [variable, in Combi.LRrule.freeSchur]
+TableauReading.disp [variable, in Combi.Combi.tableau]
+tableau_tabsh__canonical__fintype_SubFinite [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__fintype_Finite [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_SubCountable [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_Countable [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_SubChoice [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_Choice [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_SubEquality [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_Equality [definition, in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_SubType [definition, in Combi.Combi.tableau]
+tableau_is_row [lemma, in Combi.Combi.tableau]
+Tableau.disp [variable, in Combi.Combi.tableau]
+Tableau.T [variable, in Combi.Combi.tableau]
+tabnat_of_ordK [lemma, in Combi.MPoly.Schur_altdef]
+tabnat_of_ord [definition, in Combi.MPoly.Schur_altdef]
+tabnat_of_ord_subproof [lemma, in Combi.MPoly.Schur_altdef]
+tabnat_of_ord_fun [definition, in Combi.MPoly.Schur_altdef]
+tabord_of_natK [lemma, in Combi.MPoly.Schur_altdef]
+tabord_of_nat [definition, in Combi.MPoly.Schur_altdef]
+tabord_of_nat_subproof [lemma, in Combi.MPoly.Schur_altdef]
+tabord_of_nat_fun [definition, in Combi.MPoly.Schur_altdef]
+TabPair [abbreviation, in Combi.LRrule.Schensted]
+tabrowconst [definition, in Combi.Combi.tableau]
+tabrowconst_subproof [lemma, in Combi.Combi.tableau]
+tabrows [definition, in Combi.LRrule.Greene]
+tabrowsk [definition, in Combi.LRrule.Greene]
+tabrows_non0 [lemma, in Combi.LRrule.Greene]
+tabsh [record, in Combi.Combi.tableau]
+tabshP [lemma, in Combi.Combi.tableau]
+tabshval [projection, in Combi.Combi.tableau]
+tabsh_is_std [lemma, in Combi.MPoly.Schur_altdef]
+tabsh_reading_RSE [lemma, in Combi.LRrule.freeSchur]
+tabsh_reading_RSP [lemma, in Combi.LRrule.freeSchur]
+tabsh_reading_RS [definition, in Combi.LRrule.freeSchur]
+tabsh_to_wordK [lemma, in Combi.Combi.tableau]
+tabsh_readingP [lemma, in Combi.Combi.tableau]
+tabsh_reading [definition, in Combi.Combi.tableau]
+tabwordshape [definition, in Combi.LRrule.freeSchur]
+tabwordshape_col [lemma, in Combi.MPoly.Schur_mpoly]
+tabwordshape_row [lemma, in Combi.MPoly.Schur_mpoly]
+tabword_of_tuple_freeSchur [lemma, in Combi.LRrule.freeSchur]
+tabword_of_tuple_freeSchur_inj [lemma, in Combi.LRrule.freeSchur]
+tabword_of_tuple [definition, in Combi.LRrule.freeSchur]
+tab_eval_partdom [lemma, in Combi.MPoly.Schur_altdef]
+tab_eqP [lemma, in Combi.Combi.tableau]
+tab0 [lemma, in Combi.Combi.tableau]
+take_enumI [lemma, in Combi.SSRcomplements.tools]
+take_drop_langQ [lemma, in Combi.LRrule.freeSchur]
+take_prefixes [lemma, in Combi.Combi.Dyckword]
+TamariCover [section, in Combi.Combi.bintree]
+TamariCover.n [variable, in Combi.Combi.bintree]
+TamariLattice [module, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBLattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBLattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TLattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BLattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinLattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_Lattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_MeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_JoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_POrder [definition, in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_Preorder [definition, in Combi.Combi.bintree]
+TamariLattice.botETamari [lemma, in Combi.Combi.bintree]
+TamariLattice.Exports [module, in Combi.Combi.bintree]
+TamariLattice.Exports.botETamari [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.flipsz_join [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.flipsz_meet [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.right_sizes_meet [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.rotations_Tamari [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.TamariE [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.Tamari_vctleq [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.Tamari_flip [definition, in Combi.Combi.bintree]
+TamariLattice.Exports.topETamari [definition, in Combi.Combi.bintree]
+TamariLattice.flipsz_join [lemma, in Combi.Combi.bintree]
+TamariLattice.flipsz_meet [lemma, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_41 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_39 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_38 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_37 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_34 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_33 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_32 [definition, in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_29 [definition, in Combi.Combi.bintree]
+TamariLattice.leftcomb_bottom [lemma, in Combi.Combi.bintree]
+TamariLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.bintree]
+TamariLattice.Order_Le_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.Combi.bintree]
+TamariLattice.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.bintree]
+TamariLattice.rightcomb_top [lemma, in Combi.Combi.bintree]
+TamariLattice.right_sizes_meet [lemma, in Combi.Combi.bintree]
+TamariLattice.rotations_Tamari [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari [definition, in Combi.Combi.bintree]
+TamariLattice.TamariE [lemma, in Combi.Combi.bintree]
+TamariLattice.TamariLattice [section, in Combi.Combi.bintree]
+TamariLattice.TamariLattice.n [variable, in Combi.Combi.bintree]
+TamariLattice.Tamari_vctleq [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari_flip [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari_anti [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari_trans [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari_refl [lemma, in Combi.Combi.bintree]
+TamariLattice.Tamari_sumn_right_sizes [lemma, in Combi.Combi.bintree]
+TamariLattice.Tjoin [definition, in Combi.Combi.bintree]
+TamariLattice.TjoinP [lemma, in Combi.Combi.bintree]
+TamariLattice.Tmeet [definition, in Combi.Combi.bintree]
+TamariLattice.TmeetC [lemma, in Combi.Combi.bintree]
+TamariLattice.TmeetP [lemma, in Combi.Combi.bintree]
+TamariLattice.TmeetPr [lemma, in Combi.Combi.bintree]
+TamariLattice.Tmeet_proof [lemma, in Combi.Combi.bintree]
+TamariLattice.topETamari [lemma, in Combi.Combi.bintree]
+TamariP [lemma, in Combi.Combi.bintree]
+TamariVector [definition, in Combi.Combi.bintree]
+Tamari_succ [lemma, in Combi.Combi.bintree]
+Tamari_display [lemma, in Combi.Combi.bintree]
+Tamari_add_bounded [lemma, in Combi.Combi.bintree]
+Tamari_add_min [lemma, in Combi.Combi.bintree]
+Tamari_add_head [lemma, in Combi.Combi.bintree]
+Tamari_cat [lemma, in Combi.Combi.bintree]
+Tamari_take [lemma, in Combi.Combi.bintree]
+Tamari_catr [lemma, in Combi.Combi.bintree]
+Tamari_drop [lemma, in Combi.Combi.bintree]
+Tamari_consP [lemma, in Combi.Combi.bintree]
+TcastVal [section, in Combi.SymGroup.reprSn]
+TcastVal.T [variable, in Combi.SymGroup.reprSn]
+tclosure [definition, in Combi.SymGroup.presentSn]
+TClosureInvset [section, in Combi.SymGroup.weak_order]
+TClosureInvset.n0 [variable, in Combi.SymGroup.weak_order]
+tclosureP [lemma, in Combi.SymGroup.weak_order]
+tclosure_Delta [lemma, in Combi.SymGroup.weak_order]
+tclosure_sub [lemma, in Combi.SymGroup.presentSn]
+Test [section, in Combi.Combi.skewpart]
+Tests [section, in Combi.Combi.bintree]
+Tests [section, in Combi.LRrule.Schensted]
+Tests [section, in Combi.Combi.skewpart]
+Tests [section, in Combi.MPoly.MurnaghanNakayama]
+Tests [section, in Combi.MPoly.MurnaghanNakayama]
+TestsComp [section, in Combi.Combi.bintree]
+Tests.bla [variable, in Combi.Combi.bintree]
+Tests.partn [variable, in Combi.MPoly.MurnaghanNakayama]
+TextBookDefStartStop [section, in Combi.Combi.skewpart]
+TextBookDefStartStop.Hrib [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.inner [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.outer [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.partinn [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.partout [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.start [variable, in Combi.Combi.skewpart]
+TextBookDefStartStop.stop [variable, in Combi.Combi.skewpart]
+TextBookImplDef [section, in Combi.Combi.skewpart]
+TextBookImplDef.Htb [variable, in Combi.Combi.skewpart]
+TextBookImplDef.inner [variable, in Combi.Combi.skewpart]
+TextBookImplDef.outer [variable, in Combi.Combi.skewpart]
+TextBookImplDef.partinn [variable, in Combi.Combi.skewpart]
+TextBookImplDef.partout [variable, in Combi.Combi.skewpart]
+Theorem2 [lemma, in Combi.HookFormula.hook]
+therule [library]
+tinj [definition, in Combi.SymGroup.towerSn]
+tinjA [lemma, in Combi.SymGroup.towerSn]
+tinjE1 [lemma, in Combi.SymGroup.towerSn]
+tinjval [definition, in Combi.SymGroup.towerSn]
+tinjval_inj [lemma, in Combi.SymGroup.towerSn]
+tinj_morphM [lemma, in Combi.SymGroup.towerSn]
+tinj1E [lemma, in Combi.SymGroup.towerSn]
+Tm [abbreviation, in Combi.MPoly.Schur_altdef]
+toDepRSPair [lemma, in Combi.LRrule.freeSchur]
+tools [library]
+TowerMorphism [section, in Combi.SymGroup.towerSn]
+TowerMorphism.m [variable, in Combi.SymGroup.towerSn]
+TowerMorphism.n [variable, in Combi.SymGroup.towerSn]
+towerSn [library]
+towerSn_cfextprod__canonical__sesquilinear_Bilinear [definition, in Combi.SymGroup.towerSn]
+to_word_skew_reshape [lemma, in Combi.Combi.skewtab]
+to_word_yamtab [lemma, in Combi.LRrule.Yam_plact]
+to_maskK [lemma, in Combi.Combi.subseq]
+to_mask [definition, in Combi.Combi.subseq]
+to_mask_spec [lemma, in Combi.Combi.subseq]
+to_word_map_tab [lemma, in Combi.Combi.tableau]
+to_word_enum_tabsh [lemma, in Combi.Combi.tableau]
+to_word_filter_nnil [lemma, in Combi.Combi.tableau]
+to_wordK [lemma, in Combi.Combi.tableau]
+to_word_rcons [lemma, in Combi.Combi.tableau]
+to_word_cons [lemma, in Combi.Combi.tableau]
+to_word [definition, in Combi.Combi.tableau]
+to_word_map_shiftn [lemma, in Combi.LRrule.therule]
+tpermC [lemma, in Combi.SymGroup.presentSn]
+tperm_conj [lemma, in Combi.SymGroup.cycletype]
+tperm_braid [lemma, in Combi.SymGroup.presentSn]
+tpval [projection, in Combi.Combi.permuted]
+trace_seqrP [lemma, in Combi.HookFormula.hook]
+trace_seqlP [lemma, in Combi.HookFormula.hook]
+trace_corner_box [lemma, in Combi.HookFormula.hook]
+trace_seq_uniq [lemma, in Combi.HookFormula.hook]
+trace_seq [definition, in Combi.HookFormula.hook]
+trace_size_leg_length [lemma, in Combi.HookFormula.hook]
+trace_size_arm_length [lemma, in Combi.HookFormula.hook]
+transf [lemma, in Combi.LRrule.Schensted]
+transitive [projection, in ALEA.Ccpo]
+Transitive [section, in Combi.Basic.congr]
+transitive_DeltaI1 [lemma, in Combi.SymGroup.presentSn]
+Transitive.bound [variable, in Combi.Basic.congr]
+Transitive.FullKnown [section, in Combi.Basic.congr]
+Transitive.FullKnown.full [variable, in Combi.Basic.congr]
+Transitive.FullKnown.Hfull [variable, in Combi.Basic.congr]
+Transitive.Hbound [variable, in Combi.Basic.congr]
+Transitive.Hinvar [variable, in Combi.Basic.congr]
+Transitive.invar [variable, in Combi.Basic.congr]
+Transitive.rule [variable, in Combi.Basic.congr]
+Transitive.T [variable, in Combi.Basic.congr]
+Transp [section, in Combi.Combi.std]
+transP [lemma, in Combi.Basic.congr]
+Transp [section, in Combi.SymGroup.presentSn]
+Transp.Alph [variable, in Combi.Combi.std]
+Transp.disp [variable, in Combi.Combi.std]
+Transp.T [variable, in Combi.SymGroup.presentSn]
+tree_encodeK [lemma, in Combi.Combi.bintree]
+tree_decode [definition, in Combi.Combi.bintree]
+tree_encode [definition, in Combi.Combi.bintree]
+TriangularInv [section, in Combi.Basic.unitriginv]
+TriangularInv.disp [variable, in Combi.Basic.unitriginv]
+TriangularInv.M [variable, in Combi.Basic.unitriginv]
+TriangularInv.Munitrig [variable, in Combi.Basic.unitriginv]
+TriangularInv.R [variable, in Combi.Basic.unitriginv]
+TriangularInv.T [variable, in Combi.Basic.unitriginv]
+trivIs [lemma, in Combi.LRrule.Greene]
+trivIseq [definition, in Combi.LRrule.Greene]
+TrivISeq [section, in Combi.LRrule.Greene]
+trivIseq_tabcols [lemma, in Combi.LRrule.Greene]
+trivIseq_shcols [lemma, in Combi.LRrule.Greene]
+trivIseq_shrows [lemma, in Combi.LRrule.Greene]
+trivIseq_map [lemma, in Combi.LRrule.Greene]
+trivIseq_cover [lemma, in Combi.LRrule.Greene]
+trivIseq_consK [lemma, in Combi.LRrule.Greene]
+TrivISeq.T [variable, in Combi.LRrule.Greene]
+trivIsetpart [lemma, in Combi.Combi.setpartition]
+trivIset_iporbits [lemma, in Combi.SymGroup.permcent]
+trivIset_coverD1 [lemma, in Combi.LRrule.Greene_inv]
+trivIset_coverU [lemma, in Combi.LRrule.Greene_inv]
+trivIset_coverU1 [lemma, in Combi.LRrule.Greene_inv]
+trivIset_setrev [lemma, in Combi.LRrule.Greene_inv]
+trivIset_tabcolsk [lemma, in Combi.LRrule.Greene]
+trivIset_tabrowsk [lemma, in Combi.LRrule.Greene]
+trivIset_I [lemma, in Combi.LRrule.Greene]
+trivIsubseq [lemma, in Combi.LRrule.Greene]
+trivsetpart [definition, in Combi.Combi.setpartition]
+trivsetpart_subproof [lemma, in Combi.Combi.setpartition]
+trivSimpl [definition, in Combi.SymGroup.presentSn]
+triv_part [lemma, in Combi.SSRcomplements.tools]
+triv_sign_not_sim [lemma, in Combi.SymGroup.reprSn]
+triv_sign_neq [lemma, in Combi.SymGroup.reprSn]
+triv_Chi [lemma, in Combi.SymGroup.reprSn]
+triv_irr [lemma, in Combi.SymGroup.reprSn]
+triv_repr [definition, in Combi.SymGroup.reprSn]
+triv_mx_repr [lemma, in Combi.SymGroup.reprSn]
+triv_mx [definition, in Combi.SymGroup.reprSn]
+trval [projection, in Combi.Combi.bintree]
+trval [projection, in Combi.Combi.ordtree]
+tsumnE [lemma, in Combi.LRrule.implem]
+tval_tcastE [lemma, in Combi.SymGroup.reprSn]
+

U

+UDn [definition, in Combi.Combi.Dyckword]
+UDn_Dyck [lemma, in Combi.Combi.Dyckword]
+UnDn [definition, in Combi.Combi.Dyckword]
+UnDn_Dyck [lemma, in Combi.Combi.Dyckword]
+undup_step [lemma, in Combi.Basic.congr]
+undup_finType [definition, in Combi.Basic.combclass]
+unif [record, in ALEA.Qmeasure]
+unifnat [definition, in ALEA.Qmeasure]
+UnifNat [section, in ALEA.Qmeasure]
+Uniform [definition, in ALEA.Qmeasure]
+Uniform_def_ne [lemma, in ALEA.Qmeasure]
+Uniform_unif_seq_eq [lemma, in ALEA.Qmeasure]
+Uniform_in_seq [lemma, in ALEA.Qmeasure]
+Uniform_eq_out [lemma, in ALEA.Qmeasure]
+Uniform_eq_in [lemma, in ALEA.Qmeasure]
+Uniform_simpl [lemma, in ALEA.Qmeasure]
+unif_def [definition, in ALEA.Qmeasure]
+unif2fin [definition, in ALEA.Qmeasure]
+UnionPart [section, in Combi.Combi.partition]
+UnionPart.k [variable, in Combi.Combi.partition]
+UnionPart.l [variable, in Combi.Combi.partition]
+UnionPart.m [variable, in Combi.Combi.partition]
+UnionPart.n [variable, in Combi.Combi.partition]
+union_intpartnE [lemma, in Combi.Combi.partition]
+union_intpartn [definition, in Combi.Combi.partition]
+union_intpartn_subproof [lemma, in Combi.Combi.partition]
+union_intpartA [lemma, in Combi.Combi.partition]
+union_intpartC [lemma, in Combi.Combi.partition]
+union_intpartE [lemma, in Combi.Combi.partition]
+union_intpart [definition, in Combi.Combi.partition]
+union_intpart_subproof [lemma, in Combi.Combi.partition]
+union_finType [definition, in Combi.Basic.combclass]
+UniqFinType [section, in Combi.Basic.combclass]
+UniqFinType.P [variable, in Combi.Basic.combclass]
+UniqFinType.subenum [variable, in Combi.Basic.combclass]
+UniqFinType.subenumE [variable, in Combi.Basic.combclass]
+UniqFinType.subenum_uniq [variable, in Combi.Basic.combclass]
+UniqFinType.T [variable, in Combi.Basic.combclass]
+UniqFinType.TP [variable, in Combi.Basic.combclass]
+uniq_wordperm [lemma, in Combi.Combi.std]
+uniq_next [lemma, in Combi.SSRcomplements.tools]
+uniq_sum_count_mem [lemma, in Combi.SSRcomplements.tools]
+uniq_vect_n_k [lemma, in Combi.Combi.vectNK]
+uniq_step [lemma, in Combi.Basic.congr]
+uniq_finType [definition, in Combi.Basic.combclass]
+UniTriangular [section, in Combi.Basic.unitriginv]
+UniTriangular.disp [variable, in Combi.Basic.unitriginv]
+UniTriangular.R [variable, in Combi.Basic.unitriginv]
+UniTriangular.T [variable, in Combi.Basic.unitriginv]
+unitrig [definition, in Combi.Basic.unitriginv]
+unitriginv [library]
+unitrigP [lemma, in Combi.Basic.unitriginv]
+unitrig_sum1rV [lemma, in Combi.Basic.unitriginv]
+unitrig_sumrV [lemma, in Combi.Basic.unitriginv]
+unitrig_sum1lV [lemma, in Combi.Basic.unitriginv]
+unitrig_sumlV [lemma, in Combi.Basic.unitriginv]
+unitrig_sum1r [lemma, in Combi.Basic.unitriginv]
+unitrig_sumr [lemma, in Combi.Basic.unitriginv]
+unitrig_sum1l [lemma, in Combi.Basic.unitriginv]
+unitrig_suml [lemma, in Combi.Basic.unitriginv]
+unitrig1 [lemma, in Combi.Basic.unitriginv]
+unit_stable_sub [lemma, in ALEA.Qmeasure]
+unit_monotonic [lemma, in ALEA.Qmeasure]
+unit_stable_eq [lemma, in ALEA.Qmeasure]
+unlift_seqE [lemma, in Combi.LRrule.Greene]
+Unnamed_thm0 [definition, in Combi.Combi.std]
+Unnamed_thm [definition, in Combi.Combi.std]
+Unnamed_thm10 [definition, in Combi.Combi.bintree]
+Unnamed_thm9 [definition, in Combi.Combi.bintree]
+Unnamed_thm8 [definition, in Combi.Combi.bintree]
+Unnamed_thm7 [definition, in Combi.Combi.bintree]
+Unnamed_thm6 [definition, in Combi.Combi.bintree]
+Unnamed_thm5 [definition, in Combi.Combi.bintree]
+Unnamed_thm4 [definition, in Combi.Combi.bintree]
+Unnamed_thm3 [definition, in Combi.Combi.bintree]
+Unnamed_thm2 [definition, in Combi.Combi.bintree]
+Unnamed_thm1 [definition, in Combi.Combi.bintree]
+Unnamed_thm0 [definition, in Combi.Combi.bintree]
+Unnamed_thm [definition, in Combi.Combi.bintree]
+Unnamed_thm13 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm12 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm11 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm10 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm9 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm8 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm7 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm6 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm5 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm4 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm3 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm2 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm1 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm0 [definition, in Combi.LRrule.Schensted]
+Unnamed_thm [definition, in Combi.LRrule.Schensted]
+Unnamed_thm40 [definition, in Combi.Combi.skewpart]
+Unnamed_thm39 [definition, in Combi.Combi.skewpart]
+Unnamed_thm38 [definition, in Combi.Combi.skewpart]
+Unnamed_thm37 [definition, in Combi.Combi.skewpart]
+Unnamed_thm36 [definition, in Combi.Combi.skewpart]
+Unnamed_thm35 [definition, in Combi.Combi.skewpart]
+Unnamed_thm34 [definition, in Combi.Combi.skewpart]
+Unnamed_thm33 [definition, in Combi.Combi.skewpart]
+Unnamed_thm32 [definition, in Combi.Combi.skewpart]
+Unnamed_thm31 [definition, in Combi.Combi.skewpart]
+Unnamed_thm30 [definition, in Combi.Combi.skewpart]
+Unnamed_thm29 [definition, in Combi.Combi.skewpart]
+Unnamed_thm28 [definition, in Combi.Combi.skewpart]
+Unnamed_thm27 [definition, in Combi.Combi.skewpart]
+Unnamed_thm26 [definition, in Combi.Combi.skewpart]
+Unnamed_thm25 [definition, in Combi.Combi.skewpart]
+Unnamed_thm24 [definition, in Combi.Combi.skewpart]
+Unnamed_thm23 [definition, in Combi.Combi.skewpart]
+Unnamed_thm22 [definition, in Combi.Combi.skewpart]
+Unnamed_thm21 [definition, in Combi.Combi.skewpart]
+Unnamed_thm20 [definition, in Combi.Combi.skewpart]
+Unnamed_thm19 [definition, in Combi.Combi.skewpart]
+Unnamed_thm18 [definition, in Combi.Combi.skewpart]
+Unnamed_thm17 [definition, in Combi.Combi.skewpart]
+Unnamed_thm16 [definition, in Combi.Combi.skewpart]
+Unnamed_thm15 [definition, in Combi.Combi.skewpart]
+Unnamed_thm14 [definition, in Combi.Combi.skewpart]
+Unnamed_thm13 [definition, in Combi.Combi.skewpart]
+Unnamed_thm12 [definition, in Combi.Combi.skewpart]
+Unnamed_thm11 [definition, in Combi.Combi.skewpart]
+Unnamed_thm10 [definition, in Combi.Combi.skewpart]
+Unnamed_thm9 [definition, in Combi.Combi.skewpart]
+Unnamed_thm8 [definition, in Combi.Combi.skewpart]
+Unnamed_thm7 [definition, in Combi.Combi.skewpart]
+Unnamed_thm6 [definition, in Combi.Combi.skewpart]
+Unnamed_thm5 [definition, in Combi.Combi.skewpart]
+Unnamed_thm4 [definition, in Combi.Combi.skewpart]
+Unnamed_thm3 [definition, in Combi.Combi.skewpart]
+Unnamed_thm2 [definition, in Combi.Combi.skewpart]
+Unnamed_thm1 [definition, in Combi.Combi.skewpart]
+Unnamed_thm0 [definition, in Combi.Combi.skewpart]
+Unnamed_thm [definition, in Combi.Combi.skewpart]
+Unnamed_thm8 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm7 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm6 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm5 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm4 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm3 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm2 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm1 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm0 [definition, in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm [definition, in Combi.MPoly.MurnaghanNakayama]
+upoints [projection, in ALEA.Qmeasure]
+usize [definition, in ALEA.Qmeasure]
+usize_pos [lemma, in ALEA.Qmeasure]
+

V

+val_intpartn3 [lemma, in Combi.Combi.partition]
+val_intpartn2 [lemma, in Combi.Combi.partition]
+val_intpartn1 [lemma, in Combi.Combi.partition]
+val_intpartn0 [lemma, in Combi.Combi.partition]
+val_stdtabshcast [lemma, in Combi.Combi.stdtab]
+val_enum_subseqs [lemma, in Combi.Combi.subseq]
+val_omegasf [lemma, in Combi.MPoly.sympoly]
+val_descset [lemma, in Combi.Combi.composition]
+val2pos [definition, in Combi.LRrule.stdplact]
+val2posE [lemma, in Combi.LRrule.stdplact]
+val2pos_enum [lemma, in Combi.LRrule.stdplact]
+val2pos_inj [lemma, in Combi.LRrule.stdplact]
+VandermondeDet [section, in Combi.MPoly.antisym]
+VandermondeDet.n [variable, in Combi.MPoly.antisym]
+VandermondeDet.R [variable, in Combi.MPoly.antisym]
+'a_ _ [notation, in Combi.MPoly.antisym]
+Vandet [definition, in Combi.MPoly.antisym]
+Vandet_VanprodE [lemma, in Combi.MPoly.antisym]
+Vanmx [definition, in Combi.MPoly.antisym]
+Vanmx_antimE [lemma, in Combi.MPoly.antisym]
+Vanprod [section, in Combi.MPoly.antisym]
+Vanprod [definition, in Combi.MPoly.antisym]
+Vanprod_alt [lemma, in Combi.MPoly.antisym]
+Vanprod_alt_int [lemma, in Combi.MPoly.antisym]
+Vanprod_dhomog [lemma, in Combi.MPoly.antisym]
+Vanprod_neq0 [lemma, in Combi.MPoly.antisym]
+Vanprod_coeff_rho [lemma, in Combi.MPoly.antisym]
+Vanprod_anti [lemma, in Combi.MPoly.antisym]
+Vanprod.abound [variable, in Combi.MPoly.antisym]
+Vanprod.n [variable, in Combi.MPoly.antisym]
+Vanprod.R [variable, in Combi.MPoly.antisym]
+Vanprod.rbound [variable, in Combi.MPoly.antisym]
+'a_ _ [notation, in Combi.MPoly.antisym]
+'X_ _ [notation, in Combi.MPoly.antisym]
+vb_strip_lexi [lemma, in Combi.MPoly.Schur_altdef]
+vb_strip_rem_col0 [lemma, in Combi.MPoly.Schur_altdef]
+vb_strip_conjE [lemma, in Combi.Combi.skewpart]
+vb_strip_conj [lemma, in Combi.Combi.skewpart]
+vb_strip_diffP [lemma, in Combi.Combi.skewpart]
+vb_stripP [lemma, in Combi.Combi.skewpart]
+vb_strip_included [lemma, in Combi.Combi.skewpart]
+vb_strip [definition, in Combi.Combi.skewpart]
+vctleq [definition, in Combi.Combi.bintree]
+vctleqP [lemma, in Combi.Combi.bintree]
+vctleq_rotation [lemma, in Combi.Combi.bintree]
+vctleq_sumn_right_sizes [lemma, in Combi.Combi.bintree]
+vctleq_anti [lemma, in Combi.Combi.bintree]
+vctleq_trans [lemma, in Combi.Combi.bintree]
+vctleq_refl [lemma, in Combi.Combi.bintree]
+vctmin [definition, in Combi.Combi.bintree]
+vctminC [lemma, in Combi.Combi.bintree]
+vctminP [lemma, in Combi.Combi.bintree]
+vctminPl [lemma, in Combi.Combi.bintree]
+vctminPr [lemma, in Combi.Combi.bintree]
+vctmin_Tamari [lemma, in Combi.Combi.bintree]
+vct_succ [lemma, in Combi.Combi.bintree]
+vecmx_indexK [lemma, in Combi.MPoly.Cauchy]
+VectNK [section, in Combi.Combi.vectNK]
+vectNK [library]
+Vector [section, in Combi.MPoly.homogsym]
+vector_Lmodule_hasFinDim__to__vector_LSemiModule_hasFinDim [definition, in Combi.MPoly.homogsym]
+Vector.d [variable, in Combi.MPoly.homogsym]
+Vector.n0 [variable, in Combi.MPoly.homogsym]
+Vector.R [variable, in Combi.MPoly.homogsym]
+vect_0_k [lemma, in Combi.Combi.vectNK]
+vect_n_kP [lemma, in Combi.Combi.vectNK]
+vect_n_k_in [lemma, in Combi.Combi.vectNK]
+vect_n_k [definition, in Combi.Combi.vectNK]
+versions [definition, in Combi.Combi.std]
+

W

+walk_to_corner_decomp [lemma, in Combi.HookFormula.hook]
+walk_to_corner_emptyr [lemma, in Combi.HookFormula.hook]
+walk_to_corner_emptyl [lemma, in Combi.HookFormula.hook]
+walk_to_corner_inv [lemma, in Combi.HookFormula.hook]
+walk_to_corner_simpl [lemma, in Combi.HookFormula.hook]
+walk_to_corner_end_simpl [lemma, in Combi.HookFormula.hook]
+walk_to_corner0_simpl [lemma, in Combi.HookFormula.hook]
+walk_to_corner [definition, in Combi.HookFormula.hook]
+walk_to_corner_rec [definition, in Combi.HookFormula.hook]
+wcord [definition, in Combi.SymGroup.presentSn]
+wcordE [lemma, in Combi.SymGroup.presentSn]
+wcord_cons [lemma, in Combi.SymGroup.presentSn]
+WeakOrder [module, in Combi.SymGroup.weak_order]
+WeakOrder.Def [section, in Combi.SymGroup.weak_order]
+WeakOrder.Def.n0 [variable, in Combi.SymGroup.weak_order]
+_ <=R _ [notation, in Combi.SymGroup.weak_order]
+WeakOrder.Exports [module, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.lepermP [definition, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.leperm_lengthE [definition, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.leperm_length [definition, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.n [abbreviation, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.WeakOrder [section, in Combi.SymGroup.weak_order]
+WeakOrder.Exports.WeakOrder.n0 [variable, in Combi.SymGroup.weak_order]
+_ \/R _ (Combi_scope) [notation, in Combi.SymGroup.weak_order]
+_ /\R _ (Combi_scope) [notation, in Combi.SymGroup.weak_order]
+_ <R _ (Combi_scope) [notation, in Combi.SymGroup.weak_order]
+_ <=R _ (Combi_scope) [notation, in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__fintype_isFinite [definition, in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__eqtype_hasDecEq [definition, in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__choice_Choice_isCountable [definition, in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__choice_hasChoice [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_14 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_13 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_factory_10 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_9 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_8 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_7 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_6 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_factory_1 [definition, in Combi.SymGroup.weak_order]
+WeakOrder.leperm [definition, in Combi.SymGroup.weak_order]
+WeakOrder.lepermP [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.leperm_anti [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.leperm_trans [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.leperm_refl [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.leperm_lengthE [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.leperm_length [lemma, in Combi.SymGroup.weak_order]
+WeakOrder.n [abbreviation, in Combi.SymGroup.weak_order]
+WeakOrder.Order_Le_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_FinPOrder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_FinPreorder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_POrder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_Preorder [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__fintype_Finite [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__choice_Countable [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__choice_Choice [definition, in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__eqtype_Equality [definition, in Combi.SymGroup.weak_order]
+weak_order [library]
+weight [definition, in ALEA.Qmeasure]
+weight_is_unit [lemma, in ALEA.Qmeasure]
+weight_pos [projection, in ALEA.Qmeasure]
+weight_case [lemma, in ALEA.Qmeasure]
+weight_nonneg [lemma, in ALEA.Qmeasure]
+weight1_size [lemma, in ALEA.Qmeasure]
+Wikipedia_Murnaghan_Nakayama [definition, in Combi.SymGroup.Frobenius_char]
+word [abbreviation, in Combi.Basic.congr]
+word [abbreviation, in Combi.Basic.congr]
+word [abbreviation, in Combi.Basic.congr]
+word [abbreviation, in Combi.Basic.congr]
+wordcd [definition, in Combi.SymGroup.presentSn]
+wordcdE [lemma, in Combi.SymGroup.presentSn]
+wordcd_ltn [lemma, in Combi.SymGroup.presentSn]
+wordperm [definition, in Combi.Combi.std]
+wordperm_inj [lemma, in Combi.Combi.std]
+wordperm_invP [lemma, in Combi.Combi.std]
+wordperm_std [lemma, in Combi.Combi.std]
+wordperm_iota [lemma, in Combi.Combi.std]
+wp [abbreviation, in Combi.Combi.permuted]
+

Y

+Yama [section, in Combi.Combi.Yamanouchi]
+Yamanouchi [library]
+Yamanouchi_yamn__canonical__fintype_SubFinite [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__fintype_Finite [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_SubCountable [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_Countable [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_SubChoice [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_Choice [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_SubEquality [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_Equality [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_SubType [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__fintype_SubFinite [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__fintype_Finite [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_SubCountable [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_Countable [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_SubChoice [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_Choice [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_SubEquality [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_Equality [definition, in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_SubType [definition, in Combi.Combi.Yamanouchi]
+yameval [record, in Combi.Combi.Yamanouchi]
+yamevalP [lemma, in Combi.Combi.Yamanouchi]
+yamevalval [projection, in Combi.Combi.Yamanouchi]
+yamn [record, in Combi.Combi.Yamanouchi]
+yamnP [lemma, in Combi.Combi.Yamanouchi]
+yamnval [projection, in Combi.Combi.Yamanouchi]
+yamn_partition_evalseq [lemma, in Combi.Combi.Yamanouchi]
+yamn_PredEq [lemma, in Combi.Combi.Yamanouchi]
+YamOfEval [section, in Combi.Combi.Yamanouchi]
+YamOfEval.ev [variable, in Combi.Combi.Yamanouchi]
+YamOfSize [section, in Combi.Combi.Yamanouchi]
+YamOfSize.n [variable, in Combi.Combi.Yamanouchi]
+yamrow [definition, in Combi.LRrule.therule]
+yamrowP [lemma, in Combi.LRrule.therule]
+yamtab [definition, in Combi.LRrule.Yam_plact]
+yamtabP [lemma, in Combi.LRrule.Yam_plact]
+yamtab_unique [lemma, in Combi.LRrule.Yam_plact]
+yamtab_rcons [lemma, in Combi.LRrule.Yam_plact]
+yamtab_rec [definition, in Combi.LRrule.Yam_plact]
+yamtab_shift_countE [lemma, in Combi.LRrule.implem]
+yamtab_rows_countE [lemma, in Combi.LRrule.implem]
+yamtab_shift_is_row [lemma, in Combi.LRrule.implem]
+yamtab_rows_is_row [lemma, in Combi.LRrule.implem]
+yamtab_shift_dominate [lemma, in Combi.LRrule.implem]
+yamtab_rows_dominate [lemma, in Combi.LRrule.implem]
+yamtab_shift_size [lemma, in Combi.LRrule.implem]
+yamtab_rows_size [lemma, in Combi.LRrule.implem]
+yamtab_shift_included [lemma, in Combi.LRrule.implem]
+yamtab_rows_included [lemma, in Combi.LRrule.implem]
+yamtab_shiftP [lemma, in Combi.LRrule.implem]
+yamtab_rowsP [lemma, in Combi.LRrule.implem]
+yamtab_shift_drop [lemma, in Combi.LRrule.implem]
+yamtab_shift [definition, in Combi.LRrule.implem]
+yamtab_rows [definition, in Combi.LRrule.implem]
+yam_tail_non_nil [lemma, in Combi.LRrule.Schensted]
+yam_std_inj [lemma, in Combi.LRrule.Yam_plact]
+yam_plactic_shape [lemma, in Combi.LRrule.Yam_plact]
+yam_plactic_hyper [lemma, in Combi.LRrule.Yam_plact]
+yam_of_stdtabP [lemma, in Combi.Combi.stdtab]
+yam_of_stdtabK [lemma, in Combi.Combi.stdtab]
+yam_of_stdtab [definition, in Combi.Combi.stdtab]
+yam_of_stdtab_rec [definition, in Combi.Combi.stdtab]
+yam_of_rowpart [lemma, in Combi.LRrule.therule]
+yam_to_word [projection, in Combi.LRrule.implem]
+Yam_plact [library]
+Ymon [definition, in Combi.MPoly.Cauchy]
+YmonK [lemma, in Combi.MPoly.Cauchy]
+Ymon_bij [lemma, in Combi.MPoly.Cauchy]
+YoungIrrDef [section, in Combi.SymGroup.Frobenius_char]
+YoungIrrDef.n [variable, in Combi.SymGroup.Frobenius_char]
+'irrSG[ _ ] [notation, in Combi.SymGroup.Frobenius_char]
+'M[ _ ] [notation, in Combi.SymGroup.Frobenius_char]
+YoungLattice [module, in Combi.Combi.partition]
+YoungLattice.bottom_YoungE [lemma, in Combi.Combi.partition]
+YoungLattice.choice_Countable__to__choice_Choice_isCountable [definition, in Combi.Combi.partition]
+YoungLattice.choice_Countable__to__eqtype_hasDecEq [definition, in Combi.Combi.partition]
+YoungLattice.choice_Countable__to__choice_hasChoice [definition, in Combi.Combi.partition]
+YoungLattice.emptypart_bottom [lemma, in Combi.Combi.partition]
+YoungLattice.Exports [module, in Combi.Combi.partition]
+YoungLattice.Exports.bottom_YoungE [definition, in Combi.Combi.partition]
+YoungLattice.Exports.intpartYoung [abbreviation, in Combi.Combi.partition]
+YoungLattice.Exports.join_YoungE [lemma, in Combi.Combi.partition]
+YoungLattice.Exports.le_Young_sumn [definition, in Combi.Combi.partition]
+YoungLattice.Exports.le_YoungP [definition, in Combi.Combi.partition]
+YoungLattice.Exports.le_YoungE [definition, in Combi.Combi.partition]
+YoungLattice.Exports.lt_Young_sumn [definition, in Combi.Combi.partition]
+YoungLattice.Exports.meet_YoungE [lemma, in Combi.Combi.partition]
+YoungLattice.Exports.nth_join_Young [lemma, in Combi.Combi.partition]
+YoungLattice.Exports.nth_meet_Young [lemma, in Combi.Combi.partition]
+YoungLattice.Exports.size_join_Young [lemma, in Combi.Combi.partition]
+YoungLattice.Exports.size_meet_Young [lemma, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_115 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_113 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_111 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_110 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_109 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_106 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_105 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_104 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_101 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_100 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_96 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_95 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_94 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_93 [definition, in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_89 [definition, in Combi.Combi.partition]
+YoungLattice.intpartYoung [definition, in Combi.Combi.partition]
+YoungLattice.join_YoungP [lemma, in Combi.Combi.partition]
+YoungLattice.join_YoungC [lemma, in Combi.Combi.partition]
+YoungLattice.join_Young_le [lemma, in Combi.Combi.partition]
+YoungLattice.join_Young [definition, in Combi.Combi.partition]
+YoungLattice.join_Young_subproof [lemma, in Combi.Combi.partition]
+YoungLattice.join_Young_fun [definition, in Combi.Combi.partition]
+YoungLattice.le_Young_sumn [lemma, in Combi.Combi.partition]
+YoungLattice.le_YoungP [lemma, in Combi.Combi.partition]
+YoungLattice.le_YoungE [lemma, in Combi.Combi.partition]
+YoungLattice.le_Young_anti [lemma, in Combi.Combi.partition]
+YoungLattice.le_Young_trans [lemma, in Combi.Combi.partition]
+YoungLattice.le_Young_refl [lemma, in Combi.Combi.partition]
+YoungLattice.le_Young [definition, in Combi.Combi.partition]
+YoungLattice.lt_Young_sumn [lemma, in Combi.Combi.partition]
+YoungLattice.meet_YoungP [lemma, in Combi.Combi.partition]
+YoungLattice.meet_Young_le [lemma, in Combi.Combi.partition]
+YoungLattice.meet_YoungC [lemma, in Combi.Combi.partition]
+YoungLattice.meet_Young [definition, in Combi.Combi.partition]
+YoungLattice.meet_Young_subproof [lemma, in Combi.Combi.partition]
+YoungLattice.meet_Young_fun [definition, in Combi.Combi.partition]
+YoungLattice.nth_join_Young [lemma, in Combi.Combi.partition]
+YoungLattice.nth_meet_Young [lemma, in Combi.Combi.partition]
+YoungLattice.Order_Lattice_Meet_isDistrLattice__to__Order_Lattice_isDistributive [definition, in Combi.Combi.partition]
+YoungLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.Order_Le_isPOrder__to__Order_isDuallyPreorder [definition, in Combi.Combi.partition]
+YoungLattice.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [definition, in Combi.Combi.partition]
+YoungLattice.ordtype_Inhabited__to__ordtype_isInhabited [definition, in Combi.Combi.partition]
+YoungLattice.size_join_Young [lemma, in Combi.Combi.partition]
+YoungLattice.size_meet_Young [lemma, in Combi.Combi.partition]
+YoungLattice.YoungLattice [section, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BDistrLattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_DistrLattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BLattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BJoinSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BMeetSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BPOrder [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BPreorder [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_InhLattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_Lattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_MeetSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_JoinSemilattice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_InhPOrder [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_POrder [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_Preorder [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_Inhabited [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__choice_Countable [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__choice_Choice [definition, in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__eqtype_Equality [definition, in Combi.Combi.partition]
+'YL [notation, in Combi.Combi.partition]
+YoungLattice.Young_meetUl [lemma, in Combi.Combi.partition]
+YoungLattice.Young_display [lemma, in Combi.Combi.partition]
+YoungSG [definition, in Combi.SymGroup.Frobenius_char]
+Young_rule_partdom [lemma, in Combi.SymGroup.Frobenius_char]
+Young_rule [lemma, in Combi.SymGroup.Frobenius_char]
+Young_char [lemma, in Combi.SymGroup.Frobenius_char]
+

Z

+zcard [definition, in Combi.SymGroup.permcent]
+zcard_rem [lemma, in Combi.SymGroup.permcent]
+zcard_any [lemma, in Combi.SymGroup.permcent]
+zcard_nil [lemma, in Combi.SymGroup.permcent]
+zcoeff [definition, in Combi.SymGroup.towerSn]
+zcoeffE [lemma, in Combi.SymGroup.towerSn]
+

other

+_ =Br _ (bool_scope) [notation, in Combi.SymGroup.presentSn]
+_ +|+ _ (Combi_scope) [notation, in Combi.Combi.partition]
+_ / _ (Combi_scope) [notation, in Combi.Combi.partition]
+[ in _ [ _ ] , _ .-homsym ] (form_scope) [notation, in Combi.MPoly.homogsym]
+_ .-homsym (form_scope) [notation, in Combi.MPoly.homogsym]
+_ .-supp [ _ , _ ] (form_scope) [notation, in Combi.LRrule.Greene]
+[ Dyck {{ _ }} _ ] (form_scope) [notation, in Combi.Combi.Dyckword]
+[ Dyck of _ by _ ] (form_scope) [notation, in Combi.Combi.Dyckword]
+[ Dyck of _ ] (form_scope) [notation, in Combi.Combi.Dyckword]
+'s_[ _ ] (group_scope) [notation, in Combi.SymGroup.presentSn]
+'s_ _ (group_scope) [notation, in Combi.SymGroup.presentSn]
+'K ( _ , _ ) (nat_scope) [notation, in Combi.MPoly.Schur_altdef]
+'C [ _ ] (nat_scope) [notation, in Combi.Combi.multinomial]
+_ @@_ _ (O_scope) [notation, in ALEA.Ccpo]
+_ @_ _ (O_scope) [notation, in ALEA.Ccpo]
+_ <_> _ (O_scope) [notation, in ALEA.Ccpo]
+_ -c> _ (O_scope) [notation, in ALEA.Ccpo]
+mlub _ (O_scope) [notation, in ALEA.Ccpo]
+0 (O_scope) [notation, in ALEA.Ccpo]
+_ @2 _ (O_scope) [notation, in ALEA.Ccpo]
+_ @ _ (O_scope) [notation, in ALEA.Ccpo]
+_ <o> _ (O_scope) [notation, in ALEA.Ccpo]
+_ -m-> _ (O_scope) [notation, in ALEA.Ccpo]
+_ --m-> _ (O_scope) [notation, in ALEA.Ccpo]
+_ --m> _ (O_scope) [notation, in ALEA.Ccpo]
+_ -m> _ (O_scope) [notation, in ALEA.Ccpo]
+_ < _ (O_scope) [notation, in ALEA.Ccpo]
+_ >= _ (O_scope) [notation, in ALEA.Ccpo]
+_ <= _ (O_scope) [notation, in ALEA.Ccpo]
+'[ _ | _ ] _( _ , _ ) (ring_scope) [notation, in Combi.MPoly.homogsym]
+'[ _ | _ ] (ring_scope) [notation, in Combi.MPoly.homogsym]
+'K^-1 ( _ , _ ) (ring_scope) [notation, in Combi.MPoly.Schur_altdef]
+'K ( _ , _ ) (ring_scope) [notation, in Combi.MPoly.Schur_altdef]
+'1_[ _ ] (ring_scope) [notation, in Combi.SymGroup.cycletype]
+_ == _ (type_scope) [notation, in ALEA.Ccpo]
+_ (XY) [notation, in Combi.MPoly.Cauchy]
+_ (X) [notation, in Combi.MPoly.Cauchy]
+_ (Y) [notation, in Combi.MPoly.Cauchy]
+_ *h _ [notation, in Combi.MPoly.homogsym]
+_ \/T _ [notation, in Combi.Combi.bintree]
+_ /\T _ [notation, in Combi.Combi.bintree]
+_ <T _ [notation, in Combi.Combi.bintree]
+_ <=T _ [notation, in Combi.Combi.bintree]
+_ <=V _ [notation, in Combi.Combi.bintree]
+_ \o^ _ [notation, in Combi.SymGroup.towerSn]
+_ \ox _ [notation, in Combi.SymGroup.towerSn]
+_ =Pl _ [notation, in Combi.LRrule.plactic]
+tabsh[ _ ] _ [notation, in Combi.Combi.tableau]
+#{ _ } [notation, in Combi.SSRcomplements.tools]
+'e_ _ [notation, in Combi.MPoly.sympoly]
+'e[ _ ] [notation, in Combi.MPoly.sympoly]
+'he [notation, in Combi.MPoly.homogsym]
+'he[ _ ] [notation, in Combi.MPoly.homogsym]
+'hh [notation, in Combi.MPoly.homogsym]
+'hh[ _ ] [notation, in Combi.MPoly.homogsym]
+'hm [notation, in Combi.MPoly.homogsym]
+'hm[ _ ] [notation, in Combi.MPoly.homogsym]
+'hp [notation, in Combi.MPoly.homogsym]
+'hp[ _ ] [notation, in Combi.MPoly.homogsym]
+'hs [notation, in Combi.MPoly.homogsym]
+'hs[ _ ] [notation, in Combi.MPoly.homogsym]
+'h_ _ [notation, in Combi.MPoly.sympoly]
+'h[ _ ] [notation, in Combi.MPoly.sympoly]
+'II_ _ [notation, in Combi.MPoly.antisym]
+'II_ _ [notation, in Combi.SymGroup.presentSn]
+'irrSG[ _ ] [notation, in Combi.SymGroup.Frobenius_char]
+'M[ _ ] [notation, in Combi.SymGroup.Frobenius_char]
+'m[ _ ] [notation, in Combi.MPoly.sympoly]
+'P_ _ [notation, in Combi.Combi.partition]
+'p_ _ [notation, in Combi.MPoly.sympoly]
+'p[ _ ] [notation, in Combi.MPoly.sympoly]
+'SC_ _ [notation, in Combi.SymGroup.permcent]
+'SG_ _ [notation, in Combi.SymGroup.towerSn]
+'SG_ _ [notation, in Combi.SymGroup.presentSn]
+'s[ _ ] [notation, in Combi.MPoly.sympoly]
+'z_ _ [notation, in Combi.SymGroup.towerSn]
+'1z_[ _ ] [notation, in Combi.SymGroup.towerSn]
+[1/2] [notation, in ALEA.Qmeasure]
+{ homsym _ [ _ , _ ] } [notation, in Combi.MPoly.homogsym]
+{ sympoly _ [ _ ] } [notation, in Combi.MPoly.sympoly]
+{{ [notation, in Combi.Combi.Dyckword]
+}} [notation, in Combi.Combi.Dyckword]
+

+

Notation Index

+

A

+_ * _ [in Combi.SSRcomplements.tools]
+*%M [in Combi.SSRcomplements.tools]
+1 [in Combi.SSRcomplements.tools]
+_ # _ [in Combi.Combi.permuted]
+'a_ _ [in Combi.MPoly.Schur_altdef]
+'e_ _ [in Combi.MPoly.Schur_altdef]
+_ # _ [in Combi.MPoly.antisym]
+'a_ _ [in Combi.MPoly.antisym]
+_ # _ [in Combi.MPoly.Schur_altdef]
+'a_ _ [in Combi.MPoly.Schur_altdef]
+

B

+_ # _ [in Combi.MPoly.sympoly]
+'e_ _ [in Combi.MPoly.sympoly]
+'h_ _ [in Combi.MPoly.sympoly]
+'m[ _ ] [in Combi.MPoly.sympoly]
+'p_ _ [in Combi.MPoly.sympoly]
+_ * _ [in Combi.Combi.subseq]
+*%M [in Combi.Combi.subseq]
+1 [in Combi.Combi.subseq]
+

C

+'s_ _ (group_scope) [in Combi.SymGroup.presentSn]
+_ =Br _ [in Combi.SymGroup.presentSn]
+'I[ _ .. _ ] [in Combi.SymGroup.presentSn]
+'s_[ _ ] [in Combi.SymGroup.presentSn]
+_ (XY) [in Combi.MPoly.Cauchy]
+_ (X) [in Combi.MPoly.Cauchy]
+_ (Y) [in Combi.MPoly.Cauchy]
+_ *:M _ [in Combi.MPoly.Cauchy]
+\Pi _ [in Combi.MPoly.sympoly]
+'Xn_ _ [in Combi.MPoly.sympoly]
+'Xn [in Combi.MPoly.sympoly]
+'XXn_ _ [in Combi.MPoly.sympoly]
+'1_[ _ ] (ring_scope) [in Combi.SymGroup.cycletype]
+

D

+_ / _ (Combi_scope) [in Combi.Combi.partition]
+

E

+'s_[ _ ] [in Combi.SymGroup.presentSn]
+'s_ _ [in Combi.SymGroup.presentSn]
+

G

+_ .-supp (form_scope) [in Combi.LRrule.Greene]
+

H

+_ *h _ [in Combi.MPoly.homogsym]
+

I

+'a_ _ [in Combi.MPoly.Schur_altdef]
+'s_ _ [in Combi.MPoly.Schur_altdef]
+'z_ _ [in Combi.SymGroup.towerSn]
+'1z_[ _ ] [in Combi.SymGroup.towerSn]
+'pi_ _ [in Combi.MPoly.homogsym]
+'PDom_ _ [in Combi.Combi.partition]
+_ ^# [in Combi.Combi.partition]
+'PDom [in Combi.Combi.partition]
+'PDom [in Combi.Combi.partition]
+'PLexi_ _ [in Combi.Combi.partition]
+'PLexi [in Combi.Combi.partition]
+'Pr [in Combi.Combi.skewpart]
+

K

+'K ( _ , _ ) (nat_scope) [in Combi.MPoly.Schur_altdef]
+'K ( _ , _ ) (ring_scope) [in Combi.MPoly.Schur_altdef]
+

L

+'s_[ _ ] [in Combi.SymGroup.presentSn]
+'s_ _ [in Combi.SymGroup.presentSn]
+

M

+_ # _ [in Combi.MPoly.antisym]
+_ # _ [in Combi.MPoly.antisym]
+'a_ _ [in Combi.MPoly.MurnaghanNakayama]
+

P

+'P [in Combi.Combi.partition]
+'CC ( _ ) (group_scope) [in Combi.SymGroup.permcent]
+'s_[ _ ] [in Combi.SymGroup.presentSn]
+'s_ _ [in Combi.SymGroup.presentSn]
+'g_ _ [in Combi.SymGroup.presentSn]
+'g_ _ [in Combi.MPoly.sympoly]
+'g[ _ ] [in Combi.MPoly.sympoly]
+'co[ _ ]_ _ [in Combi.MPoly.sympoly]
+'co[ _ ] [in Combi.MPoly.sympoly]
+'gA_ _ [in Combi.MPoly.sympoly]
+'gA[ _ ] [in Combi.MPoly.sympoly]
+'gB_ _ [in Combi.MPoly.sympoly]
+'gB[ _ ] [in Combi.MPoly.sympoly]
+

R

+_ =Br _ [in Combi.SymGroup.presentSn]
+'s_[ _ ] [in Combi.SymGroup.presentSn]
+'s_ _ [in Combi.SymGroup.presentSn]
+'CRef [in Combi.Combi.composition]
+'s_ _ [in Combi.MPoly.Schur_altdef]
+

S

+'[ _ | _ ] (ring_scope) [in Combi.MPoly.homogsym]
+'[ _ | _ ] (ring_scope) [in Combi.MPoly.homogsym]
+_ (X) [in Combi.MPoly.Cauchy]
+_ (Y) [in Combi.MPoly.Cauchy]
+'hpC[ _ ] [in Combi.MPoly.Cauchy]
+'hsC[ _ ] [in Combi.MPoly.Cauchy]
+'a_ _ [in Combi.MPoly.Schur_altdef]
+'s_[ _ ] [in Combi.MPoly.Schur_altdef]
+'s[ _ ] [in Combi.MPoly.sympoly]
+_ <=R _ [in Combi.SSRcomplements.sorted]
+_ ^~ [in Combi.MPoly.sympoly]
+_ ^~ [in Combi.MPoly.sympoly]
+

V

+'a_ _ [in Combi.MPoly.antisym]
+'a_ _ [in Combi.MPoly.antisym]
+'X_ _ [in Combi.MPoly.antisym]
+

W

+_ <=R _ [in Combi.SymGroup.weak_order]
+_ \/R _ (Combi_scope) [in Combi.SymGroup.weak_order]
+_ /\R _ (Combi_scope) [in Combi.SymGroup.weak_order]
+_ <R _ (Combi_scope) [in Combi.SymGroup.weak_order]
+_ <=R _ (Combi_scope) [in Combi.SymGroup.weak_order]
+

Y

+'irrSG[ _ ] [in Combi.SymGroup.Frobenius_char]
+'M[ _ ] [in Combi.SymGroup.Frobenius_char]
+'YL [in Combi.Combi.partition]
+

other

+_ =Br _ (bool_scope) [in Combi.SymGroup.presentSn]
+_ +|+ _ (Combi_scope) [in Combi.Combi.partition]
+_ / _ (Combi_scope) [in Combi.Combi.partition]
+[ in _ [ _ ] , _ .-homsym ] (form_scope) [in Combi.MPoly.homogsym]
+_ .-homsym (form_scope) [in Combi.MPoly.homogsym]
+_ .-supp [ _ , _ ] (form_scope) [in Combi.LRrule.Greene]
+[ Dyck {{ _ }} _ ] (form_scope) [in Combi.Combi.Dyckword]
+[ Dyck of _ by _ ] (form_scope) [in Combi.Combi.Dyckword]
+[ Dyck of _ ] (form_scope) [in Combi.Combi.Dyckword]
+'s_[ _ ] (group_scope) [in Combi.SymGroup.presentSn]
+'s_ _ (group_scope) [in Combi.SymGroup.presentSn]
+'K ( _ , _ ) (nat_scope) [in Combi.MPoly.Schur_altdef]
+'C [ _ ] (nat_scope) [in Combi.Combi.multinomial]
+_ @@_ _ (O_scope) [in ALEA.Ccpo]
+_ @_ _ (O_scope) [in ALEA.Ccpo]
+_ <_> _ (O_scope) [in ALEA.Ccpo]
+_ -c> _ (O_scope) [in ALEA.Ccpo]
+mlub _ (O_scope) [in ALEA.Ccpo]
+0 (O_scope) [in ALEA.Ccpo]
+_ @2 _ (O_scope) [in ALEA.Ccpo]
+_ @ _ (O_scope) [in ALEA.Ccpo]
+_ <o> _ (O_scope) [in ALEA.Ccpo]
+_ -m-> _ (O_scope) [in ALEA.Ccpo]
+_ --m-> _ (O_scope) [in ALEA.Ccpo]
+_ --m> _ (O_scope) [in ALEA.Ccpo]
+_ -m> _ (O_scope) [in ALEA.Ccpo]
+_ < _ (O_scope) [in ALEA.Ccpo]
+_ >= _ (O_scope) [in ALEA.Ccpo]
+_ <= _ (O_scope) [in ALEA.Ccpo]
+'[ _ | _ ] _( _ , _ ) (ring_scope) [in Combi.MPoly.homogsym]
+'[ _ | _ ] (ring_scope) [in Combi.MPoly.homogsym]
+'K^-1 ( _ , _ ) (ring_scope) [in Combi.MPoly.Schur_altdef]
+'K ( _ , _ ) (ring_scope) [in Combi.MPoly.Schur_altdef]
+'1_[ _ ] (ring_scope) [in Combi.SymGroup.cycletype]
+_ == _ (type_scope) [in ALEA.Ccpo]
+_ (XY) [in Combi.MPoly.Cauchy]
+_ (X) [in Combi.MPoly.Cauchy]
+_ (Y) [in Combi.MPoly.Cauchy]
+_ *h _ [in Combi.MPoly.homogsym]
+_ \/T _ [in Combi.Combi.bintree]
+_ /\T _ [in Combi.Combi.bintree]
+_ <T _ [in Combi.Combi.bintree]
+_ <=T _ [in Combi.Combi.bintree]
+_ <=V _ [in Combi.Combi.bintree]
+_ \o^ _ [in Combi.SymGroup.towerSn]
+_ \ox _ [in Combi.SymGroup.towerSn]
+_ =Pl _ [in Combi.LRrule.plactic]
+tabsh[ _ ] _ [in Combi.Combi.tableau]
+#{ _ } [in Combi.SSRcomplements.tools]
+'e_ _ [in Combi.MPoly.sympoly]
+'e[ _ ] [in Combi.MPoly.sympoly]
+'he [in Combi.MPoly.homogsym]
+'he[ _ ] [in Combi.MPoly.homogsym]
+'hh [in Combi.MPoly.homogsym]
+'hh[ _ ] [in Combi.MPoly.homogsym]
+'hm [in Combi.MPoly.homogsym]
+'hm[ _ ] [in Combi.MPoly.homogsym]
+'hp [in Combi.MPoly.homogsym]
+'hp[ _ ] [in Combi.MPoly.homogsym]
+'hs [in Combi.MPoly.homogsym]
+'hs[ _ ] [in Combi.MPoly.homogsym]
+'h_ _ [in Combi.MPoly.sympoly]
+'h[ _ ] [in Combi.MPoly.sympoly]
+'II_ _ [in Combi.MPoly.antisym]
+'II_ _ [in Combi.SymGroup.presentSn]
+'irrSG[ _ ] [in Combi.SymGroup.Frobenius_char]
+'M[ _ ] [in Combi.SymGroup.Frobenius_char]
+'m[ _ ] [in Combi.MPoly.sympoly]
+'P_ _ [in Combi.Combi.partition]
+'p_ _ [in Combi.MPoly.sympoly]
+'p[ _ ] [in Combi.MPoly.sympoly]
+'SC_ _ [in Combi.SymGroup.permcent]
+'SG_ _ [in Combi.SymGroup.towerSn]
+'SG_ _ [in Combi.SymGroup.presentSn]
+'s[ _ ] [in Combi.MPoly.sympoly]
+'z_ _ [in Combi.SymGroup.towerSn]
+'1z_[ _ ] [in Combi.SymGroup.towerSn]
+[1/2] [in ALEA.Qmeasure]
+{ homsym _ [ _ , _ ] } [in Combi.MPoly.homogsym]
+{ sympoly _ [ _ ] } [in Combi.MPoly.sympoly]
+{{ [in Combi.Combi.Dyckword]
+}} [in Combi.Combi.Dyckword]
+

+

Module Index

+

B

+Builders_1.Builders_Export_5 [in Combi.Basic.ordtype]
+Builders_1.Super [in Combi.Basic.ordtype]
+Builders_1 [in Combi.Basic.ordtype]
+Builders_6.Builders_Export_10 [in Combi.Combi.composition]
+Builders_6.Super [in Combi.Combi.composition]
+Builders_6 [in Combi.Combi.composition]
+Builders_1.Builders_Export_5 [in Combi.Combi.composition]
+Builders_1.Super [in Combi.Combi.composition]
+Builders_1 [in Combi.Combi.composition]
+

E

+Example1 [in Combi.Basic.combclass]
+Example2 [in Combi.Basic.combclass]
+Example3 [in Combi.Basic.combclass]
+

I

+Inhabited [in Combi.Basic.ordtype]
+InhabitedElpiOperations [in Combi.Basic.ordtype]
+Inhabited.Exports [in Combi.Basic.ordtype]
+InhFinite [in Combi.Basic.ordtype]
+InhFiniteElpiOperations [in Combi.Basic.ordtype]
+InhFinite.Exports [in Combi.Basic.ordtype]
+InhFinLattice [in Combi.Basic.ordtype]
+InhFinLatticeElpiOperations [in Combi.Basic.ordtype]
+InhFinLattice.Exports [in Combi.Basic.ordtype]
+InhFinOrder [in Combi.Basic.ordtype]
+InhFinOrderElpiOperations [in Combi.Basic.ordtype]
+InhFinOrder.Exports [in Combi.Basic.ordtype]
+InhFinPOrder [in Combi.Basic.ordtype]
+InhFinPOrderElpiOperations [in Combi.Basic.ordtype]
+InhFinPOrder.Exports [in Combi.Basic.ordtype]
+InhLattice [in Combi.Basic.ordtype]
+InhLatticeElpiOperations [in Combi.Basic.ordtype]
+InhLattice.Exports [in Combi.Basic.ordtype]
+InhOrder [in Combi.Basic.ordtype]
+InhOrderElpiOperations [in Combi.Basic.ordtype]
+InhOrder.Exports [in Combi.Basic.ordtype]
+InhPOrder [in Combi.Basic.ordtype]
+InhPOrderElpiOperations [in Combi.Basic.ordtype]
+InhPOrder.Exports [in Combi.Basic.ordtype]
+InhTBLattice [in Combi.Basic.ordtype]
+InhTBLatticeElpiOperations [in Combi.Basic.ordtype]
+InhTBLattice.Exports [in Combi.Basic.ordtype]
+IntPartNDom [in Combi.Combi.partition]
+IntPartNDom.Exports [in Combi.Combi.partition]
+IntPartNLexi [in Combi.Combi.partition]
+IntPartNLexi.Exports [in Combi.Combi.partition]
+isInhabited [in Combi.Basic.ordtype]
+isInhabitedType [in Combi.Basic.ordtype]
+isInhabitedType.Exports [in Combi.Basic.ordtype]
+isInhabited.Exports [in Combi.Basic.ordtype]
+IsoBottom [in Combi.Combi.composition]
+IsoBottom.Exports [in Combi.Combi.composition]
+IsoTop [in Combi.Combi.composition]
+IsoTop.Exports [in Combi.Combi.composition]
+

L

+LeqGeqOrder [in Combi.SSRcomplements.sorted]
+

N

+NoSetContainingBoth [in Combi.LRrule.Greene_inv]
+

P

+PermLattice [in Combi.SymGroup.weak_order]
+PermLattice.Exports [in Combi.SymGroup.weak_order]
+

R

+RefinementOrder [in Combi.Combi.composition]
+RefinementOrder.Exports [in Combi.Combi.composition]
+RefinmentOrder [in Combi.Combi.setpartition]
+RefinmentOrder.Exports [in Combi.Combi.setpartition]
+

S

+SetContainingBothLeft [in Combi.LRrule.Greene_inv]
+Swap [in Combi.LRrule.Greene_inv]
+SymPolyHomogKey [in Combi.MPoly.homogsym]
+

T

+TamariLattice [in Combi.Combi.bintree]
+TamariLattice.Exports [in Combi.Combi.bintree]
+

W

+WeakOrder [in Combi.SymGroup.weak_order]
+WeakOrder.Exports [in Combi.SymGroup.weak_order]
+

Y

+YoungLattice [in Combi.Combi.partition]
+YoungLattice.Exports [in Combi.Combi.partition]
+

+

Variable Index

+

A

+AbelianBigOp.idx [in Combi.SSRcomplements.tools]
+AbelianBigOp.op [in Combi.SSRcomplements.tools]
+AbelianBigOp.R [in Combi.SSRcomplements.tools]
+ActOnTuple.n [in Combi.Combi.permuted]
+ActOnTuple.T [in Combi.Combi.permuted]
+ActOnTuple.w [in Combi.Combi.permuted]
+AllLeqLtn.disp [in Combi.Basic.ordtype]
+AllLeqLtn.T [in Combi.Basic.ordtype]
+Alternant.HasIncr.add_mpart_mesym [in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.d [in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.h [in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.k [in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr.la [in Combi.MPoly.Schur_altdef]
+Alternant.n [in Combi.MPoly.Schur_altdef]
+Alternant.R [in Combi.MPoly.Schur_altdef]
+AlternIDomain.Hchar [in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hpanti [in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hphomog [in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.Hpn0 [in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial.p [in Combi.MPoly.antisym]
+AlternIDomain.n [in Combi.MPoly.antisym]
+AlternIDomain.R [in Combi.MPoly.antisym]
+AlternStraighten.n0 [in Combi.MPoly.Schur_altdef]
+AlternStraighten.R [in Combi.MPoly.Schur_altdef]
+AppendNth.disp [in Combi.Combi.stdtab]
+AppendNth.T [in Combi.Combi.stdtab]
+Assoc.m [in Combi.SymGroup.towerSn]
+Assoc.n [in Combi.SymGroup.towerSn]
+Assoc.p [in Combi.SymGroup.towerSn]
+

B

+BalToDyck.Hbal1 [in Combi.Combi.Dyckword]
+BalToDyck.w [in Combi.Combi.Dyckword]
+Bases.n [in Combi.MPoly.sympoly]
+Bases.R [in Combi.MPoly.sympoly]
+Bigop.idx [in Combi.Combi.subseq]
+Bigop.op [in Combi.Combi.subseq]
+Bigop.R [in Combi.Combi.subseq]
+Bigop.T [in Combi.Combi.subseq]
+Bigsums.A [in ALEA.Qmeasure]
+BigTrivISeq.idx [in Combi.LRrule.Greene]
+BigTrivISeq.op [in Combi.LRrule.Greene]
+BigTrivISeq.R [in Combi.LRrule.Greene]
+BigTrivISeq.T [in Combi.LRrule.Greene]
+BijectionExtTab.Hmu [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.Hstrip [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.Hsz [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.la [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.m [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.mu [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.n [in Combi.MPoly.Schur_altdef]
+BijectionExtTab.s [in Combi.MPoly.Schur_altdef]
+BijFiberedSet.Defs.HcardEq [in Combi.Combi.fibered_set]
+BijFiberedSet.Defs.U [in Combi.Combi.fibered_set]
+BijFiberedSet.Defs.V [in Combi.Combi.fibered_set]
+BijFiberedSet.I [in Combi.Combi.fibered_set]
+BoxInSkew.inner [in Combi.Combi.partition]
+BoxInSkew.outer [in Combi.Combi.partition]
+BraidRed.n [in Combi.SymGroup.presentSn]
+Builders_1.Builders_1.fresh_name_2 [in Combi.Basic.ordtype]
+Builders_1.Builders_1.T [in Combi.Basic.ordtype]
+Builders_6.Builders_6.fresh_name_7 [in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_Order_Preorder_isDuallyPOrder [in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_Order_isDuallyPreorder [in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_eqtype_hasDecEq [in Combi.Combi.composition]
+Builders_6.Builders_6.local_mixin_choice_hasChoice [in Combi.Combi.composition]
+Builders_6.Builders_6.T [in Combi.Combi.composition]
+Builders_6.Builders_6.disp [in Combi.Combi.composition]
+Builders_1.Builders_1.fresh_name_2 [in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_Order_Preorder_isDuallyPOrder [in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_Order_isDuallyPreorder [in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_eqtype_hasDecEq [in Combi.Combi.composition]
+Builders_1.Builders_1.local_mixin_choice_hasChoice [in Combi.Combi.composition]
+Builders_1.Builders_1.T [in Combi.Combi.composition]
+Builders_1.Builders_1.disp [in Combi.Combi.composition]
+

C

+CanPorbit.s [in Combi.SymGroup.cycletype]
+CanPorbit.T [in Combi.SymGroup.cycletype]
+CanWord.n0 [in Combi.SymGroup.presentSn]
+Cast.d1 [in Combi.MPoly.sympoly]
+Cast.d2 [in Combi.MPoly.sympoly]
+Cast.eq_d [in Combi.MPoly.sympoly]
+Cast.la [in Combi.MPoly.sympoly]
+Cast.n0 [in Combi.MPoly.sympoly]
+Cast.R [in Combi.MPoly.sympoly]
+Cast.T [in Combi.LRrule.Greene]
+CategoricalSystems.R [in Combi.MPoly.sympoly]
+CauchyKernelField.R [in Combi.MPoly.Cauchy]
+CauchyKernel.Big.idx [in Combi.MPoly.Cauchy]
+CauchyKernel.Big.op [in Combi.MPoly.Cauchy]
+CauchyKernel.Big.R [in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam.d [in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam.famYinv_fun [in Combi.MPoly.Cauchy]
+CauchyKernel.d [in Combi.MPoly.Cauchy]
+CauchyKernel.m0 [in Combi.MPoly.Cauchy]
+CauchyKernel.n0 [in Combi.MPoly.Cauchy]
+CauchyKernel.R [in Combi.MPoly.Cauchy]
+CauchyKernel.vecmx_index [in Combi.MPoly.Cauchy]
+CFExtProdDefs.aT [in Combi.SymGroup.towerSn]
+CFExtProdDefs.G [in Combi.SymGroup.towerSn]
+CFExtProdDefs.gT [in Combi.SymGroup.towerSn]
+CFExtProdDefs.H [in Combi.SymGroup.towerSn]
+CFExtProdTheory.aT [in Combi.SymGroup.towerSn]
+CFExtProdTheory.G [in Combi.SymGroup.towerSn]
+CFExtProdTheory.gT [in Combi.SymGroup.towerSn]
+CFExtProdTheory.H [in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.n1 [in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.n2 [in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.rG [in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd.rH [in Combi.SymGroup.towerSn]
+ChangeBaseMonomial.n [in Combi.MPoly.sympoly]
+ChangeBaseMonomial.R [in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum.nvar0 [in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum.R [in Combi.MPoly.sympoly]
+ChangeBasis.HandE.E [in Combi.MPoly.sympoly]
+ChangeBasis.HandE.E0 [in Combi.MPoly.sympoly]
+ChangeBasis.HandE.H [in Combi.MPoly.sympoly]
+ChangeBasis.HandE.Hanti [in Combi.MPoly.sympoly]
+ChangeBasis.HandE.H0 [in Combi.MPoly.sympoly]
+ChangeBasis.n0 [in Combi.MPoly.sympoly]
+ChangeBasis.R [in Combi.MPoly.sympoly]
+ChangeField.d [in Combi.MPoly.homogsym]
+ChangeField.mor [in Combi.MPoly.homogsym]
+ChangeField.n0 [in Combi.MPoly.homogsym]
+ChangeField.R [in Combi.MPoly.homogsym]
+ChangeField.S [in Combi.MPoly.homogsym]
+ChangeNVar.d [in Combi.MPoly.homogsym]
+ChangeNVar.d [in Combi.MPoly.sympoly]
+ChangeNVar.Hd [in Combi.MPoly.homogsym]
+ChangeNVar.Hd [in Combi.MPoly.sympoly]
+ChangeNVar.m0 [in Combi.MPoly.homogsym]
+ChangeNVar.m0 [in Combi.MPoly.sympoly]
+ChangeNVar.n0 [in Combi.MPoly.homogsym]
+ChangeNVar.n0 [in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen.Gen [in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen.Hcnvargen [in Combi.MPoly.sympoly]
+ChangeNVar.R [in Combi.MPoly.homogsym]
+ChangeNVar.R [in Combi.MPoly.sympoly]
+CharDotProduct.G [in Combi.SymGroup.Frobenius_char]
+CharDotProduct.gT [in Combi.SymGroup.Frobenius_char]
+classGroup.aT [in Combi.SymGroup.towerSn]
+classGroup.G [in Combi.SymGroup.towerSn]
+classGroup.gT [in Combi.SymGroup.towerSn]
+classGroup.H [in Combi.SymGroup.towerSn]
+Codes.FinType.n [in Combi.SymGroup.presentSn]
+CommutativeImage.n [in Combi.LRrule.freeSchur]
+CommutativeImage.R [in Combi.LRrule.freeSchur]
+CompOfn.n [in Combi.Combi.composition]
+CongruenceClosure.Alph [in Combi.Basic.congr]
+CongruenceClosure.Hinvar_congr [in Combi.Basic.congr]
+CongruenceClosure.Hsym [in Combi.Basic.congr]
+CongruenceClosure.inv [in Combi.Basic.congr]
+CongruenceClosure.rule [in Combi.Basic.congr]
+CongruenceFacts.Alph [in Combi.Basic.congr]
+CongruenceFacts.Hcongr [in Combi.Basic.congr]
+CongruenceFacts.Hequiv [in Combi.Basic.congr]
+CongruenceFacts.r [in Combi.Basic.congr]
+ConjTab.disp [in Combi.Combi.stdtab]
+ConjTab.T [in Combi.Combi.stdtab]
+Conj.d1 [in Combi.LRrule.freeSchur]
+Conj.d2 [in Combi.LRrule.freeSchur]
+ConnectCompl.e [in Combi.Combi.skewpart]
+ConnectCompl.T [in Combi.Combi.skewpart]
+Connected4.inner [in Combi.Combi.skewpart]
+Connected4.outer [in Combi.Combi.skewpart]
+Coord.d [in Combi.MPoly.homogsym]
+Coord.Hd [in Combi.MPoly.homogsym]
+Coord.n0 [in Combi.MPoly.homogsym]
+Coord.R [in Combi.MPoly.homogsym]
+CoversFinPOrder.disp [in Combi.Basic.ordtype]
+CoversFinPOrder.T [in Combi.Basic.ordtype]
+CoverSurgery.N [in Combi.LRrule.Greene_inv]
+CoverSurgery.P [in Combi.LRrule.Greene_inv]
+CoverSurgery.Q [in Combi.LRrule.Greene_inv]
+CoverSurgery.S [in Combi.LRrule.Greene_inv]
+CutK.T [in Combi.Combi.vectNK]
+Cut3.match3 [in Combi.Combi.vectNK]
+Cut3.T [in Combi.Combi.vectNK]
+CycleTypeConj.T [in Combi.SymGroup.cycletype]
+CycleType.CFunIndicator.ct [in Combi.SymGroup.cycletype]
+CycleType.Classes.ct [in Combi.SymGroup.cycletype]
+CycleType.T [in Combi.SymGroup.cycletype]
+

D

+DefsFiber.eqct [in Combi.SymGroup.cycletype]
+DefsFiber.s [in Combi.SymGroup.cycletype]
+DefsFiber.t [in Combi.SymGroup.cycletype]
+DefsFiber.U [in Combi.SymGroup.cycletype]
+DefsFiber.V [in Combi.SymGroup.cycletype]
+DefsKostkaMon.d [in Combi.MPoly.Schur_altdef]
+DefsKostkaMon.la [in Combi.MPoly.Schur_altdef]
+DefsKostkaMon.n [in Combi.MPoly.Schur_altdef]
+Defs.Alph [in Combi.LRrule.shuffle]
+Defs.Alph [in Combi.LRrule.plactic]
+Defs.disp [in Combi.LRrule.plactic]
+Defs.S [in Combi.Combi.setpartition]
+Defs.T [in Combi.Combi.setpartition]
+Defs.word [in Combi.LRrule.shuffle]
+Defs.word [in Combi.LRrule.plactic]
+DefTrivSign.d [in Combi.SymGroup.reprSn]
+DefTrivSign.n [in Combi.SymGroup.reprSn]
+DefType.d [in Combi.MPoly.homogsym]
+DefType.Hvar [in Combi.MPoly.homogsym]
+DefType.n [in Combi.MPoly.homogsym]
+DefType.n [in Combi.MPoly.sympoly]
+DefType.R [in Combi.MPoly.homogsym]
+DefType.R [in Combi.MPoly.sympoly]
+Depend.Hsym [in Combi.Basic.congr]
+Depend.inv [in Combi.Basic.congr]
+Depend.rule [in Combi.Basic.congr]
+Depend.T [in Combi.Basic.congr]
+DescSet.n [in Combi.Combi.composition]
+Dominate.disp [in Combi.Combi.skewtab]
+Dominate.disp [in Combi.Combi.tableau]
+Dominate.T [in Combi.Combi.skewtab]
+Dominate.T [in Combi.Combi.tableau]
+Duality.Alph [in Combi.LRrule.Greene_inv]
+Duality.disp [in Combi.LRrule.Greene_inv]
+Duality.k [in Combi.LRrule.Greene_inv]
+Duality.w [in Combi.LRrule.Greene_inv]
+Duality.word [in Combi.LRrule.Greene_inv]
+DualRule.Alph [in Combi.LRrule.plactic]
+DualRule.disp [in Combi.LRrule.plactic]
+DualRule.word [in Combi.LRrule.plactic]
+Dual.hb_instance_111.T [in Combi.Basic.ordtype]
+Dual.hb_instance_111.d [in Combi.Basic.ordtype]
+Dual.hb_instance_100.T [in Combi.Basic.ordtype]
+Dual.hb_instance_100.d [in Combi.Basic.ordtype]
+Dual.hb_instance_91.T [in Combi.Basic.ordtype]
+Dual.hb_instance_91.d [in Combi.Basic.ordtype]
+Dual.hb_instance_80.T [in Combi.Basic.ordtype]
+Dual.hb_instance_80.d [in Combi.Basic.ordtype]
+Dual.hb_instance_69.T [in Combi.Basic.ordtype]
+Dual.hb_instance_69.d [in Combi.Basic.ordtype]
+Dual.hb_instance_60.T [in Combi.Basic.ordtype]
+Dual.hb_instance_60.d [in Combi.Basic.ordtype]
+Dual.hb_instance_53.T [in Combi.Basic.ordtype]
+Dual.hb_instance_53.d [in Combi.Basic.ordtype]
+Dual.hb_instance_48.T [in Combi.Basic.ordtype]
+DyckSetInd.P [in Combi.Combi.Dyckword]
+DyckSetInd.Pcons [in Combi.Combi.Dyckword]
+DyckSetInd.Pnil [in Combi.Combi.Dyckword]
+DyckToBal.HDyck [in Combi.Combi.Dyckword]
+DyckToBal.Hrt [in Combi.Combi.Dyckword]
+DyckToBal.rt [in Combi.Combi.Dyckword]
+DyckToBal.w [in Combi.Combi.Dyckword]
+DyckWordRotationBijection.n [in Combi.Combi.Dyckword]
+

E

+ElemTransp.n0 [in Combi.SymGroup.presentSn]
+EltrConj.n [in Combi.SymGroup.reprSn]
+EltrP.i [in Combi.MPoly.antisym]
+EltrP.n [in Combi.MPoly.antisym]
+Empty.T [in Combi.Combi.setpartition]
+EnumFintype.P [in Combi.Basic.combclass]
+EnumFintype.subenum [in Combi.Basic.combclass]
+EnumFintype.subenumP [in Combi.Basic.combclass]
+EnumFintype.subenum_countE [in Combi.Basic.combclass]
+EnumFintype.T [in Combi.Basic.combclass]
+EnumFintype.TP [in Combi.Basic.combclass]
+EqInvAltDef.disp1 [in Combi.Combi.std]
+EqInvAltDef.disp2 [in Combi.Combi.std]
+EqInvAltDef.disp3 [in Combi.Combi.std]
+EqInvAltDef.S [in Combi.Combi.std]
+EqInvAltDef.T [in Combi.Combi.std]
+EqInvAltDef.U [in Combi.Combi.std]
+EqInvDef.disp1 [in Combi.Combi.std]
+EqInvDef.disp2 [in Combi.Combi.std]
+EqInvDef.disp3 [in Combi.Combi.std]
+EqInvDef.S [in Combi.Combi.std]
+EqInvDef.T [in Combi.Combi.std]
+EqInvDef.U [in Combi.Combi.std]
+EqInvPosRemBig.disp1 [in Combi.Combi.std]
+EqInvPosRemBig.disp2 [in Combi.Combi.std]
+EqInvPosRemBig.S [in Combi.Combi.std]
+EqInvPosRemBig.T [in Combi.Combi.std]
+Examples.u [in Combi.Combi.std]
+Examples.v [in Combi.Combi.std]
+

F

+FastImplem.ComRing.R [in Combi.MPoly.MurnaghanNakayama]
+FastImplem.n0 [in Combi.MPoly.MurnaghanNakayama]
+FilterLeqGeq.disp [in Combi.Combi.skewtab]
+FilterLeqGeq.T [in Combi.Combi.skewtab]
+FindCorner.EndsAt.Alpha [in Combi.HookFormula.hook]
+FindCorner.EndsAt.Beta [in Combi.HookFormula.hook]
+FindCorner.EndsAt.Hcorn [in Combi.HookFormula.hook]
+FindCorner.Formula.alpha [in Combi.HookFormula.hook]
+FindCorner.Formula.R [in Combi.HookFormula.hook]
+FindCorner.Formula.T [in Combi.HookFormula.hook]
+FindCorner.p [in Combi.HookFormula.hook]
+FindCorner.Theorem2.Alpha [in Combi.HookFormula.hook]
+FindCorner.Theorem2.Beta [in Combi.HookFormula.hook]
+FindCorner.Theorem2.Hcorn [in Combi.HookFormula.hook]
+FindCorner.Theorem2.Hpartc' [in Combi.HookFormula.hook]
+FindCorner.Theorem2.p' [in Combi.HookFormula.hook]
+FinerCard.finPQ [in Combi.Combi.setpartition]
+FinerCard.P [in Combi.Combi.setpartition]
+FinerCard.Q [in Combi.Combi.setpartition]
+FinerCard.S [in Combi.Combi.setpartition]
+FinerCard.T [in Combi.Combi.setpartition]
+FiniteDistributions.A [in ALEA.Qmeasure]
+FiniteDistributions.p [in ALEA.Qmeasure]
+FinSet.T [in Combi.SSRcomplements.tools]
+FinType.d [in Combi.Combi.tableau]
+FinType.disp [in Combi.Combi.tableau]
+FinType.hb_instance_22.n [in Combi.Combi.ordtree]
+FinType.n [in Combi.Combi.permuted]
+FinType.sh [in Combi.Combi.tableau]
+FinType.T [in Combi.Combi.subseq]
+FinType.T [in Combi.Combi.tableau]
+FinType.T [in Combi.Combi.permuted]
+FinType.tabsh_enum [in Combi.Combi.tableau]
+FinType.w [in Combi.Combi.subseq]
+FinType.w [in Combi.Combi.permuted]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.w [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.HTriple [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij.Q [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.Hdrop [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.Htake [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.w [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.T [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop.disp [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.Hsh2 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.Hsh1 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.T2 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.U2 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.T1 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.U1 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.d1 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.d2 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.P1 [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.P2 [in Combi.LRrule.freeSchur]
+FreeSchur.Degree.d [in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.d1 [in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.d2 [in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.Q1 [in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule.Q2 [in Combi.LRrule.freeSchur]
+FreeSchur.n0 [in Combi.LRrule.freeSchur]
+FreeSchur.R [in Combi.LRrule.freeSchur]
+

G

+Generators.gen [in Combi.MPoly.sympoly]
+Generators.gen_homog [in Combi.MPoly.sympoly]
+Generators.n [in Combi.MPoly.sympoly]
+Generators.R [in Combi.MPoly.sympoly]
+GreeneCat.Alph [in Combi.LRrule.Greene]
+GreeneCat.HR [in Combi.LRrule.Greene]
+GreeneCat.lsplit [in Combi.LRrule.Greene]
+GreeneCat.M [in Combi.LRrule.Greene]
+GreeneCat.N [in Combi.LRrule.Greene]
+GreeneCat.R [in Combi.LRrule.Greene]
+GreeneCat.rsplit [in Combi.LRrule.Greene]
+GreeneCat.V [in Combi.LRrule.Greene]
+GreeneCat.W [in Combi.LRrule.Greene]
+GreeneDef.Alph [in Combi.LRrule.Greene]
+GreeneDef.HR [in Combi.LRrule.Greene]
+GreeneDef.N [in Combi.LRrule.Greene]
+GreeneDef.R [in Combi.LRrule.Greene]
+GreeneDef.wt [in Combi.LRrule.Greene]
+GreeneInj.R1 [in Combi.LRrule.Greene]
+GreeneInj.R2 [in Combi.LRrule.Greene]
+GreeneInj.T1 [in Combi.LRrule.Greene]
+GreeneInj.T2 [in Combi.LRrule.Greene]
+GreeneInvariantsDual.Alph [in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual.disp [in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual.word [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.Alph [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.disp [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.k [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.u [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.v1 [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.v2 [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.w [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule.word [in Combi.LRrule.Greene_inv]
+GreeneInvariants.Alph [in Combi.LRrule.Greene_inv]
+GreeneInvariants.disp [in Combi.LRrule.Greene_inv]
+GreeneInvariants.word [in Combi.LRrule.Greene_inv]
+GreenEqShape.d1 [in Combi.LRrule.Greene_inv]
+GreenEqShape.d2 [in Combi.LRrule.Greene_inv]
+GreenEqShape.S [in Combi.LRrule.Greene_inv]
+GreenEqShape.T [in Combi.LRrule.Greene_inv]
+GreeneRec.Alph [in Combi.LRrule.Greene]
+GreeneRec.cast_set_tab [in Combi.LRrule.Greene]
+GreeneRec.disp [in Combi.LRrule.Greene]
+GreeneRec.Induction.t [in Combi.LRrule.Greene]
+GreeneRec.Induction.t0 [in Combi.LRrule.Greene]
+GreeneRec.sym_cast [in Combi.LRrule.Greene]
+GreeneSeq.Alph [in Combi.LRrule.Greene]
+GreeneSeq.HnegR [in Combi.LRrule.Greene]
+GreeneSeq.HR [in Combi.LRrule.Greene]
+GreeneSeq.negR [in Combi.LRrule.Greene]
+GreeneSeq.R [in Combi.LRrule.Greene]
+GreeneTab.Alph [in Combi.LRrule.Greene]
+GreeneTab.disp [in Combi.LRrule.Greene]
+

H

+hb_instance_31.d [in Combi.Combi.partition]
+hb_instance_22.n [in Combi.Combi.bintree]
+hb_instance_43.n [in Combi.Basic.ordtype]
+hb_instance_32.T' [in Combi.Basic.ordtype]
+hb_instance_32.T [in Combi.Basic.ordtype]
+hb_instance_27.T [in Combi.Basic.ordtype]
+hb_instance_22.T [in Combi.Basic.ordtype]
+hb_instance_17.n [in Combi.Basic.ordtype]
+HomogSymLModType.d [in Combi.MPoly.homogsym]
+HomogSymLModType.n [in Combi.MPoly.homogsym]
+HomogSymLModType.R [in Combi.MPoly.homogsym]
+HomogSymProd.c [in Combi.MPoly.homogsym]
+HomogSymProd.d [in Combi.MPoly.homogsym]
+HomogSymProd.n [in Combi.MPoly.homogsym]
+HomogSymProd.R [in Combi.MPoly.homogsym]
+HomSymField.d [in Combi.MPoly.homogsym]
+HomSymField.Hd [in Combi.MPoly.homogsym]
+HomSymField.n0 [in Combi.MPoly.homogsym]
+HomSymField.R [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.d [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.Hla [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.Hlla [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.la [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons.l0 [in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.d1 [in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.d2 [in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.la [in Combi.MPoly.homogsym]
+HomSymProdGen.Merge.mu [in Combi.MPoly.homogsym]
+HomSymProdGen.n0 [in Combi.MPoly.homogsym]
+HomSymProdGen.R [in Combi.MPoly.homogsym]
+

I

+Identity.n [in Combi.HookFormula.Frobenius_ident]
+IdomainSchurSym.n0 [in Combi.MPoly.Schur_altdef]
+IdomainSchurSym.R [in Combi.MPoly.Schur_altdef]
+ImsetInj.f [in Combi.SSRcomplements.tools]
+ImsetInj.f_inj [in Combi.SSRcomplements.tools]
+ImsetInj.T [in Combi.SSRcomplements.tools]
+ImsetInj.T1 [in Combi.SSRcomplements.tools]
+ImsetInj.T2 [in Combi.SSRcomplements.tools]
+IncrMap.disp1 [in Combi.LRrule.plactic]
+IncrMap.disp1 [in Combi.Combi.tableau]
+IncrMap.disp2 [in Combi.LRrule.plactic]
+IncrMap.disp2 [in Combi.Combi.tableau]
+IncrMap.F [in Combi.LRrule.plactic]
+IncrMap.F [in Combi.Combi.tableau]
+IncrMap.Hincr [in Combi.LRrule.plactic]
+IncrMap.T1 [in Combi.LRrule.plactic]
+IncrMap.T1 [in Combi.Combi.tableau]
+IncrMap.T2 [in Combi.LRrule.plactic]
+IncrMap.T2 [in Combi.Combi.tableau]
+IncrMap.u [in Combi.LRrule.plactic]
+IncrMap.v [in Combi.LRrule.plactic]
+Induction.m [in Combi.SymGroup.towerSn]
+Induction.n [in Combi.SymGroup.towerSn]
+InHomSym.d [in Combi.MPoly.homogsym]
+InHomSym.n0 [in Combi.MPoly.homogsym]
+InHomSym.R [in Combi.MPoly.homogsym]
+IntpartnCons.d [in Combi.Combi.partition]
+IntpartnCons.Hla [in Combi.Combi.partition]
+IntpartnCons.Hlla [in Combi.Combi.partition]
+IntpartnCons.la [in Combi.Combi.partition]
+IntpartnCons.l0 [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom.d [in Combi.Combi.partition]
+IntPartNDom.IntPartNTopBottom.d [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi.d [in Combi.Combi.partition]
+IntPartN.la [in Combi.Combi.skewpart]
+IntPartN.m [in Combi.Combi.skewpart]
+IntPartN.nbox [in Combi.Combi.skewpart]
+IntPartN.val_id [in Combi.Combi.skewpart]
+InvarContHom.Alph [in Combi.Basic.congr]
+InvarContHom.Hcongr [in Combi.Basic.congr]
+InvarContHom.Hhom [in Combi.Basic.congr]
+InvarContHom.rule [in Combi.Basic.congr]
+InvarContHom.szinvar [in Combi.Basic.congr]
+InvarContMultHom.Alph [in Combi.Basic.congr]
+InvarContMultHom.Hcongr [in Combi.Basic.congr]
+InvarContMultHom.Hmulthom [in Combi.Basic.congr]
+InvarContMultHom.Hsym [in Combi.Basic.congr]
+InvarContMultHom.rule [in Combi.Basic.congr]
+InvSet.n [in Combi.SymGroup.presentSn]
+isInhabitedType.isInhabitedType.T [in Combi.Basic.ordtype]
+isInhabited.isInhabited.T [in Combi.Basic.ordtype]
+IsoBottom.IsoBottom.disp [in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_Order_Preorder_isDuallyPOrder [in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_Order_isDuallyPreorder [in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_eqtype_hasDecEq [in Combi.Combi.composition]
+IsoBottom.IsoBottom.local_mixin_choice_hasChoice [in Combi.Combi.composition]
+IsoBottom.IsoBottom.T [in Combi.Combi.composition]
+IsoTop.IsoTop.disp [in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_Order_Preorder_isDuallyPOrder [in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_Order_isDuallyPreorder [in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_eqtype_hasDecEq [in Combi.Combi.composition]
+IsoTop.IsoTop.local_mixin_choice_hasChoice [in Combi.Combi.composition]
+IsoTop.IsoTop.T [in Combi.Combi.composition]
+

K

+KostkaEq.d [in Combi.MPoly.Schur_altdef]
+KostkaEq.la [in Combi.MPoly.Schur_altdef]
+Kostka.d [in Combi.MPoly.Schur_altdef]
+KsuppInj.Hinv [in Combi.LRrule.stdplact]
+KsuppInj.s [in Combi.LRrule.stdplact]
+KsuppInj.t [in Combi.LRrule.stdplact]
+

L

+Length.n0 [in Combi.SymGroup.presentSn]
+Length.PartCode.is_partcode [in Combi.SymGroup.presentSn]
+Length.PartCode.word_of_partcocode [in Combi.SymGroup.presentSn]
+LEPermTheory.n0 [in Combi.SymGroup.weak_order]
+LinRepr.G [in Combi.SymGroup.reprSn]
+LinRepr.gT [in Combi.SymGroup.reprSn]
+LRrule_Pieri.R [in Combi.MPoly.sympoly]
+LRrule_Pieri.n0 [in Combi.MPoly.sympoly]
+LRTriple.Alph [in Combi.LRrule.shuffle]
+LRTriple.disp [in Combi.LRrule.shuffle]
+LRTriple.word [in Combi.LRrule.shuffle]
+LR.d1 [in Combi.LRrule.therule]
+LR.d1 [in Combi.LRrule.implem]
+LR.d2 [in Combi.LRrule.therule]
+LR.d2 [in Combi.LRrule.implem]
+LR.n0 [in Combi.LRrule.implem]
+LR.Pieri.n0 [in Combi.LRrule.therule]
+LR.Pieri.R [in Combi.LRrule.therule]
+LR.P1 [in Combi.LRrule.implem]
+LR.P2 [in Combi.LRrule.implem]
+LR.R [in Combi.LRrule.implem]
+LR.TheRule.n0 [in Combi.LRrule.therule]
+LR.TheRule.OneCoeff.Hincl [in Combi.LRrule.therule]
+LR.TheRule.OneCoeff.P [in Combi.LRrule.therule]
+LR.TheRule.P1 [in Combi.LRrule.therule]
+LR.TheRule.P2 [in Combi.LRrule.therule]
+LR.TheRule.R [in Combi.LRrule.therule]
+

M

+MaxPerm.n [in Combi.SymGroup.presentSn]
+MaxSeq.disp [in Combi.Basic.ordtype]
+MaxSeq.T [in Combi.Basic.ordtype]
+MeasureProp.A [in ALEA.Qmeasure]
+MeasureProp.m [in ALEA.Qmeasure]
+MinDropEq.T [in Combi.Combi.skewpart]
+MNRule.n0 [in Combi.MPoly.MurnaghanNakayama]
+MNRule.R [in Combi.MPoly.MurnaghanNakayama]
+MonDistrib.A [in ALEA.Qmeasure]
+MonDistrib.B [in ALEA.Qmeasure]
+MonomPart.n [in Combi.MPoly.antisym]
+MPoESymHomog.n [in Combi.MPoly.sympoly]
+MPoESymHomog.R [in Combi.MPoly.sympoly]
+MPolySym.n [in Combi.MPoly.antisym]
+MPolySym.R [in Combi.MPoly.antisym]
+MultAlternSymp.n0 [in Combi.MPoly.MurnaghanNakayama]
+MultAlternSymp.R [in Combi.MPoly.MurnaghanNakayama]
+MultinomCompl.n [in Combi.MPoly.sympoly]
+MultinomCompl.R [in Combi.MPoly.sympoly]
+MultSymsSympIDomain.n0 [in Combi.MPoly.MurnaghanNakayama]
+MultSymsSymp.n0 [in Combi.MPoly.MurnaghanNakayama]
+MultSymsSymp.R [in Combi.MPoly.MurnaghanNakayama]
+

N

+NonEmpty.Bump.HRow [in Combi.LRrule.Schensted]
+NonEmpty.Bump.l [in Combi.LRrule.Schensted]
+NonEmpty.Bump.Row [in Combi.LRrule.Schensted]
+NonEmpty.disp [in Combi.LRrule.Schensted]
+NonEmpty.Insert.HRow [in Combi.LRrule.Schensted]
+NonEmpty.Insert.l [in Combi.LRrule.Schensted]
+NonEmpty.Insert.Row [in Combi.LRrule.Schensted]
+NonEmpty.T [in Combi.LRrule.Schensted]
+NoSetContainingBoth.Case.a [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.Alph [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.b [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.disp [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.HnoBoth [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.k [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.P [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.Px [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.R [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.u [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.v [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.word [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.x [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Case.y [in Combi.LRrule.Greene_inv]
+NThAddRibbon.lesmin [in Combi.Combi.skewpart]
+NThAddRibbon.less [in Combi.Combi.skewpart]
+NThAddRibbon.lessz [in Combi.Combi.skewpart]
+NThAddRibbon.rem [in Combi.Combi.skewpart]
+NThAddRibbon.sh [in Combi.Combi.skewpart]
+NThAddRibbon.start [in Combi.Combi.skewpart]
+NThAddRibbon.stop [in Combi.Combi.skewpart]
+NThAddRibbon.sztd [in Combi.Combi.skewpart]
+NVar.Character.Hn [in Combi.SymGroup.Frobenius_char]
+NVar.Character.n [in Combi.SymGroup.Frobenius_char]
+NVar.Defs.Hn [in Combi.SymGroup.Frobenius_char]
+NVar.Defs.n [in Combi.SymGroup.Frobenius_char]
+NVar.nvar0 [in Combi.SymGroup.Frobenius_char]
+

O

+OfSize.n [in Combi.Combi.ordtree]
+OmegaHomSym.d [in Combi.MPoly.homogsym]
+OmegaHomSym.n0 [in Combi.MPoly.homogsym]
+OmegaHomSym.R [in Combi.MPoly.homogsym]
+OmegaProd.n0 [in Combi.MPoly.homogsym]
+OmegaProd.R [in Combi.MPoly.homogsym]
+Omega.d [in Combi.MPoly.sympoly]
+Omega.n0 [in Combi.MPoly.sympoly]
+Omega.R [in Combi.MPoly.sympoly]
+OperDistr.A [in ALEA.Qmeasure]
+OperDistr.B [in ALEA.Qmeasure]
+OperDistr.C [in ALEA.Qmeasure]
+OperDistr.MuBool.m [in ALEA.Qmeasure]
+OrdTableau.d [in Combi.Combi.tableau]
+OrdTableau.Hszs [in Combi.Combi.tableau]
+OrdTableau.n [in Combi.Combi.tableau]
+OrdTableau.sh [in Combi.Combi.tableau]
+OutEval.outev [in Combi.LRrule.implem]
+OutEval.tsumn [in Combi.LRrule.implem]
+

P

+PackedSpec.eval [in Combi.LRrule.implem]
+PackedSpec.inner [in Combi.LRrule.implem]
+PackedSpec.outer [in Combi.LRrule.implem]
+PartOfn.n [in Combi.Combi.partition]
+PermComp.T [in Combi.SSRcomplements.permcomp]
+PermCycles.CM.s [in Combi.SymGroup.permcent]
+PermCycles.T [in Combi.SymGroup.permcent]
+PermCycles.T [in Combi.SymGroup.cycles]
+PermEq.Alph [in Combi.Combi.std]
+PermEq.disp [in Combi.Combi.std]
+PermLattice.Exports.PermLattice.n0 [in Combi.SymGroup.weak_order]
+PermLattice.PermLattice.n0 [in Combi.SymGroup.weak_order]
+PermOfInvSetEltr.n0 [in Combi.SymGroup.presentSn]
+Permuted.SizeN.n [in Combi.Combi.permuted]
+Permuted.T [in Combi.Combi.permuted]
+PlactDual.Alph [in Combi.LRrule.plactic]
+PlactDual.disp [in Combi.LRrule.plactic]
+PlactDual.word [in Combi.LRrule.plactic]
+PorbitBijection.CM [in Combi.SymGroup.cycletype]
+PorbitBijection.s [in Combi.SymGroup.cycletype]
+PorbitBijection.t [in Combi.SymGroup.cycletype]
+PorbitBijection.U [in Combi.SymGroup.cycletype]
+PorbitBijection.V [in Combi.SymGroup.cycletype]
+PreimPartition.D [in Combi.Combi.Dyckword]
+PreimPartition.f [in Combi.Combi.Dyckword]
+PreimPartition.rT [in Combi.Combi.Dyckword]
+PreimPartition.T [in Combi.Combi.Dyckword]
+PresentationSn.eltrG [in Combi.SymGroup.presentSn]
+PresentationSn.gT [in Combi.SymGroup.presentSn]
+PresentationSn.n [in Combi.SymGroup.presentSn]
+ProdGen.co [in Combi.MPoly.sympoly]
+ProdGen.Defs.gen [in Combi.MPoly.sympoly]
+ProdGen.Defs.gen_homog [in Combi.MPoly.sympoly]
+ProdGen.gA [in Combi.MPoly.sympoly]
+ProdGen.gB [in Combi.MPoly.sympoly]
+ProdGen.n [in Combi.MPoly.sympoly]
+ProdGen.R [in Combi.MPoly.sympoly]
+

Q

+QTableau.disp [in Combi.LRrule.Schensted]
+QTableau.T [in Combi.LRrule.Schensted]
+

R

+RCons.T [in Combi.Combi.subseq]
+Recursion.HPnil [in Combi.Combi.ordtree]
+Recursion.IHforest [in Combi.Combi.ordtree]
+Recursion.IHtree [in Combi.Combi.ordtree]
+Recursion.P [in Combi.Combi.ordtree]
+Recursion.PF [in Combi.Combi.ordtree]
+Reduced.n [in Combi.SymGroup.presentSn]
+RefinementOrder.RefinementOrder.n [in Combi.Combi.composition]
+RefinmentOrder.Exports.Finer.S [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.Finer.T [in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder.S [in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder.T [in Combi.Combi.setpartition]
+RemoveBig.Alph [in Combi.LRrule.plactic]
+RemoveBig.disp [in Combi.LRrule.plactic]
+RemoveBig.disp [in Combi.Basic.ordtype]
+RemoveBig.T [in Combi.Basic.ordtype]
+RemoveBig.word [in Combi.LRrule.plactic]
+RemoveBig.Z [in Combi.Basic.ordtype]
+Restriction.m [in Combi.SymGroup.towerSn]
+Restriction.n [in Combi.SymGroup.towerSn]
+RestrIntervBig.Alph [in Combi.LRrule.plactic]
+RestrIntervBig.disp [in Combi.LRrule.plactic]
+RestrIntervBig.L [in Combi.LRrule.plactic]
+RestrIntervBig.word [in Combi.LRrule.plactic]
+RestrIntervSmall.Alph [in Combi.LRrule.plactic]
+RestrIntervSmall.disp [in Combi.LRrule.plactic]
+RestrIntervSmall.word [in Combi.LRrule.plactic]
+RevConj.Alph [in Combi.LRrule.Greene_inv]
+RevConj.d [in Combi.LRrule.Greene_inv]
+Rev.Alph [in Combi.LRrule.Greene]
+Rev.Alph [in Combi.LRrule.plactic]
+Rev.disp [in Combi.LRrule.plactic]
+Rev.word [in Combi.LRrule.plactic]
+RibbonOn.Hrib [in Combi.Combi.skewpart]
+RibbonOn.inner [in Combi.Combi.skewpart]
+RibbonOn.outer [in Combi.Combi.skewpart]
+RibbonOn.partinn [in Combi.Combi.skewpart]
+RibbonOn.partout [in Combi.Combi.skewpart]
+RibbonOn.start [in Combi.Combi.skewpart]
+RibbonOn.stop [in Combi.Combi.skewpart]
+RingSchurSym.n0 [in Combi.MPoly.Schur_altdef]
+RingSchurSym.R [in Combi.MPoly.Schur_altdef]
+RowsAndCols.Alph [in Combi.LRrule.plactic]
+RowsAndCols.disp [in Combi.LRrule.plactic]
+RowsAndCols.word [in Combi.LRrule.plactic]
+Rows.disp [in Combi.Combi.tableau]
+Rows.T [in Combi.Combi.tableau]
+RSToPlactic.Alph [in Combi.LRrule.plactic]
+RSToPlactic.disp [in Combi.LRrule.plactic]
+RSToPlactic.word [in Combi.LRrule.plactic]
+

S

+ScalarChange.mor [in Combi.MPoly.antisym]
+ScalarChange.mor [in Combi.MPoly.sympoly]
+ScalarChange.n [in Combi.MPoly.antisym]
+ScalarChange.n0 [in Combi.MPoly.sympoly]
+ScalarChange.R [in Combi.MPoly.antisym]
+ScalarChange.R [in Combi.MPoly.sympoly]
+ScalarChange.S [in Combi.MPoly.antisym]
+ScalarChange.S [in Combi.MPoly.sympoly]
+ScalarProduct.d [in Combi.MPoly.homogsym]
+ScalarProduct.Hd [in Combi.MPoly.homogsym]
+ScalarProduct.n0 [in Combi.MPoly.homogsym]
+Scalar.d [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_73.mu [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_73.la [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_67.p [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_62.la [in Combi.MPoly.Cauchy]
+Scalar.Hd [in Combi.MPoly.Cauchy]
+Scalar.n0 [in Combi.MPoly.Cauchy]
+SchurAlternantDef.n0 [in Combi.MPoly.Schur_altdef]
+SchurAlternantDef.R [in Combi.MPoly.Schur_altdef]
+SchurComRingType.n0 [in Combi.MPoly.Schur_mpoly]
+SchurComRingType.R [in Combi.MPoly.Schur_mpoly]
+Schur.n0 [in Combi.MPoly.Schur_mpoly]
+Schur.n0 [in Combi.MPoly.sympoly]
+Schur.R [in Combi.MPoly.Schur_mpoly]
+Schur.R [in Combi.MPoly.sympoly]
+SeqLemmas.T [in Combi.SSRcomplements.tools]
+SetAct.aT [in Combi.SSRcomplements.permcomp]
+SetAct.D [in Combi.SSRcomplements.permcomp]
+SetAct.rT [in Combi.SSRcomplements.permcomp]
+SetAct.to [in Combi.SSRcomplements.permcomp]
+SetContainingBothLeft.Case.a [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Alph [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.b [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.Hposb [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.HT [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn.T [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BNotIn.HbNin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.c [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.disp [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Hposa [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Hposc [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.HRabc [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.HS [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.k [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.P [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.Px [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.R [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.S [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.u [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.v [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.word [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.x [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.x' [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.y [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis.Alph [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis.disp [in Combi.LRrule.Greene_inv]
+SetPartitionShape.T [in Combi.Combi.partition]
+SetPartition.T [in Combi.SSRcomplements.tools]
+SimpleRecursion.IHtree [in Combi.Combi.ordtree]
+SimpleRecursion.P [in Combi.Combi.ordtree]
+Singleton.T [in Combi.Combi.setpartition]
+Singleton.x [in Combi.Combi.setpartition]
+SizeN.n [in Combi.Combi.bintree]
+Size.n [in Combi.Combi.bintree]
+Sn.n [in Combi.SymGroup.cycletype]
+Sorted.Hanti [in Combi.SSRcomplements.sorted]
+Sorted.R [in Combi.SSRcomplements.sorted]
+Sorted.Rrefl [in Combi.SSRcomplements.sorted]
+Sorted.Rtrans [in Combi.SSRcomplements.sorted]
+Sorted.T [in Combi.SSRcomplements.sorted]
+Sorted.Z [in Combi.SSRcomplements.sorted]
+Spec.disp1 [in Combi.Combi.std]
+Spec.disp2 [in Combi.Combi.std]
+Spec.d1 [in Combi.LRrule.implem]
+Spec.d2 [in Combi.LRrule.implem]
+Spec.hgt [in Combi.Combi.skewpart]
+Spec.Hincl [in Combi.LRrule.implem]
+Spec.Hret [in Combi.Combi.skewpart]
+Spec.nbox [in Combi.Combi.skewpart]
+Spec.P [in Combi.LRrule.implem]
+Spec.partsh [in Combi.Combi.skewpart]
+Spec.pos [in Combi.Combi.skewpart]
+Spec.P1 [in Combi.LRrule.implem]
+Spec.P2 [in Combi.LRrule.implem]
+Spec.res [in Combi.Combi.skewpart]
+Spec.S [in Combi.Combi.std]
+Spec.sh [in Combi.Combi.skewpart]
+Spec.T [in Combi.Combi.std]
+SRel.T [in Combi.SymGroup.presentSn]
+SSRComplFinset.aT [in Combi.SSRcomplements.tools]
+SSRComplFinset.f [in Combi.SSRcomplements.tools]
+SSRComplFinset.rT [in Combi.SSRcomplements.tools]
+StabilityProperties.A [in ALEA.Qmeasure]
+StabilityProperties.m [in ALEA.Qmeasure]
+StabilityProperties.Mstable_sub [in ALEA.Qmeasure]
+Standardisation.Alph [in Combi.Combi.std]
+Standardisation.disp [in Combi.Combi.std]
+StdCombClass.n [in Combi.Combi.std]
+StdKostka.d [in Combi.MPoly.Schur_altdef]
+StdKostka.la [in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.Hd [in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.n [in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar.td [in Combi.MPoly.Schur_altdef]
+StdRS.Alph [in Combi.LRrule.stdplact]
+StdRS.disp [in Combi.LRrule.stdplact]
+StdtabCombClass.n [in Combi.Combi.stdtab]
+StdtabCombClass.stdtabn_enum [in Combi.Combi.stdtab]
+StdtabnOfStdtabsh.n [in Combi.Combi.stdtab]
+StdtabnOfStdtabsh.sh [in Combi.Combi.stdtab]
+StdtabOfShape.sh [in Combi.Combi.stdtab]
+StdtabOfShape.stdtabsh_enum [in Combi.Combi.stdtab]
+StdTakeDrop.disp1 [in Combi.Combi.std]
+StdTakeDrop.disp2 [in Combi.Combi.std]
+StdTakeDrop.S [in Combi.Combi.std]
+StdTakeDrop.T [in Combi.Combi.std]
+SubseqSortedIn.leT [in Combi.Combi.subseq]
+SubseqSortedIn.T [in Combi.Combi.subseq]
+SubseqSorted.leT [in Combi.Combi.subseq]
+SubseqSorted.T [in Combi.Combi.subseq]
+SubtypesDisjointUnion.FI [in Combi.Basic.combclass]
+SubtypesDisjointUnion.Hpart [in Combi.Basic.combclass]
+SubtypesDisjointUnion.HPTi [in Combi.Basic.combclass]
+SubtypesDisjointUnion.P [in Combi.Basic.combclass]
+SubtypesDisjointUnion.Pi [in Combi.Basic.combclass]
+SubtypesDisjointUnion.PI [in Combi.Basic.combclass]
+SubtypesDisjointUnion.T [in Combi.Basic.combclass]
+SubtypesDisjointUnion.TI [in Combi.Basic.combclass]
+SubtypesDisjointUnion.TP [in Combi.Basic.combclass]
+SubtypesDisjointUnion.TPi [in Combi.Basic.combclass]
+SubtypesDisjointUnion.TPI [in Combi.Basic.combclass]
+SubtypesDisjointUnion.union_enum [in Combi.Basic.combclass]
+SubUndup.P [in Combi.Basic.combclass]
+SubUndup.subenum [in Combi.Basic.combclass]
+SubUndup.subenumP [in Combi.Basic.combclass]
+SubUndup.subenum_in [in Combi.Basic.combclass]
+SubUndup.T [in Combi.Basic.combclass]
+SubUndup.TP [in Combi.Basic.combclass]
+Swap.Swap.Alph [in Combi.LRrule.Greene_inv]
+Swap.Swap.disp [in Combi.LRrule.Greene_inv]
+Swap.Swap.l0 [in Combi.LRrule.Greene_inv]
+Swap.Swap.l1 [in Combi.LRrule.Greene_inv]
+Swap.Swap.R [in Combi.LRrule.Greene_inv]
+Swap.Swap.u [in Combi.LRrule.Greene_inv]
+Swap.Swap.v [in Combi.LRrule.Greene_inv]
+Swap.Swap.word [in Combi.LRrule.Greene_inv]
+Swap.Swap.x [in Combi.LRrule.Greene_inv]
+SymheSymsInt.d [in Combi.MPoly.sympoly]
+SymheSymsInt.n [in Combi.MPoly.sympoly]
+SymheSyms.d [in Combi.MPoly.sympoly]
+SymheSyms.n [in Combi.MPoly.sympoly]
+SymheSyms.R [in Combi.MPoly.sympoly]
+SymPolF.m [in Combi.MPoly.sympoly]
+SymPolF.R [in Combi.MPoly.sympoly]
+SymPolyComRingType.n [in Combi.MPoly.sympoly]
+SymPolyComRingType.R [in Combi.MPoly.sympoly]
+SymPolyIdomainType.n [in Combi.MPoly.sympoly]
+SymPolyIdomainType.R [in Combi.MPoly.sympoly]
+SymPolyRingType.n [in Combi.MPoly.sympoly]
+SymPolyRingType.R [in Combi.MPoly.sympoly]
+SymsSymmInt.d [in Combi.MPoly.sympoly]
+SymsSymmInt.n [in Combi.MPoly.sympoly]
+SymsSymm.d [in Combi.MPoly.sympoly]
+SymsSymm.n [in Combi.MPoly.sympoly]
+SymsSymm.R [in Combi.MPoly.sympoly]
+

T

+TableauReading.A [in Combi.LRrule.freeSchur]
+TableauReading.A [in Combi.Combi.tableau]
+TableauReading.disp [in Combi.LRrule.freeSchur]
+TableauReading.disp [in Combi.Combi.tableau]
+Tableau.disp [in Combi.Combi.tableau]
+Tableau.T [in Combi.Combi.tableau]
+TamariCover.n [in Combi.Combi.bintree]
+TamariLattice.TamariLattice.n [in Combi.Combi.bintree]
+TcastVal.T [in Combi.SymGroup.reprSn]
+TClosureInvset.n0 [in Combi.SymGroup.weak_order]
+Tests.bla [in Combi.Combi.bintree]
+Tests.partn [in Combi.MPoly.MurnaghanNakayama]
+TextBookDefStartStop.Hrib [in Combi.Combi.skewpart]
+TextBookDefStartStop.inner [in Combi.Combi.skewpart]
+TextBookDefStartStop.outer [in Combi.Combi.skewpart]
+TextBookDefStartStop.partinn [in Combi.Combi.skewpart]
+TextBookDefStartStop.partout [in Combi.Combi.skewpart]
+TextBookDefStartStop.start [in Combi.Combi.skewpart]
+TextBookDefStartStop.stop [in Combi.Combi.skewpart]
+TextBookImplDef.Htb [in Combi.Combi.skewpart]
+TextBookImplDef.inner [in Combi.Combi.skewpart]
+TextBookImplDef.outer [in Combi.Combi.skewpart]
+TextBookImplDef.partinn [in Combi.Combi.skewpart]
+TextBookImplDef.partout [in Combi.Combi.skewpart]
+TowerMorphism.m [in Combi.SymGroup.towerSn]
+TowerMorphism.n [in Combi.SymGroup.towerSn]
+Transitive.bound [in Combi.Basic.congr]
+Transitive.FullKnown.full [in Combi.Basic.congr]
+Transitive.FullKnown.Hfull [in Combi.Basic.congr]
+Transitive.Hbound [in Combi.Basic.congr]
+Transitive.Hinvar [in Combi.Basic.congr]
+Transitive.invar [in Combi.Basic.congr]
+Transitive.rule [in Combi.Basic.congr]
+Transitive.T [in Combi.Basic.congr]
+Transp.Alph [in Combi.Combi.std]
+Transp.disp [in Combi.Combi.std]
+Transp.T [in Combi.SymGroup.presentSn]
+TriangularInv.disp [in Combi.Basic.unitriginv]
+TriangularInv.M [in Combi.Basic.unitriginv]
+TriangularInv.Munitrig [in Combi.Basic.unitriginv]
+TriangularInv.R [in Combi.Basic.unitriginv]
+TriangularInv.T [in Combi.Basic.unitriginv]
+TrivISeq.T [in Combi.LRrule.Greene]
+

U

+UnionPart.k [in Combi.Combi.partition]
+UnionPart.l [in Combi.Combi.partition]
+UnionPart.m [in Combi.Combi.partition]
+UnionPart.n [in Combi.Combi.partition]
+UniqFinType.P [in Combi.Basic.combclass]
+UniqFinType.subenum [in Combi.Basic.combclass]
+UniqFinType.subenumE [in Combi.Basic.combclass]
+UniqFinType.subenum_uniq [in Combi.Basic.combclass]
+UniqFinType.T [in Combi.Basic.combclass]
+UniqFinType.TP [in Combi.Basic.combclass]
+UniTriangular.disp [in Combi.Basic.unitriginv]
+UniTriangular.R [in Combi.Basic.unitriginv]
+UniTriangular.T [in Combi.Basic.unitriginv]
+

V

+VandermondeDet.n [in Combi.MPoly.antisym]
+VandermondeDet.R [in Combi.MPoly.antisym]
+Vanprod.abound [in Combi.MPoly.antisym]
+Vanprod.n [in Combi.MPoly.antisym]
+Vanprod.R [in Combi.MPoly.antisym]
+Vanprod.rbound [in Combi.MPoly.antisym]
+Vector.d [in Combi.MPoly.homogsym]
+Vector.n0 [in Combi.MPoly.homogsym]
+Vector.R [in Combi.MPoly.homogsym]
+

W

+WeakOrder.Def.n0 [in Combi.SymGroup.weak_order]
+WeakOrder.Exports.WeakOrder.n0 [in Combi.SymGroup.weak_order]
+

Y

+YamOfEval.ev [in Combi.Combi.Yamanouchi]
+YamOfSize.n [in Combi.Combi.Yamanouchi]
+YoungIrrDef.n [in Combi.SymGroup.Frobenius_char]
+

+

Library Index

+

A

+antisym
+

B

+bintree
+

C

+Cauchy
+Ccpo
+combclass
+composition
+congr
+cycles
+cycletype
+

D

+Dyckword
+

E

+Erdos_Szekeres
+extract
+

F

+fibered_set
+freeSchur
+Frobenius_char
+Frobenius_ident
+

G

+Greene
+Greene_inv
+

H

+homogsym
+hook
+

I

+implem
+

M

+Misc
+multinomial
+MurnaghanNakayama
+

O

+ordcast
+ordtree
+ordtype
+

P

+partition
+permcent
+permcomp
+permuted
+plactic
+presentSn
+

Q

+Qmeasure
+

R

+reprSn
+

S

+Schensted
+Schur_altdef
+Schur_mpoly
+setpartition
+shuffle
+skewpart
+skewtab
+sorted
+std
+stdplact
+stdtab
+subseq
+sympoly
+

T

+tableau
+therule
+tools
+towerSn
+

U

+unitriginv
+

V

+vectNK
+

W

+weak_order
+

Y

+Yamanouchi
+Yam_plact
+

+

Lemma Index

+

A

+abelian_cycle_dec [in Combi.SymGroup.cycles]
+abelian_perm_dec [in Combi.SymGroup.cycles]
+abelian_disjoint_psupports [in Combi.SymGroup.cycles]
+addE [in Combi.LRrule.implem]
+add_corner_decr_nth [in Combi.Combi.partition]
+add_mesymK [in Combi.MPoly.Schur_altdef]
+add_mesymE [in Combi.MPoly.Schur_altdef]
+add_mpart_mesymP [in Combi.MPoly.Schur_altdef]
+add_ribbon_intpartnP [in Combi.Combi.skewpart]
+add_ribbon_intpartnE [in Combi.Combi.skewpart]
+add_ribbon_intpartn_spec [in Combi.Combi.skewpart]
+add_ribbonP [in Combi.Combi.skewpart]
+add_ribbon_onP [in Combi.Combi.skewpart]
+add_ribbon_height [in Combi.Combi.skewpart]
+add_ribbon_on_remP [in Combi.Combi.skewpart]
+allLeqCons [in Combi.Basic.ordtype]
+allLeqConsE [in Combi.Basic.ordtype]
+allLeqE [in Combi.Basic.ordtype]
+allLeqP [in Combi.Basic.ordtype]
+allLeq_to_word_tl [in Combi.Combi.stdtab]
+allLeq_to_word_hd [in Combi.Combi.stdtab]
+allLeq_is_row_rcons [in Combi.LRrule.plactic]
+allLeq_posbig [in Combi.Basic.ordtype]
+allLeq_last [in Combi.Basic.ordtype]
+allLeq_rconsK [in Combi.Basic.ordtype]
+allLeq_rev [in Combi.Basic.ordtype]
+allLeq_catE [in Combi.Basic.ordtype]
+allLeq_consK [in Combi.Basic.ordtype]
+allLtnCons [in Combi.Basic.ordtype]
+allLtnConsE [in Combi.Basic.ordtype]
+allLtnW [in Combi.Basic.ordtype]
+allLtn_std_rec [in Combi.Combi.std]
+allLtn_posbig [in Combi.Basic.ordtype]
+allLtn_last [in Combi.Basic.ordtype]
+allLtn_rconsK [in Combi.Basic.ordtype]
+allLtn_rev [in Combi.Basic.ordtype]
+allLtn_catE [in Combi.Basic.ordtype]
+allLtn_consK [in Combi.Basic.ordtype]
+allLtn_notin [in Combi.Basic.ordtype]
+all_leqzip_refl [in Combi.Combi.bintree]
+all_in_shuffler [in Combi.LRrule.shuffle]
+all_in_shufflel [in Combi.LRrule.shuffle]
+all_size_shuffle [in Combi.LRrule.shuffle]
+all_allLtn_cat [in Combi.Combi.skewtab]
+all_ltn_nth_tabsh [in Combi.Combi.tableau]
+all_permuted [in Combi.Combi.permuted]
+all_unionP [in Combi.Basic.combclass]
+all_subenum [in Combi.Basic.combclass]
+all_partsums [in Combi.Combi.composition]
+alt_add1_0 [in Combi.MPoly.antisym]
+alt_alternate [in Combi.MPoly.antisym]
+alt_rho_non0 [in Combi.MPoly.antisym]
+alt_uniq_non0 [in Combi.MPoly.antisym]
+alt_detE [in Combi.MPoly.antisym]
+alt_anti [in Combi.MPoly.antisym]
+alt_homog [in Combi.MPoly.antisym]
+alt_straight_add_ribbon [in Combi.MPoly.Schur_altdef]
+alt_straight_add_ribbon0 [in Combi.MPoly.Schur_altdef]
+alt_straight_step [in Combi.MPoly.Schur_altdef]
+alt_uniq [in Combi.MPoly.Schur_altdef]
+alt_SchurE [in Combi.MPoly.Schur_altdef]
+alt_mpart_syme [in Combi.MPoly.Schur_altdef]
+alt_syme [in Combi.MPoly.Schur_altdef]
+antisym_submod_closed [in Combi.MPoly.antisym]
+antisym_zmod [in Combi.MPoly.antisym]
+antisym_smalln [in Combi.MPoly.antisym]
+antisym_pchar2 [in Combi.MPoly.antisym]
+antisym_key [in Combi.MPoly.antisym]
+anti_anti [in Combi.MPoly.antisym]
+append_nth_conj_tab [in Combi.Combi.stdtab]
+append_nth_injl [in Combi.Combi.stdtab]
+append_nth_remn [in Combi.Combi.stdtab]
+app2_simpl [in ALEA.Ccpo]
+arm_length_incr_nth_nrow [in Combi.HookFormula.hook]
+arm_length_incr_nth_row [in Combi.HookFormula.hook]
+arm_length_corner_box [in Combi.HookFormula.hook]
+arm_length_ltl [in Combi.HookFormula.hook]
+arm_length_ler [in Combi.HookFormula.hook]
+auxbijP [in Combi.LRrule.Yam_plact]
+auxbij_inj [in Combi.LRrule.Yam_plact]
+

B

+bal_of_DyckK [in Combi.Combi.Dyckword]
+bal_of_DyckP [in Combi.Combi.Dyckword]
+basis_homsym [in Combi.MPoly.homogsym]
+behead_hookpartn [in Combi.Combi.partition]
+behead_incr_nth [in Combi.LRrule.Schensted]
+belast_behead_rcons [in Combi.SSRcomplements.tools]
+beq_nat_neq [in ALEA.Misc]
+bigcup_seq_cover [in Combi.LRrule.Greene]
+bigop_trivIseq [in Combi.LRrule.Greene]
+big_mxvec_index [in Combi.MPoly.Cauchy]
+big_seq_sub [in Combi.SSRcomplements.tools]
+big_enum_box_in [in Combi.Combi.partition]
+big_enum_box_skew [in Combi.Combi.partition]
+big_box_skew2 [in Combi.Combi.partition]
+big_box_skew [in Combi.Combi.partition]
+big_subseqs_undup_cond [in Combi.Combi.subseq]
+big_subseqs_undup [in Combi.Combi.subseq]
+big_subseqs_cons_cond [in Combi.Combi.subseq]
+big_subseqs_cons [in Combi.Combi.subseq]
+big_subseqs0 [in Combi.Combi.subseq]
+big_subseqs_cond [in Combi.Combi.subseq]
+big_subseqs [in Combi.Combi.subseq]
+bijLRyamP [in Combi.LRrule.therule]
+bijLR_image [in Combi.LRrule.therule]
+bijLR_inj [in Combi.LRrule.therule]
+bijLR_surj [in Combi.LRrule.therule]
+bijLR_LRsupport [in Combi.LRrule.therule]
+bijRS [in Combi.LRrule.Schensted]
+bijRStab [in Combi.LRrule.Schensted]
+bij_LRsupportP [in Combi.LRrule.freeSchur]
+binomial_sumn_iota [in Combi.SSRcomplements.tools]
+bintreeszP [in Combi.Combi.bintree]
+bintree_of_join_Dyck [in Combi.Combi.Dyckword]
+bintree_of_nil_Dyck [in Combi.Combi.Dyckword]
+bintree_of_DyckK [in Combi.Combi.Dyckword]
+bintree_of_Dyck_spec [in Combi.Combi.Dyckword]
+bin_to_ordtreeK [in Combi.Combi.ordtree]
+bin_to_forestK [in Combi.Combi.ordtree]
+bisimul_instab [in Combi.LRrule.plactic]
+boolRP [in Combi.MPoly.sympoly]
+bool_of_braceK [in Combi.Combi.Dyckword]
+BoxIn_subproof [in Combi.Combi.partition]
+box_in_incr_nth [in Combi.Combi.partition]
+box_inP [in Combi.Combi.partition]
+box_skewP [in Combi.Combi.partition]
+brace_of_boolK [in Combi.Combi.Dyckword]
+braidCP [in Combi.SymGroup.presentSn]
+braidredE [in Combi.SymGroup.presentSn]
+braidred_size_decr [in Combi.SymGroup.presentSn]
+braidred_to_canword [in Combi.SymGroup.presentSn]
+braidred_inscode_path [in Combi.SymGroup.presentSn]
+braidred_catl [in Combi.SymGroup.presentSn]
+braidrule_homog [in Combi.SymGroup.presentSn]
+braidrule_sym [in Combi.SymGroup.presentSn]
+braidww [in Combi.SymGroup.presentSn]
+braid_to_canword [in Combi.SymGroup.presentSn]
+braid_ltn_lineC [in Combi.SymGroup.presentSn]
+braid_pred_lineC [in Combi.SymGroup.presentSn]
+braid_reduced [in Combi.SymGroup.presentSn]
+braid_prods [in Combi.SymGroup.presentSn]
+braid_rev [in Combi.SymGroup.presentSn]
+braid_is_congr [in Combi.SymGroup.presentSn]
+braid_trans [in Combi.SymGroup.presentSn]
+braid_ltrans [in Combi.SymGroup.presentSn]
+braid_sym [in Combi.SymGroup.presentSn]
+braid_refl [in Combi.SymGroup.presentSn]
+braid_equiv [in Combi.SymGroup.presentSn]
+braid_abaP [in Combi.SymGroup.presentSn]
+Builders_6.isotop [in Combi.Combi.composition]
+Builders_1.isobottom [in Combi.Combi.composition]
+bumped_lt [in Combi.LRrule.plactic]
+bumprowinvK [in Combi.LRrule.Schensted]
+bumprow_rcons [in Combi.LRrule.Schensted]
+bumprow_count [in Combi.LRrule.Schensted]
+bumprow_size [in Combi.LRrule.Schensted]
+bump_dominateK [in Combi.LRrule.Schensted]
+bump_dominate [in Combi.LRrule.Schensted]
+bump_bumprowE [in Combi.LRrule.Schensted]
+bump_tail [in Combi.LRrule.Schensted]
+bump_nil [in Combi.LRrule.Schensted]
+bump_size_ins [in Combi.LRrule.Schensted]
+bump_inspos_lt_size [in Combi.LRrule.Schensted]
+bump_insposE [in Combi.LRrule.Schensted]
+bump_mininspredE [in Combi.LRrule.Schensted]
+bump_bumprow_rconsE [in Combi.LRrule.plactic]
+

C

+canporbitE [in Combi.SymGroup.cycletype]
+canporbitP [in Combi.SymGroup.cycletype]
+canporbit_cymap [in Combi.SymGroup.cycletype]
+canwordE [in Combi.SymGroup.presentSn]
+canwordP [in Combi.SymGroup.presentSn]
+canword_eltr [in Combi.SymGroup.presentSn]
+canword_path_npos [in Combi.SymGroup.presentSn]
+canword_straightenE [in Combi.SymGroup.presentSn]
+canword_reduced [in Combi.SymGroup.presentSn]
+canword1 [in Combi.SymGroup.presentSn]
+card_class_of_part [in Combi.SymGroup.permcent]
+card_class_perm [in Combi.SymGroup.permcent]
+card_cent1_perm [in Combi.SymGroup.permcent]
+card_stab_iporbits [in Combi.SymGroup.permcent]
+card_porbitgrpE [in Combi.SymGroup.permcent]
+card_stdwordn [in Combi.Combi.std]
+card_box_in [in Combi.Combi.partition]
+card_box_skew [in Combi.Combi.partition]
+card_intpartn [in Combi.Combi.partition]
+card_bintreesz [in Combi.Combi.bintree]
+card_seq [in Combi.LRrule.Greene]
+card_eq_eval [in Combi.MPoly.Schur_altdef]
+card_setdiff [in Combi.MPoly.Schur_altdef]
+card_stdtabsh_rat_rec [in Combi.HookFormula.hook]
+card_yam_stdtabE [in Combi.HookFormula.hook]
+card_yama0 [in Combi.HookFormula.hook]
+card_yama_rec [in Combi.HookFormula.hook]
+card_LRtab_set_shapeE [in Combi.LRrule.freeSchur]
+card_LRtab_set_leq [in Combi.LRrule.freeSchur]
+card_Dyck_hsz [in Combi.Combi.Dyckword]
+card_bal_Dyck_hsz [in Combi.Combi.Dyckword]
+card_preim_Dyck_of_bal [in Combi.Combi.Dyckword]
+card_bal_hsz [in Combi.Combi.Dyckword]
+card_bintreesz_dyck [in Combi.Combi.Dyckword]
+card_preim_nth [in Combi.Combi.Dyckword]
+card_preim_partition [in Combi.Combi.Dyckword]
+card_stdtabsh_conj_part [in Combi.Combi.stdtab]
+card_classes_perm [in Combi.SymGroup.cycletype]
+card_classCT_neq0 [in Combi.SymGroup.cycletype]
+card_psupport_conjg [in Combi.SymGroup.cycletype]
+card_pred_card_porbits [in Combi.SymGroup.cycletype]
+card_psupport_noteq1 [in Combi.SymGroup.cycles]
+card_ordtreesz [in Combi.Combi.ordtree]
+card_stpn_shape_hook [in Combi.HookFormula.Frobenius_ident]
+card_stpn_shape [in Combi.HookFormula.Frobenius_ident]
+card_preim_part_of_compn [in Combi.Combi.permuted]
+card_permuted_multinomial_subset [in Combi.Combi.permuted]
+card_permuted_multinomial [in Combi.Combi.permuted]
+card_permuted [in Combi.Combi.permuted]
+card_permuted_seq_sub [in Combi.Combi.permuted]
+card_permuted_prod [in Combi.Combi.permuted]
+card_stab_tuple [in Combi.Combi.permuted]
+card_Delta [in Combi.SymGroup.presentSn]
+card_codesz [in Combi.SymGroup.presentSn]
+card_unionE [in Combi.Basic.combclass]
+card_subE [in Combi.Basic.combclass]
+card_descset [in Combi.Combi.composition]
+card_intcompn [in Combi.Combi.composition]
+cast_conj_inpart [in Combi.Combi.partition]
+cast_intpartn_bij [in Combi.Combi.partition]
+cast_intpartn_inj [in Combi.Combi.partition]
+cast_intpartnKV [in Combi.Combi.partition]
+cast_intpartnK [in Combi.Combi.partition]
+cast_intpartn_id [in Combi.Combi.partition]
+cast_intpartnE [in Combi.Combi.partition]
+cast_enum [in Combi.LRrule.stdplact]
+cast_set_inj [in Combi.SSRcomplements.ordcast]
+cast_map_cond [in Combi.SSRcomplements.ordcast]
+cast_lshift [in Combi.SymGroup.towerSn]
+cast_rshift [in Combi.SymGroup.towerSn]
+cast_cycle_typeSN [in Combi.SymGroup.cycletype]
+cast_IirrS2 [in Combi.SymGroup.reprSn]
+Catalan_binS [in Combi.Combi.bintree]
+Catalan_bin0 [in Combi.Combi.bintree]
+Catalan_bin_leqE [in Combi.Combi.bintree]
+Catalan_binE [in Combi.Combi.Dyckword]
+catlangM [in Combi.LRrule.freeSchur]
+catleft_rotations [in Combi.Combi.bintree]
+cat_leftA [in Combi.Combi.bintree]
+cat_left_Node [in Combi.Combi.bintree]
+cat_leftt0 [in Combi.Combi.bintree]
+cat_left0t [in Combi.Combi.bintree]
+cat_tuple_inj [in Combi.LRrule.freeSchur]
+cat3_equiv_cut3 [in Combi.Combi.vectNK]
+Cauchy_homsymp_zhomsymp [in Combi.MPoly.Cauchy]
+Cauchy_kernel_coeff_homog [in Combi.MPoly.Cauchy]
+Cauchy_kernel_coeff_symmetric [in Combi.MPoly.Cauchy]
+Cauchy_kernel_symmetric [in Combi.MPoly.Cauchy]
+Cauchy_homsyms_homsyms [in Combi.MPoly.Cauchy]
+Cauchy_homsymm_homsymh [in Combi.MPoly.Cauchy]
+Cauchy_symm_symh [in Combi.MPoly.Cauchy]
+Cauchy_kernel_dhomog [in Combi.MPoly.Cauchy]
+cent1_permE [in Combi.SymGroup.permcent]
+cent1_stab_iporbit_porbitgrpS [in Combi.SymGroup.permcent]
+cent1_act_on_iporbits [in Combi.SymGroup.permcent]
+cent1_act_on_porbits [in Combi.SymGroup.permcent]
+cent1_act_porbit [in Combi.SymGroup.permcent]
+cfdotr_ncfuniCT [in Combi.SymGroup.towerSn]
+cfdot_Ind_ncfuniCT [in Combi.SymGroup.towerSn]
+cfdot_Ind_cfuniCT [in Combi.SymGroup.towerSn]
+cfdot_classfun_part [in Combi.SymGroup.towerSn]
+cfdot_is_dot [in Combi.SymGroup.Frobenius_char]
+cfdot_is_hermitian [in Combi.SymGroup.Frobenius_char]
+cfdot_is_bilinear [in Combi.SymGroup.Frobenius_char]
+cfextprodBl [in Combi.SymGroup.towerSn]
+cfextprodBr [in Combi.SymGroup.towerSn]
+cfextprodDl [in Combi.SymGroup.towerSn]
+cfextprodDr [in Combi.SymGroup.towerSn]
+cfextprodMnl [in Combi.SymGroup.towerSn]
+cfextprodMnr [in Combi.SymGroup.towerSn]
+cfextprodNl [in Combi.SymGroup.towerSn]
+cfextprodNr [in Combi.SymGroup.towerSn]
+cfextprodZl [in Combi.SymGroup.towerSn]
+cfextprodZr [in Combi.SymGroup.towerSn]
+cfextprod_cfuni [in Combi.SymGroup.towerSn]
+cfextprod_char [in Combi.SymGroup.towerSn]
+cfextprod_suml [in Combi.SymGroup.towerSn]
+cfextprod_sumr [in Combi.SymGroup.towerSn]
+cfextprod_is_bilinear [in Combi.SymGroup.towerSn]
+cfextprod_subproof [in Combi.SymGroup.towerSn]
+cfextprod0l [in Combi.SymGroup.towerSn]
+cfextprod0r [in Combi.SymGroup.towerSn]
+cfRepr_extprod [in Combi.SymGroup.towerSn]
+cfRepr_sign2 [in Combi.SymGroup.reprSn]
+cfRepr_signed [in Combi.SymGroup.reprSn]
+cfRepr_sign [in Combi.SymGroup.reprSn]
+cfRepr_trivE [in Combi.SymGroup.reprSn]
+cfRepr_triv [in Combi.SymGroup.reprSn]
+cfRepr1_lin_char [in Combi.SymGroup.reprSn]
+cfuniCTE [in Combi.SymGroup.cycletype]
+cfuniCTnE [in Combi.SymGroup.cycletype]
+cfuni_Res [in Combi.SymGroup.towerSn]
+cfuni_tinj [in Combi.SymGroup.towerSn]
+changeTdropP [in Combi.LRrule.freeSchur]
+changeTtakeP [in Combi.LRrule.freeSchur]
+changeUdropP [in Combi.LRrule.freeSchur]
+changeUtakeP [in Combi.LRrule.freeSchur]
+changeUTK [in Combi.LRrule.freeSchur]
+charfun_simplr [in Combi.HookFormula.hook]
+charfun_simpll [in Combi.HookFormula.hook]
+charSG0 [in Combi.SymGroup.reprSn]
+charSG1 [in Combi.SymGroup.reprSn]
+char_mpoly [in Combi.MPoly.antisym]
+choose_one_countE [in Combi.LRrule.implem]
+classCTP [in Combi.SymGroup.cycletype]
+classCT_inj [in Combi.SymGroup.cycletype]
+classes_of_permP [in Combi.SymGroup.cycletype]
+classXE [in Combi.SymGroup.towerSn]
+classXI [in Combi.SymGroup.towerSn]
+class_disj [in Combi.SymGroup.towerSn]
+class_double_neg [in ALEA.Misc]
+class_exc [in ALEA.Misc]
+class_and [in ALEA.Misc]
+class_orc [in ALEA.Misc]
+class_false [in ALEA.Misc]
+class_neg [in ALEA.Misc]
+class_braid1 [in Combi.SymGroup.presentSn]
+cnvarhomsyme [in Combi.MPoly.homogsym]
+cnvarhomsymh [in Combi.MPoly.homogsym]
+cnvarhomsymm [in Combi.MPoly.homogsym]
+cnvarhomsymp [in Combi.MPoly.homogsym]
+cnvarhomsyms [in Combi.MPoly.homogsym]
+cnvarhomsym_is_linear [in Combi.MPoly.homogsym]
+cnvarhomsym_subproof [in Combi.MPoly.homogsym]
+cnvarsym_geq_trans [in Combi.MPoly.sympoly]
+cnvarsym_leq_trans [in Combi.MPoly.sympoly]
+cnvarsym_id [in Combi.MPoly.sympoly]
+cnvarsym_is_monoid_morphism [in Combi.MPoly.sympoly]
+cnvarsym_is_linear [in Combi.MPoly.sympoly]
+cnvarsym_subproof [in Combi.MPoly.sympoly]
+cnvar_symm [in Combi.MPoly.sympoly]
+cnvar_syms [in Combi.MPoly.sympoly]
+cnvar_prodsymp [in Combi.MPoly.sympoly]
+cnvar_prodsymh [in Combi.MPoly.sympoly]
+cnvar_prodsyme [in Combi.MPoly.sympoly]
+cnvar_prodgen [in Combi.MPoly.sympoly]
+cnvar_symp [in Combi.MPoly.sympoly]
+cnvar_symh [in Combi.MPoly.sympoly]
+cnvar_syme [in Combi.MPoly.sympoly]
+cnvar_leq_symeE [in Combi.MPoly.sympoly]
+cocodeE [in Combi.SymGroup.presentSn]
+cocodeP [in Combi.SymGroup.presentSn]
+cocode_straightenE [in Combi.SymGroup.presentSn]
+cocode_rec_cat [in Combi.SymGroup.presentSn]
+cocode2P [in Combi.SymGroup.presentSn]
+codeszP [in Combi.SymGroup.presentSn]
+code_ltn_size [in Combi.SymGroup.presentSn]
+coeffXdiff [in Combi.MPoly.antisym]
+coeff_prodXdiff [in Combi.MPoly.antisym]
+coeff_symh_to_symp [in Combi.MPoly.sympoly]
+coeff_prodgen_cast [in Combi.MPoly.sympoly]
+colcompnP [in Combi.Combi.composition]
+colpartnE [in Combi.Combi.partition]
+colpartnP [in Combi.Combi.partition]
+colpartn_subproof [in Combi.Combi.partition]
+commtuple_morph [in Combi.LRrule.freeSchur]
+commute_cyclic [in Combi.SymGroup.permcent]
+commword_morph [in Combi.LRrule.freeSchur]
+compS [in ALEA.Misc]
+CompSpec_rect [in ALEA.Misc]
+comp_symbe [in Combi.MPoly.homogsym]
+comp_monotonic_left [in ALEA.Ccpo]
+comp_monotonic_right [in ALEA.Ccpo]
+comp_simpl [in ALEA.Ccpo]
+comp0 [in ALEA.Misc]
+comp0 [in Combi.Combi.composition]
+congrruleP [in Combi.Basic.congr]
+congrrule_sym [in Combi.Basic.congr]
+congrrule_invar [in Combi.Basic.congr]
+congrrule_is_congr [in Combi.Basic.congr]
+congr_RS [in Combi.LRrule.plactic]
+congr_bump [in Combi.LRrule.plactic]
+congr_row_2 [in Combi.LRrule.plactic]
+congr_row_1 [in Combi.LRrule.plactic]
+congr_cat [in Combi.Basic.congr]
+congr_catr [in Combi.Basic.congr]
+congr_catl [in Combi.Basic.congr]
+congr_rcons [in Combi.Basic.congr]
+congr_cons [in Combi.Basic.congr]
+conjbijK [in Combi.SymGroup.cycletype]
+conjbijP [in Combi.SymGroup.cycletype]
+conjg_porbits_homog [in Combi.SymGroup.cycletype]
+conjg_porbits_stab [in Combi.SymGroup.cycletype]
+conjg_cycle [in Combi.SymGroup.cycletype]
+conj_porbitgrp [in Combi.SymGroup.permcent]
+conj_hookpartn [in Combi.Combi.partition]
+conj_colpartn [in Combi.Combi.partition]
+conj_rowpartn [in Combi.Combi.partition]
+conj_intpartnK [in Combi.Combi.partition]
+conj_intpartnP [in Combi.Combi.partition]
+conj_intpartK [in Combi.Combi.partition]
+conj_leqE [in Combi.Combi.partition]
+conj_ltnE [in Combi.Combi.partition]
+conj_part_incr_nth [in Combi.Combi.partition]
+conj_partK [in Combi.Combi.partition]
+conj_part_ind [in Combi.Combi.partition]
+conj_nseq [in Combi.Combi.partition]
+conj_stdtabsh_bij [in Combi.Combi.stdtab]
+conj_stdtabshP [in Combi.Combi.stdtab]
+conj_stdtabnP [in Combi.Combi.stdtab]
+conj_tabK [in Combi.Combi.stdtab]
+conj_tab_shapeK [in Combi.Combi.stdtab]
+conj_permP [in Combi.SymGroup.cycletype]
+connect_from_sym [in Combi.Combi.skewpart]
+connect_rev [in Combi.Combi.skewpart]
+conn4_sym [in Combi.Combi.skewpart]
+cons_in_map_cons [in Combi.SSRcomplements.tools]
+cons_head_behead [in Combi.SSRcomplements.tools]
+cons_TamariP [in Combi.Combi.bintree]
+continuous_comp [in ALEA.Ccpo]
+continuous_sym [in ALEA.Ccpo]
+continuous_continuous2 [in ALEA.Ccpo]
+continuous_eq_compat [in ALEA.Ccpo]
+continuous2_app2 [in ALEA.Ccpo]
+continuous2_comp2 [in ALEA.Ccpo]
+continuous2_comp [in ALEA.Ccpo]
+continuous2_sym [in ALEA.Ccpo]
+continuous2_right [in ALEA.Ccpo]
+continuous2_left [in ALEA.Ccpo]
+continuous2_continuous [in ALEA.Ccpo]
+continuous2_eq_compat [in ALEA.Ccpo]
+continuous2_app [in ALEA.Ccpo]
+cont_app_simpl [in ALEA.Ccpo]
+coord_zsymspsp [in Combi.MPoly.Cauchy]
+coord_zsympsps [in Combi.MPoly.Cauchy]
+coord_symbp [in Combi.MPoly.homogsym]
+coord_symbs [in Combi.MPoly.homogsym]
+coord_symbh [in Combi.MPoly.homogsym]
+coord_symbe [in Combi.MPoly.homogsym]
+coord_symbm [in Combi.MPoly.homogsym]
+coord_map_homsym [in Combi.MPoly.homogsym]
+corner_hook_length1 [in Combi.HookFormula.hook]
+corner_leg_length0 [in Combi.HookFormula.hook]
+corner_arm_length0 [in Combi.HookFormula.hook]
+corner_box_conj_part [in Combi.HookFormula.hook]
+corner_box_in_part [in Combi.HookFormula.hook]
+Corollary4 [in Combi.HookFormula.hook]
+Corollary4_eq [in Combi.HookFormula.hook]
+count_mem_iota [in Combi.SSRcomplements.tools]
+count_rcons [in Combi.SSRcomplements.tools]
+count_set_of_card [in Combi.Combi.partition]
+count_mem_vect_n_k_eq_1 [in Combi.Combi.vectNK]
+count_RS [in Combi.LRrule.Schensted]
+count_instab [in Combi.LRrule.Schensted]
+count_mem_height0 [in Combi.Combi.Dyckword]
+count_gt_dominate [in Combi.Combi.tableau]
+count_mem_LRyamtab_list [in Combi.LRrule.implem]
+count_unionP [in Combi.Basic.combclass]
+coversEV [in Combi.Basic.ordtype]
+coversP [in Combi.Basic.ordtype]
+covers_Tamari [in Combi.Combi.bintree]
+covers_permP [in Combi.SymGroup.weak_order]
+covers_rind [in Combi.Basic.ordtype]
+covers_path [in Combi.Basic.ordtype]
+covers_connect [in Combi.Basic.ordtype]
+covers_ind [in Combi.Basic.ordtype]
+covers_dual [in Combi.Basic.ordtype]
+cover_iporbits [in Combi.SymGroup.permcent]
+cover_cast [in Combi.SSRcomplements.ordcast]
+cover_tabcols [in Combi.LRrule.Greene]
+cover_tabcols_rec [in Combi.LRrule.Greene]
+cover_nil [in Combi.LRrule.Greene]
+cover_setpart [in Combi.Combi.setpartition]
+co_hprXYE [in Combi.MPoly.Cauchy]
+co_hpYE [in Combi.MPoly.Cauchy]
+co_hpXY_is_zmod_morphism [in Combi.MPoly.Cauchy]
+co_hp_hp [in Combi.MPoly.Cauchy]
+co_hp_is_scalar [in Combi.MPoly.Cauchy]
+co_hp_is_zmod_morphism [in Combi.MPoly.Cauchy]
+cshift_simpl [in ALEA.Ccpo]
+CTpartnK [in Combi.SymGroup.cycletype]
+CTpartn_colpartn [in Combi.SymGroup.cycletype]
+cutcover [in Combi.LRrule.Greene]
+cut_k_flatten [in Combi.Combi.vectNK]
+cycleij_in [in Combi.SymGroup.presentSn]
+cycleij_inS [in Combi.SymGroup.presentSn]
+cycleij_gt [in Combi.SymGroup.presentSn]
+cycleij_lt [in Combi.SymGroup.presentSn]
+cycleij_j [in Combi.SymGroup.presentSn]
+cycle_type_tinjC [in Combi.SymGroup.towerSn]
+cycle_type_tinj [in Combi.SymGroup.towerSn]
+cycle_typeSn_permCT [in Combi.SymGroup.cycletype]
+cycle_typeSn1 [in Combi.SymGroup.cycletype]
+cycle_type_tpermP [in Combi.SymGroup.cycletype]
+cycle_type_tperm [in Combi.SymGroup.cycletype]
+cycle_type_eq [in Combi.SymGroup.cycletype]
+cycle_dec_conjg [in Combi.SymGroup.cycletype]
+cycle_type_cyclic [in Combi.SymGroup.cycletype]
+cycle_type_conjg [in Combi.SymGroup.cycletype]
+cycle_typeV [in Combi.SymGroup.cycletype]
+cycle_type1 [in Combi.SymGroup.cycletype]
+cycle_type_subproof [in Combi.SymGroup.cycletype]
+cycle_decP [in Combi.SymGroup.cycles]
+cycle_decE [in Combi.SymGroup.cycles]
+cycle_cyclic [in Combi.SymGroup.cycles]
+cycle_type_eltr [in Combi.SymGroup.reprSn]
+cyclicP [in Combi.SymGroup.cycles]
+cyclic_conjg [in Combi.SymGroup.cycletype]
+cyclic_dec [in Combi.SymGroup.cycles]
+cymapcan_perm [in Combi.SymGroup.cycletype]
+cymapcan_aux [in Combi.SymGroup.cycletype]
+cymapE [in Combi.SymGroup.cycletype]
+cymapK [in Combi.SymGroup.cycletype]
+cymapP [in Combi.SymGroup.cycletype]
+cymap_comp [in Combi.SymGroup.cycletype]
+cymap_id [in Combi.SymGroup.cycletype]
+

D

+decomp_cf_triv [in Combi.SymGroup.towerSn]
+decr_nth_intpartE [in Combi.Combi.partition]
+decr_nthK [in Combi.Combi.partition]
+dec_exists_lt [in ALEA.Misc]
+dec_sig_lt [in ALEA.Misc]
+DeltaP [in Combi.SymGroup.presentSn]
+del_rem_corner [in Combi.Combi.partition]
+depth_tree_eq2P [in Combi.Combi.ordtree]
+depth_tree_eq1 [in Combi.Combi.ordtree]
+depth_ordtree_pos [in Combi.Combi.ordtree]
+depth_ordtreeE [in Combi.Combi.ordtree]
+descsetK [in Combi.Combi.composition]
+descset_bij [in Combi.Combi.composition]
+descset_inj [in Combi.Combi.composition]
+det_unitrig [in Combi.Basic.unitriginv]
+dhomog_of_sym_is_linear [in Combi.MPoly.homogsym]
+diag_shift [in ALEA.Ccpo]
+diag_le_compat [in ALEA.Ccpo]
+diffX_neq0 [in Combi.MPoly.antisym]
+diff_shapeK [in Combi.Combi.partition]
+diff_shape_pad0 [in Combi.Combi.partition]
+diff_shape_eq [in Combi.Combi.partition]
+diff_nth_sumn_take [in Combi.Combi.composition]
+dim_homsym [in Combi.MPoly.homogsym]
+dim_cfReprSG [in Combi.SymGroup.Frobenius_char]
+dim_irrSG [in Combi.SymGroup.Frobenius_char]
+disjoint_psupport_dprodE [in Combi.SymGroup.permcent]
+disjoint_imset [in Combi.SSRcomplements.tools]
+disjoint_cover [in Combi.LRrule.Greene_inv]
+disjoint_inj_rec [in Combi.LRrule.Greene]
+disjoint_inj [in Combi.LRrule.Greene]
+disjoint_psupports_conjg [in Combi.SymGroup.cycletype]
+disjoint_psupports_porbits [in Combi.SymGroup.cycles]
+disjoint_psupports_cycles [in Combi.SymGroup.cycles]
+disjoint_psupports_of_decomp [in Combi.SymGroup.cycles]
+disjoint_cycle_dec [in Combi.SymGroup.cycles]
+disjoint_perm_dec [in Combi.SymGroup.cycles]
+disjoint_psupport_subset [in Combi.SymGroup.cycles]
+disj_perm_of_setpart [in Combi.SymGroup.cycletype]
+div_central_binomial [in Combi.Combi.Dyckword]
+dominant_eq [in Combi.MPoly.antisym]
+dominant_mpart [in Combi.MPoly.sympoly]
+dominateK_inspos [in Combi.LRrule.Schensted]
+dominateP [in Combi.Combi.tableau]
+dominate_inspos [in Combi.LRrule.Schensted]
+dominate_tl [in Combi.Combi.tableau]
+dominate_head [in Combi.Combi.tableau]
+dominate_cut [in Combi.Combi.tableau]
+dominate_take [in Combi.Combi.tableau]
+dominate_rcons [in Combi.Combi.tableau]
+dominate_trans [in Combi.Combi.tableau]
+dominate_recE [in Combi.Combi.tableau]
+double_lub_shift [in ALEA.Ccpo]
+double_lub_diag [in ALEA.Ccpo]
+drop_enumI [in Combi.SSRcomplements.tools]
+drop_nilE [in Combi.SSRcomplements.tools]
+dvdn_zcard_fact [in Combi.SymGroup.permcent]
+dvdn_prodfact [in Combi.Combi.multinomial]
+dvdn_card_permuted [in Combi.Combi.permuted]
+dyckE [in Combi.Combi.Dyckword]
+DyckP [in Combi.Combi.Dyckword]
+Dyck_of_balK [in Combi.Combi.Dyckword]
+Dyck_of_Dyck_hsz [in Combi.Combi.Dyckword]
+Dyck_of_balP [in Combi.Combi.Dyckword]
+Dyck_of_bintreeK [in Combi.Combi.Dyckword]
+Dyck_size_even [in Combi.Combi.Dyckword]
+Dyck_ind [in Combi.Combi.Dyckword]
+Dyck_cut_ex [in Combi.Combi.Dyckword]
+Dyck_word_OwCw [in Combi.Combi.Dyckword]
+Dyck_word_flatten [in Combi.Combi.Dyckword]
+Dyck_word_cat [in Combi.Combi.Dyckword]
+Dyck_word_OwC [in Combi.Combi.Dyckword]
+Dyck_wordP [in Combi.Combi.Dyckword]
+Dyck_prefixP [in Combi.Combi.Dyckword]
+

E

+eltrD_ord [in Combi.SymGroup.presentSn]
+eltrK [in Combi.SymGroup.presentSn]
+eltrL_ord [in Combi.SymGroup.presentSn]
+eltrpK [in Combi.MPoly.antisym]
+eltrR_ord [in Combi.SymGroup.presentSn]
+eltrV [in Combi.SymGroup.presentSn]
+eltr_conj [in Combi.SymGroup.reprSn]
+eltr_ind [in Combi.SymGroup.presentSn]
+eltr_genSn [in Combi.SymGroup.presentSn]
+eltr_exchange [in Combi.SymGroup.presentSn]
+eltr_comm [in Combi.SymGroup.presentSn]
+eltr_braid [in Combi.SymGroup.presentSn]
+eltr2 [in Combi.SymGroup.presentSn]
+empty_intpartP [in Combi.Combi.partition]
+ends_at_rem_cornerE [in Combi.HookFormula.hook]
+enumIMN [in Combi.LRrule.Greene]
+enumIsize_to_word [in Combi.LRrule.Greene]
+enum_stdwordn_uniq [in Combi.Combi.std]
+enum_stdwordnE [in Combi.Combi.std]
+enum_box_in_uniq [in Combi.Combi.partition]
+enum_box_skewE [in Combi.Combi.partition]
+enum_box_skew_uniq [in Combi.Combi.partition]
+enum_intpartnE [in Combi.Combi.partition]
+enum_partnP [in Combi.Combi.partition]
+enum_partn_countE [in Combi.Combi.partition]
+enum_partn_allP [in Combi.Combi.partition]
+enum_partnsE [in Combi.Combi.partition]
+enum_partns_countE [in Combi.Combi.partition]
+enum_partns_allP [in Combi.Combi.partition]
+enum_partnskE [in Combi.Combi.partition]
+enum_partnsk_countE [in Combi.Combi.partition]
+enum_partnsk_allP [in Combi.Combi.partition]
+enum_bintreesz_countE [in Combi.Combi.bintree]
+enum_bintreesz_uniq [in Combi.Combi.bintree]
+enum_bintreeszP [in Combi.Combi.bintree]
+enum_bintreeszE [in Combi.Combi.bintree]
+enum_bintreesz0 [in Combi.Combi.bintree]
+enum_bintreesz_leqE [in Combi.Combi.bintree]
+enum_bintreesz_leq_leqE [in Combi.Combi.bintree]
+enum_cast_ord [in Combi.SSRcomplements.ordcast]
+enum_traceP [in Combi.HookFormula.hook]
+enum_trace_uniq [in Combi.HookFormula.hook]
+enum_stdtabshE [in Combi.Combi.stdtab]
+enum_subseqsE [in Combi.Combi.subseq]
+enum_ord_sorted [in Combi.SSRcomplements.sorted]
+enum_ord_sorted_ltn [in Combi.SSRcomplements.sorted]
+enum_ordtreesz_countE [in Combi.Combi.ordtree]
+enum_ordtreesz_uniq [in Combi.Combi.ordtree]
+enum_ordtreeszP [in Combi.Combi.ordtree]
+enum_yamnE [in Combi.Combi.Yamanouchi]
+enum_yamevalE [in Combi.Combi.Yamanouchi]
+enum_yameval_countE [in Combi.Combi.Yamanouchi]
+enum_yamevalP [in Combi.Combi.Yamanouchi]
+enum_setpart_set1 [in Combi.Combi.setpartition]
+enum_setpart_set0 [in Combi.Combi.setpartition]
+enum_codeszE [in Combi.SymGroup.presentSn]
+enum_codesz_countE [in Combi.SymGroup.presentSn]
+enum_codeszP [in Combi.SymGroup.presentSn]
+enum_unionE [in Combi.Basic.combclass]
+enum_sub_undupE [in Combi.Basic.combclass]
+enum_subE [in Combi.Basic.combclass]
+enum_intcompnE [in Combi.Combi.composition]
+enum_compnP [in Combi.Combi.composition]
+enum_compn_countE [in Combi.Combi.composition]
+enum_compn_allP [in Combi.Combi.composition]
+enum_compnE [in Combi.Combi.composition]
+enum_compn_any [in Combi.Combi.composition]
+enum_compn_rec_any [in Combi.Combi.composition]
+enum0 [in Combi.SSRcomplements.tools]
+eqevalP [in Combi.MPoly.Schur_altdef]
+equivalence_partitionE [in Combi.Combi.setpartition]
+equiv_rtrans [in Combi.Basic.congr]
+equi_fbbij [in Combi.Combi.fibered_set]
+eq_inv_transp [in Combi.Combi.std]
+eq_inv_catr [in Combi.Combi.std]
+eq_inv_catl [in Combi.Combi.std]
+eq_inv_std [in Combi.Combi.std]
+eq_inv_rembig [in Combi.Combi.std]
+eq_inv_posbig [in Combi.Combi.std]
+eq_inv_allgtnX [in Combi.Combi.std]
+eq_inv_allgt_imply [in Combi.Combi.std]
+eq_inv_rconsK [in Combi.Combi.std]
+eq_inv_consK [in Combi.Combi.std]
+eq_inv_inversionP [in Combi.Combi.std]
+eq_invP [in Combi.Combi.std]
+eq_inv_size [in Combi.Combi.std]
+eq_inv_trans [in Combi.Combi.std]
+eq_inv_sym [in Combi.Combi.std]
+eq_inv_nil [in Combi.Combi.std]
+eq_inv_refl [in Combi.Combi.std]
+eq_from_nth_notin [in Combi.SSRcomplements.tools]
+eq_bintreeP [in Combi.Combi.bintree]
+eq_inv_is_skew_tableau_reshape [in Combi.Combi.skewtab]
+eq_inv_is_skew_tableau_reshape_size [in Combi.Combi.skewtab]
+eq_inv_skew_dominate [in Combi.Combi.skewtab]
+eq_Greene_rel [in Combi.LRrule.Greene]
+eq_Greene_rel_t [in Combi.LRrule.Greene]
+eq_from_shape_get_tab [in Combi.Combi.stdtab]
+eq_inv_is_row [in Combi.Combi.stdtab]
+eq_ord_equiv [in ALEA.Ccpo]
+eq_in_porbit [in Combi.SymGroup.cycles]
+eq_ordtreeP [in Combi.Combi.ordtree]
+eq_interv_part [in Combi.Combi.skewpart]
+eq_seqE [in Combi.Combi.permuted]
+eq_mnm1 [in Combi.MPoly.sympoly]
+Erdos_Szekeres [in Combi.Erdos_Szekeres.Erdos_Szekeres]
+er_eqE [in Combi.MPoly.homogsym]
+Esym [in Combi.MPoly.homogsym]
+esympolyf_eval_is_monoid_morphism [in Combi.MPoly.sympoly]
+esympolyf_eval_is_linear [in Combi.MPoly.sympoly]
+evalE [in Combi.MPoly.Schur_altdef]
+evalseq_hyper_yam [in Combi.Combi.Yamanouchi]
+evalseq_decr_yam [in Combi.Combi.Yamanouchi]
+evalseq_eq_size [in Combi.Combi.Yamanouchi]
+evalseq_cons [in Combi.Combi.Yamanouchi]
+evalseq_countE [in Combi.Combi.Yamanouchi]
+evalseq0 [in Combi.Combi.Yamanouchi]
+evalXY_homog [in Combi.MPoly.Cauchy]
+evalXY_XE [in Combi.MPoly.Cauchy]
+evalXY_is_linear [in Combi.MPoly.Cauchy]
+evalXY_is_monoid_morphism [in Combi.MPoly.Cauchy]
+evalXY_is_zmod_morphism [in Combi.MPoly.Cauchy]
+eval_ext_tab [in Combi.MPoly.Schur_altdef]
+eval_res_tab [in Combi.MPoly.Schur_altdef]
+eval_yameval [in Combi.Combi.Yamanouchi]
+Example1.all_isOne [in Combi.Basic.combclass]
+Example1.card_isOne [in Combi.Basic.combclass]
+Example1.enum_isOne [in Combi.Basic.combclass]
+Example1.isOne_count_1 [in Combi.Basic.combclass]
+Example2.all_isoneE [in Combi.Basic.combclass]
+Example2.card_isOne [in Combi.Basic.combclass]
+Example2.enum_isOne [in Combi.Basic.combclass]
+Example2.isOne_uniq [in Combi.Basic.combclass]
+Example3.all_isOne [in Combi.Basic.combclass]
+Example3.card_isOne [in Combi.Basic.combclass]
+Example3.enum_isOne [in Combi.Basic.combclass]
+Example3.isOne_in [in Combi.Basic.combclass]
+excluded_middle [in ALEA.Misc]
+exc_intro_class [in ALEA.Misc]
+exc_intro [in ALEA.Misc]
+exists_minh [in Combi.Combi.Dyckword]
+exist_size_Sch [in Combi.LRrule.Schensted]
+expand_prod_add1_seq [in Combi.HookFormula.hook]
+expg_tinj_rshift [in Combi.SymGroup.towerSn]
+expg_tinj_lshift [in Combi.SymGroup.towerSn]
+expg_prod_of_disjoint [in Combi.SymGroup.cycles]
+expri2 [in Combi.MPoly.sympoly]
+expUmpartE [in Combi.MPoly.sympoly]
+expUmpartNE [in Combi.MPoly.sympoly]
+exp1sumnDsize [in Combi.MPoly.sympoly]
+extlsplit [in Combi.LRrule.Greene]
+extprod_mx_repr_subproof [in Combi.SymGroup.towerSn]
+extractIE [in Combi.LRrule.Greene]
+extractmaskE [in Combi.LRrule.Greene]
+extractS [in Combi.LRrule.Greene]
+extract_rev_set [in Combi.LRrule.Greene_inv]
+extract_cut [in Combi.LRrule.Greene_inv]
+extract_tabcols_rec [in Combi.LRrule.Greene]
+extract_tabrows_rec [in Combi.LRrule.Greene]
+extract_tabrows_0 [in Combi.LRrule.Greene]
+extract0 [in Combi.LRrule.Greene]
+extract1 [in Combi.LRrule.Greene]
+extract2 [in Combi.LRrule.Greene]
+extrsplit [in Combi.LRrule.Greene]
+extsubsI [in Combi.LRrule.Greene]
+extsubsIm [in Combi.LRrule.Greene]
+extsubsm [in Combi.LRrule.Greene]
+ext_tab_inj [in Combi.MPoly.Schur_altdef]
+ext_tabK [in Combi.MPoly.Schur_altdef]
+ext_tab_subproof [in Combi.MPoly.Schur_altdef]
+ex_set_setpart_shape [in Combi.Combi.partition]
+ex_setpart_shape [in Combi.Combi.partition]
+ex_subset_card [in Combi.Combi.partition]
+ex_dropeq [in Combi.Combi.skewpart]
+

F

+factor_Dyck_seq [in Combi.Combi.Dyckword]
+factor_Dyck [in Combi.Combi.Dyckword]
+famYinv_subproof [in Combi.MPoly.Cauchy]
+famY_bij [in Combi.MPoly.Cauchy]
+famY_subproof [in Combi.MPoly.Cauchy]
+fbbijK [in Combi.Combi.fibered_set]
+fbbijP [in Combi.Combi.fibered_set]
+fbbij_in_fiber [in Combi.Combi.fibered_set]
+fbset_fbbij [in Combi.Combi.fibered_set]
+FcharE [in Combi.SymGroup.Frobenius_char]
+FcharK [in Combi.SymGroup.Frobenius_char]
+FcharNvar [in Combi.SymGroup.Frobenius_char]
+Fchar_irrSGE [in Combi.SymGroup.Frobenius_char]
+Fchar_ind_morph [in Combi.SymGroup.Frobenius_char]
+Fchar_inv_isometry [in Combi.SymGroup.Frobenius_char]
+Fchar_isometry [in Combi.SymGroup.Frobenius_char]
+Fchar_sign [in Combi.SymGroup.Frobenius_char]
+Fchar_inv_homsymp [in Combi.SymGroup.Frobenius_char]
+Fchar_triv [in Combi.SymGroup.Frobenius_char]
+Fchar_invK [in Combi.SymGroup.Frobenius_char]
+Fchar_inv_is_linear [in Combi.SymGroup.Frobenius_char]
+Fchar_ncfuniCT [in Combi.SymGroup.Frobenius_char]
+Fchar_is_linear [in Combi.SymGroup.Frobenius_char]
+Fchar_invE [in Combi.SymGroup.Frobenius_char]
+fcomp_simpl [in ALEA.Ccpo]
+fcomp2_simpl [in ALEA.Ccpo]
+fcontm_fcont_comp_simpl [in ALEA.Ccpo]
+fcont_compn_com [in ALEA.Ccpo]
+fcont_compn_Sn_simpl [in ALEA.Ccpo]
+fcont_SEQ_simpl [in ALEA.Ccpo]
+fcont_continuous [in ALEA.Ccpo]
+fcont_eq_compat2 [in ALEA.Ccpo]
+fcont_le_compat2 [in ALEA.Ccpo]
+fcont_COMP_simpl [in ALEA.Ccpo]
+fcont_Comp_simpl [in ALEA.Ccpo]
+fcont_comp_le_compat [in ALEA.Ccpo]
+fcont_comp_simpl [in ALEA.Ccpo]
+fcont_lub_simpl [in ALEA.Ccpo]
+fcont_app_continuous [in ALEA.Ccpo]
+fcont_app_simpl [in ALEA.Ccpo]
+fcont_eq [in ALEA.Ccpo]
+fcont_le [in ALEA.Ccpo]
+fcont_eq_elim [in ALEA.Ccpo]
+fcont_eq_intro [in ALEA.Ccpo]
+fcont_le_elim [in ALEA.Ccpo]
+fcont_le_intro [in ALEA.Ccpo]
+fcont2_comp_simpl [in ALEA.Ccpo]
+fcpo_lub_simpl [in ALEA.Ccpo]
+feq_trans [in ALEA.Misc]
+feq_sym [in ALEA.Misc]
+feq_refl [in ALEA.Misc]
+fiber_slporbitE [in Combi.SymGroup.cycletype]
+fif_continuous2 [in ALEA.Ccpo]
+fif_continuous_right [in ALEA.Ccpo]
+fif_continuous_left2 [in ALEA.Ccpo]
+fif_continuous_left [in ALEA.Ccpo]
+Fif_continuous_left [in ALEA.Ccpo]
+Fif_continuous_right [in ALEA.Ccpo]
+Fif_simpl [in ALEA.Ccpo]
+filtergtn_LRsupport [in Combi.LRrule.therule]
+filterleq_LRsupport [in Combi.LRrule.therule]
+filter_gt_RS [in Combi.LRrule.shuffle]
+filter_le_first_row0 [in Combi.Combi.skewtab]
+filter_gt_first_row0 [in Combi.Combi.skewtab]
+filter_le_dominate [in Combi.Combi.skewtab]
+filter_ext_tab [in Combi.MPoly.Schur_altdef]
+filter_to_word [in Combi.Combi.tableau]
+filter_gt_dominate [in Combi.Combi.tableau]
+filter_le_row [in Combi.Combi.tableau]
+filter_gt_row [in Combi.Combi.tableau]
+filter_le_to_word [in Combi.LRrule.therule]
+filter_gt_to_word [in Combi.LRrule.therule]
+filter_gt_shiftn [in Combi.LRrule.therule]
+filter_le_shiftn [in Combi.LRrule.therule]
+find_append_nth [in Combi.Combi.stdtab]
+Finite_in_seq [in ALEA.Qmeasure]
+Finite_eq_out [in ALEA.Qmeasure]
+Finite_eq_in [in ALEA.Qmeasure]
+Finite_simpl [in ALEA.Qmeasure]
+finite_stable_sub [in ALEA.Qmeasure]
+finite_simpl [in ALEA.Qmeasure]
+finite_stdtabn [in Combi.Combi.stdtab]
+finite_stdtabsh [in Combi.Combi.stdtab]
+finite_tabsh [in Combi.Combi.tableau]
+finite_unionP [in Combi.Basic.combclass]
+finite_sub_undupP [in Combi.Basic.combclass]
+finite_subP [in Combi.Basic.combclass]
+finord_wf_down [in Combi.Basic.ordtype]
+finord_wf [in Combi.Basic.ordtype]
+fixp_le_lub [in ALEA.Ccpo]
+fixp_ind_rel [in ALEA.Ccpo]
+fixp_ind [in ALEA.Ccpo]
+fixp_double [in ALEA.Ccpo]
+FIXP_compn [in ALEA.Ccpo]
+FIXP_comp [in ALEA.Ccpo]
+FIXP_comp_com [in ALEA.Ccpo]
+FIXP_inv [in ALEA.Ccpo]
+FIXP_eq [in ALEA.Ccpo]
+FIXP_eq_compat [in ALEA.Ccpo]
+FIXP_le_compat [in ALEA.Ccpo]
+FIXP_simpl [in ALEA.Ccpo]
+Fixp_cont_simpl [in ALEA.Ccpo]
+fixp_continuous_eq [in ALEA.Ccpo]
+fixp_continuous [in ALEA.Ccpo]
+Fixp_simpl [in ALEA.Ccpo]
+fixp_le_compat [in ALEA.Ccpo]
+fixp_inv [in ALEA.Ccpo]
+fixp_eq [in ALEA.Ccpo]
+fixp_le [in ALEA.Ccpo]
+flatten_equiv_cut_k [in Combi.Combi.vectNK]
+flipK [in Combi.Combi.bintree]
+flipszK [in Combi.Combi.bintree]
+flipsz_leftcomb [in Combi.Combi.bintree]
+flipsz_rightcomb [in Combi.Combi.bintree]
+flipsz_subproof [in Combi.Combi.bintree]
+Flip_simpl [in ALEA.Qmeasure]
+flip_false [in ALEA.Qmeasure]
+flip_true [in ALEA.Qmeasure]
+flip_prob [in ALEA.Qmeasure]
+flip_stable_sub [in ALEA.Qmeasure]
+flip_leftcomb [in Combi.Combi.bintree]
+flip_rightcomb [in Combi.Combi.bintree]
+fmon_lub_simpl [in ALEA.Ccpo]
+fmon_cte_comp [in ALEA.Ccpo]
+fmon_le_compat2 [in ALEA.Ccpo]
+fmon_diag_simpl [in ALEA.Ccpo]
+fmon_eq [in ALEA.Ccpo]
+fmon_le [in ALEA.Ccpo]
+fnatO_elim [in ALEA.Ccpo]
+foldr_join_Dyck_inj [in Combi.Combi.Dyckword]
+foldr_maxn [in Combi.Combi.Yamanouchi]
+ford_eq_intro [in ALEA.Ccpo]
+ford_eq_elim [in ALEA.Ccpo]
+ford_le_intro [in ALEA.Ccpo]
+ford_le_elim [in ALEA.Ccpo]
+forest_to_bintreeK [in Combi.Combi.ordtree]
+Formula1 [in Combi.HookFormula.hook]
+freeSchurP [in Combi.LRrule.freeSchur]
+free_LR_rule_alternate [in Combi.LRrule.freeSchur]
+free_LR_rule [in Combi.LRrule.freeSchur]
+Frobenius_char [in Combi.SymGroup.Frobenius_char]
+Frobenius_char_coord [in Combi.SymGroup.Frobenius_char]
+Frobenius_char_homsymdot [in Combi.SymGroup.Frobenius_char]
+Frobenius_ident_rat [in Combi.HookFormula.Frobenius_ident]
+Frobenius_ident [in Combi.HookFormula.Frobenius_ident]
+from_vct0 [in Combi.Combi.bintree]
+from_vct_cat [in Combi.Combi.bintree]
+from_vctK [in Combi.Combi.bintree]
+from_vct_cat_leftE [in Combi.Combi.bintree]
+from_vct_accE [in Combi.Combi.bintree]
+from_vct_fuelE [in Combi.Combi.bintree]
+from_vct_acc_nil [in Combi.Combi.bintree]
+from_left_cat [in Combi.Combi.bintree]
+from_leftK [in Combi.Combi.bintree]
+from_revdualK [in Combi.LRrule.plactic]
+from_descsetK [in Combi.Combi.composition]
+from_descset_spec [in Combi.Combi.composition]
+fs_porbitP [in Combi.SymGroup.cycletype]
+full_bound [in Combi.Basic.congr]
+fun2_mon2 [in ALEA.Ccpo]
+

G

+genclassE [in Combi.Basic.congr]
+gencongrP [in Combi.Basic.congr]
+gencongr_generic_ind [in Combi.Basic.congr]
+gencongr_unique [in Combi.Basic.congr]
+gencongr_imply [in Combi.Basic.congr]
+gencongr_invar [in Combi.Basic.congr]
+gencongr_ind [in Combi.Basic.congr]
+gencongr_min [in Combi.Basic.congr]
+gencongr_is_congr [in Combi.Basic.congr]
+gencongr_equiv [in Combi.Basic.congr]
+genfun_length [in Combi.SymGroup.presentSn]
+get_conj_tab [in Combi.Combi.stdtab]
+get_tab_append_nth [in Combi.Combi.stdtab]
+get_map_tab [in Combi.Combi.tableau]
+get_tab_default [in Combi.Combi.tableau]
+Greene_rel_one [in Combi.Erdos_Szekeres.Erdos_Szekeres]
+Greene_invseq [in Combi.LRrule.stdplact]
+Greene_std [in Combi.LRrule.stdplact]
+Greene_col_eq_shape_RS [in Combi.LRrule.Greene_inv]
+Greene_row_eq_shape_RS [in Combi.LRrule.Greene_inv]
+Greene_col_RS [in Combi.LRrule.Greene_inv]
+Greene_col_invar_plactic [in Combi.LRrule.Greene_inv]
+Greene_row_RS [in Combi.LRrule.Greene_inv]
+Greene_row_invar_plactic [in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact2i [in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact2 [in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact1i [in Combi.LRrule.Greene_inv]
+Greene_col_invar_plact1 [in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact2i [in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact2 [in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact1i [in Combi.LRrule.Greene_inv]
+Greene_row_invar_plact1 [in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact2 [in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact1i [in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact1 [in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact2i [in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact2i [in Combi.LRrule.Greene_inv]
+Greene_col_leq_plact1 [in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact2 [in Combi.LRrule.Greene_inv]
+Greene_row_leq_plact1i [in Combi.LRrule.Greene_inv]
+Greene_col_dual [in Combi.LRrule.Greene_inv]
+Greene_row_dual [in Combi.LRrule.Greene_inv]
+Greene_col_tab_eq_shape [in Combi.LRrule.Greene]
+Greene_row_tab_eq_shape [in Combi.LRrule.Greene]
+Greene_col_tab [in Combi.LRrule.Greene]
+Greene_row_tab [in Combi.LRrule.Greene]
+Greene_rel_rev [in Combi.LRrule.Greene]
+Greene_rel_uniq [in Combi.LRrule.Greene]
+Greene_rel_seq [in Combi.LRrule.Greene]
+Greene_rel_cat [in Combi.LRrule.Greene]
+Greene_rel_t_cat [in Combi.LRrule.Greene]
+Greene_rel_t_uniq [in Combi.LRrule.Greene]
+Greene_rel_t_cast [in Combi.LRrule.Greene]
+Greene_rel_t_0 [in Combi.LRrule.Greene]
+Greene_rel_t_sup [in Combi.LRrule.Greene]
+Greene_rel_t_inf [in Combi.LRrule.Greene]
+grel_neig4_sym [in Combi.Combi.skewpart]
+gtn_braidC [in Combi.SymGroup.presentSn]
+gt_trans [in Combi.LRrule.Greene]
+

H

+hasincr0 [in Combi.MPoly.Schur_altdef]
+HBeta [in Combi.HookFormula.hook]
+HBeta' [in Combi.HookFormula.hook]
+hb_strip_rowE [in Combi.Combi.skewtab]
+hb_strip_shape_res_tab [in Combi.MPoly.Schur_altdef]
+hb_strip_conjE [in Combi.Combi.skewpart]
+hb_strip_conj [in Combi.Combi.skewpart]
+hb_stripP [in Combi.Combi.skewpart]
+hb_strip_size [in Combi.Combi.skewpart]
+hb_strip_included [in Combi.Combi.skewpart]
+Hcrn [in Combi.HookFormula.hook]
+head_pad [in Combi.Combi.partition]
+head_tableau_non_nil [in Combi.LRrule.Schensted]
+head_leq_invbumped [in Combi.LRrule.Schensted]
+head_lt_invins [in Combi.LRrule.Schensted]
+head_instab [in Combi.LRrule.Schensted]
+head_ins_lt_bumped [in Combi.LRrule.Schensted]
+head_leq_last_sorted [in Combi.SSRcomplements.sorted]
+head_filter_gt_tab [in Combi.Combi.tableau]
+head_revcode [in Combi.SymGroup.presentSn]
+head_row_skew_yam [in Combi.LRrule.implem]
+height_take_leq [in Combi.Combi.Dyckword]
+height_nseq [in Combi.Combi.Dyckword]
+height_rev [in Combi.Combi.Dyckword]
+height_drop [in Combi.Combi.Dyckword]
+height_cat [in Combi.Combi.Dyckword]
+height_rcons [in Combi.Combi.Dyckword]
+height_cons [in Combi.Combi.Dyckword]
+height_nil [in Combi.Combi.Dyckword]
+Hinvst [in Combi.LRrule.stdplact]
+Hinvts [in Combi.LRrule.stdplact]
+homogsym_vecaxiom [in Combi.MPoly.homogsym]
+homog_X_mPo_gen [in Combi.MPoly.sympoly]
+homog_symmE [in Combi.MPoly.sympoly]
+homsymdotBl [in Combi.MPoly.homogsym]
+homsymdotBr [in Combi.MPoly.homogsym]
+homsymdotC [in Combi.MPoly.homogsym]
+homsymdotDl [in Combi.MPoly.homogsym]
+homsymdotDr [in Combi.MPoly.homogsym]
+homsymdotE [in Combi.MPoly.homogsym]
+homsymdotMnl [in Combi.MPoly.homogsym]
+homsymdotMnr [in Combi.MPoly.homogsym]
+homsymdotNl [in Combi.MPoly.homogsym]
+homsymdotNr [in Combi.MPoly.homogsym]
+homsymdotpp [in Combi.MPoly.homogsym]
+homsymdotss [in Combi.MPoly.Cauchy]
+homsymdotZl [in Combi.MPoly.homogsym]
+homsymdotZr [in Combi.MPoly.homogsym]
+homsymdot_omegasf [in Combi.MPoly.homogsym]
+homsymdot_is_dot [in Combi.MPoly.homogsym]
+homsymdot_sumr [in Combi.MPoly.homogsym]
+homsymdot_suml [in Combi.MPoly.homogsym]
+homsymdot_is_hermitian [in Combi.MPoly.homogsym]
+homsymdot_is_bilinear [in Combi.MPoly.homogsym]
+homsymdot0l [in Combi.MPoly.homogsym]
+homsymdot0r [in Combi.MPoly.homogsym]
+homsymE [in Combi.MPoly.homogsym]
+homsyme_character [in Combi.SymGroup.Frobenius_char]
+homsymh_character [in Combi.SymGroup.Frobenius_char]
+homsymmE [in Combi.MPoly.homogsym]
+homsymprodBl [in Combi.MPoly.homogsym]
+homsymprodBr [in Combi.MPoly.homogsym]
+homsymprodDl [in Combi.MPoly.homogsym]
+homsymprodDr [in Combi.MPoly.homogsym]
+homsymprodMnl [in Combi.MPoly.homogsym]
+homsymprodMnr [in Combi.MPoly.homogsym]
+homsymprodNl [in Combi.MPoly.homogsym]
+homsymprodNr [in Combi.MPoly.homogsym]
+homsymprodZl [in Combi.MPoly.homogsym]
+homsymprodZr [in Combi.MPoly.homogsym]
+homsymprod_h1p [in Combi.MPoly.homogsym]
+homsymprod_h1e [in Combi.MPoly.homogsym]
+homsymprod_h1h [in Combi.MPoly.homogsym]
+homsymprod_hp [in Combi.MPoly.homogsym]
+homsymprod_he [in Combi.MPoly.homogsym]
+homsymprod_hh [in Combi.MPoly.homogsym]
+homsymprod_suml [in Combi.MPoly.homogsym]
+homsymprod_sumr [in Combi.MPoly.homogsym]
+homsymprod_is_bilinear [in Combi.MPoly.homogsym]
+homsymprod_subproof [in Combi.MPoly.homogsym]
+homsymprod0l [in Combi.MPoly.homogsym]
+homsymprod0r [in Combi.MPoly.homogsym]
+homsymp_orthogonal [in Combi.MPoly.homogsym]
+homsyms_orthonormal [in Combi.MPoly.Cauchy]
+homsyms_homsympM [in Combi.MPoly.MurnaghanNakayama]
+homsym_is_dhomog [in Combi.MPoly.homogsym]
+homsym_is_linear [in Combi.MPoly.homogsym]
+homsym_inj [in Combi.MPoly.homogsym]
+HookLengthFormula [in Combi.HookFormula.hook]
+HookLengthFormula_rat [in Combi.HookFormula.hook]
+hookpartnE [in Combi.Combi.partition]
+hookpartnP [in Combi.Combi.partition]
+hookpartnPE [in Combi.Combi.partition]
+hookpartn_row [in Combi.Combi.partition]
+hookpartn_col [in Combi.Combi.partition]
+hookpartn_subproof [in Combi.Combi.partition]
+hook_length_prod_div [in Combi.HookFormula.hook]
+hook_length_prod_nat [in Combi.HookFormula.hook]
+hook_length_prod_non0 [in Combi.HookFormula.hook]
+hook_length_last_rectangle [in Combi.HookFormula.hook]
+hook_boxes_empty [in Combi.HookFormula.hook]
+hook_length_pred [in Combi.HookFormula.hook]
+hook_length_incr_nth [in Combi.HookFormula.hook]
+hook_length_incr_nth_col [in Combi.HookFormula.hook]
+hook_length_incr_nth_row [in Combi.HookFormula.hook]
+hook_length_corner_box [in Combi.HookFormula.hook]
+hook_length1_corner [in Combi.HookFormula.hook]
+hook_length_ltr [in Combi.HookFormula.hook]
+hook_length_ltl [in Combi.HookFormula.hook]
+hook_length_conj_part [in Combi.HookFormula.hook]
+hook_length_geq1 [in Combi.HookFormula.hook]
+Hp [in Combi.HookFormula.hook]
+Hpart' [in Combi.HookFormula.hook]
+Hszrcons [in Combi.MPoly.Schur_altdef]
+hyper_stdtabnP [in Combi.LRrule.freeSchur]
+hyper_stdtabP [in Combi.LRrule.freeSchur]
+hyper_stdtabsh_subproof [in Combi.Combi.stdtab]
+hyper_yam_of_eval [in Combi.Combi.Yamanouchi]
+hyper_yamP [in Combi.Combi.Yamanouchi]
+

I

+Id_simpl [in ALEA.Ccpo]
+ieqi1F [in Combi.SymGroup.presentSn]
+ieqi2F [in Combi.SymGroup.presentSn]
+if_else_not [in ALEA.Misc]
+if_then_not [in ALEA.Misc]
+if_else [in ALEA.Misc]
+if_then [in ALEA.Misc]
+if_beq_nat_nat_eq_dec [in ALEA.Misc]
+image_map_finer [in Combi.Combi.setpartition]
+Imonotonic [in ALEA.Ccpo]
+Imonotonic2 [in ALEA.Ccpo]
+Imon_simpl [in ALEA.Ccpo]
+imon_simpl [in ALEA.Ccpo]
+imon2_simpl [in ALEA.Ccpo]
+Imon2_simpl [in ALEA.Ccpo]
+imsetD [in Combi.SSRcomplements.tools]
+imset_trivIset [in Combi.SSRcomplements.tools]
+imset_inj [in Combi.SSRcomplements.tools]
+imset_transversal_preim [in Combi.Combi.Dyckword]
+imset_classCT [in Combi.SymGroup.cycletype]
+incl [in Combi.Combi.skewpart]
+includedP [in Combi.Combi.partition]
+included_pad0 [in Combi.Combi.partition]
+included_conj_partE [in Combi.Combi.partition]
+included_conj_part [in Combi.Combi.partition]
+included_anti [in Combi.Combi.partition]
+included_sumnE [in Combi.Combi.partition]
+included_incr_nth_inner [in Combi.Combi.partition]
+included_decr_nth [in Combi.Combi.partition]
+included_incr_nth [in Combi.Combi.partition]
+included_trans [in Combi.Combi.partition]
+included_refl [in Combi.Combi.partition]
+included_behead [in Combi.Combi.partition]
+included_shape_filter_gt [in Combi.Combi.skewtab]
+included_shape_filter_gt_tab [in Combi.LRrule.therule]
+included_add_ribbon [in Combi.Combi.skewpart]
+incr_first_n_nthC [in Combi.Combi.partition]
+incr_nthK [in Combi.Combi.partition]
+incr_nth_injl [in Combi.Combi.stdtab]
+incr_equiv [in Combi.SSRcomplements.sorted]
+incr_tab [in Combi.Combi.tableau]
+incr_nth_size [in Combi.Combi.Yamanouchi]
+index_invstd [in Combi.LRrule.shuffle]
+index_sfilterleq [in Combi.LRrule.shuffle]
+index_leq_filter [in Combi.LRrule.shuffle]
+indporbitP [in Combi.SymGroup.cycletype]
+indporbit_cymap [in Combi.SymGroup.cycletype]
+inh_chooseE [in Combi.Basic.ordtype]
+inh_xchooseE [in Combi.Basic.ordtype]
+inj_strict_mon [in ALEA.Ccpo]
+inordi [in Combi.SymGroup.presentSn]
+inordi_neq_i1 [in Combi.SymGroup.presentSn]
+inordi1 [in Combi.SymGroup.presentSn]
+inord_predS [in Combi.SymGroup.presentSn]
+inord1i [in Combi.SymGroup.presentSn]
+inporbits_im [in Combi.SymGroup.permcent]
+inporbits1 [in Combi.SymGroup.permcent]
+inscodeP [in Combi.SymGroup.presentSn]
+insE [in Combi.LRrule.Schensted]
+insposE [in Combi.LRrule.Schensted]
+inspos_leq_exP [in Combi.LRrule.Schensted]
+inspos_lt_size_ins [in Combi.LRrule.Schensted]
+inspos_leq_size [in Combi.LRrule.Schensted]
+inspos_rcons [in Combi.LRrule.plactic]
+inspredN_lt_inspos [in Combi.LRrule.Schensted]
+inspred_mininspred [in Combi.LRrule.Schensted]
+inspred_inspos [in Combi.LRrule.Schensted]
+inspred_any_bump [in Combi.LRrule.Schensted]
+insrowE [in Combi.LRrule.Schensted]
+insrow_head_lt [in Combi.LRrule.Schensted]
+instabnrowE [in Combi.LRrule.Schensted]
+instabnrowinvK [in Combi.LRrule.Schensted]
+instab_non_nil [in Combi.LRrule.Schensted]
+insub_wordcdK [in Combi.SymGroup.presentSn]
+ins_bumprowE [in Combi.LRrule.Schensted]
+ins_non_nil [in Combi.LRrule.Schensted]
+ins_leq [in Combi.LRrule.Schensted]
+ins_head_lt [in Combi.LRrule.Schensted]
+intcompnP [in Combi.Combi.composition]
+intcompn_behead_sub_proof [in Combi.MPoly.sympoly]
+intcompn_cons_sub_proof [in Combi.MPoly.sympoly]
+intcompn_castE [in Combi.Combi.composition]
+intcompn_sumn [in Combi.Combi.composition]
+intcompn0 [in Combi.Combi.composition]
+intcompn1 [in Combi.Combi.composition]
+intcompn2 [in Combi.Combi.composition]
+intcompP [in Combi.Combi.composition]
+IntPartNDom.botEintpartndom [in Combi.Combi.partition]
+IntPartNDom.double_minn [in Combi.Combi.partition]
+IntPartNDom.from_parttupleK [in Combi.Combi.partition]
+IntPartNDom.from_parttupleP [in Combi.Combi.partition]
+IntPartNDom.is_parttupleP [in Combi.Combi.partition]
+IntPartNDom.join_intpartnE [in Combi.Combi.partition]
+IntPartNDom.join_intpartnP [in Combi.Combi.partition]
+IntPartNDom.leEpartdom [in Combi.Combi.partition]
+IntPartNDom.le_meet_intpartn [in Combi.Combi.partition]
+IntPartNDom.meet_intpartnP [in Combi.Combi.partition]
+IntPartNDom.meet_intpartnC [in Combi.Combi.partition]
+IntPartNDom.nth_parttuple_minn [in Combi.Combi.partition]
+IntPartNDom.nth_parttuple [in Combi.Combi.partition]
+IntPartNDom.partdom_colpartn [in Combi.Combi.partition]
+IntPartNDom.partdom_rowpartn [in Combi.Combi.partition]
+IntPartNDom.partdom_conj_intpartn [in Combi.Combi.partition]
+IntPartNDom.partdom_display [in Combi.Combi.partition]
+IntPartNDom.partdom_antisym [in Combi.Combi.partition]
+IntPartNDom.parttupleK [in Combi.Combi.partition]
+IntPartNDom.parttupleP [in Combi.Combi.partition]
+IntPartNDom.parttuplePK [in Combi.Combi.partition]
+IntPartNDom.parttuple_minnP [in Combi.Combi.partition]
+IntPartNDom.parttuple_minnC [in Combi.Combi.partition]
+IntPartNDom.sumn_take_pardom_meet [in Combi.Combi.partition]
+IntPartNDom.sumn_take_part_fromtuple [in Combi.Combi.partition]
+IntPartNDom.sum_diff_tuple [in Combi.Combi.partition]
+IntPartNDom.sum_diff [in Combi.Combi.partition]
+IntPartNDom.take_intpartn_over [in Combi.Combi.partition]
+IntPartNDom.topEintpartndom [in Combi.Combi.partition]
+IntPartNLexi.botEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.colpartn_bot [in Combi.Combi.partition]
+IntPartNLexi.leEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.ltEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.rowpartn_top [in Combi.Combi.partition]
+IntPartNLexi.topEintpartnlexi [in Combi.Combi.partition]
+intpartnP [in Combi.Combi.partition]
+intpartn_cons [in Combi.Combi.partition]
+intpartn_nth0 [in Combi.Combi.partition]
+intpartn_count_leq2E [in Combi.Combi.partition]
+intpartn_leq [in Combi.Combi.partition]
+intpartn_leq_head [in Combi.Combi.partition]
+intpartn_sorted [in Combi.Combi.partition]
+intpartn0 [in Combi.Combi.partition]
+intpartn1 [in Combi.Combi.partition]
+intpartn2 [in Combi.Combi.partition]
+intpartn3 [in Combi.Combi.partition]
+intpartP [in Combi.Combi.partition]
+intpart_of_monP [in Combi.MPoly.homogsym]
+intpart_rem_corner_ind [in Combi.Combi.partition]
+intpart_sumn_take_inj [in Combi.Combi.partition]
+intpart_eqP [in Combi.Combi.partition]
+intpart_sorted [in Combi.Combi.partition]
+invar_rewrite_path [in Combi.Basic.congr]
+invar_step [in Combi.Basic.congr]
+invar_undupE [in Combi.Basic.congr]
+invbumprowK [in Combi.LRrule.Schensted]
+invbump_dom [in Combi.LRrule.Schensted]
+invbump_geq_head [in Combi.LRrule.Schensted]
+invinstabnrowK [in Combi.LRrule.Schensted]
+invseqE [in Combi.Combi.std]
+invseqK [in Combi.LRrule.stdplact]
+invseqRSE [in Combi.LRrule.stdplact]
+invseqRSPQE [in Combi.LRrule.stdplact]
+invseq_invstd [in Combi.Combi.std]
+invseq_nthE [in Combi.Combi.std]
+invseq_is_std [in Combi.Combi.std]
+invseq_sym [in Combi.Combi.std]
+invsetK [in Combi.SymGroup.presentSn]
+invsetP [in Combi.SymGroup.presentSn]
+invset_eltrR [in Combi.SymGroup.presentSn]
+invset_eltrL [in Combi.SymGroup.presentSn]
+invset_inj [in Combi.SymGroup.presentSn]
+invset_maxpermMl [in Combi.SymGroup.presentSn]
+invset_maxpermMr [in Combi.SymGroup.presentSn]
+invset_maxperm [in Combi.SymGroup.presentSn]
+invset_permV [in Combi.SymGroup.presentSn]
+invset_Delta [in Combi.SymGroup.presentSn]
+invstdK [in Combi.Combi.std]
+invstdRSE [in Combi.LRrule.stdplact]
+invstd_inj [in Combi.Combi.std]
+invstd_is_std [in Combi.Combi.std]
+invstd_cat_in_shsh [in Combi.LRrule.shuffle]
+invstd_catleq [in Combi.LRrule.shuffle]
+invstd_catgtn [in Combi.LRrule.shuffle]
+inv_weight_pos [in ALEA.Qmeasure]
+in_seq_sum [in ALEA.Qmeasure]
+in_std_ltn_size [in Combi.Combi.std]
+in_homsym_comp_symbe [in Combi.MPoly.homogsym]
+in_homsymE [in Combi.MPoly.homogsym]
+in_homsym_is_linear [in Combi.MPoly.homogsym]
+in_conj_part [in Combi.Combi.partition]
+in_conj_part_impl [in Combi.Combi.partition]
+in_part_is_part [in Combi.Combi.partition]
+in_part_le [in Combi.Combi.partition]
+in_part_non0 [in Combi.Combi.partition]
+in_shape_size [in Combi.Combi.partition]
+in_skew_nil [in Combi.Combi.partition]
+in_shape_nil [in Combi.Combi.partition]
+in_skew_in [in Combi.Combi.partition]
+in_skew_out [in Combi.Combi.partition]
+in_skewE [in Combi.Combi.partition]
+in_vect_n_k [in Combi.Combi.vectNK]
+in_hook_boxesP [in Combi.HookFormula.hook]
+in_hook_shape [in Combi.HookFormula.hook]
+in_porbit_setP [in Combi.SymGroup.cycles]
+in_psupport [in Combi.SymGroup.cycles]
+in_shape_tab [in Combi.Combi.tableau]
+in_shape_tab_size [in Combi.Combi.tableau]
+in_maxL [in Combi.Basic.ordtype]
+Iord_app [in ALEA.Ccpo]
+iotagtnk [in Combi.LRrule.Greene]
+iota_geq [in Combi.SSRcomplements.tools]
+iota_ltn [in Combi.SSRcomplements.tools]
+iota_hookE [in Combi.HookFormula.hook]
+iota_cut_i [in Combi.SymGroup.presentSn]
+irev_w [in Combi.LRrule.Greene_inv]
+irrSGP [in Combi.SymGroup.Frobenius_char]
+irrSG_char_int [in Combi.SymGroup.Frobenius_char]
+irrSG_irr [in Combi.SymGroup.Frobenius_char]
+irrSG_orthonormal [in Combi.SymGroup.Frobenius_char]
+irr_S2 [in Combi.SymGroup.reprSn]
+isantisymP [in Combi.MPoly.antisym]
+isantisym_alt [in Combi.MPoly.antisym]
+isantisym_mlead_rho [in Combi.MPoly.antisym]
+isantisym_mlead_iota [in Combi.MPoly.antisym]
+isantisym_msupp_uniq [in Combi.MPoly.antisym]
+isantisym_eltrP [in Combi.MPoly.antisym]
+isantisym_msupp [in Combi.MPoly.antisym]
+isantisym_tpermP [in Combi.MPoly.antisym]
+isglb_decr_lift [in ALEA.Ccpo]
+isglb_decr_ext [in ALEA.Ccpo]
+isglb_eq_compat_right [in ALEA.Ccpo]
+isglb_eq_compat_left [in ALEA.Ccpo]
+isglb_eq_compat [in ALEA.Ccpo]
+isglb_le [in ALEA.Ccpo]
+ishift_le_compat [in ALEA.Ccpo]
+ishift_simpl [in ALEA.Ccpo]
+islub_lub [in ALEA.Ccpo]
+islub_mlub [in ALEA.Ccpo]
+islub_fun_intro [in ALEA.Ccpo]
+islub_unique [in ALEA.Ccpo]
+islub_unique_eq [in ALEA.Ccpo]
+islub_decr [in ALEA.Ccpo]
+islub_exch [in ALEA.Ccpo]
+islub_incr_lift [in ALEA.Ccpo]
+islub_incr_ext [in ALEA.Ccpo]
+islub_eq_compat_right [in ALEA.Ccpo]
+islub_eq_compat_left [in ALEA.Ccpo]
+islub_eq_compat [in ALEA.Ccpo]
+isom_tinj [in Combi.SymGroup.towerSn]
+isperm_of_porbit [in Combi.SymGroup.cycletype]
+issym_eltrP [in Combi.MPoly.antisym]
+issym_tpermP [in Combi.MPoly.antisym]
+is_std_wordpermP [in Combi.Combi.std]
+is_stdP [in Combi.Combi.std]
+is_homsym_submod_closed [in Combi.MPoly.homogsym]
+is_part_decr_nth_part [in Combi.Combi.partition]
+is_add_corner_conj_part [in Combi.Combi.partition]
+is_part_conj [in Combi.Combi.partition]
+is_part_incr_first_n [in Combi.Combi.partition]
+is_part_nseq1 [in Combi.Combi.partition]
+is_part_decr_nth [in Combi.Combi.partition]
+is_rem_cornerP [in Combi.Combi.partition]
+is_part_incr_nth [in Combi.Combi.partition]
+is_part_incr_nth_size [in Combi.Combi.partition]
+is_part_rem_trail0 [in Combi.Combi.partition]
+is_part_catr [in Combi.Combi.partition]
+is_part_catl [in Combi.Combi.partition]
+is_part_rconsK [in Combi.Combi.partition]
+is_part_subseq [in Combi.Combi.partition]
+is_part_behead [in Combi.Combi.partition]
+is_part_consK [in Combi.Combi.partition]
+is_part_sortedE [in Combi.Combi.partition]
+is_part_ijP [in Combi.Combi.partition]
+is_partP [in Combi.Combi.partition]
+is_col_dual [in Combi.LRrule.Greene_inv]
+is_row_dual [in Combi.LRrule.Greene_inv]
+is_dominant_nth_partm [in Combi.MPoly.antisym]
+is_dominant_partm [in Combi.MPoly.antisym]
+is_stdtab_of_n_LRtriple [in Combi.LRrule.shuffle]
+is_tableau_std [in Combi.Combi.skewtab]
+is_tableau_reshape_std [in Combi.Combi.skewtab]
+is_skew_tableau_reshape_std [in Combi.Combi.skewtab]
+is_skew_tableau_skew_reshape_pad0 [in Combi.Combi.skewtab]
+is_skew_tableau_filter_le [in Combi.Combi.skewtab]
+is_skew_tableau_filter_le_tmp [in Combi.Combi.skewtab]
+is_skew_tableau_pad0 [in Combi.Combi.skewtab]
+is_skew_tableau0 [in Combi.Combi.skewtab]
+is_skew_tableauP [in Combi.Combi.skewtab]
+is_part_skew_yam [in Combi.Combi.skewtab]
+is_skew_yamE [in Combi.Combi.skewtab]
+is_stdtab_RStabmap2 [in Combi.LRrule.Schensted]
+is_tableau_instabnrowinv1 [in Combi.LRrule.Schensted]
+is_yam_RSmap2 [in Combi.LRrule.Schensted]
+is_tableau_RSmap1 [in Combi.LRrule.Schensted]
+is_rem_corner_instabnrow [in Combi.LRrule.Schensted]
+is_row_invins [in Combi.LRrule.Schensted]
+is_tableau_RS [in Combi.LRrule.Schensted]
+is_tableau_instab [in Combi.LRrule.Schensted]
+is_row_Sch [in Combi.LRrule.Schensted]
+is_row_ins [in Combi.LRrule.Schensted]
+is_trace_corner_nil [in Combi.HookFormula.hook]
+is_trace_in_shape [in Combi.HookFormula.hook]
+is_trace_in_in_shape [in Combi.HookFormula.hook]
+is_trace_lastl [in Combi.HookFormula.hook]
+is_trace_lastr [in Combi.HookFormula.hook]
+is_trace_tlr [in Combi.HookFormula.hook]
+is_trace_tll [in Combi.HookFormula.hook]
+is_yam_plactic [in Combi.LRrule.Yam_plact]
+is_part_incr_nth1E [in Combi.LRrule.Yam_plact]
+is_part_incr_nthE [in Combi.LRrule.Yam_plact]
+is_stdtab_conj [in Combi.Combi.stdtab]
+is_part_shape_deg [in Combi.Combi.stdtab]
+is_row_stdE [in Combi.Combi.stdtab]
+is_stdtab_remn [in Combi.Combi.stdtab]
+is_tab_append_nth_size [in Combi.Combi.stdtab]
+is_tab_append_nth_size_alternative_proof [in Combi.Combi.stdtab]
+is_tableau_filter_gt [in Combi.Combi.tableau]
+is_tableau_getP [in Combi.Combi.tableau]
+is_tableau_sorted_dominate [in Combi.Combi.tableau]
+is_part_sht [in Combi.Combi.tableau]
+is_tableau_catr [in Combi.Combi.tableau]
+is_tableau_catl [in Combi.Combi.tableau]
+is_tableau_rconsK [in Combi.Combi.tableau]
+is_tableauP [in Combi.Combi.tableau]
+is_row_set_nth [in Combi.Combi.tableau]
+is_row_yamrow [in Combi.LRrule.therule]
+is_skew_reshape_tableauP [in Combi.LRrule.therule]
+is_skew_tableau_map_shiftn [in Combi.LRrule.therule]
+is_yam_cat_any [in Combi.Combi.Yamanouchi]
+is_add_corner_yam [in Combi.Combi.Yamanouchi]
+is_rem_corner_yam [in Combi.Combi.Yamanouchi]
+is_yam_decr [in Combi.Combi.Yamanouchi]
+is_yam_catr [in Combi.Combi.Yamanouchi]
+is_yam_tl [in Combi.Combi.Yamanouchi]
+is_part_eval_yam [in Combi.Combi.Yamanouchi]
+is_yam_ijP [in Combi.Combi.Yamanouchi]
+is_yamP [in Combi.Combi.Yamanouchi]
+is_finer_card [in Combi.Combi.setpartition]
+is_invset_tclosureU [in Combi.SymGroup.weak_order]
+is_part_of_add_ribbon [in Combi.Combi.skewpart]
+is_part_add_ribbon [in Combi.Combi.skewpart]
+is_part_add_ribbon_on [in Combi.Combi.skewpart]
+is_code_straighten [in Combi.SymGroup.presentSn]
+is_invset_Delta [in Combi.SymGroup.presentSn]
+is_code_rconsK [in Combi.SymGroup.presentSn]
+is_code_rcons [in Combi.SymGroup.presentSn]
+is_codeP [in Combi.SymGroup.presentSn]
+is_part_pad0 [in Combi.LRrule.implem]
+is_comp_cat [in Combi.Combi.composition]
+is_comp_rcons [in Combi.Combi.composition]
+is_comp_cons [in Combi.Combi.composition]
+is_comp1 [in Combi.Combi.composition]
+is_compP [in Combi.Combi.composition]
+iterO_simpl [in ALEA.Ccpo]
+iterS_simpl [in ALEA.Ccpo]
+IterS_simpl [in ALEA.Ccpo]
+iter_continuous_eq [in ALEA.Ccpo]
+iter_continuous [in ALEA.Ccpo]
+iter_incr [in ALEA.Ccpo]
+i1eqiF [in Combi.SymGroup.presentSn]
+i2eqiF [in Combi.SymGroup.presentSn]
+

J

+joingU1 [in Combi.SymGroup.presentSn]
+join_tab_skew [in Combi.Combi.skewtab]
+join_tab_filter [in Combi.Combi.skewtab]
+join_Dyck_inj [in Combi.Combi.Dyckword]
+join_stdtab_in_shuffle [in Combi.LRrule.therule]
+join_stdtab [in Combi.LRrule.therule]
+

K

+KostkaInv_unitrig [in Combi.MPoly.Schur_altdef]
+KostkaMon_partdom [in Combi.MPoly.Schur_altdef]
+KostkaMon_sumeval [in Combi.MPoly.Schur_altdef]
+KostkaStd [in Combi.MPoly.Schur_altdef]
+Kostka_unitrig [in Combi.MPoly.Schur_altdef]
+Kostka_recE [in Combi.MPoly.Schur_altdef]
+Kostka_rec_size0 [in Combi.MPoly.Schur_altdef]
+Kostka_ind [in Combi.MPoly.Schur_altdef]
+Kostka_diag [in Combi.MPoly.Schur_altdef]
+Kostka_partdom [in Combi.MPoly.Schur_altdef]
+Kostka_size0 [in Combi.MPoly.Schur_altdef]
+Kostka_sumnE [in Combi.MPoly.Schur_altdef]
+Kostka_any [in Combi.MPoly.Schur_altdef]
+Kostka_mnmwiden [in Combi.MPoly.Schur_altdef]
+Kostka_Coeff [in Combi.MPoly.Schur_altdef]
+Kostka0 [in Combi.MPoly.Schur_altdef]
+ksuppCol_inj_plact2i [in Combi.LRrule.Greene_inv]
+ksuppCol_inj_plact1 [in Combi.LRrule.Greene_inv]
+ksuppRow_inj_plact2 [in Combi.LRrule.Greene_inv]
+ksuppRow_inj_plact1i [in Combi.LRrule.Greene_inv]
+ksupp_inj_invseq [in Combi.LRrule.stdplact]
+ksupp_inj_stdI [in Combi.LRrule.stdplact]
+ksupp_inj_std [in Combi.LRrule.stdplact]
+ksupp_gt_tabcolsk [in Combi.LRrule.Greene]
+ksupp_leqX_tabrowsk [in Combi.LRrule.Greene]
+ksupp_inj_rev [in Combi.LRrule.Greene]
+ksupp_cast [in Combi.LRrule.Greene]
+ksupp0 [in Combi.LRrule.Greene]
+

L

+langQE [in Combi.LRrule.shuffle]
+last_behead_rcons [in Combi.SSRcomplements.tools]
+last_incr_nth_non0 [in Combi.Combi.partition]
+last_rot_pfminh [in Combi.Combi.Dyckword]
+last_big_append_nth [in Combi.Combi.stdtab]
+last_bigP [in Combi.Combi.stdtab]
+last_ins_lt [in Combi.LRrule.plactic]
+last_yam [in Combi.Combi.Yamanouchi]
+law1 [in ALEA.Qmeasure]
+law2 [in ALEA.Qmeasure]
+law3 [in ALEA.Qmeasure]
+lcast_com [in Combi.LRrule.Greene]
+LeafP [in Combi.Combi.bintree]
+leftcomb_rotations [in Combi.Combi.bintree]
+left_branchK [in Combi.Combi.bintree]
+leg_length_corner_box [in Combi.HookFormula.hook]
+leg_length_lel [in Combi.HookFormula.hook]
+leg_length_ltr [in Combi.HookFormula.hook]
+leL_geLdualE [in Combi.LRrule.plactic]
+Lemma3 [in Combi.HookFormula.hook]
+lengthKL [in Combi.SymGroup.presentSn]
+lengthKR [in Combi.SymGroup.presentSn]
+lengthM [in Combi.SymGroup.presentSn]
+lengthME [in Combi.SymGroup.presentSn]
+lengthV [in Combi.SymGroup.presentSn]
+length_eq1 [in Combi.SymGroup.presentSn]
+length_eq0 [in Combi.SymGroup.presentSn]
+length_permcd [in Combi.SymGroup.presentSn]
+length_prods [in Combi.SymGroup.presentSn]
+length_eltr [in Combi.SymGroup.presentSn]
+length_descR [in Combi.SymGroup.presentSn]
+length_sub1R [in Combi.SymGroup.presentSn]
+length_add1R [in Combi.SymGroup.presentSn]
+length_descL [in Combi.SymGroup.presentSn]
+length_sub1L [in Combi.SymGroup.presentSn]
+length_add1L [in Combi.SymGroup.presentSn]
+length_maxpermMl [in Combi.SymGroup.presentSn]
+length_maxpermMr [in Combi.SymGroup.presentSn]
+length_maxpermE [in Combi.SymGroup.presentSn]
+length_maxperm [in Combi.SymGroup.presentSn]
+length_max [in Combi.SymGroup.presentSn]
+length1 [in Combi.SymGroup.presentSn]
+leperm_invset [in Combi.SymGroup.weak_order]
+leperm_succ [in Combi.SymGroup.weak_order]
+leperm_factorP [in Combi.SymGroup.weak_order]
+leperm_maxperm [in Combi.SymGroup.weak_order]
+leperm_maxpermMl [in Combi.SymGroup.weak_order]
+leperm1p [in Combi.SymGroup.weak_order]
+leq_sumn_in [in Combi.SSRcomplements.tools]
+leq_addE [in Combi.SSRcomplements.tools]
+leq_head_sumn [in Combi.Combi.partition]
+leq_Greene [in Combi.LRrule.Greene]
+less [in Combi.Combi.skewpart]
+lessz [in Combi.Combi.skewpart]
+let_indep_distr [in ALEA.Qmeasure]
+let_indep [in ALEA.Qmeasure]
+le_intpartndomlexi [in Combi.Combi.partition]
+le_mlub [in ALEA.Ccpo]
+le_isglb [in ALEA.Ccpo]
+le_Ole [in ALEA.Ccpo]
+linvseqK [in Combi.LRrule.stdplact]
+linvseqP [in Combi.Combi.std]
+linvseq_sizeP [in Combi.Combi.std]
+linvseq_subset_iota [in Combi.Combi.std]
+linvseq_ltn_szt [in Combi.Combi.std]
+lin_char_Sn [in Combi.SymGroup.reprSn]
+lin_char_reprP [in Combi.SymGroup.reprSn]
+LRcoeffE [in Combi.LRrule.implem]
+LRcoeffP [in Combi.LRrule.implem]
+LRcoeff_computeP [in Combi.LRrule.therule]
+LRrule_langQ_alternate [in Combi.LRrule.shuffle]
+LRrule_langQ [in Combi.LRrule.shuffle]
+lrshift_recF [in Combi.LRrule.Greene]
+LRsupport_conj [in Combi.LRrule.freeSchur]
+LRtab_coeff_conj [in Combi.LRrule.freeSchur]
+LRtab_coeff_shapeE [in Combi.LRrule.freeSchur]
+LRtab_coeffP [in Combi.LRrule.freeSchur]
+LRtab_set_included [in Combi.LRrule.therule]
+LRtab_coeffP [in Combi.LRrule.implem]
+LRtripleP [in Combi.LRrule.shuffle]
+LRtriple_conj [in Combi.LRrule.shuffle]
+LRtriple_cat_equiv [in Combi.LRrule.shuffle]
+LRtriple_cat_langQ [in Combi.LRrule.shuffle]
+LRtriple_fastE [in Combi.LRrule.shuffle]
+LRyamtabP [in Combi.LRrule.implem]
+LRyamtab_spec_recip [in Combi.LRrule.implem]
+LRyamtab_all [in Combi.LRrule.implem]
+LRyamtab_eval [in Combi.LRrule.implem]
+LRyamtab_skew_tableau [in Combi.LRrule.implem]
+LRyamtab_shape [in Combi.LRrule.implem]
+LRyamtab_included [in Combi.LRrule.implem]
+LRyamtab_yam [in Combi.LRrule.implem]
+LRyamtab_list_countE [in Combi.LRrule.implem]
+LRyamtab_list_skew_tableau0 [in Combi.LRrule.implem]
+LRyamtab_list_shape0 [in Combi.LRrule.implem]
+LRyamtab_list_size [in Combi.LRrule.implem]
+LRyamtab_list_pad0 [in Combi.LRrule.implem]
+LRyamtab_list_included [in Combi.LRrule.implem]
+LRyamtab_list_recP [in Combi.LRrule.implem]
+LRyam_coeff_colpartn [in Combi.LRrule.therule]
+LRyam_coeff_rowpart [in Combi.LRrule.therule]
+LRyam_coeffP [in Combi.LRrule.therule]
+LRyam_coeffE [in Combi.LRrule.therule]
+LRyam_spec_recip [in Combi.LRrule.implem]
+LR_rule_irrSG [in Combi.SymGroup.Frobenius_char]
+LR_rule_tab [in Combi.LRrule.freeSchur]
+lshift_in_rshift_recF [in Combi.LRrule.Greene]
+lshift_recP [in Combi.LRrule.Greene]
+lsplit_rec_tab [in Combi.LRrule.Greene]
+ltcovers [in Combi.Basic.ordtype]
+ltL_gtLdualE [in Combi.LRrule.plactic]
+ltnPred [in Combi.HookFormula.hook]
+ltn_braidC [in Combi.SymGroup.presentSn]
+ltperm_invset [in Combi.SymGroup.weak_order]
+ltperm_length [in Combi.SymGroup.weak_order]
+lt_intpartndomlexi [in Combi.Combi.partition]
+lt_bumped [in Combi.LRrule.Schensted]
+lt_inspos_nth [in Combi.LRrule.Schensted]
+lub_le_fixp [in ALEA.Ccpo]
+lub_cont2_app2_eq [in ALEA.Ccpo]
+lub_app2_eq [in ALEA.Ccpo]
+lub_comp_eq [in ALEA.Ccpo]
+lub_app2_le [in ALEA.Ccpo]
+lub_comp_le [in ALEA.Ccpo]
+lub_fcpo_mon [in ALEA.Ccpo]
+lub_mon_fcpo [in ALEA.Ccpo]
+lub_ishift [in ALEA.Ccpo]
+lub_Olt [in ALEA.Ccpo]
+lub_seq_eq [in ALEA.Ccpo]
+lub_eq_lift [in ALEA.Ccpo]
+lub_le_lift [in ALEA.Ccpo]
+lub_lift_left [in ALEA.Ccpo]
+lub_lift_right [in ALEA.Ccpo]
+lub_cte [in ALEA.Ccpo]
+

M

+map_homsymbs [in Combi.MPoly.homogsym]
+map_homsymbp [in Combi.MPoly.homogsym]
+map_homsymbh [in Combi.MPoly.homogsym]
+map_homsymbe [in Combi.MPoly.homogsym]
+map_homsymbm [in Combi.MPoly.homogsym]
+map_homsyms [in Combi.MPoly.homogsym]
+map_homsymp [in Combi.MPoly.homogsym]
+map_homsymh [in Combi.MPoly.homogsym]
+map_homsyme [in Combi.MPoly.homogsym]
+map_homsymm [in Combi.MPoly.homogsym]
+map_homsym_is_scalable [in Combi.MPoly.homogsym]
+map_homsym_is_zmod_morphism [in Combi.MPoly.homogsym]
+map_sympoly_d_homog [in Combi.MPoly.homogsym]
+map_filter_comp [in Combi.SSRcomplements.tools]
+map_mpolyX [in Combi.MPoly.antisym]
+map_finer_pblock [in Combi.Combi.setpartition]
+map_finer_in [in Combi.Combi.setpartition]
+map_finer_subset [in Combi.Combi.setpartition]
+map_syms [in Combi.MPoly.sympoly]
+map_symp_prod [in Combi.MPoly.sympoly]
+map_symp [in Combi.MPoly.sympoly]
+map_symh_prod [in Combi.MPoly.sympoly]
+map_symh [in Combi.MPoly.sympoly]
+map_syme_prod [in Combi.MPoly.sympoly]
+map_syme [in Combi.MPoly.sympoly]
+map_symm [in Combi.MPoly.sympoly]
+map_sympoly_is_scalable [in Combi.MPoly.sympoly]
+map_sympoly_is_monoid_morphism [in Combi.MPoly.sympoly]
+map_sympoly_is_zmod_morphism [in Combi.MPoly.sympoly]
+map_mpoly_issym [in Combi.MPoly.sympoly]
+mask_injP [in Combi.Combi.subseq]
+mask1E [in Combi.Combi.subseq]
+maxLb [in Combi.Basic.ordtype]
+maxLP [in Combi.Basic.ordtype]
+maxLPt [in Combi.Basic.ordtype]
+maxL_iota_n [in Combi.Basic.ordtype]
+maxL_iota [in Combi.Basic.ordtype]
+maxL_LbR [in Combi.Basic.ordtype]
+maxL_perm [in Combi.Basic.ordtype]
+maxL_cat [in Combi.Basic.ordtype]
+maxpermK [in Combi.SymGroup.presentSn]
+maxpermV [in Combi.SymGroup.presentSn]
+maxXL [in Combi.Basic.ordtype]
+mcoeffXU [in Combi.MPoly.sympoly]
+mcoeff_symbs [in Combi.MPoly.homogsym]
+mcoeff_alt [in Combi.MPoly.antisym]
+mcoeff_arbound [in Combi.MPoly.antisym]
+mcoeff_alt_SchurE [in Combi.MPoly.Schur_altdef]
+mcoeff_symm [in Combi.MPoly.sympoly]
+mcoeff_symm_pol [in Combi.MPoly.sympoly]
+mcoeff_symh [in Combi.MPoly.sympoly]
+mcoeff_symh_pol [in Combi.MPoly.sympoly]
+mcoeff_symh_pol_bound [in Combi.MPoly.sympoly]
+mdeg_tnth_monsY [in Combi.MPoly.Cauchy]
+mdeg_monX [in Combi.MPoly.Cauchy]
+mdeg_rho [in Combi.MPoly.antisym]
+mdeg_mpart [in Combi.MPoly.antisym]
+mem_std [in Combi.Combi.std]
+mem_drop_enumI [in Combi.SSRcomplements.tools]
+mem_take_enumI [in Combi.SSRcomplements.tools]
+mem_takeP [in Combi.SSRcomplements.tools]
+mem_enum_box_in [in Combi.Combi.partition]
+mem_enum_box_skew [in Combi.Combi.partition]
+mem_enum_bintreesz [in Combi.Combi.bintree]
+mem_cast [in Combi.SSRcomplements.ordcast]
+mem_shsh [in Combi.LRrule.shuffle]
+mem_sfilterleqK [in Combi.LRrule.shuffle]
+mem_shuffle [in Combi.LRrule.shuffle]
+mem_shuffle_pred [in Combi.LRrule.shuffle]
+mem_shuffle_predU [in Combi.LRrule.shuffle]
+mem_shape_vect_n_k [in Combi.Combi.vectNK]
+mem_enum_seqE [in Combi.LRrule.Greene]
+mem_RSclass [in Combi.LRrule.Schensted]
+mem_prefixesP [in Combi.Combi.Dyckword]
+mem_preim_partition [in Combi.Combi.Dyckword]
+mem_to_word [in Combi.Combi.tableau]
+mem_enum_ordtreesz [in Combi.Combi.ordtree]
+mem_pblock_setpart [in Combi.Combi.setpartition]
+mem_setpart_pblock [in Combi.Combi.setpartition]
+mem_eq_pblock [in Combi.Combi.setpartition]
+mem_pblockC [in Combi.Combi.setpartition]
+mem_enum_permuted [in Combi.Combi.permuted]
+mem_fiber [in Combi.Combi.fibered_set]
+mem_Delta [in Combi.SymGroup.presentSn]
+mem_partsum_gt [in Combi.Combi.composition]
+mem_partsum_non0 [in Combi.Combi.composition]
+merge_cons [in Combi.Combi.partition]
+merge_sortedE [in Combi.Combi.partition]
+merge_is_part [in Combi.Combi.partition]
+mesymlm_rbound [in Combi.MPoly.antisym]
+mesym_SchurE [in Combi.MPoly.Schur_mpoly]
+mfun2_simpl [in ALEA.Ccpo]
+mindropeqC [in Combi.Combi.skewpart]
+mindropeq_non0 [in Combi.Combi.skewpart]
+mindropeq_nthP [in Combi.Combi.skewpart]
+mindropeq_cons_neq [in Combi.Combi.skewpart]
+mindropeq_cons_eq [in Combi.Combi.skewpart]
+mindropeq_nil [in Combi.Combi.skewpart]
+mindropeq_leq [in Combi.Combi.skewpart]
+mindropeq_eq [in Combi.Combi.skewpart]
+mindropeq0 [in Combi.Combi.skewpart]
+minhE [in Combi.Combi.Dyckword]
+minhP [in Combi.Combi.Dyckword]
+minh_rrw [in Combi.Combi.Dyckword]
+minh_neg [in Combi.Combi.Dyckword]
+minSS [in Combi.SSRcomplements.tools]
+Minvl [in Combi.Basic.unitriginv]
+Minvr [in Combi.Basic.unitriginv]
+Minv_lincombr [in Combi.Basic.unitriginv]
+Minv_lincombl [in Combi.Basic.unitriginv]
+Minv_unitrig [in Combi.Basic.unitriginv]
+Minv_uni [in Combi.Basic.unitriginv]
+Minv_trig [in Combi.Basic.unitriginv]
+mk_isglb [in ALEA.Ccpo]
+mlead_antisym_sorted [in Combi.MPoly.antisym]
+Mlet_assoc [in ALEA.Qmeasure]
+Mlet_ext [in ALEA.Qmeasure]
+Mlet_unit [in ALEA.Qmeasure]
+Mlet_eq_compat [in ALEA.Qmeasure]
+MLet_simpl [in ALEA.Qmeasure]
+Mlet_le_compat [in ALEA.Qmeasure]
+Mlet_simpl_eq [in ALEA.Qmeasure]
+Mlet_simpl [in ALEA.Qmeasure]
+mlub_lift_left [in ALEA.Ccpo]
+mlub_lift_right [in ALEA.Ccpo]
+mlub_le [in ALEA.Ccpo]
+mlub_eq_compat [in ALEA.Ccpo]
+mlub_le_compat [in ALEA.Ccpo]
+mnm_perm [in Combi.MPoly.antisym]
+mnm_n0E [in Combi.MPoly.sympoly]
+MN_coeffE [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_homogP [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_recP [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_consE [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_notincl [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec_szE [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_homogP [in Combi.MPoly.MurnaghanNakayama]
+MN_coeffP [in Combi.MPoly.MurnaghanNakayama]
+MN_coeffP_int [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_consE [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff0 [in Combi.MPoly.MurnaghanNakayama]
+monotonic_sym [in ALEA.Ccpo]
+monotonic_intro [in ALEA.Ccpo]
+monotonic2_sym [in ALEA.Ccpo]
+monsYK [in Combi.MPoly.Cauchy]
+monsY_bij [in Combi.MPoly.Cauchy]
+mons2mE [in Combi.MPoly.Schur_mpoly]
+mon_ord_equiv_simpl [in ALEA.Ccpo]
+mon_fun_eq [in ALEA.Ccpo]
+mon_fun_eq_monotonic [in ALEA.Ccpo]
+mon_eq_compat [in ALEA.Ccpo]
+mon_le_compat [in ALEA.Ccpo]
+mon_simpl [in ALEA.Ccpo]
+mon2_fun2 [in ALEA.Ccpo]
+mon2_le_compat [in ALEA.Ccpo]
+mon2_simpl [in ALEA.Ccpo]
+mon2_elim2 [in ALEA.Ccpo]
+mon2_elim1 [in ALEA.Ccpo]
+morph_eltr [in Combi.SymGroup.presentSn]
+mpartE [in Combi.MPoly.antisym]
+mpartK [in Combi.MPoly.antisym]
+mpartS [in Combi.MPoly.Schur_altdef]
+mpart_partm_perm [in Combi.MPoly.antisym]
+mpart_is_dominant [in Combi.MPoly.antisym]
+mpart0 [in Combi.MPoly.antisym]
+mseq_lift_right_left [in ALEA.Ccpo]
+mseq_lift_right_le_compat [in ALEA.Ccpo]
+mseq_lift_right_simpl [in ALEA.Ccpo]
+mseq_lift_left_le_compat [in ALEA.Ccpo]
+mseq_lift_left_simpl [in ALEA.Ccpo]
+mshift_continuous2 [in ALEA.Ccpo]
+mshift_le_compat [in ALEA.Ccpo]
+mshift_simpl [in ALEA.Ccpo]
+mshift2_eq [in ALEA.Ccpo]
+Mstable_linear [in ALEA.Qmeasure]
+Mstable_mull [in ALEA.Qmeasure]
+Mstable_divi [in ALEA.Qmeasure]
+Mstable_muli [in ALEA.Qmeasure]
+Mstable_addi [in ALEA.Qmeasure]
+Mstable_divn [in ALEA.Qmeasure]
+Mstable_subn [in ALEA.Qmeasure]
+Mstable_addn [in ALEA.Qmeasure]
+Mstable_add [in ALEA.Qmeasure]
+Mstable_opp [in ALEA.Qmeasure]
+Mstable_eq [in ALEA.Qmeasure]
+Mstable0 [in ALEA.Qmeasure]
+msuppX1 [in Combi.MPoly.antisym]
+msym_pihomog [in Combi.MPoly.homogsym]
+msym_map_mpoly [in Combi.MPoly.antisym]
+msym_fundamental_symh_un [in Combi.MPoly.sympoly]
+multinomialE [in Combi.Combi.multinomial]
+multinomial_filter_neq0 [in Combi.Combi.multinomial]
+multinomial_cat [in Combi.Combi.multinomial]
+multinomial_factd [in Combi.Combi.multinomial]
+multinomial_nseq1 [in Combi.Combi.multinomial]
+multinomial_nseq [in Combi.Combi.multinomial]
+multinomial_fact [in Combi.Combi.multinomial]
+multinomial0 [in Combi.Combi.multinomial]
+multinomial1 [in Combi.Combi.multinomial]
+multinomial2 [in Combi.Combi.multinomial]
+mult_altern_pmap [in Combi.MPoly.MurnaghanNakayama]
+mult_altern_oapp [in Combi.MPoly.MurnaghanNakayama]
+mult_altern_symp_pol [in Combi.MPoly.MurnaghanNakayama]
+mult_syme_U [in Combi.MPoly.sympoly]
+mult_symh_powersum [in Combi.MPoly.sympoly]
+mult_symh_U [in Combi.MPoly.sympoly]
+mul_ek_pk [in Combi.MPoly.sympoly]
+mul_ek_p1 [in Combi.MPoly.sympoly]
+Munit_eq_compat [in ALEA.Qmeasure]
+Munit_simpl_eq [in ALEA.Qmeasure]
+Munit_simpl [in ALEA.Qmeasure]
+Murnaghan_NakayamaCT [in Combi.SymGroup.Frobenius_char]
+Murnaghan_Nakayama_char [in Combi.SymGroup.Frobenius_char]
+mu_pos_cond [in ALEA.Qmeasure]
+mu_bool_cond [in ALEA.Qmeasure]
+mu_in_seq [in ALEA.Qmeasure]
+mu_stable_sum [in ALEA.Qmeasure]
+mu_random_sum [in ALEA.Qmeasure]
+mu_uniform_sum [in ALEA.Qmeasure]
+mu_bool_negb1 [in ALEA.Qmeasure]
+mu_bool_negb [in ALEA.Qmeasure]
+mu_bool_negb0 [in ALEA.Qmeasure]
+mu_bool_impl1 [in ALEA.Qmeasure]
+mu_bool_impl [in ALEA.Qmeasure]
+mu_bool_0le [in ALEA.Qmeasure]
+mu_bool_le1 [in ALEA.Qmeasure]
+mu_eq_compat [in ALEA.Qmeasure]
+mu_le_compat [in ALEA.Qmeasure]
+mu_stable_inv_inv [in ALEA.Qmeasure]
+mu_stable_mulr [in ALEA.Qmeasure]
+mu_cte [in ALEA.Qmeasure]
+mu_stable_le1 [in ALEA.Qmeasure]
+mu_stable_pos [in ALEA.Qmeasure]
+mu_add_zero [in ALEA.Qmeasure]
+mu_stable_mull [in ALEA.Qmeasure]
+mu_stable_add [in ALEA.Qmeasure]
+mu_stable_inv [in ALEA.Qmeasure]
+mu_one_eq [in ALEA.Qmeasure]
+mu_zero_eq [in ALEA.Qmeasure]
+mu_zero [in ALEA.Qmeasure]
+mu_stable_eq [in ALEA.Qmeasure]
+mu_monotonic [in ALEA.Qmeasure]
+mu_walk_to_corner_is_trace [in Combi.HookFormula.hook]
+mwmwgt_homogP [in Combi.MPoly.sympoly]
+mxvec_indexK [in Combi.MPoly.Cauchy]
+

N

+nat_monotonic_inv [in ALEA.Ccpo]
+nat_monotonic [in ALEA.Ccpo]
+nat_mx_repr [in Combi.SymGroup.reprSn]
+nat_compare_specT [in ALEA.Misc]
+nbump_bumprowE [in Combi.LRrule.Schensted]
+nbump_ins_rconsE [in Combi.LRrule.Schensted]
+nbump_size_ins [in Combi.LRrule.Schensted]
+nbump_inspos_eq_size [in Combi.LRrule.Schensted]
+nbump_insposE [in Combi.LRrule.Schensted]
+nbump_mininspredE [in Combi.LRrule.Schensted]
+nbump_bumprow_rconsE [in Combi.LRrule.plactic]
+ncfuniCT_Ind [in Combi.SymGroup.towerSn]
+ncfuniCT_gen [in Combi.SymGroup.towerSn]
+neig4boxE [in Combi.Combi.skewpart]
+neig4box_sym [in Combi.Combi.skewpart]
+neig4_sym [in Combi.Combi.skewpart]
+Neq_lt_0 [in ALEA.Misc]
+neq0zcard [in Combi.SymGroup.permcent]
+neq0zcoeff [in Combi.SymGroup.towerSn]
+Newton_syme [in Combi.MPoly.sympoly]
+Newton_syme1 [in Combi.MPoly.sympoly]
+Newton_symh1 [in Combi.MPoly.sympoly]
+Newton_symh [in Combi.MPoly.sympoly]
+NirrSn [in Combi.SymGroup.reprSn]
+NirrS2 [in Combi.SymGroup.reprSn]
+Nlt0_le1 [in ALEA.Misc]
+nohasincr_setdiff [in Combi.MPoly.Schur_altdef]
+non_decr_equiv [in Combi.SSRcomplements.sorted]
+NoSetContainingBoth.extract_swap_set [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.Hcast [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.ksupp_Q [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.size_cover_Q [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_set_inj [in Combi.LRrule.Greene_inv]
+notbump [in Combi.LRrule.Schensted]
+notin0_part [in Combi.Combi.partition]
+not_hasincr_part [in Combi.MPoly.Schur_altdef]
+not_and_elim_right [in ALEA.Misc]
+not_and_elim_left [in ALEA.Misc]
+Npos [in ALEA.Misc]
+Nsucc_pred_pos [in ALEA.Misc]
+nth_sizeu2 [in Combi.Combi.std]
+nth_sizeu1 [in Combi.Combi.std]
+nth_sizeu [in Combi.Combi.std]
+nth_transp [in Combi.Combi.std]
+nth_set_nth_any [in Combi.SSRcomplements.tools]
+nth_set_nth_expand [in Combi.SSRcomplements.tools]
+nth_diff_shape [in Combi.Combi.partition]
+nth_pad [in Combi.Combi.partition]
+nth_conjE [in Combi.Combi.partition]
+nth_incr_first_n [in Combi.Combi.partition]
+nth_decr_nth_neq [in Combi.Combi.partition]
+nth_decr_nth [in Combi.Combi.partition]
+nth_rem_trail0 [in Combi.Combi.partition]
+nth_part_non0 [in Combi.Combi.partition]
+nth_std_pos [in Combi.LRrule.stdplact]
+nth_vctmin [in Combi.Combi.bintree]
+nth_add_setdiff [in Combi.MPoly.Schur_altdef]
+nth_lt_inspos [in Combi.LRrule.Schensted]
+nth_inspos_ins [in Combi.LRrule.Schensted]
+nth_pfminh [in Combi.Combi.Dyckword]
+nth_evalseq [in Combi.Combi.Yamanouchi]
+nth_add_ribbon_stop_lt [in Combi.Combi.skewpart]
+nth_add_ribbon_in [in Combi.Combi.skewpart]
+nth_add_ribbon_start [in Combi.Combi.skewpart]
+nth_add_ribbon_lt_start [in Combi.Combi.skewpart]
+nth_inspos [in Combi.Basic.ordtype]
+nth_rembig [in Combi.Basic.ordtype]
+nth_lt_posbig [in Combi.Basic.ordtype]
+nth_posbig [in Combi.Basic.ordtype]
+N2Nat_inj_pos [in ALEA.Misc]
+N2Nat_inj_le [in ALEA.Misc]
+N2Nat_inj_lt [in ALEA.Misc]
+N2Nat_le_mono [in ALEA.Misc]
+N2Nat_lt_mono [in ALEA.Misc]
+

O

+odd_cycle_type [in Combi.SymGroup.cycletype]
+odd_size_permE [in Combi.SymGroup.presentSn]
+odd_eltr [in Combi.SymGroup.presentSn]
+Oeq_trans [in ALEA.Ccpo]
+Oeq_le_sym [in ALEA.Ccpo]
+Oeq_le [in ALEA.Ccpo]
+Oeq_sym [in ALEA.Ccpo]
+Oeq_refl_eq [in ALEA.Ccpo]
+Oeq_refl [in ALEA.Ccpo]
+Ole_not_lt [in ALEA.Ccpo]
+Ole_lt_trans [in ALEA.Ccpo]
+Ole_notle_lt [in ALEA.Ccpo]
+Ole_diff_lt [in ALEA.Ccpo]
+Ole_eq_left [in ALEA.Ccpo]
+Ole_eq_right [in ALEA.Ccpo]
+Ole_eq_compat [in ALEA.Ccpo]
+Ole_antisym [in ALEA.Ccpo]
+Ole_refl [in ALEA.Ccpo]
+Ole_trans [in ALEA.Ccpo]
+Ole_refl_eq_inv [in ALEA.Ccpo]
+Ole_refl_eq [in ALEA.Ccpo]
+Olt_le_trans [in ALEA.Ccpo]
+Olt_antirefl [in ALEA.Ccpo]
+Olt_trans [in ALEA.Ccpo]
+Olt_notle [in ALEA.Ccpo]
+Olt_le [in ALEA.Ccpo]
+Olt_neq_rev [in ALEA.Ccpo]
+Olt_neq [in ALEA.Ccpo]
+Olt_eq_compat [in ALEA.Ccpo]
+omegahomsym_rmorph [in Combi.MPoly.homogsym]
+omegahomsym_is_linear [in Combi.MPoly.homogsym]
+omegahomsym_subproof [in Combi.MPoly.homogsym]
+omegasfK [in Combi.MPoly.sympoly]
+omegasf_syms [in Combi.MPoly.sympoly]
+omegasf_prodsymp [in Combi.MPoly.sympoly]
+omegasf_prodsymh [in Combi.MPoly.sympoly]
+omegasf_prodsyme [in Combi.MPoly.sympoly]
+omegasf_compsymh [in Combi.MPoly.sympoly]
+omegasf_sympolyf_eval [in Combi.MPoly.sympoly]
+omegasf_homogE [in Combi.MPoly.sympoly]
+omegasf_homog [in Combi.MPoly.sympoly]
+omegasf_symp [in Combi.MPoly.sympoly]
+omegasf_symh [in Combi.MPoly.sympoly]
+omegasf_syme [in Combi.MPoly.sympoly]
+omegasf_is_monoid_morphism [in Combi.MPoly.sympoly]
+omegasf_is_linear [in Combi.MPoly.sympoly]
+omegasf_is_symmetric [in Combi.MPoly.sympoly]
+omega_homsymp [in Combi.MPoly.homogsym]
+omega_homsyms [in Combi.MPoly.homogsym]
+omega_homsyme [in Combi.MPoly.homogsym]
+omega_homsymh [in Combi.MPoly.homogsym]
+omega_Fchar [in Combi.SymGroup.Frobenius_char]
+omega_Fchar_inv [in Combi.SymGroup.Frobenius_char]
+one_letter_included [in Combi.LRrule.implem]
+one_letter_choicesP [in Combi.LRrule.implem]
+orc_intro [in ALEA.Misc]
+orc_right [in ALEA.Misc]
+orc_left [in ALEA.Misc]
+order_cyclic [in Combi.SymGroup.cycles]
+OrdNode_inj [in Combi.Combi.ordtree]
+OrdSetoid [in ALEA.Ccpo]
+ordtreeszP [in Combi.Combi.ordtree]
+ord_to_bintreeK [in Combi.Combi.ordtree]
+ord0_in_map_liftF [in Combi.LRrule.Greene]
+outer_shape_pad0 [in Combi.Combi.partition]
+outer_shapeK [in Combi.Combi.partition]
+outputSpecP [in Combi.LRrule.implem]
+outputSpec_count_mem [in Combi.LRrule.implem]
+out_perm_prod [in Combi.SymGroup.cycles]
+

P

+partdomP [in Combi.Combi.partition]
+partdomshP [in Combi.Combi.partition]
+partdomsh_merge [in Combi.Combi.partition]
+partdomsh_merge1 [in Combi.Combi.partition]
+partdomsh_cons2E [in Combi.Combi.partition]
+partdomsh_cons2 [in Combi.Combi.partition]
+partdomsh_add [in Combi.Combi.partition]
+partdom_union_intpart [in Combi.Combi.partition]
+partdom_union_intpartr [in Combi.Combi.partition]
+partdom_union_intpartl [in Combi.Combi.partition]
+partdom_consK [in Combi.Combi.partition]
+partdom_anti [in Combi.Combi.partition]
+partdom_trans [in Combi.Combi.partition]
+partdom_refl [in Combi.Combi.partition]
+partdom_nil [in Combi.Combi.partition]
+partition_psupport [in Combi.SymGroup.cycles]
+partition_porbits [in Combi.SymGroup.cycles]
+partmE [in Combi.MPoly.antisym]
+partmK [in Combi.MPoly.antisym]
+partmP [in Combi.MPoly.antisym]
+partm_permK [in Combi.MPoly.antisym]
+partnCTE [in Combi.SymGroup.cycletype]
+partnCTK [in Combi.SymGroup.cycletype]
+partnCT_congr [in Combi.SymGroup.cycletype]
+partsums_cons [in Combi.Combi.composition]
+partsums_cat [in Combi.Combi.composition]
+part_rem_corner_ind [in Combi.Combi.partition]
+part_nseq1P [in Combi.Combi.partition]
+part_includedP [in Combi.Combi.partition]
+part_sumn_rectangle [in Combi.Combi.partition]
+part_eqP [in Combi.Combi.partition]
+part_leq_head [in Combi.Combi.partition]
+part_head_non0 [in Combi.Combi.partition]
+part_head0F [in Combi.Combi.partition]
+part_yam_of_stdtab [in Combi.Combi.stdtab]
+part_rcons_ind [in Combi.Combi.Yamanouchi]
+part_set1_eq [in Combi.Combi.setpartition]
+part_ordinal1 [in Combi.Combi.setpartition]
+part_ordinal0 [in Combi.Combi.setpartition]
+part_sumn_count [in Combi.MPoly.sympoly]
+part_of_comp_subproof [in Combi.Combi.composition]
+part_is_comp [in Combi.Combi.composition]
+part0 [in Combi.Combi.partition]
+path_braidred_catl [in Combi.SymGroup.presentSn]
+pblock_trivsetpart [in Combi.Combi.setpartition]
+pblock_setpart1 [in Combi.Combi.setpartition]
+pblock_notin [in Combi.Combi.setpartition]
+pblock_in [in Combi.Combi.setpartition]
+pchar0_algC [in Combi.MPoly.Cauchy]
+pchar0_rat [in Combi.MPoly.Cauchy]
+pchar0_algC [in Combi.MPoly.homogsym]
+pchar0_algC [in Combi.SymGroup.Frobenius_char]
+pchar0_rat [in Combi.SymGroup.Frobenius_char]
+permCTP [in Combi.SymGroup.cycletype]
+permCT_colpartn [in Combi.SymGroup.cycletype]
+permCT_colpartn_card [in Combi.SymGroup.cycletype]
+permCT_exists [in Combi.SymGroup.cycletype]
+permcyclesC [in Combi.SymGroup.permcent]
+permcyclesK [in Combi.SymGroup.permcent]
+permcyclesM [in Combi.SymGroup.permcent]
+permcyclesP [in Combi.SymGroup.permcent]
+permcycles_inj [in Combi.SymGroup.permcent]
+permKP [in Combi.SSRcomplements.permcomp]
+PermLattice.Exports.bottom_perm [in Combi.SymGroup.weak_order]
+PermLattice.Exports.invset_join [in Combi.SymGroup.weak_order]
+PermLattice.Exports.perm_join_meetE [in Combi.SymGroup.weak_order]
+PermLattice.Exports.top_perm [in Combi.SymGroup.weak_order]
+PermLattice.infperm_is_meet [in Combi.SymGroup.weak_order]
+PermLattice.invset_supperm [in Combi.SymGroup.weak_order]
+PermLattice.suppermC [in Combi.SymGroup.weak_order]
+PermLattice.suppermP [in Combi.SymGroup.weak_order]
+PermLattice.suppermPl [in Combi.SymGroup.weak_order]
+PermLattice.suppermPr [in Combi.SymGroup.weak_order]
+PermLattice.supperm_is_join [in Combi.SymGroup.weak_order]
+permutedact_is_action [in Combi.Combi.permuted]
+permutedact_subproof [in Combi.Combi.permuted]
+permutedP [in Combi.Combi.permuted]
+permuted_action_trans [in Combi.Combi.permuted]
+perm_size_uniq [in Combi.Combi.std]
+perm_stdE [in Combi.Combi.std]
+perm_of_std [in Combi.Combi.std]
+perm_std [in Combi.Combi.std]
+perm_union_intpartn [in Combi.Combi.partition]
+perm_union_intpart [in Combi.Combi.partition]
+perm_enum_basis [in Combi.MPoly.Schur_mpoly]
+perm_smalln [in Combi.MPoly.antisym]
+perm_mpart_partm [in Combi.MPoly.antisym]
+perm_partm [in Combi.MPoly.antisym]
+perm_mpart [in Combi.MPoly.antisym]
+perm_sfilterleq [in Combi.LRrule.shuffle]
+perm_sfiltergtn [in Combi.LRrule.shuffle]
+perm_shiftn_std [in Combi.LRrule.shuffle]
+perm_shuffle [in Combi.LRrule.shuffle]
+perm_join_tab [in Combi.Combi.skewtab]
+perm_KostkaMon [in Combi.MPoly.Schur_altdef]
+perm_RS [in Combi.LRrule.Schensted]
+perm_tabword_of_tuple [in Combi.LRrule.freeSchur]
+perm_commword [in Combi.LRrule.freeSchur]
+perm_multinomial [in Combi.Combi.multinomial]
+perm_append_nth [in Combi.Combi.stdtab]
+perm_of_setpartE [in Combi.SymGroup.cycletype]
+perm_of_porbitE [in Combi.SymGroup.cycletype]
+perm_of_porbit_subproof [in Combi.SymGroup.cycletype]
+perm_decE [in Combi.SymGroup.cycles]
+perm_evalseq [in Combi.Combi.Yamanouchi]
+perm_display [in Combi.SymGroup.weak_order]
+perm_eq_permuted_tuple [in Combi.Combi.permuted]
+perm_invar_congr [in Combi.Basic.congr]
+perm_invar [in Combi.Basic.congr]
+perm_bound [in Combi.Basic.congr]
+perm_rembig [in Combi.Basic.ordtype]
+perm_allLtnE [in Combi.Basic.ordtype]
+perm_allLtn [in Combi.Basic.ordtype]
+perm_allLeqE [in Combi.Basic.ordtype]
+perm_allLeq [in Combi.Basic.ordtype]
+perm_on_prods_length [in Combi.SymGroup.presentSn]
+perm_on_prods_length_ord [in Combi.SymGroup.presentSn]
+perm_on_prods [in Combi.SymGroup.presentSn]
+perm_on_cocode_recP [in Combi.SymGroup.presentSn]
+perm_of_invsetK [in Combi.SymGroup.presentSn]
+perm_of_relP [in Combi.SymGroup.presentSn]
+pfminhE [in Combi.Combi.Dyckword]
+pfminhP [in Combi.Combi.Dyckword]
+pfminh_rrw [in Combi.Combi.Dyckword]
+pfminh_pos [in Combi.Combi.Dyckword]
+pfminh_min [in Combi.Combi.Dyckword]
+pfminh_size [in Combi.Combi.Dyckword]
+Pieri_colpartn [in Combi.LRrule.therule]
+Pieri_rowpartn [in Combi.LRrule.therule]
+pihomog_sym [in Combi.MPoly.homogsym]
+pihomog_mPo [in Combi.MPoly.sympoly]
+plactic_RS [in Combi.LRrule.Greene_inv]
+plactic_shapeRS [in Combi.LRrule.Greene_inv]
+plactic_shapeRS_row_proof [in Combi.LRrule.Greene_inv]
+plactic_filter_lt [in Combi.LRrule.plactic]
+plactic_filter_le [in Combi.LRrule.plactic]
+plactic_filter_gt [in Combi.LRrule.plactic]
+plactic_filter_ge [in Combi.LRrule.plactic]
+plactruleP [in Combi.LRrule.plactic]
+plactrule_homog [in Combi.LRrule.plactic]
+plactrule_sym [in Combi.LRrule.plactic]
+plact_changeUT [in Combi.LRrule.freeSchur]
+plact_changeUT_drop [in Combi.LRrule.freeSchur]
+plact_changeUT_take [in Combi.LRrule.freeSchur]
+plact_from_yam [in Combi.LRrule.Yam_plact]
+plact_map_in_incr [in Combi.LRrule.plactic]
+plact_from_dualE [in Combi.LRrule.plactic]
+plact_dualE [in Combi.LRrule.plactic]
+plact_from_revdual [in Combi.LRrule.plactic]
+plact_revdual [in Combi.LRrule.plactic]
+plact_uniq_revE [in Combi.LRrule.plactic]
+plact_uniq_rev [in Combi.LRrule.plactic]
+plact_col [in Combi.LRrule.plactic]
+plact_row [in Combi.LRrule.plactic]
+plact_homog [in Combi.LRrule.plactic]
+plact_is_congr [in Combi.LRrule.plactic]
+plact_trans [in Combi.LRrule.plactic]
+plact_ltrans [in Combi.LRrule.plactic]
+plact_sym [in Combi.LRrule.plactic]
+plact_refl [in Combi.LRrule.plactic]
+plact_equiv [in Combi.LRrule.plactic]
+plact1dual [in Combi.LRrule.plactic]
+plact1I [in Combi.LRrule.plactic]
+plact1idual [in Combi.LRrule.plactic]
+plact1iP [in Combi.LRrule.plactic]
+plact1i_homog [in Combi.LRrule.plactic]
+plact1P [in Combi.LRrule.plactic]
+plact1_ge [in Combi.LRrule.plactic]
+plact1_homog [in Combi.LRrule.plactic]
+plact2dual [in Combi.LRrule.plactic]
+plact2I [in Combi.LRrule.plactic]
+plact2idual [in Combi.LRrule.plactic]
+plact2iP [in Combi.LRrule.plactic]
+plact2i_homog [in Combi.LRrule.plactic]
+plact2P [in Combi.LRrule.plactic]
+plact2_ge [in Combi.LRrule.plactic]
+plact2_homog [in Combi.LRrule.plactic]
+polXY_scaleAr [in Combi.MPoly.Cauchy]
+polXY_scaleAl [in Combi.MPoly.Cauchy]
+polX_XY_is_linear [in Combi.MPoly.Cauchy]
+polX_XY_is_monoid_morphism [in Combi.MPoly.Cauchy]
+polX_XY_is_zmod_morphism [in Combi.MPoly.Cauchy]
+polyXY_scale [in Combi.MPoly.Cauchy]
+polyX_inj [in Combi.MPoly.antisym]
+polY_XY_is_linear [in Combi.MPoly.Cauchy]
+polY_XY_is_monoid_morphism [in Combi.MPoly.Cauchy]
+polY_XY_is_zmod_morphism [in Combi.MPoly.Cauchy]
+porbitgrpE [in Combi.SymGroup.permcent]
+porbitPb [in Combi.SymGroup.cycletype]
+porbits_tinj [in Combi.SymGroup.towerSn]
+porbits_tperm [in Combi.SymGroup.cycletype]
+porbits_perm_of_setpart [in Combi.SymGroup.cycletype]
+porbits_of_set [in Combi.SymGroup.cycletype]
+porbits_conjg [in Combi.SymGroup.cycletype]
+porbit_permcycles [in Combi.SymGroup.permcent]
+porbit_tinj_rshift [in Combi.SymGroup.towerSn]
+porbit_tinj_lshift [in Combi.SymGroup.towerSn]
+porbit_tpermR [in Combi.SymGroup.cycletype]
+porbit_tpermL [in Combi.SymGroup.cycletype]
+porbit_tpermD [in Combi.SymGroup.cycletype]
+porbit_set_of_set [in Combi.SymGroup.cycletype]
+porbit_conjg [in Combi.SymGroup.cycletype]
+porbit_cymap [in Combi.SymGroup.cycletype]
+porbit_cymapcan [in Combi.SymGroup.cycletype]
+porbit_set_of_disjoint [in Combi.SymGroup.cycles]
+porbit_set_restr [in Combi.SymGroup.cycles]
+porbit_restr_perm [in Combi.SymGroup.cycles]
+porbit_set_astabs [in Combi.SymGroup.cycles]
+porbit_set_eq0 [in Combi.SymGroup.cycles]
+porbit_mod [in Combi.SymGroup.cycles]
+porbit_fix [in Combi.SymGroup.cycles]
+posbigE [in Combi.Basic.ordtype]
+posbig_invseq [in Combi.LRrule.stdplact]
+posbig_take_dropE [in Combi.Basic.ordtype]
+posbig_size [in Combi.Basic.ordtype]
+posbig_size_cons [in Combi.Basic.ordtype]
+pqpair_inj [in Combi.LRrule.Schensted]
+predi_eltrpE [in Combi.MPoly.antisym]
+predi_eltrp [in Combi.MPoly.antisym]
+predLR_bij_LRsupport [in Combi.LRrule.freeSchur]
+pred_LRtriple_conj [in Combi.LRrule.shuffle]
+pred_LRtriple_fast_filter_gt [in Combi.LRrule.shuffle]
+pred_LRtriple_fast_bijLRyam [in Combi.LRrule.therule]
+pred0_std [in Combi.LRrule.shuffle]
+preimset_trivIset [in Combi.SSRcomplements.tools]
+preim_Dyck_of_balE [in Combi.Combi.Dyckword]
+presentation_S4 [in Combi.SymGroup.presentSn]
+presentation_S3 [in Combi.SymGroup.presentSn]
+presentation_S2 [in Combi.SymGroup.presentSn]
+presentation_Sn_eltr [in Combi.SymGroup.presentSn]
+prob_choose_corner_ends_at [in Combi.HookFormula.hook]
+prodsK [in Combi.SymGroup.presentSn]
+prodsV [in Combi.SymGroup.presentSn]
+prods_straighten [in Combi.SymGroup.presentSn]
+prods_reducesE [in Combi.SymGroup.presentSn]
+prods_wordcd_inj [in Combi.SymGroup.presentSn]
+prods_codesz_bij [in Combi.SymGroup.presentSn]
+prods_iota_ltmi [in Combi.SymGroup.presentSn]
+prods_iota_mi [in Combi.SymGroup.presentSn]
+prod_sympXY [in Combi.MPoly.Cauchy]
+prod_conjg [in Combi.SymGroup.towerSn]
+prod_hook_length_quot_row [in Combi.HookFormula.hook]
+prod_of_disjoint [in Combi.SymGroup.cycles]
+prod_homog [in Combi.MPoly.sympoly]
+prod_prodgen [in Combi.MPoly.sympoly]
+prod_gen_cast [in Combi.MPoly.sympoly]
+prod_gen_colpartn [in Combi.MPoly.sympoly]
+prod_genM [in Combi.MPoly.sympoly]
+prod_gen0 [in Combi.MPoly.sympoly]
+prod_gen_homog [in Combi.MPoly.sympoly]
+psupports_perm_of_porbit [in Combi.SymGroup.cycletype]
+psupport_perm_of_porbit [in Combi.SymGroup.cycletype]
+psupport_conjg [in Combi.SymGroup.cycletype]
+psupport_of_disjoint [in Combi.SymGroup.cycles]
+psupport_disjointC [in Combi.SymGroup.cycles]
+psupport_cycle_dec [in Combi.SymGroup.cycles]
+psupport_restr_perm [in Combi.SymGroup.cycles]
+psupport_restr_perm_incl [in Combi.SymGroup.cycles]
+psupport_card_porbit [in Combi.SymGroup.cycles]
+psupport_stable [in Combi.SymGroup.cycles]
+psupport_eq0 [in Combi.SymGroup.cycles]
+psupport_perm_on [in Combi.SymGroup.cycles]
+psupport_expg [in Combi.SymGroup.cycles]
+psupport1 [in Combi.SymGroup.cycles]
+pyampair_inj [in Combi.LRrule.Schensted]
+

R

+Random_in_range [in ALEA.Qmeasure]
+Random_eq_out [in ALEA.Qmeasure]
+Random_eq_in [in ALEA.Qmeasure]
+Random_simpl [in ALEA.Qmeasure]
+rcast_com [in Combi.LRrule.Greene]
+rconsK [in Combi.SSRcomplements.tools]
+rcons_nilF [in Combi.SSRcomplements.tools]
+rcons_set_nth [in Combi.SSRcomplements.tools]
+rcons_bal_of_Dyck [in Combi.Combi.Dyckword]
+rcons_rcons [in Combi.LRrule.plactic]
+rec_tree [in Combi.Combi.ordtree]
+reducedM [in Combi.SymGroup.presentSn]
+reducedP [in Combi.SymGroup.presentSn]
+reduced_braid [in Combi.SymGroup.presentSn]
+reduced_wcord [in Combi.SymGroup.presentSn]
+reduced_rconsK [in Combi.SymGroup.presentSn]
+reduced_consK [in Combi.SymGroup.presentSn]
+reduced_catl [in Combi.SymGroup.presentSn]
+reduced_catr [in Combi.SymGroup.presentSn]
+reduced_sprod_code [in Combi.SymGroup.presentSn]
+reduced_revE [in Combi.SymGroup.presentSn]
+reduced_rev [in Combi.SymGroup.presentSn]
+reduced_iiF [in Combi.SymGroup.presentSn]
+reduced_nil [in Combi.SymGroup.presentSn]
+reduceP [in Combi.SymGroup.presentSn]
+reducesP [in Combi.SymGroup.presentSn]
+reduces_catl [in Combi.SymGroup.presentSn]
+RefinementOrder.botEcompnref [in Combi.Combi.composition]
+RefinementOrder.compnref_rev [in Combi.Combi.composition]
+RefinementOrder.compnref_display [in Combi.Combi.composition]
+RefinementOrder.descset_colcompn [in Combi.Combi.composition]
+RefinementOrder.descset_rowcompn [in Combi.Combi.composition]
+RefinementOrder.descset_join [in Combi.Combi.composition]
+RefinementOrder.descset_meet [in Combi.Combi.composition]
+RefinementOrder.descset_mono [in Combi.Combi.composition]
+RefinementOrder.leEcompnref [in Combi.Combi.composition]
+RefinementOrder.topEcompnref [in Combi.Combi.composition]
+RefinmentOrder.Exports.is_finer_subpartP [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.join_finerE [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.join_finer_eq_in_S [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.setpart_topE [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.setpart_bottomE [in Combi.Combi.setpartition]
+RefinmentOrder.is_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_setpart_anti [in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_trans [in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_refl [in Combi.Combi.setpartition]
+RefinmentOrder.is_finer_pblockP [in Combi.Combi.setpartition]
+RefinmentOrder.join_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.join_finerC [in Combi.Combi.setpartition]
+RefinmentOrder.join_finer_equivalence [in Combi.Combi.setpartition]
+RefinmentOrder.le_join_finer [in Combi.Combi.setpartition]
+RefinmentOrder.le_meet_finer [in Combi.Combi.setpartition]
+RefinmentOrder.meet_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.meet_finerC [in Combi.Combi.setpartition]
+RefinmentOrder.meet_finer_subproof [in Combi.Combi.setpartition]
+RefinmentOrder.mem_meet_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.setpartfiner_display [in Combi.Combi.setpartition]
+RefinmentOrder.setpart_conn [in Combi.Combi.setpartition]
+RefinmentOrder.setpart1_bottom [in Combi.Combi.setpartition]
+RefinmentOrder.trivsetpart_top [in Combi.Combi.setpartition]
+rembigE [in Combi.Basic.ordtype]
+rembigP [in Combi.Basic.ordtype]
+rembig_ins_std [in Combi.Combi.std]
+rembig_RS [in Combi.LRrule.plactic]
+rembig_RS_last_big [in Combi.LRrule.plactic]
+rembig_plactcongr [in Combi.LRrule.plactic]
+rembig_plact [in Combi.LRrule.plactic]
+rembig_plact2i [in Combi.LRrule.plactic]
+rembig_plact2 [in Combi.LRrule.plactic]
+rembig_plact1i [in Combi.LRrule.plactic]
+rembig_plact1 [in Combi.LRrule.plactic]
+rembig_iota [in Combi.Basic.ordtype]
+rembig_uniq [in Combi.Basic.ordtype]
+rembig_subseq [in Combi.Basic.ordtype]
+rembig_rev_uniq [in Combi.Basic.ordtype]
+rembig_eq_permR [in Combi.Basic.ordtype]
+rembig_eq_permL [in Combi.Basic.ordtype]
+rembig_cat [in Combi.Basic.ordtype]
+rembig_catL [in Combi.Basic.ordtype]
+rembig_catR [in Combi.Basic.ordtype]
+remnP [in Combi.Combi.stdtab]
+rem_corner_conj_part [in Combi.Combi.partition]
+rem_corner_incr_first_nE [in Combi.Combi.partition]
+rem_corner_incr_first_n [in Combi.Combi.partition]
+rem_corners_uniq [in Combi.Combi.partition]
+rem_corner_incr_nth [in Combi.Combi.partition]
+rem_irr [in Combi.SymGroup.Frobenius_char]
+rem_irr1 [in Combi.SymGroup.Frobenius_char]
+repr_S2 [in Combi.SymGroup.reprSn]
+repr1 [in Combi.SymGroup.reprSn]
+repr1_S1 [in Combi.SymGroup.reprSn]
+repr1_S0 [in Combi.SymGroup.reprSn]
+reshape_index_walk_to [in Combi.HookFormula.hook]
+restr_perm_porbits [in Combi.SymGroup.permcent]
+restr_perm_genC [in Combi.SymGroup.permcent]
+restr_perm_inj [in Combi.SymGroup.cycles]
+restr_perm_psupportE [in Combi.SymGroup.cycles]
+restr_perm_neq [in Combi.SymGroup.cycles]
+res_tab_inj [in Combi.MPoly.Schur_altdef]
+res_tabK [in Combi.MPoly.Schur_altdef]
+res_tabP [in Combi.MPoly.Schur_altdef]
+revdualK [in Combi.LRrule.plactic]
+revsetK [in Combi.LRrule.Greene]
+rev_is_ksupp_col [in Combi.LRrule.Greene_inv]
+rev_is_ksupp_row [in Combi.LRrule.Greene_inv]
+rev_enum [in Combi.LRrule.Greene_inv]
+rev_ksuppKV [in Combi.LRrule.Greene_inv]
+rev_ksuppK [in Combi.LRrule.Greene_inv]
+rev_set_inj [in Combi.LRrule.Greene_inv]
+rev_ord_cast_inj [in Combi.LRrule.Greene_inv]
+rev_intcompnK [in Combi.Combi.composition]
+rev_intcompn_spec [in Combi.Combi.composition]
+rewrite_path_min [in Combi.Basic.congr]
+rewrite_path_sym [in Combi.Basic.congr]
+rewrite_path_trans [in Combi.Basic.congr]
+rho_uniq [in Combi.MPoly.antisym]
+rho_iota [in Combi.MPoly.antisym]
+ribbonE [in Combi.Combi.skewpart]
+ribbonP [in Combi.Combi.skewpart]
+ribbontb_start_subproof [in Combi.Combi.skewpart]
+ribbontb_stop_ltn [in Combi.Combi.skewpart]
+ribbon_textbookE [in Combi.Combi.skewpart]
+ribbon_textbook_on [in Combi.Combi.skewpart]
+ribbon_on_no_square [in Combi.Combi.skewpart]
+ribbon_on_conn4_skew [in Combi.Combi.skewpart]
+ribbon_on_conn4 [in Combi.Combi.skewpart]
+ribbon_on_conn4_box_ex [in Combi.Combi.skewpart]
+ribbon_on_box_exP [in Combi.Combi.skewpart]
+ribbon_on_box_exE [in Combi.Combi.skewpart]
+ribbon_addE [in Combi.Combi.skewpart]
+ribbon_on_addE [in Combi.Combi.skewpart]
+ribbon_startrem [in Combi.Combi.skewpart]
+ribbon_on_startrem [in Combi.Combi.skewpart]
+ribbon_on_startrem_acc [in Combi.Combi.skewpart]
+ribbon_on0_startrem [in Combi.Combi.skewpart]
+ribbon_sumn_diffE [in Combi.Combi.skewpart]
+ribbon_sumn_lt [in Combi.Combi.skewpart]
+ribbon_included [in Combi.Combi.skewpart]
+ribbon_from_included [in Combi.Combi.skewpart]
+ribbon_mindropeq [in Combi.Combi.skewpart]
+ribbon_from_mindropeq [in Combi.Combi.skewpart]
+ribbon_fromP [in Combi.Combi.skewpart]
+ribbon_on_inj [in Combi.Combi.skewpart]
+ribbon_on_sumn [in Combi.Combi.skewpart]
+ribbon_on_startE [in Combi.Combi.skewpart]
+ribbon_on_stopE [in Combi.Combi.skewpart]
+ribbon_on_mindropeq [in Combi.Combi.skewpart]
+ribbon_on_height [in Combi.Combi.skewpart]
+ribbon_on_stop_lt [in Combi.Combi.skewpart]
+ribbon_on_start_le [in Combi.Combi.skewpart]
+ribbon_on_included [in Combi.Combi.skewpart]
+ribbon_on_is_skew [in Combi.Combi.skewpart]
+ribbon_on_nth_leq [in Combi.Combi.skewpart]
+ribbon_on_start_stop [in Combi.Combi.skewpart]
+ribbon_onSS [in Combi.Combi.skewpart]
+ribbon_noneq [in Combi.Combi.skewpart]
+ribbon_from_noneq [in Combi.Combi.skewpart]
+ribbon_consK [in Combi.Combi.skewpart]
+ribbon_from_impl [in Combi.Combi.skewpart]
+rightcomb_rotationsE [in Combi.Combi.bintree]
+rightcomb_rotations [in Combi.Combi.bintree]
+right_sizesK [in Combi.Combi.bintree]
+right_sizesP [in Combi.Combi.bintree]
+right_sizes_cat_left [in Combi.Combi.bintree]
+right_sizes_from_left [in Combi.Combi.bintree]
+right_sizes_left_comb [in Combi.Combi.bintree]
+rotationP [in Combi.Combi.bintree]
+rotations_right_sizesP [in Combi.Combi.bintree]
+rotations_add_bounded [in Combi.Combi.bintree]
+rotations_add [in Combi.Combi.bintree]
+rotations_add_head [in Combi.Combi.bintree]
+rotations_vctleq_impl [in Combi.Combi.bintree]
+rotations_neq [in Combi.Combi.bintree]
+rotations_flip [in Combi.Combi.bintree]
+rotations_flip_impl [in Combi.Combi.bintree]
+rotations_right_sub [in Combi.Combi.bintree]
+rotations_left_sub [in Combi.Combi.bintree]
+rot_is_Dyck [in Combi.Combi.Dyckword]
+rot_pfminhE [in Combi.Combi.Dyckword]
+rowcompnP [in Combi.Combi.composition]
+rowpartnE [in Combi.Combi.partition]
+rowpartnSE [in Combi.Combi.partition]
+rowpartn_subproof [in Combi.Combi.partition]
+rowpartn0E [in Combi.Combi.partition]
+row_hb_strip [in Combi.Combi.skewtab]
+row_dominate [in Combi.Combi.tableau]
+row_lt_by_pos [in Combi.Combi.tableau]
+row_free1 [in Combi.SymGroup.reprSn]
+rrw_bal1 [in Combi.Combi.Dyckword]
+RSclassE [in Combi.LRrule.Schensted]
+RSclassP [in Combi.LRrule.Schensted]
+RSclass_countE [in Combi.LRrule.Schensted]
+rshift_in_lshift_recF [in Combi.LRrule.Greene]
+rshift_recP [in Combi.LRrule.Greene]
+RSinvstdE [in Combi.LRrule.stdplact]
+RSmapE [in Combi.LRrule.Schensted]
+RSmapinv2K [in Combi.LRrule.Schensted]
+RSmapK [in Combi.LRrule.Schensted]
+RSmap_std [in Combi.LRrule.stdplact]
+RSmap_spec [in Combi.LRrule.Schensted]
+RSpairyamQ [in Combi.LRrule.freeSchur]
+RSperm [in Combi.LRrule.Schensted]
+rsplit_rec_tab [in Combi.LRrule.Greene]
+RSstdE [in Combi.LRrule.Schensted]
+RStabE [in Combi.LRrule.freeSchur]
+RStabinvK [in Combi.LRrule.Schensted]
+RStabK [in Combi.LRrule.Schensted]
+RStabmapE [in Combi.LRrule.Schensted]
+RSTabmapstdE [in Combi.LRrule.stdplact]
+RStabmap_std [in Combi.LRrule.stdplact]
+RStabmap_spec [in Combi.LRrule.Schensted]
+rsymrel_total [in Combi.SymGroup.presentSn]
+rsymrel_trans [in Combi.SymGroup.presentSn]
+rsymrel_anti [in Combi.SymGroup.presentSn]
+rsymrel_refl [in Combi.SymGroup.presentSn]
+rsym_invsetP [in Combi.SymGroup.presentSn]
+rsym_invset_total [in Combi.SymGroup.presentSn]
+rsym_invset_trans [in Combi.SymGroup.presentSn]
+rsym_invset_anti [in Combi.SymGroup.presentSn]
+rsym_invset_refl [in Combi.SymGroup.presentSn]
+RS_rev_uniq [in Combi.LRrule.Greene_inv]
+RS_tabE [in Combi.LRrule.Greene_inv]
+RS_yam [in Combi.LRrule.Yam_plact]
+RS_yam_RS [in Combi.LRrule.Yam_plact]
+rtransP [in Combi.Basic.congr]
+rtrans_min [in Combi.Basic.congr]
+rtrans_ind [in Combi.Basic.congr]
+rule_gencongr [in Combi.Basic.congr]
+rule_congrrule [in Combi.Basic.congr]
+rule_rtrans [in Combi.Basic.congr]
+

S

+same_connect_rev [in Combi.Combi.skewpart]
+scale_polXYE [in Combi.MPoly.Cauchy]
+scale_polXYDl [in Combi.MPoly.Cauchy]
+scale_polXYDr [in Combi.MPoly.Cauchy]
+scale_polXY1m [in Combi.MPoly.Cauchy]
+scale_polXYA [in Combi.MPoly.Cauchy]
+SchurE [in Combi.LRrule.freeSchur]
+Schur_rowpartn [in Combi.MPoly.Schur_mpoly]
+Schur_oversize [in Combi.MPoly.Schur_mpoly]
+Schur_tabsh_readingE [in Combi.MPoly.Schur_mpoly]
+Schur_homog [in Combi.MPoly.Schur_altdef]
+Schur_sym [in Combi.MPoly.Schur_altdef]
+Schur_sym_idomain [in Combi.MPoly.Schur_altdef]
+Schur_cast [in Combi.MPoly.Schur_altdef]
+Schur_freeSchurE [in Combi.LRrule.freeSchur]
+Schur0 [in Combi.MPoly.Schur_mpoly]
+Schur1 [in Combi.MPoly.Schur_mpoly]
+Schutzenberger_shuffle_plact [in Combi.LRrule.shuffle]
+Sch_max_size [in Combi.LRrule.Schensted]
+Sch_leq_last [in Combi.LRrule.Schensted]
+Sch_exists [in Combi.LRrule.Schensted]
+Sch_size [in Combi.LRrule.Schensted]
+Sch_rcons [in Combi.LRrule.Schensted]
+Sch_plact [in Combi.LRrule.plactic]
+seq_masks_uniq [in Combi.Combi.subseq]
+setactC [in Combi.SSRcomplements.permcomp]
+setactI [in Combi.SSRcomplements.permcomp]
+setactU [in Combi.SSRcomplements.permcomp]
+setactU1 [in Combi.SSRcomplements.permcomp]
+setact0 [in Combi.SSRcomplements.permcomp]
+setact1 [in Combi.SSRcomplements.permcomp]
+SetContainingBothLeft.coverS1T1 [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.cover_bin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.disjointS1T1 [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.enumUltV [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.exists_Qy [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.exists_Q_noboth [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_bT [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_Tb [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_cS [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_Sa [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_T1E [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_S1E [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_swap_setSE [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.extract_SE [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.inPQE [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ksupp_bin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ksupp_bnotin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posaS1_bin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posaT1_bin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posa_inTF [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posbinSF [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posb_inSF [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posc_inTF [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posc_subproof [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbin_noboth [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbnotin_noboth [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RabcGtnX [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RabcLeqX [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.size_cover_bnotin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_T1 [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_S1 [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_swap_set_S [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.sorted_extract_swap_set [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.ST_cover_disjoint [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.S1_subsST [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.tnth_posc [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.trivIset_Qbin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.TSneq [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.TS_disjoint [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.T1_subsST [in Combi.LRrule.Greene_inv]
+setpartP [in Combi.Combi.setpartition]
+setpart_shape_imset [in Combi.Combi.partition]
+setpart_shape_union [in Combi.Combi.partition]
+setpart_shapeP [in Combi.Combi.partition]
+setpart_set1_eq_set1 [in Combi.Combi.setpartition]
+setpart_set0_eq_set0 [in Combi.Combi.setpartition]
+setpart_eq [in Combi.Combi.setpartition]
+setpart_subset [in Combi.Combi.setpartition]
+setpart_non0 [in Combi.Combi.setpartition]
+setpart1_subproof [in Combi.Combi.setpartition]
+setU1E [in Combi.SSRcomplements.tools]
+set_nth_non_nil [in Combi.SSRcomplements.tools]
+set_nth_rcons [in Combi.SSRcomplements.tools]
+set_head_default [in Combi.SSRcomplements.tools]
+set_nil [in Combi.LRrule.Greene]
+set_nth_LxR [in Combi.LRrule.plactic]
+set1_disjoint [in Combi.SSRcomplements.tools]
+sfiltergtn_invstd [in Combi.LRrule.shuffle]
+sfiltergtn_is_std [in Combi.LRrule.shuffle]
+sfilterleqE [in Combi.LRrule.shuffle]
+sfilterleqK [in Combi.LRrule.shuffle]
+sfilterleq_invstd [in Combi.LRrule.shuffle]
+sfilterleq_is_std [in Combi.LRrule.shuffle]
+sfilterleq_LRsupport_skew [in Combi.LRrule.therule]
+sfilterleq_LRsupportP [in Combi.LRrule.therule]
+shaped_hyper_stdtabnP [in Combi.LRrule.freeSchur]
+shape_invseq [in Combi.LRrule.stdplact]
+shape_RS_std [in Combi.LRrule.stdplact]
+shape_RS_revdual [in Combi.LRrule.Greene_inv]
+shape_join_tab_skew_reshape [in Combi.Combi.skewtab]
+shape_join_tab [in Combi.Combi.skewtab]
+shape_inner_filter_le [in Combi.Combi.skewtab]
+shape_skew_reshape [in Combi.Combi.skewtab]
+shape_tabcols [in Combi.LRrule.Greene]
+shape_res_tab_subproof [in Combi.MPoly.Schur_altdef]
+shape_RStabmapE [in Combi.LRrule.Schensted]
+shape_instabnrowinv1 [in Combi.LRrule.Schensted]
+shape_RSmap_eq [in Combi.LRrule.Schensted]
+shape_instabnrow [in Combi.LRrule.Schensted]
+shape_dropRS [in Combi.LRrule.freeSchur]
+shape_takeRS [in Combi.LRrule.freeSchur]
+shape_bij_LRsupport [in Combi.LRrule.freeSchur]
+shape_hyper_stdtabnP [in Combi.LRrule.freeSchur]
+shape_RS_yam [in Combi.LRrule.Yam_plact]
+shape_yamtab [in Combi.LRrule.Yam_plact]
+shape_conj_tab [in Combi.Combi.stdtab]
+shape_deg_stdtabn_of_sh [in Combi.Combi.stdtab]
+shape_stdtabsh [in Combi.Combi.stdtab]
+shape_yam_of_stdtab [in Combi.Combi.stdtab]
+shape_stdtab_of_yam [in Combi.Combi.stdtab]
+shape_append_nth [in Combi.Combi.stdtab]
+shape_map_tab [in Combi.Combi.tableau]
+shape_tabsh [in Combi.Combi.tableau]
+shape_bijLR [in Combi.LRrule.therule]
+shape0 [in Combi.Combi.stdtab]
+shcol_cards [in Combi.LRrule.Greene]
+shiftinv_pos_homo [in Combi.Basic.ordtype]
+shiftinv_posK [in Combi.Basic.ordtype]
+shiftn_skew_dominate [in Combi.LRrule.therule]
+shiftuK [in Combi.LRrule.shuffle]
+shift_plactcongr [in Combi.LRrule.shuffle]
+shift_le_compat [in ALEA.Ccpo]
+shift_simpl [in ALEA.Ccpo]
+shift_pos_mono [in Combi.Basic.ordtype]
+shift_posK [in Combi.Basic.ordtype]
+shrows_cards [in Combi.LRrule.Greene]
+shsh_sfilterleq [in Combi.LRrule.shuffle]
+shsh_sfiltergtn [in Combi.LRrule.shuffle]
+shuffleC [in Combi.LRrule.shuffle]
+shuffle_nil [in Combi.LRrule.shuffle]
+signed_mx_repr [in Combi.SymGroup.reprSn]
+sign_Chi2 [in Combi.SymGroup.reprSn]
+sign_char2 [in Combi.SymGroup.reprSn]
+sign_charP [in Combi.SymGroup.reprSn]
+sign_char_subproof [in Combi.SymGroup.reprSn]
+sign_irr [in Combi.SymGroup.reprSn]
+sign_mx_repr [in Combi.SymGroup.reprSn]
+SimpleCalculation [in Combi.HookFormula.hook]
+sizeIk [in Combi.LRrule.Greene]
+size_invstd [in Combi.Combi.std]
+size_invseq [in Combi.Combi.std]
+size_leq_invseq [in Combi.Combi.std]
+size_all_leq [in Combi.Combi.std]
+size_std [in Combi.Combi.std]
+size_std_rec [in Combi.Combi.std]
+size_sdtn [in Combi.Combi.std]
+size_enum_partn [in Combi.Combi.partition]
+size_enum_partns [in Combi.Combi.partition]
+size_enum_partnsk [in Combi.Combi.partition]
+size_hookpartn [in Combi.Combi.partition]
+size_rowpartn [in Combi.Combi.partition]
+size_colpartn [in Combi.Combi.partition]
+size_diff_shape [in Combi.Combi.partition]
+size_included [in Combi.Combi.partition]
+size_conj_part [in Combi.Combi.partition]
+size_incr_first_n [in Combi.Combi.partition]
+size_rem_trail0 [in Combi.Combi.partition]
+size_part [in Combi.Combi.partition]
+size_vctmin [in Combi.Combi.bintree]
+size_from_vct [in Combi.Combi.bintree]
+size_from_vct_acc [in Combi.Combi.bintree]
+size_right_sizes [in Combi.Combi.bintree]
+size_rotations [in Combi.Combi.bintree]
+size_flip [in Combi.Combi.bintree]
+size_leftcomb [in Combi.Combi.bintree]
+size_rightcomb [in Combi.Combi.bintree]
+size_left_branch [in Combi.Combi.bintree]
+size_cat_left [in Combi.Combi.bintree]
+size_from_left [in Combi.Combi.bintree]
+size_mem_enum_bintreeszP [in Combi.Combi.bintree]
+size_enum_bintreeszE [in Combi.Combi.bintree]
+size_enum_bintreesz [in Combi.Combi.bintree]
+size_Catalan_bin_leq [in Combi.Combi.bintree]
+size_tree_eq0 [in Combi.Combi.bintree]
+size_cover_rev [in Combi.LRrule.Greene_inv]
+size_rev_ksupp [in Combi.LRrule.Greene_inv]
+size_partm [in Combi.MPoly.antisym]
+size_langQ [in Combi.LRrule.shuffle]
+size_sfilterleq_cat [in Combi.LRrule.shuffle]
+size_sfiltergtn_cat [in Combi.LRrule.shuffle]
+size_sfilterleq [in Combi.LRrule.shuffle]
+size_sfiltergtn [in Combi.LRrule.shuffle]
+size_shuffle [in Combi.LRrule.shuffle]
+size_cut_k [in Combi.Combi.vectNK]
+size_join_tab [in Combi.Combi.skewtab]
+size_skew_reshape [in Combi.Combi.skewtab]
+size_row_extract [in Combi.LRrule.Greene]
+size_cover_tabcolsk [in Combi.LRrule.Greene]
+size_cover_tabrows [in Combi.LRrule.Greene]
+size_tabcols_cons [in Combi.LRrule.Greene]
+size_to_word_cons [in Combi.LRrule.Greene]
+size_tabrows [in Combi.LRrule.Greene]
+size_shrows [in Combi.LRrule.Greene]
+size_tabcols [in Combi.LRrule.Greene]
+size_shcols [in Combi.LRrule.Greene]
+size_shcols_cons [in Combi.LRrule.Greene]
+size_cover_inj [in Combi.LRrule.Greene]
+size_extract [in Combi.LRrule.Greene]
+size_coverI [in Combi.LRrule.Greene]
+size_RS [in Combi.LRrule.Schensted]
+size_instab [in Combi.LRrule.Schensted]
+size_invins [in Combi.LRrule.Schensted]
+size_RSmap2 [in Combi.LRrule.Schensted]
+size_ndec_Sch [in Combi.LRrule.Schensted]
+size_ins_non_0 [in Combi.LRrule.Schensted]
+size_ins_sup [in Combi.LRrule.Schensted]
+size_ins_inf [in Combi.LRrule.Schensted]
+size_tracel [in Combi.HookFormula.hook]
+size_tracer [in Combi.HookFormula.hook]
+size_hook_box_indices [in Combi.HookFormula.hook]
+size_RSmapinv2_yam [in Combi.LRrule.freeSchur]
+size_RS_tuple [in Combi.LRrule.freeSchur]
+size_bal_of_Dyck [in Combi.Combi.Dyckword]
+size_Dyck_of_bal [in Combi.Combi.Dyckword]
+size_UDn [in Combi.Combi.Dyckword]
+size_UnDn [in Combi.Combi.Dyckword]
+size_bintree_of_Dyck [in Combi.Combi.Dyckword]
+size_Dyck_of_bintree [in Combi.Combi.Dyckword]
+size_foldr_join_Dyck [in Combi.Combi.Dyckword]
+size_count_braceE [in Combi.Combi.Dyckword]
+size_conj_tab [in Combi.Combi.stdtab]
+size_tab_stdtabn [in Combi.Combi.stdtab]
+size_yam_of_stdtab [in Combi.Combi.stdtab]
+size_notin_stdtab_of_yam [in Combi.Combi.stdtab]
+size_yam_of_stdtab_rec [in Combi.Combi.stdtab]
+size_tab_remn [in Combi.Combi.stdtab]
+size_stdtab_of_yam [in Combi.Combi.stdtab]
+size_append_nth [in Combi.Combi.stdtab]
+size_cycle_type [in Combi.SymGroup.cycletype]
+size_revdual [in Combi.LRrule.plactic]
+size_plact [in Combi.LRrule.plactic]
+size_tabsh [in Combi.Combi.tableau]
+size_to_word [in Combi.Combi.tableau]
+size_zip2 [in Combi.LRrule.therule]
+size_leq_skew_reshape [in Combi.LRrule.therule]
+size_mem_enum_ordtreeszP [in Combi.Combi.ordtree]
+size_ord_to_bintree [in Combi.Combi.ordtree]
+size_bin_to_ordtree [in Combi.Combi.ordtree]
+size_tree_eq1 [in Combi.Combi.ordtree]
+size_ordtree_pos [in Combi.Combi.ordtree]
+size_ordtreeE [in Combi.Combi.ordtree]
+size_yamn [in Combi.Combi.Yamanouchi]
+size_yameval [in Combi.Combi.Yamanouchi]
+size_hyper_yam [in Combi.Combi.Yamanouchi]
+size_add_ribbon [in Combi.Combi.skewpart]
+size_count_mem_undup [in Combi.Combi.permuted]
+size_permuted_seq [in Combi.Combi.permuted]
+size_permuted_tuple [in Combi.Combi.permuted]
+size_invar_congr [in Combi.Basic.congr]
+size_invar_refl [in Combi.Basic.congr]
+size_invar [in Combi.Basic.congr]
+size_bound [in Combi.Basic.congr]
+size_rembig [in Combi.Basic.ordtype]
+size_straighten [in Combi.SymGroup.presentSn]
+size_inscode [in Combi.SymGroup.presentSn]
+size_braid [in Combi.SymGroup.presentSn]
+size_canword [in Combi.SymGroup.presentSn]
+size_cocode [in Combi.SymGroup.presentSn]
+size_cocode_rec [in Combi.SymGroup.presentSn]
+size_codesz [in Combi.SymGroup.presentSn]
+size_wordcd [in Combi.SymGroup.presentSn]
+size_LRyamtab_listE [in Combi.LRrule.implem]
+size_mpart_in_supp [in Combi.MPoly.sympoly]
+size_partsums [in Combi.Combi.composition]
+size_comp [in Combi.Combi.composition]
+skew_reshapeK [in Combi.Combi.skewtab]
+skew_dominate_cut [in Combi.Combi.skewtab]
+skew_dominate_consl [in Combi.Combi.skewtab]
+skew_dominate_no_overlap [in Combi.Combi.skewtab]
+skew_dominate_take [in Combi.Combi.skewtab]
+skew_dominate0 [in Combi.Combi.skewtab]
+skew_yam_included [in Combi.Combi.skewtab]
+skew_yam_catK [in Combi.Combi.skewtab]
+skew_yam_consK [in Combi.Combi.skewtab]
+skew_yam_catrK [in Combi.Combi.skewtab]
+skew_yam_cat [in Combi.Combi.skewtab]
+skew_nil_yamE [in Combi.Combi.skewtab]
+skew_yam_nil [in Combi.Combi.skewtab]
+sorted_geq_nth0E [in Combi.Combi.partition]
+sorted_geq_count_leq2E [in Combi.Combi.partition]
+sorted_std_extract [in Combi.LRrule.stdplact]
+sorted_gt_tabcols [in Combi.LRrule.Greene]
+sorted_leqX_tabrows [in Combi.LRrule.Greene]
+sorted_extract_cond [in Combi.LRrule.Greene]
+sorted_leq_last [in Combi.HookFormula.hook]
+sorted_in_leq_last [in Combi.HookFormula.hook]
+sorted_center [in Combi.LRrule.plactic]
+sorted_subseq_iota_rcons [in Combi.Combi.subseq]
+sorted_subseq_inP [in Combi.Combi.subseq]
+sorted_subseqP [in Combi.Combi.subseq]
+sorted_sumn_iotaE [in Combi.SSRcomplements.sorted]
+sorted_ltn_ind [in Combi.SSRcomplements.sorted]
+sorted_lt_by_pos [in Combi.SSRcomplements.sorted]
+sorted_last [in Combi.SSRcomplements.sorted]
+sorted_cons [in Combi.SSRcomplements.sorted]
+sorted_strictP [in Combi.SSRcomplements.sorted]
+sorted_rcons [in Combi.SSRcomplements.sorted]
+sorted_rconsK [in Combi.SSRcomplements.sorted]
+sorted_consK [in Combi.SSRcomplements.sorted]
+sorted_is_part [in Combi.LRrule.implem]
+sorted_ltn_partsums [in Combi.Combi.composition]
+sorted2P [in Combi.SSRcomplements.sorted]
+splitsetK [in Combi.LRrule.Greene]
+split_rec_cover [in Combi.LRrule.Greene]
+split_recabK [in Combi.LRrule.Greene]
+srelE [in Combi.SymGroup.presentSn]
+stable_intro [in ALEA.Ccpo]
+stab_iporbitsE [in Combi.SymGroup.permcent]
+stab_iporbitsE_prod [in Combi.SymGroup.permcent]
+stab_iporbits_map_inj [in Combi.SymGroup.permcent]
+stab_iporbits_homog [in Combi.SymGroup.permcent]
+stab_iporbits_stab [in Combi.SymGroup.permcent]
+stab_porbit [in Combi.SymGroup.permcent]
+stab_tuple_dprod [in Combi.Combi.permuted]
+stab_tuple_prod [in Combi.Combi.permuted]
+startremE [in Combi.Combi.skewpart]
+startrem_accP [in Combi.Combi.skewpart]
+startrem_accE [in Combi.Combi.skewpart]
+startrem_leq [in Combi.Combi.skewpart]
+startrem_leq_size [in Combi.Combi.skewpart]
+startrem_leq_pos [in Combi.Combi.skewpart]
+startrem_acc_geq [in Combi.Combi.skewpart]
+startrem0P [in Combi.Combi.skewpart]
+star_stable_sub [in ALEA.Qmeasure]
+star_le_compat [in ALEA.Qmeasure]
+star_monotonic [in ALEA.Qmeasure]
+star_stable_eq [in ALEA.Qmeasure]
+star_simpl [in ALEA.Qmeasure]
+stdP [in Combi.Combi.std]
+stdtabnP [in Combi.Combi.stdtab]
+stdtabn_of_sh_subproof [in Combi.Combi.stdtab]
+stdtabP [in Combi.Combi.stdtab]
+stdtabshP [in Combi.Combi.stdtab]
+stdtabsh_eval_to_word [in Combi.MPoly.Schur_altdef]
+stdtab_get_tabNE [in Combi.Combi.stdtab]
+stdtab_of_yamK [in Combi.Combi.stdtab]
+stdtab_of_yam_inj [in Combi.Combi.stdtab]
+stdtab_of_yam_nil [in Combi.Combi.stdtab]
+stdtab_of_yamP [in Combi.Combi.stdtab]
+stdwordnP [in Combi.Combi.std]
+std_cutabc [in Combi.Combi.std]
+std_transp [in Combi.Combi.std]
+std_drop_std [in Combi.Combi.std]
+std_take_std [in Combi.Combi.std]
+std_rconsK [in Combi.Combi.std]
+std_eq_invP [in Combi.Combi.std]
+std_specP [in Combi.Combi.std]
+std_spec_uniq [in Combi.Combi.std]
+std_eq_inv [in Combi.Combi.std]
+std_stdE [in Combi.Combi.std]
+std_std [in Combi.Combi.std]
+std_posbig [in Combi.Combi.std]
+std_rembig [in Combi.Combi.std]
+std_is_std [in Combi.Combi.std]
+std_max [in Combi.Combi.std]
+std_uniq [in Combi.Combi.std]
+std_perm [in Combi.Combi.std]
+std_rcons_shiftinv [in Combi.LRrule.stdplact]
+std_plact [in Combi.LRrule.stdplact]
+std_plact2 [in Combi.LRrule.stdplact]
+std_plact1 [in Combi.LRrule.stdplact]
+std_shsh [in Combi.LRrule.shuffle]
+std_of_yam [in Combi.Combi.stdtab]
+step_closed [in Combi.Basic.congr]
+step_mem [in Combi.Basic.congr]
+straighten_path_npos [in Combi.SymGroup.presentSn]
+subdescsetP [in Combi.Combi.composition]
+subdescset_partsumP [in Combi.Combi.composition]
+subenum_countE [in Combi.Basic.combclass]
+subseqsP [in Combi.Combi.subseq]
+subseqs_masks_uniq [in Combi.Combi.subseq]
+Subseqs_maskK [in Combi.Combi.subseq]
+subseq_take [in Combi.LRrule.Greene]
+subseq_rcons_neq [in Combi.Combi.subseq]
+subseq_rcons_eq [in Combi.Combi.subseq]
+subseq_partsumE [in Combi.Combi.composition]
+subset_imsetK [in Combi.SSRcomplements.tools]
+subset_abc [in Combi.LRrule.plactic]
+subset_set1 [in Combi.Combi.setpartition]
+subset_s_trans_s [in Combi.Basic.congr]
+subset_undup_step [in Combi.Basic.congr]
+subset_step [in Combi.Basic.congr]
+subType_unionE [in Combi.Basic.combclass]
+subType_seqP [in Combi.Basic.combclass]
+sub_enumE [in Combi.Basic.combclass]
+succ_neq0 [in ALEA.Qmeasure]
+sumndiff [in Combi.MPoly.Schur_altdef]
+sumnpSPE [in Combi.HookFormula.hook]
+sumn_sort [in Combi.SSRcomplements.tools]
+sumn_pred1_iota [in Combi.SSRcomplements.tools]
+sumn_nth_le [in Combi.SSRcomplements.tools]
+sumn_drop [in Combi.SSRcomplements.tools]
+sumn_take [in Combi.SSRcomplements.tools]
+sumn_mapE [in Combi.SSRcomplements.tools]
+sumn_map_condE [in Combi.SSRcomplements.tools]
+sumn_take_merge [in Combi.Combi.partition]
+sumn_union_part [in Combi.Combi.partition]
+sumn_intpartn [in Combi.Combi.partition]
+sumn_diff_shape [in Combi.Combi.partition]
+sumn_diff_shape_eq [in Combi.Combi.partition]
+sumn_included [in Combi.Combi.partition]
+sumn_take_inj [in Combi.Combi.partition]
+sumn_take_leq [in Combi.Combi.partition]
+sumn_conj_part [in Combi.Combi.partition]
+sumn_incr_first_n [in Combi.Combi.partition]
+sumn_decr_nth [in Combi.Combi.partition]
+sumn_incr_nth [in Combi.Combi.partition]
+sumn_rem_trail0 [in Combi.Combi.partition]
+sumn_right_sizes_gt [in Combi.Combi.bintree]
+sumn_partm [in Combi.MPoly.antisym]
+sumn_mpart [in Combi.MPoly.antisym]
+sumn_eval [in Combi.MPoly.Schur_altdef]
+sumn_shape_stdtabnE [in Combi.Combi.stdtab]
+sumn_diff_shape_intpartE [in Combi.LRrule.therule]
+sumn_add_ribbon [in Combi.Combi.skewpart]
+sumn_mapS [in Combi.Combi.skewpart]
+sumn_cocode [in Combi.SymGroup.presentSn]
+sum_take [in Combi.SSRcomplements.tools]
+sum_minn [in Combi.SSRcomplements.tools]
+sum_conj [in Combi.Combi.partition]
+sum_count_mem [in Combi.Basic.combclass]
+sum_syme_symh [in Combi.MPoly.sympoly]
+sum_symh_syme [in Combi.MPoly.sympoly]
+sum_symmE [in Combi.MPoly.sympoly]
+Swap.enum_cut [in Combi.LRrule.Greene_inv]
+Swap.pos0_subproof [in Combi.LRrule.Greene_inv]
+Swap.pos01F [in Combi.LRrule.Greene_inv]
+Swap.pos1_subproof [in Combi.LRrule.Greene_inv]
+Swap.size_cut_sizeu [in Combi.LRrule.Greene_inv]
+Swap.swapL [in Combi.LRrule.Greene_inv]
+Swap.swapR [in Combi.LRrule.Greene_inv]
+Swap.swap_size_cover [in Combi.LRrule.Greene_inv]
+Swap.swap_cover [in Combi.LRrule.Greene_inv]
+Swap.swap_set_inj [in Combi.LRrule.Greene_inv]
+Swap.swap_set_invol [in Combi.LRrule.Greene_inv]
+Swap.swap_inj [in Combi.LRrule.Greene_inv]
+Swap.swap_invol [in Combi.LRrule.Greene_inv]
+Swap.swap0 [in Combi.LRrule.Greene_inv]
+Swap.swap1 [in Combi.LRrule.Greene_inv]
+Swap.tnth_pos1 [in Combi.LRrule.Greene_inv]
+Swap.tnth_pos0 [in Combi.LRrule.Greene_inv]
+symbeE [in Combi.MPoly.homogsym]
+symbe_free [in Combi.MPoly.homogsym]
+symbe_basis [in Combi.MPoly.homogsym]
+symbhE [in Combi.MPoly.homogsym]
+symbh_free [in Combi.MPoly.homogsym]
+symbh_basis [in Combi.MPoly.homogsym]
+symbmE [in Combi.MPoly.homogsym]
+symbm_basis [in Combi.MPoly.homogsym]
+symbm_free [in Combi.MPoly.homogsym]
+symbpE [in Combi.MPoly.homogsym]
+symbp_free [in Combi.MPoly.homogsym]
+symbp_basis [in Combi.MPoly.homogsym]
+symbsE [in Combi.MPoly.homogsym]
+symbs_free [in Combi.MPoly.homogsym]
+symbs_basis [in Combi.MPoly.homogsym]
+symE [in Combi.MPoly.sympoly]
+syme_syms_partdom [in Combi.MPoly.sympoly]
+syme_syms [in Combi.MPoly.sympoly]
+syme_syms_partdom_int [in Combi.MPoly.sympoly]
+syme_syms_int [in Combi.MPoly.sympoly]
+syme_to_symh [in Combi.MPoly.sympoly]
+syme_symhE [in Combi.MPoly.sympoly]
+syme_cast [in Combi.MPoly.sympoly]
+syme_to_symm [in Combi.MPoly.sympoly]
+syme_homog [in Combi.MPoly.sympoly]
+syme_geqnE [in Combi.MPoly.sympoly]
+syme_sym [in Combi.MPoly.sympoly]
+syme0 [in Combi.MPoly.sympoly]
+syme1 [in Combi.MPoly.sympoly]
+symHE_intpartn [in Combi.MPoly.sympoly]
+symHE_intcompn [in Combi.MPoly.sympoly]
+symHE_prod_intcomp [in Combi.MPoly.sympoly]
+symHE_rec [in Combi.MPoly.sympoly]
+symhe1E [in Combi.MPoly.sympoly]
+symh_basisE [in Combi.MPoly.Schur_mpoly]
+symh_to_symp [in Combi.MPoly.sympoly]
+symh_to_symp_intpartn [in Combi.MPoly.sympoly]
+symh_to_symp_prod_partsum [in Combi.MPoly.sympoly]
+symh_syms_partdom [in Combi.MPoly.sympoly]
+symh_syms [in Combi.MPoly.sympoly]
+symh_syms_partdom_int [in Combi.MPoly.sympoly]
+symh_syms_int [in Combi.MPoly.sympoly]
+symh_to_syme [in Combi.MPoly.sympoly]
+symh_symeE [in Combi.MPoly.sympoly]
+symh_cast [in Combi.MPoly.sympoly]
+symh_to_symm [in Combi.MPoly.sympoly]
+symh_homog [in Combi.MPoly.sympoly]
+symh_sym [in Combi.MPoly.sympoly]
+symh_pol_any [in Combi.MPoly.sympoly]
+symh0 [in Combi.MPoly.sympoly]
+symmX [in Combi.MPoly.Cauchy]
+symm_syms_partdom [in Combi.MPoly.sympoly]
+symm_syms [in Combi.MPoly.sympoly]
+symm_syms_partdom_int [in Combi.MPoly.sympoly]
+symm_syms_int [in Combi.MPoly.sympoly]
+symm_cast [in Combi.MPoly.sympoly]
+symm_unique0 [in Combi.MPoly.sympoly]
+symm_unique [in Combi.MPoly.sympoly]
+symm_homog [in Combi.MPoly.sympoly]
+symm_oversize [in Combi.MPoly.sympoly]
+symm_sym [in Combi.MPoly.sympoly]
+symm0 [in Combi.MPoly.sympoly]
+sympe1E [in Combi.MPoly.sympoly]
+sympolP [in Combi.MPoly.sympoly]
+sympolyfK [in Combi.MPoly.sympoly]
+sympolyfP [in Combi.MPoly.sympoly]
+sympolyf_evalX [in Combi.MPoly.sympoly]
+sympolyf_evalK [in Combi.MPoly.sympoly]
+sympolyf_evalE [in Combi.MPoly.sympoly]
+sympolyf_is_monoid_morphism [in Combi.MPoly.sympoly]
+sympolyf_is_linear [in Combi.MPoly.sympoly]
+SymPolyHomogKey.homogsym1_key [in Combi.MPoly.homogsym]
+sympol_is_monoid_morphism [in Combi.MPoly.sympoly]
+sympol_is_linear [in Combi.MPoly.sympoly]
+sympol_inj [in Combi.MPoly.sympoly]
+sympXY [in Combi.MPoly.Cauchy]
+symp_cast [in Combi.MPoly.sympoly]
+symp_to_symm [in Combi.MPoly.sympoly]
+symp_homog [in Combi.MPoly.sympoly]
+symp_sym [in Combi.MPoly.sympoly]
+symp0 [in Combi.MPoly.sympoly]
+syms_prod_sympM [in Combi.MPoly.MurnaghanNakayama]
+syms_prod_sympM_int [in Combi.MPoly.MurnaghanNakayama]
+syms_sympM [in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_pmap [in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_oapp [in Combi.MPoly.MurnaghanNakayama]
+syms_sympM_oapp_int [in Combi.MPoly.MurnaghanNakayama]
+syms_syme_partdom [in Combi.MPoly.sympoly]
+syms_syme [in Combi.MPoly.sympoly]
+syms_symh_partdom [in Combi.MPoly.sympoly]
+syms_symh [in Combi.MPoly.sympoly]
+syms_syme_partdom_int [in Combi.MPoly.sympoly]
+syms_syme_int [in Combi.MPoly.sympoly]
+syms_symh_partdom_int [in Combi.MPoly.sympoly]
+syms_symh_int [in Combi.MPoly.sympoly]
+syms_symm_partdom [in Combi.MPoly.sympoly]
+syms_symm [in Combi.MPoly.sympoly]
+syms_symm_partdom_int [in Combi.MPoly.sympoly]
+syms_symm_int [in Combi.MPoly.sympoly]
+syms_symeM [in Combi.MPoly.sympoly]
+syms_symhM [in Combi.MPoly.sympoly]
+syms_symsM [in Combi.MPoly.sympoly]
+syms_cast [in Combi.MPoly.sympoly]
+syms_oversize [in Combi.MPoly.sympoly]
+syms_colpartn [in Combi.MPoly.sympoly]
+syms_rowpartn [in Combi.MPoly.sympoly]
+syms_homog [in Combi.MPoly.sympoly]
+syms0 [in Combi.MPoly.sympoly]
+syms1 [in Combi.MPoly.sympoly]
+sym_VanprodM [in Combi.MPoly.antisym]
+sym_antiE [in Combi.MPoly.antisym]
+sym_antisym_char_not2 [in Combi.MPoly.antisym]
+sym_anti [in Combi.MPoly.antisym]
+sym_smalln [in Combi.MPoly.antisym]
+sym_fundamental_symh [in Combi.MPoly.sympoly]
+sym_fundamental_symh_homog [in Combi.MPoly.sympoly]
+sym_fundamental_homog [in Combi.MPoly.sympoly]
+sym_symmE [in Combi.MPoly.sympoly]
+sztd [in Combi.Combi.skewpart]
+

T

+tabcols_cons [in Combi.LRrule.Greene]
+tabcol_cut [in Combi.LRrule.Greene]
+tableau_is_row [in Combi.Combi.tableau]
+tabnat_of_ordK [in Combi.MPoly.Schur_altdef]
+tabnat_of_ord_subproof [in Combi.MPoly.Schur_altdef]
+tabord_of_natK [in Combi.MPoly.Schur_altdef]
+tabord_of_nat_subproof [in Combi.MPoly.Schur_altdef]
+tabrowconst_subproof [in Combi.Combi.tableau]
+tabrows_non0 [in Combi.LRrule.Greene]
+tabshP [in Combi.Combi.tableau]
+tabsh_is_std [in Combi.MPoly.Schur_altdef]
+tabsh_reading_RSE [in Combi.LRrule.freeSchur]
+tabsh_reading_RSP [in Combi.LRrule.freeSchur]
+tabsh_to_wordK [in Combi.Combi.tableau]
+tabsh_readingP [in Combi.Combi.tableau]
+tabwordshape_col [in Combi.MPoly.Schur_mpoly]
+tabwordshape_row [in Combi.MPoly.Schur_mpoly]
+tabword_of_tuple_freeSchur [in Combi.LRrule.freeSchur]
+tabword_of_tuple_freeSchur_inj [in Combi.LRrule.freeSchur]
+tab_eval_partdom [in Combi.MPoly.Schur_altdef]
+tab_eqP [in Combi.Combi.tableau]
+tab0 [in Combi.Combi.tableau]
+take_enumI [in Combi.SSRcomplements.tools]
+take_drop_langQ [in Combi.LRrule.freeSchur]
+take_prefixes [in Combi.Combi.Dyckword]
+TamariLattice.botETamari [in Combi.Combi.bintree]
+TamariLattice.flipsz_join [in Combi.Combi.bintree]
+TamariLattice.flipsz_meet [in Combi.Combi.bintree]
+TamariLattice.leftcomb_bottom [in Combi.Combi.bintree]
+TamariLattice.rightcomb_top [in Combi.Combi.bintree]
+TamariLattice.right_sizes_meet [in Combi.Combi.bintree]
+TamariLattice.rotations_Tamari [in Combi.Combi.bintree]
+TamariLattice.TamariE [in Combi.Combi.bintree]
+TamariLattice.Tamari_vctleq [in Combi.Combi.bintree]
+TamariLattice.Tamari_flip [in Combi.Combi.bintree]
+TamariLattice.Tamari_anti [in Combi.Combi.bintree]
+TamariLattice.Tamari_trans [in Combi.Combi.bintree]
+TamariLattice.Tamari_refl [in Combi.Combi.bintree]
+TamariLattice.Tamari_sumn_right_sizes [in Combi.Combi.bintree]
+TamariLattice.TjoinP [in Combi.Combi.bintree]
+TamariLattice.TmeetC [in Combi.Combi.bintree]
+TamariLattice.TmeetP [in Combi.Combi.bintree]
+TamariLattice.TmeetPr [in Combi.Combi.bintree]
+TamariLattice.Tmeet_proof [in Combi.Combi.bintree]
+TamariLattice.topETamari [in Combi.Combi.bintree]
+TamariP [in Combi.Combi.bintree]
+Tamari_succ [in Combi.Combi.bintree]
+Tamari_display [in Combi.Combi.bintree]
+Tamari_add_bounded [in Combi.Combi.bintree]
+Tamari_add_min [in Combi.Combi.bintree]
+Tamari_add_head [in Combi.Combi.bintree]
+Tamari_cat [in Combi.Combi.bintree]
+Tamari_take [in Combi.Combi.bintree]
+Tamari_catr [in Combi.Combi.bintree]
+Tamari_drop [in Combi.Combi.bintree]
+Tamari_consP [in Combi.Combi.bintree]
+tclosureP [in Combi.SymGroup.weak_order]
+tclosure_Delta [in Combi.SymGroup.weak_order]
+tclosure_sub [in Combi.SymGroup.presentSn]
+Theorem2 [in Combi.HookFormula.hook]
+tinjA [in Combi.SymGroup.towerSn]
+tinjE1 [in Combi.SymGroup.towerSn]
+tinjval_inj [in Combi.SymGroup.towerSn]
+tinj_morphM [in Combi.SymGroup.towerSn]
+tinj1E [in Combi.SymGroup.towerSn]
+toDepRSPair [in Combi.LRrule.freeSchur]
+to_word_skew_reshape [in Combi.Combi.skewtab]
+to_word_yamtab [in Combi.LRrule.Yam_plact]
+to_maskK [in Combi.Combi.subseq]
+to_mask_spec [in Combi.Combi.subseq]
+to_word_map_tab [in Combi.Combi.tableau]
+to_word_enum_tabsh [in Combi.Combi.tableau]
+to_word_filter_nnil [in Combi.Combi.tableau]
+to_wordK [in Combi.Combi.tableau]
+to_word_rcons [in Combi.Combi.tableau]
+to_word_cons [in Combi.Combi.tableau]
+to_word_map_shiftn [in Combi.LRrule.therule]
+tpermC [in Combi.SymGroup.presentSn]
+tperm_conj [in Combi.SymGroup.cycletype]
+tperm_braid [in Combi.SymGroup.presentSn]
+trace_seqrP [in Combi.HookFormula.hook]
+trace_seqlP [in Combi.HookFormula.hook]
+trace_corner_box [in Combi.HookFormula.hook]
+trace_seq_uniq [in Combi.HookFormula.hook]
+trace_size_leg_length [in Combi.HookFormula.hook]
+trace_size_arm_length [in Combi.HookFormula.hook]
+transf [in Combi.LRrule.Schensted]
+transitive_DeltaI1 [in Combi.SymGroup.presentSn]
+transP [in Combi.Basic.congr]
+tree_encodeK [in Combi.Combi.bintree]
+trivIs [in Combi.LRrule.Greene]
+trivIseq_tabcols [in Combi.LRrule.Greene]
+trivIseq_shcols [in Combi.LRrule.Greene]
+trivIseq_shrows [in Combi.LRrule.Greene]
+trivIseq_map [in Combi.LRrule.Greene]
+trivIseq_cover [in Combi.LRrule.Greene]
+trivIseq_consK [in Combi.LRrule.Greene]
+trivIsetpart [in Combi.Combi.setpartition]
+trivIset_iporbits [in Combi.SymGroup.permcent]
+trivIset_coverD1 [in Combi.LRrule.Greene_inv]
+trivIset_coverU [in Combi.LRrule.Greene_inv]
+trivIset_coverU1 [in Combi.LRrule.Greene_inv]
+trivIset_setrev [in Combi.LRrule.Greene_inv]
+trivIset_tabcolsk [in Combi.LRrule.Greene]
+trivIset_tabrowsk [in Combi.LRrule.Greene]
+trivIset_I [in Combi.LRrule.Greene]
+trivIsubseq [in Combi.LRrule.Greene]
+trivsetpart_subproof [in Combi.Combi.setpartition]
+triv_part [in Combi.SSRcomplements.tools]
+triv_sign_not_sim [in Combi.SymGroup.reprSn]
+triv_sign_neq [in Combi.SymGroup.reprSn]
+triv_Chi [in Combi.SymGroup.reprSn]
+triv_irr [in Combi.SymGroup.reprSn]
+triv_mx_repr [in Combi.SymGroup.reprSn]
+tsumnE [in Combi.LRrule.implem]
+tval_tcastE [in Combi.SymGroup.reprSn]
+

U

+UDn_Dyck [in Combi.Combi.Dyckword]
+UnDn_Dyck [in Combi.Combi.Dyckword]
+undup_step [in Combi.Basic.congr]
+Uniform_def_ne [in ALEA.Qmeasure]
+Uniform_unif_seq_eq [in ALEA.Qmeasure]
+Uniform_in_seq [in ALEA.Qmeasure]
+Uniform_eq_out [in ALEA.Qmeasure]
+Uniform_eq_in [in ALEA.Qmeasure]
+Uniform_simpl [in ALEA.Qmeasure]
+union_intpartnE [in Combi.Combi.partition]
+union_intpartn_subproof [in Combi.Combi.partition]
+union_intpartA [in Combi.Combi.partition]
+union_intpartC [in Combi.Combi.partition]
+union_intpartE [in Combi.Combi.partition]
+union_intpart_subproof [in Combi.Combi.partition]
+uniq_wordperm [in Combi.Combi.std]
+uniq_next [in Combi.SSRcomplements.tools]
+uniq_sum_count_mem [in Combi.SSRcomplements.tools]
+uniq_vect_n_k [in Combi.Combi.vectNK]
+uniq_step [in Combi.Basic.congr]
+unitrigP [in Combi.Basic.unitriginv]
+unitrig_sum1rV [in Combi.Basic.unitriginv]
+unitrig_sumrV [in Combi.Basic.unitriginv]
+unitrig_sum1lV [in Combi.Basic.unitriginv]
+unitrig_sumlV [in Combi.Basic.unitriginv]
+unitrig_sum1r [in Combi.Basic.unitriginv]
+unitrig_sumr [in Combi.Basic.unitriginv]
+unitrig_sum1l [in Combi.Basic.unitriginv]
+unitrig_suml [in Combi.Basic.unitriginv]
+unitrig1 [in Combi.Basic.unitriginv]
+unit_stable_sub [in ALEA.Qmeasure]
+unit_monotonic [in ALEA.Qmeasure]
+unit_stable_eq [in ALEA.Qmeasure]
+unlift_seqE [in Combi.LRrule.Greene]
+usize_pos [in ALEA.Qmeasure]
+

V

+val_intpartn3 [in Combi.Combi.partition]
+val_intpartn2 [in Combi.Combi.partition]
+val_intpartn1 [in Combi.Combi.partition]
+val_intpartn0 [in Combi.Combi.partition]
+val_stdtabshcast [in Combi.Combi.stdtab]
+val_enum_subseqs [in Combi.Combi.subseq]
+val_omegasf [in Combi.MPoly.sympoly]
+val_descset [in Combi.Combi.composition]
+val2posE [in Combi.LRrule.stdplact]
+val2pos_enum [in Combi.LRrule.stdplact]
+val2pos_inj [in Combi.LRrule.stdplact]
+Vandet_VanprodE [in Combi.MPoly.antisym]
+Vanmx_antimE [in Combi.MPoly.antisym]
+Vanprod_alt [in Combi.MPoly.antisym]
+Vanprod_alt_int [in Combi.MPoly.antisym]
+Vanprod_dhomog [in Combi.MPoly.antisym]
+Vanprod_neq0 [in Combi.MPoly.antisym]
+Vanprod_coeff_rho [in Combi.MPoly.antisym]
+Vanprod_anti [in Combi.MPoly.antisym]
+vb_strip_lexi [in Combi.MPoly.Schur_altdef]
+vb_strip_rem_col0 [in Combi.MPoly.Schur_altdef]
+vb_strip_conjE [in Combi.Combi.skewpart]
+vb_strip_conj [in Combi.Combi.skewpart]
+vb_strip_diffP [in Combi.Combi.skewpart]
+vb_stripP [in Combi.Combi.skewpart]
+vb_strip_included [in Combi.Combi.skewpart]
+vctleqP [in Combi.Combi.bintree]
+vctleq_rotation [in Combi.Combi.bintree]
+vctleq_sumn_right_sizes [in Combi.Combi.bintree]
+vctleq_anti [in Combi.Combi.bintree]
+vctleq_trans [in Combi.Combi.bintree]
+vctleq_refl [in Combi.Combi.bintree]
+vctminC [in Combi.Combi.bintree]
+vctminP [in Combi.Combi.bintree]
+vctminPl [in Combi.Combi.bintree]
+vctminPr [in Combi.Combi.bintree]
+vctmin_Tamari [in Combi.Combi.bintree]
+vct_succ [in Combi.Combi.bintree]
+vecmx_indexK [in Combi.MPoly.Cauchy]
+vect_0_k [in Combi.Combi.vectNK]
+vect_n_kP [in Combi.Combi.vectNK]
+vect_n_k_in [in Combi.Combi.vectNK]
+

W

+walk_to_corner_decomp [in Combi.HookFormula.hook]
+walk_to_corner_emptyr [in Combi.HookFormula.hook]
+walk_to_corner_emptyl [in Combi.HookFormula.hook]
+walk_to_corner_inv [in Combi.HookFormula.hook]
+walk_to_corner_simpl [in Combi.HookFormula.hook]
+walk_to_corner_end_simpl [in Combi.HookFormula.hook]
+walk_to_corner0_simpl [in Combi.HookFormula.hook]
+wcordE [in Combi.SymGroup.presentSn]
+wcord_cons [in Combi.SymGroup.presentSn]
+WeakOrder.lepermP [in Combi.SymGroup.weak_order]
+WeakOrder.leperm_anti [in Combi.SymGroup.weak_order]
+WeakOrder.leperm_trans [in Combi.SymGroup.weak_order]
+WeakOrder.leperm_refl [in Combi.SymGroup.weak_order]
+WeakOrder.leperm_lengthE [in Combi.SymGroup.weak_order]
+WeakOrder.leperm_length [in Combi.SymGroup.weak_order]
+weight_is_unit [in ALEA.Qmeasure]
+weight_case [in ALEA.Qmeasure]
+weight_nonneg [in ALEA.Qmeasure]
+weight1_size [in ALEA.Qmeasure]
+wordcdE [in Combi.SymGroup.presentSn]
+wordcd_ltn [in Combi.SymGroup.presentSn]
+wordperm_inj [in Combi.Combi.std]
+wordperm_invP [in Combi.Combi.std]
+wordperm_std [in Combi.Combi.std]
+wordperm_iota [in Combi.Combi.std]
+

Y

+yamevalP [in Combi.Combi.Yamanouchi]
+yamnP [in Combi.Combi.Yamanouchi]
+yamn_partition_evalseq [in Combi.Combi.Yamanouchi]
+yamn_PredEq [in Combi.Combi.Yamanouchi]
+yamrowP [in Combi.LRrule.therule]
+yamtabP [in Combi.LRrule.Yam_plact]
+yamtab_unique [in Combi.LRrule.Yam_plact]
+yamtab_rcons [in Combi.LRrule.Yam_plact]
+yamtab_shift_countE [in Combi.LRrule.implem]
+yamtab_rows_countE [in Combi.LRrule.implem]
+yamtab_shift_is_row [in Combi.LRrule.implem]
+yamtab_rows_is_row [in Combi.LRrule.implem]
+yamtab_shift_dominate [in Combi.LRrule.implem]
+yamtab_rows_dominate [in Combi.LRrule.implem]
+yamtab_shift_size [in Combi.LRrule.implem]
+yamtab_rows_size [in Combi.LRrule.implem]
+yamtab_shift_included [in Combi.LRrule.implem]
+yamtab_rows_included [in Combi.LRrule.implem]
+yamtab_shiftP [in Combi.LRrule.implem]
+yamtab_rowsP [in Combi.LRrule.implem]
+yamtab_shift_drop [in Combi.LRrule.implem]
+yam_tail_non_nil [in Combi.LRrule.Schensted]
+yam_std_inj [in Combi.LRrule.Yam_plact]
+yam_plactic_shape [in Combi.LRrule.Yam_plact]
+yam_plactic_hyper [in Combi.LRrule.Yam_plact]
+yam_of_stdtabP [in Combi.Combi.stdtab]
+yam_of_stdtabK [in Combi.Combi.stdtab]
+yam_of_rowpart [in Combi.LRrule.therule]
+YmonK [in Combi.MPoly.Cauchy]
+Ymon_bij [in Combi.MPoly.Cauchy]
+YoungLattice.bottom_YoungE [in Combi.Combi.partition]
+YoungLattice.emptypart_bottom [in Combi.Combi.partition]
+YoungLattice.Exports.join_YoungE [in Combi.Combi.partition]
+YoungLattice.Exports.meet_YoungE [in Combi.Combi.partition]
+YoungLattice.Exports.nth_join_Young [in Combi.Combi.partition]
+YoungLattice.Exports.nth_meet_Young [in Combi.Combi.partition]
+YoungLattice.Exports.size_join_Young [in Combi.Combi.partition]
+YoungLattice.Exports.size_meet_Young [in Combi.Combi.partition]
+YoungLattice.join_YoungP [in Combi.Combi.partition]
+YoungLattice.join_YoungC [in Combi.Combi.partition]
+YoungLattice.join_Young_le [in Combi.Combi.partition]
+YoungLattice.join_Young_subproof [in Combi.Combi.partition]
+YoungLattice.le_Young_sumn [in Combi.Combi.partition]
+YoungLattice.le_YoungP [in Combi.Combi.partition]
+YoungLattice.le_YoungE [in Combi.Combi.partition]
+YoungLattice.le_Young_anti [in Combi.Combi.partition]
+YoungLattice.le_Young_trans [in Combi.Combi.partition]
+YoungLattice.le_Young_refl [in Combi.Combi.partition]
+YoungLattice.lt_Young_sumn [in Combi.Combi.partition]
+YoungLattice.meet_YoungP [in Combi.Combi.partition]
+YoungLattice.meet_Young_le [in Combi.Combi.partition]
+YoungLattice.meet_YoungC [in Combi.Combi.partition]
+YoungLattice.meet_Young_subproof [in Combi.Combi.partition]
+YoungLattice.nth_join_Young [in Combi.Combi.partition]
+YoungLattice.nth_meet_Young [in Combi.Combi.partition]
+YoungLattice.size_join_Young [in Combi.Combi.partition]
+YoungLattice.size_meet_Young [in Combi.Combi.partition]
+YoungLattice.Young_meetUl [in Combi.Combi.partition]
+YoungLattice.Young_display [in Combi.Combi.partition]
+Young_rule_partdom [in Combi.SymGroup.Frobenius_char]
+Young_rule [in Combi.SymGroup.Frobenius_char]
+Young_char [in Combi.SymGroup.Frobenius_char]
+

Z

+zcard_rem [in Combi.SymGroup.permcent]
+zcard_any [in Combi.SymGroup.permcent]
+zcard_nil [in Combi.SymGroup.permcent]
+zcoeffE [in Combi.SymGroup.towerSn]
+

+

Constructor Index

+

B

+BinLeaf [in Combi.Combi.bintree]
+BinNode [in Combi.Combi.bintree]
+

C

+Close [in Combi.Combi.Dyckword]
+continuous2_intro [in ALEA.Ccpo]
+cont_intro [in ALEA.Ccpo]
+CycleDecSpec [in Combi.SymGroup.cycles]
+

G

+GenCongr [in Combi.Basic.congr]
+

I

+IsInvset [in Combi.SymGroup.presentSn]
+

L

+LRTriple [in Combi.LRrule.shuffle]
+

M

+monotonic_def [in ALEA.Ccpo]
+monotonic2_intro [in ALEA.Ccpo]
+

O

+Open [in Combi.Combi.Dyckword]
+OrdNode [in Combi.Combi.ordtree]
+

R

+RefinmentOrder.MeetSpec [in Combi.Combi.setpartition]
+RelatSn [in Combi.SymGroup.presentSn]
+Rew [in Combi.Basic.congr]
+

S

+stable_def [in ALEA.Ccpo]
+stable2_intro [in ALEA.Ccpo]
+StdSpec [in Combi.Combi.std]
+

+

Projection Index

+

B

+bound [in Combi.Basic.congr]
+box_skewval [in Combi.Combi.partition]
+

C

+carrier [in Combi.Combi.fibered_set]
+cdval [in Combi.SymGroup.presentSn]
+cnval [in Combi.Combi.composition]
+coeff [in ALEA.Qmeasure]
+continuous2_intro [in ALEA.Ccpo]
+cont_intro [in ALEA.Ccpo]
+cval [in Combi.Combi.composition]
+

D

+Dbot [in ALEA.Ccpo]
+dyckword [in Combi.Combi.Dyckword]
+D0 [in ALEA.Ccpo]
+

E

+elt [in Combi.Combi.fibered_set]
+eval_eq [in Combi.LRrule.implem]
+eval_part [in Combi.LRrule.implem]
+Example1.one [in Combi.Basic.combclass]
+Example2.one [in Combi.Basic.combclass]
+Example3.one [in Combi.Basic.combclass]
+

F

+fbfun [in Combi.Combi.fibered_set]
+fbset [in Combi.Combi.fibered_set]
+fcontinuous [in ALEA.Ccpo]
+fcontm [in ALEA.Ccpo]
+fmonotonic [in ALEA.Ccpo]
+fmont [in ALEA.Ccpo]
+fs [in Combi.SymGroup.cycletype]
+fs_homog [in Combi.SymGroup.cycletype]
+fs_stab [in Combi.SymGroup.cycletype]
+

H

+Hbound [in Combi.Basic.congr]
+Hinvar_all [in Combi.Basic.congr]
+Hinvar_refl [in Combi.Basic.congr]
+homsym [in Combi.MPoly.homogsym]
+

I

+incl [in Combi.LRrule.implem]
+Inhabited.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+Inhabited.class [in Combi.Basic.ordtype]
+Inhabited.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+Inhabited.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+Inhabited.sort [in Combi.Basic.ordtype]
+InhFinite.choice_Choice_isCountable_mixin [in Combi.Basic.ordtype]
+InhFinite.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhFinite.class [in Combi.Basic.ordtype]
+InhFinite.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhFinite.fintype_isFinite_mixin [in Combi.Basic.ordtype]
+InhFinite.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhFinite.sort [in Combi.Basic.ordtype]
+InhFinLattice.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhFinLattice.choice_Choice_isCountable_mixin [in Combi.Basic.ordtype]
+InhFinLattice.class [in Combi.Basic.ordtype]
+InhFinLattice.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhFinLattice.fintype_isFinite_mixin [in Combi.Basic.ordtype]
+InhFinLattice.Order_POrder_isJoinSemilattice_mixin [in Combi.Basic.ordtype]
+InhFinLattice.Order_POrder_isMeetSemilattice_mixin [in Combi.Basic.ordtype]
+InhFinLattice.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhFinLattice.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhFinLattice.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhFinLattice.sort [in Combi.Basic.ordtype]
+InhFinOrder.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhFinOrder.choice_Choice_isCountable_mixin [in Combi.Basic.ordtype]
+InhFinOrder.class [in Combi.Basic.ordtype]
+InhFinOrder.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhFinOrder.fintype_isFinite_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_DistrLattice_isTotal_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_Lattice_isDistributive_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_POrder_isJoinSemilattice_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_POrder_isMeetSemilattice_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhFinOrder.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhFinOrder.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhFinOrder.sort [in Combi.Basic.ordtype]
+InhFinPOrder.choice_Choice_isCountable_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.class [in Combi.Basic.ordtype]
+InhFinPOrder.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.fintype_isFinite_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhFinPOrder.sort [in Combi.Basic.ordtype]
+InhLattice.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhLattice.class [in Combi.Basic.ordtype]
+InhLattice.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhLattice.Order_POrder_isJoinSemilattice_mixin [in Combi.Basic.ordtype]
+InhLattice.Order_POrder_isMeetSemilattice_mixin [in Combi.Basic.ordtype]
+InhLattice.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhLattice.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhLattice.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhLattice.sort [in Combi.Basic.ordtype]
+InhOrder.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhOrder.class [in Combi.Basic.ordtype]
+InhOrder.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_DistrLattice_isTotal_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_Lattice_isDistributive_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_POrder_isJoinSemilattice_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_POrder_isMeetSemilattice_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhOrder.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhOrder.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhOrder.sort [in Combi.Basic.ordtype]
+InhPOrder.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhPOrder.class [in Combi.Basic.ordtype]
+InhPOrder.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhPOrder.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhPOrder.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhPOrder.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhPOrder.sort [in Combi.Basic.ordtype]
+InhTBLattice.choice_hasChoice_mixin [in Combi.Basic.ordtype]
+InhTBLattice.class [in Combi.Basic.ordtype]
+InhTBLattice.eqtype_hasDecEq_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_POrder_isJoinSemilattice_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_POrder_isMeetSemilattice_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_Preorder_isDuallyPOrder_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_hasBottom_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_hasTop_mixin [in Combi.Basic.ordtype]
+InhTBLattice.Order_isDuallyPreorder_mixin [in Combi.Basic.ordtype]
+InhTBLattice.ordtype_isInhabited_mixin [in Combi.Basic.ordtype]
+InhTBLattice.sort [in Combi.Basic.ordtype]
+inner_part [in Combi.LRrule.implem]
+invar [in Combi.Basic.congr]
+isInhabitedType.x [in Combi.Basic.ordtype]
+isInhabited.inh_ex [in Combi.Basic.ordtype]
+islub_le [in ALEA.Ccpo]
+IsoBottom.disp' [in Combi.Combi.composition]
+IsoBottom.f [in Combi.Combi.composition]
+IsoBottom.f_mono [in Combi.Combi.composition]
+IsoBottom.f_can [in Combi.Combi.composition]
+IsoBottom.f' [in Combi.Combi.composition]
+IsoBottom.f'_can [in Combi.Combi.composition]
+IsoBottom.T' [in Combi.Combi.composition]
+IsoTop.disp' [in Combi.Combi.composition]
+IsoTop.f [in Combi.Combi.composition]
+IsoTop.f_mono [in Combi.Combi.composition]
+IsoTop.f_can [in Combi.Combi.composition]
+IsoTop.f' [in Combi.Combi.composition]
+IsoTop.f'_can [in Combi.Combi.composition]
+IsoTop.T' [in Combi.Combi.composition]
+is_dyckword [in Combi.Combi.Dyckword]
+

L

+le_lub [in ALEA.Ccpo]
+le_islub [in ALEA.Ccpo]
+lub [in ALEA.Ccpo]
+lub_le [in ALEA.Ccpo]
+

M

+monotonic_def [in ALEA.Ccpo]
+monotonic2_intro [in ALEA.Ccpo]
+mu [in ALEA.Qmeasure]
+mu_prob [in ALEA.Qmeasure]
+mu_stable_sub [in ALEA.Qmeasure]
+

O

+Oeq [in ALEA.Ccpo]
+Ole [in ALEA.Ccpo]
+order_rel [in ALEA.Ccpo]
+order_eq [in ALEA.Ccpo]
+outer_part [in Combi.LRrule.implem]
+

P

+pnval [in Combi.Combi.partition]
+points [in ALEA.Qmeasure]
+pqpair [in Combi.LRrule.Schensted]
+pval [in Combi.Combi.partition]
+pyampair [in Combi.LRrule.Schensted]
+

R

+reflexive [in ALEA.Ccpo]
+

S

+SetContainingBothLeft.Hba [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hbax [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hbc [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Hxba [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Rtrans [in Combi.LRrule.Greene_inv]
+setpartval [in Combi.Combi.setpartition]
+shaps_eq [in Combi.LRrule.implem]
+skew [in Combi.LRrule.implem]
+stable_def [in ALEA.Ccpo]
+stable2_intro [in ALEA.Ccpo]
+stdtabnval [in Combi.Combi.stdtab]
+stdtabshval [in Combi.Combi.stdtab]
+stdwordnval [in Combi.Combi.std]
+subseqsval [in Combi.Combi.subseq]
+sumn_eq [in Combi.LRrule.implem]
+sympol [in Combi.MPoly.sympoly]
+

T

+tabshval [in Combi.Combi.tableau]
+tpval [in Combi.Combi.permuted]
+transitive [in ALEA.Ccpo]
+trval [in Combi.Combi.bintree]
+trval [in Combi.Combi.ordtree]
+

U

+upoints [in ALEA.Qmeasure]
+

W

+weight_pos [in ALEA.Qmeasure]
+

Y

+yamevalval [in Combi.Combi.Yamanouchi]
+yamnval [in Combi.Combi.Yamanouchi]
+yam_to_word [in Combi.LRrule.implem]
+

+

Inductive Index

+

B

+bintree [in Combi.Combi.bintree]
+brace [in Combi.Combi.Dyckword]
+

C

+continuous [in ALEA.Ccpo]
+continuous2 [in ALEA.Ccpo]
+cycle_dec_spec [in Combi.SymGroup.cycles]
+

G

+Generated_EquivCongruence [in Combi.Basic.congr]
+

I

+is_invset [in Combi.SymGroup.presentSn]
+

L

+LRtriple [in Combi.LRrule.shuffle]
+

M

+monotonic [in ALEA.Ccpo]
+monotonic2 [in ALEA.Ccpo]
+

O

+ordtree [in Combi.Combi.ordtree]
+

R

+RefinmentOrder.meet_spec [in Combi.Combi.setpartition]
+relat_Sn [in Combi.SymGroup.presentSn]
+rewrite_path [in Combi.Basic.congr]
+

S

+stable [in ALEA.Ccpo]
+stable2 [in ALEA.Ccpo]
+std_spec [in Combi.Combi.std]
+

+

Section Index

+

A

+AbelianBigOp [in Combi.SSRcomplements.tools]
+ActOnTuple [in Combi.Combi.permuted]
+AllLeqLtn [in Combi.Basic.ordtype]
+Alternant [in Combi.MPoly.Schur_altdef]
+Alternant.HasIncr [in Combi.MPoly.Schur_altdef]
+AlternIDomain [in Combi.MPoly.antisym]
+AlternIDomain.LeadingMonomial [in Combi.MPoly.antisym]
+AlternStraighten [in Combi.MPoly.Schur_altdef]
+AppendNth [in Combi.Combi.stdtab]
+Assoc [in Combi.SymGroup.towerSn]
+

B

+BalToDyck [in Combi.Combi.Dyckword]
+Bases [in Combi.MPoly.sympoly]
+Bigop [in Combi.Combi.subseq]
+Bigsums [in ALEA.Qmeasure]
+BigTrivISeq [in Combi.LRrule.Greene]
+BijBinTrees [in Combi.Combi.Dyckword]
+Bijection [in Combi.Combi.stdtab]
+BijectionExtTab [in Combi.MPoly.Schur_altdef]
+Bijection.StdTabInd [in Combi.Combi.stdtab]
+BijFiberedSet [in Combi.Combi.fibered_set]
+BijFiberedSet.Defs [in Combi.Combi.fibered_set]
+BoxInSkew [in Combi.Combi.partition]
+BraidRed [in Combi.SymGroup.presentSn]
+Builders_1.Builders_1.Builders_1 [in Combi.Basic.ordtype]
+Builders_6.Builders_6.Builders_6 [in Combi.Combi.composition]
+Builders_1.Builders_1.Builders_1 [in Combi.Combi.composition]
+

C

+CanPorbit [in Combi.SymGroup.cycletype]
+CanWord [in Combi.SymGroup.presentSn]
+Cast [in Combi.LRrule.Greene]
+Cast [in Combi.MPoly.sympoly]
+Casts [in Combi.SSRcomplements.ordcast]
+CategoricalSystems [in Combi.MPoly.sympoly]
+CauchyKernel [in Combi.MPoly.Cauchy]
+CauchyKernelField [in Combi.MPoly.Cauchy]
+CauchyKernel.Big [in Combi.MPoly.Cauchy]
+CauchyKernel.BijectionFam [in Combi.MPoly.Cauchy]
+CFExtProdDefs [in Combi.SymGroup.towerSn]
+CFExtProdTheory [in Combi.SymGroup.towerSn]
+CFExtProdTheory.ReprExtProd [in Combi.SymGroup.towerSn]
+ChangeBaseMonomial [in Combi.MPoly.sympoly]
+ChangeBasis [in Combi.MPoly.sympoly]
+ChangeBasisSymhPowerSum [in Combi.MPoly.sympoly]
+ChangeBasis.HandE [in Combi.MPoly.sympoly]
+ChangeField [in Combi.MPoly.homogsym]
+ChangeNVar [in Combi.MPoly.homogsym]
+ChangeNVar [in Combi.MPoly.sympoly]
+ChangeNVar.ProdGen [in Combi.MPoly.sympoly]
+CharDotProduct [in Combi.SymGroup.Frobenius_char]
+classGroup [in Combi.SymGroup.towerSn]
+Codes [in Combi.SymGroup.presentSn]
+Codes.FinType [in Combi.SymGroup.presentSn]
+Combi [in Combi.SymGroup.presentSn]
+CommutativeImage [in Combi.LRrule.freeSchur]
+CompOfn [in Combi.Combi.composition]
+CongruenceClosure [in Combi.Basic.congr]
+CongruenceFacts [in Combi.Basic.congr]
+Conj [in Combi.LRrule.freeSchur]
+ConjTab [in Combi.Combi.stdtab]
+ConnectCompl [in Combi.Combi.skewpart]
+Connected4 [in Combi.Combi.skewpart]
+Coord [in Combi.MPoly.homogsym]
+CoversFinPOrder [in Combi.Basic.ordtype]
+CoverSurgery [in Combi.LRrule.Greene_inv]
+CutK [in Combi.Combi.vectNK]
+Cut3 [in Combi.Combi.vectNK]
+CycleType [in Combi.SymGroup.cycletype]
+CycleTypeConj [in Combi.SymGroup.cycletype]
+CycleType.CFunIndicator [in Combi.SymGroup.cycletype]
+CycleType.Classes [in Combi.SymGroup.cycletype]
+CycleType.Permofcycletype [in Combi.SymGroup.cycletype]
+CycleType.TPerm [in Combi.SymGroup.cycletype]
+

D

+Defs [in Combi.Combi.partition]
+Defs [in Combi.LRrule.shuffle]
+Defs [in Combi.Combi.Dyckword]
+Defs [in Combi.LRrule.plactic]
+Defs [in Combi.Combi.setpartition]
+Defs [in Combi.Combi.composition]
+DefsFiber [in Combi.SymGroup.cycletype]
+DefsKostkaMon [in Combi.MPoly.Schur_altdef]
+DefTrivSign [in Combi.SymGroup.reprSn]
+DefType [in Combi.MPoly.homogsym]
+DefType [in Combi.MPoly.sympoly]
+Depend [in Combi.Basic.congr]
+DescSet [in Combi.Combi.composition]
+Dominate [in Combi.Combi.skewtab]
+Dominate [in Combi.Combi.tableau]
+Dual [in Combi.Basic.ordtype]
+Duality [in Combi.LRrule.Greene_inv]
+DualRule [in Combi.LRrule.plactic]
+Dual.hb_instance_111.hb_instance_111 [in Combi.Basic.ordtype]
+Dual.hb_instance_100.hb_instance_100 [in Combi.Basic.ordtype]
+Dual.hb_instance_91.hb_instance_91 [in Combi.Basic.ordtype]
+Dual.hb_instance_80.hb_instance_80 [in Combi.Basic.ordtype]
+Dual.hb_instance_69.hb_instance_69 [in Combi.Basic.ordtype]
+Dual.hb_instance_60.hb_instance_60 [in Combi.Basic.ordtype]
+Dual.hb_instance_53.hb_instance_53 [in Combi.Basic.ordtype]
+Dual.hb_instance_48.hb_instance_48 [in Combi.Basic.ordtype]
+DyckFactor [in Combi.Combi.Dyckword]
+DyckSetInd [in Combi.Combi.Dyckword]
+DyckToBal [in Combi.Combi.Dyckword]
+DyckType [in Combi.Combi.Dyckword]
+DyckWordRotationBijection [in Combi.Combi.Dyckword]
+

E

+ElemTransp [in Combi.SymGroup.presentSn]
+EltrConj [in Combi.SymGroup.reprSn]
+EltrP [in Combi.MPoly.antisym]
+Empty [in Combi.Combi.setpartition]
+EnumFintype [in Combi.Basic.combclass]
+EqInvAltDef [in Combi.Combi.std]
+EqInvDef [in Combi.Combi.std]
+EqInvPosRemBig [in Combi.Combi.std]
+EqInvSkewTab [in Combi.Combi.skewtab]
+Examples [in Combi.Combi.std]
+

F

+FastImplem [in Combi.MPoly.MurnaghanNakayama]
+FastImplem.ComRing [in Combi.MPoly.MurnaghanNakayama]
+FilterLeqGeq [in Combi.Combi.skewtab]
+FindCorner [in Combi.HookFormula.hook]
+FindCorner.EndsAt [in Combi.HookFormula.hook]
+FindCorner.Formula [in Combi.HookFormula.hook]
+FindCorner.Theorem2 [in Combi.HookFormula.hook]
+FinerCard [in Combi.Combi.setpartition]
+FiniteDistributions [in ALEA.Qmeasure]
+FinSet [in Combi.SSRcomplements.tools]
+FinType [in Combi.Combi.subseq]
+FinType [in Combi.Combi.tableau]
+FinType [in Combi.Combi.ordtree]
+FinType [in Combi.Combi.permuted]
+FinType.hb_instance_22.hb_instance_22 [in Combi.Combi.ordtree]
+FreeSchur [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.DefBij [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT.TakeDrop [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport.ChangeUT [in Combi.LRrule.freeSchur]
+FreeSchur.Coeffs.Bij_LRsupport [in Combi.LRrule.freeSchur]
+FreeSchur.Degree [in Combi.LRrule.freeSchur]
+FreeSchur.FreeLRrule [in Combi.LRrule.freeSchur]
+

G

+Generators [in Combi.MPoly.sympoly]
+GreeneCat [in Combi.LRrule.Greene]
+GreeneDef [in Combi.LRrule.Greene]
+GreeneInj [in Combi.LRrule.Greene]
+GreeneInvariants [in Combi.LRrule.Greene_inv]
+GreeneInvariantsDual [in Combi.LRrule.Greene_inv]
+GreeneInvariantsRule [in Combi.LRrule.Greene_inv]
+GreenEqShape [in Combi.LRrule.Greene_inv]
+GreeneRec [in Combi.LRrule.Greene]
+GreeneRec.Induction [in Combi.LRrule.Greene]
+GreeneSeq [in Combi.LRrule.Greene]
+GreeneTab [in Combi.LRrule.Greene]
+

H

+hb_instance_31.hb_instance_31 [in Combi.Combi.partition]
+hb_instance_22.hb_instance_22 [in Combi.Combi.bintree]
+hb_instance_43.hb_instance_43 [in Combi.Basic.ordtype]
+hb_instance_32.hb_instance_32 [in Combi.Basic.ordtype]
+hb_instance_27.hb_instance_27 [in Combi.Basic.ordtype]
+hb_instance_22.hb_instance_22 [in Combi.Basic.ordtype]
+hb_instance_17.hb_instance_17 [in Combi.Basic.ordtype]
+HomogSymLModType [in Combi.MPoly.homogsym]
+HomogSymProd [in Combi.MPoly.homogsym]
+HomSymField [in Combi.MPoly.homogsym]
+HomSymProdGen [in Combi.MPoly.homogsym]
+HomSymProdGen.Cons [in Combi.MPoly.homogsym]
+HomSymProdGen.Merge [in Combi.MPoly.homogsym]
+

I

+Identity [in Combi.HookFormula.Frobenius_ident]
+IdomainSchurSym [in Combi.MPoly.Schur_altdef]
+ImsetInj [in Combi.SSRcomplements.tools]
+IncrMap [in Combi.LRrule.plactic]
+IncrMap [in Combi.Combi.tableau]
+Induction [in Combi.SymGroup.towerSn]
+InHomSym [in Combi.MPoly.homogsym]
+IntPartN [in Combi.Combi.skewpart]
+IntpartnCons [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom [in Combi.Combi.partition]
+IntPartNDom.IntPartNTopBottom [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi [in Combi.Combi.partition]
+InvarContHom [in Combi.Basic.congr]
+InvarContMultHom [in Combi.Basic.congr]
+InvSeq [in Combi.Combi.std]
+InvSet [in Combi.SymGroup.presentSn]
+isInhabitedType.isInhabitedType.isInhabitedType [in Combi.Basic.ordtype]
+isInhabited.isInhabited.isInhabited [in Combi.Basic.ordtype]
+IsoBottom.IsoBottom.IsoBottom [in Combi.Combi.composition]
+IsoTop.IsoTop.IsoTop [in Combi.Combi.composition]
+

K

+Kostka [in Combi.MPoly.Schur_altdef]
+KostkaEq [in Combi.MPoly.Schur_altdef]
+KostkaRec [in Combi.MPoly.Schur_altdef]
+KsuppInj [in Combi.LRrule.stdplact]
+

L

+Length [in Combi.SymGroup.presentSn]
+Length.PartCode [in Combi.SymGroup.presentSn]
+LEPermTheory [in Combi.SymGroup.weak_order]
+LinRepr [in Combi.SymGroup.reprSn]
+LR [in Combi.LRrule.therule]
+LR [in Combi.LRrule.implem]
+LRrule_Pieri [in Combi.MPoly.sympoly]
+LRTriple [in Combi.LRrule.shuffle]
+LR.Pieri [in Combi.LRrule.therule]
+LR.TheRule [in Combi.LRrule.therule]
+LR.TheRule.OneCoeff [in Combi.LRrule.therule]
+

M

+MaxPerm [in Combi.SymGroup.presentSn]
+MaxSeq [in Combi.Basic.ordtype]
+MeasureProp [in ALEA.Qmeasure]
+MinDropEq [in Combi.Combi.skewpart]
+MNRule [in Combi.MPoly.MurnaghanNakayama]
+MonDistrib [in ALEA.Qmeasure]
+MonomPart [in Combi.MPoly.antisym]
+MPoESymHomog [in Combi.MPoly.sympoly]
+MPolySym [in Combi.MPoly.antisym]
+MultAlternSymp [in Combi.MPoly.MurnaghanNakayama]
+MultinomCompl [in Combi.MPoly.sympoly]
+MultSymsSymp [in Combi.MPoly.MurnaghanNakayama]
+MultSymsSympIDomain [in Combi.MPoly.MurnaghanNakayama]
+

N

+NonEmpty [in Combi.LRrule.Schensted]
+NonEmpty.Bijection [in Combi.LRrule.Schensted]
+NonEmpty.Bump [in Combi.LRrule.Schensted]
+NonEmpty.Classes [in Combi.LRrule.Schensted]
+NonEmpty.Dominate [in Combi.LRrule.Schensted]
+NonEmpty.Insert [in Combi.LRrule.Schensted]
+NonEmpty.Inverse [in Combi.LRrule.Schensted]
+NonEmpty.InverseBump [in Combi.LRrule.Schensted]
+NonEmpty.Schensted [in Combi.LRrule.Schensted]
+NonEmpty.Statistics [in Combi.LRrule.Schensted]
+NonEmpty.Tableaux [in Combi.LRrule.Schensted]
+NoSetContainingBoth.Case [in Combi.LRrule.Greene_inv]
+NThAddRibbon [in Combi.Combi.skewpart]
+NVar [in Combi.SymGroup.Frobenius_char]
+NVar.Character [in Combi.SymGroup.Frobenius_char]
+NVar.Defs [in Combi.SymGroup.Frobenius_char]
+

O

+OfSize [in Combi.Combi.ordtree]
+Omega [in Combi.MPoly.sympoly]
+OmegaHomSym [in Combi.MPoly.homogsym]
+OmegaProd [in Combi.MPoly.homogsym]
+OperDistr [in ALEA.Qmeasure]
+OperDistr.MuBool [in ALEA.Qmeasure]
+OrdTableau [in Combi.Combi.tableau]
+OutEval [in Combi.LRrule.implem]
+

P

+PackedSpec [in Combi.LRrule.implem]
+PartOfn [in Combi.Combi.partition]
+PermComp [in Combi.SSRcomplements.permcomp]
+PermCycles [in Combi.SymGroup.permcent]
+PermCycles [in Combi.SymGroup.cycles]
+PermCycles.CM [in Combi.SymGroup.permcent]
+PermEq [in Combi.Combi.std]
+PermLattice.Exports.PermLattice [in Combi.SymGroup.weak_order]
+PermLattice.PermLattice [in Combi.SymGroup.weak_order]
+PermOfInvSetEltr [in Combi.SymGroup.presentSn]
+Permuted [in Combi.Combi.permuted]
+Permuted.SizeN [in Combi.Combi.permuted]
+PlactDual [in Combi.LRrule.plactic]
+PorbitBijection [in Combi.SymGroup.cycletype]
+PreimPartition [in Combi.Combi.Dyckword]
+PresentationSn [in Combi.SymGroup.presentSn]
+ProdGen [in Combi.MPoly.sympoly]
+ProdGen.Defs [in Combi.MPoly.sympoly]
+

Q

+QTableau [in Combi.LRrule.Schensted]
+

R

+RCons [in Combi.Combi.subseq]
+Recursion [in Combi.Combi.ordtree]
+Reduced [in Combi.SymGroup.presentSn]
+RefinementOrder.RefinementOrder [in Combi.Combi.composition]
+RefinmentOrder.Exports.Finer [in Combi.Combi.setpartition]
+RefinmentOrder.RefinmentOrder [in Combi.Combi.setpartition]
+RemoveBig [in Combi.LRrule.plactic]
+RemoveBig [in Combi.Basic.ordtype]
+Restriction [in Combi.SymGroup.towerSn]
+RestrIntervBig [in Combi.LRrule.plactic]
+RestrIntervSmall [in Combi.LRrule.plactic]
+Rev [in Combi.LRrule.Greene]
+Rev [in Combi.LRrule.plactic]
+RevConj [in Combi.LRrule.Greene_inv]
+RibbonOn [in Combi.Combi.skewpart]
+RingSchurSym [in Combi.MPoly.Schur_altdef]
+Rows [in Combi.Combi.tableau]
+RowsAndCols [in Combi.LRrule.plactic]
+RSToPlactic [in Combi.LRrule.plactic]
+

S

+Scalar [in Combi.MPoly.Cauchy]
+ScalarChange [in Combi.MPoly.antisym]
+ScalarChange [in Combi.MPoly.sympoly]
+ScalarProduct [in Combi.MPoly.homogsym]
+Scalar.hb_instance_73.hb_instance_73 [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_67.hb_instance_67 [in Combi.MPoly.Cauchy]
+Scalar.hb_instance_62.hb_instance_62 [in Combi.MPoly.Cauchy]
+Schur [in Combi.MPoly.Schur_mpoly]
+Schur [in Combi.MPoly.sympoly]
+SchurAlternantDef [in Combi.MPoly.Schur_altdef]
+SchurComRingType [in Combi.MPoly.Schur_mpoly]
+SeqLemmas [in Combi.SSRcomplements.tools]
+SetAct [in Combi.SSRcomplements.permcomp]
+SetContainingBothLeft.Case [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BIn [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Case.BNotIn [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.RelHypothesis [in Combi.LRrule.Greene_inv]
+SetPartition [in Combi.SSRcomplements.tools]
+SetPartitionShape [in Combi.Combi.partition]
+ShiftedShuffle [in Combi.LRrule.shuffle]
+SimpleRecursion [in Combi.Combi.ordtree]
+Singleton [in Combi.Combi.setpartition]
+Size [in Combi.Combi.bintree]
+SizeN [in Combi.Combi.bintree]
+Sn [in Combi.SymGroup.cycletype]
+Sorted [in Combi.SSRcomplements.sorted]
+Spec [in Combi.Combi.std]
+Spec [in Combi.Combi.skewpart]
+Spec [in Combi.LRrule.implem]
+SRel [in Combi.SymGroup.presentSn]
+SSRComplFinset [in Combi.SSRcomplements.tools]
+StabilityProperties [in ALEA.Qmeasure]
+Standardisation [in Combi.Combi.std]
+StandardWords [in Combi.Combi.std]
+StdCombClass [in Combi.Combi.std]
+StdKostka [in Combi.MPoly.Schur_altdef]
+StdKostka.Nvar [in Combi.MPoly.Schur_altdef]
+StdRS [in Combi.LRrule.stdplact]
+StdtabCombClass [in Combi.Combi.stdtab]
+StdtabnOfStdtabsh [in Combi.Combi.stdtab]
+StdtabOfShape [in Combi.Combi.stdtab]
+StdTakeDrop [in Combi.Combi.std]
+SubseqSorted [in Combi.Combi.subseq]
+SubseqSortedIn [in Combi.Combi.subseq]
+SubtypesDisjointUnion [in Combi.Basic.combclass]
+SubUndup [in Combi.Basic.combclass]
+Swap.Swap [in Combi.LRrule.Greene_inv]
+SymheSyms [in Combi.MPoly.sympoly]
+SymheSymsInt [in Combi.MPoly.sympoly]
+SymPolF [in Combi.MPoly.sympoly]
+SymPolyComRingType [in Combi.MPoly.sympoly]
+SymPolyIdomainType [in Combi.MPoly.sympoly]
+SymPolyRingType [in Combi.MPoly.sympoly]
+SymsSymm [in Combi.MPoly.sympoly]
+SymsSymmInt [in Combi.MPoly.sympoly]
+

T

+Tableau [in Combi.Combi.tableau]
+TableauReading [in Combi.LRrule.freeSchur]
+TableauReading [in Combi.Combi.tableau]
+TamariCover [in Combi.Combi.bintree]
+TamariLattice.TamariLattice [in Combi.Combi.bintree]
+TcastVal [in Combi.SymGroup.reprSn]
+TClosureInvset [in Combi.SymGroup.weak_order]
+Test [in Combi.Combi.skewpart]
+Tests [in Combi.Combi.bintree]
+Tests [in Combi.LRrule.Schensted]
+Tests [in Combi.Combi.skewpart]
+Tests [in Combi.MPoly.MurnaghanNakayama]
+Tests [in Combi.MPoly.MurnaghanNakayama]
+TestsComp [in Combi.Combi.bintree]
+TextBookDefStartStop [in Combi.Combi.skewpart]
+TextBookImplDef [in Combi.Combi.skewpart]
+TowerMorphism [in Combi.SymGroup.towerSn]
+Transitive [in Combi.Basic.congr]
+Transitive.FullKnown [in Combi.Basic.congr]
+Transp [in Combi.Combi.std]
+Transp [in Combi.SymGroup.presentSn]
+TriangularInv [in Combi.Basic.unitriginv]
+TrivISeq [in Combi.LRrule.Greene]
+

U

+UnifNat [in ALEA.Qmeasure]
+UnionPart [in Combi.Combi.partition]
+UniqFinType [in Combi.Basic.combclass]
+UniTriangular [in Combi.Basic.unitriginv]
+

V

+VandermondeDet [in Combi.MPoly.antisym]
+Vanprod [in Combi.MPoly.antisym]
+VectNK [in Combi.Combi.vectNK]
+Vector [in Combi.MPoly.homogsym]
+

W

+WeakOrder.Def [in Combi.SymGroup.weak_order]
+WeakOrder.Exports.WeakOrder [in Combi.SymGroup.weak_order]
+

Y

+Yama [in Combi.Combi.Yamanouchi]
+YamOfEval [in Combi.Combi.Yamanouchi]
+YamOfSize [in Combi.Combi.Yamanouchi]
+YoungIrrDef [in Combi.SymGroup.Frobenius_char]
+YoungLattice.YoungLattice [in Combi.Combi.partition]
+

+

Instance Index

+

A

+app_mon [in ALEA.Qmeasure]
+

C

+comp_monotonic2 [in ALEA.Ccpo]
+cont_app_monotonic [in ALEA.Ccpo]
+cont0 [in ALEA.Ccpo]
+cont2_continuous [in ALEA.Ccpo]
+cshift_continuous2 [in ALEA.Ccpo]
+

F

+Fcontm_continuous [in ALEA.Ccpo]
+fcontm_monotonic [in ALEA.Ccpo]
+fcont_continuous2 [in ALEA.Ccpo]
+fcont_Comp_continuous2 [in ALEA.Ccpo]
+fcont_comp_continuous [in ALEA.Ccpo]
+fcont_cpo [in ALEA.Ccpo]
+fcont_lub_continuous [in ALEA.Ccpo]
+fcont_ord [in ALEA.Ccpo]
+fcpo [in ALEA.Ccpo]
+Fif_continuous2 [in ALEA.Ccpo]
+fif_mon2 [in ALEA.Ccpo]
+finite_mon [in ALEA.Qmeasure]
+Fixp_cont_continuous [in ALEA.Ccpo]
+fixp_monotonic [in ALEA.Ccpo]
+flip_mon [in ALEA.Qmeasure]
+fmono [in ALEA.Ccpo]
+fmonotonic2 [in ALEA.Ccpo]
+fmon_cpo [in ALEA.Ccpo]
+fmon2_mon [in ALEA.Ccpo]
+ford [in ALEA.Ccpo]
+fstable [in ALEA.Ccpo]
+fstable2 [in ALEA.Ccpo]
+fun2_monotonic [in ALEA.Ccpo]
+

I

+Id_mon [in ALEA.Ccpo]
+ishift_continuous [in ALEA.Ccpo]
+ishift_mon [in ALEA.Ccpo]
+iter_monotonic [in ALEA.Ccpo]
+iter_mon [in ALEA.Ccpo]
+

L

+lub_continuous [in ALEA.Ccpo]
+lub_shift_mon [in ALEA.Ccpo]
+lub_mon [in ALEA.Ccpo]
+

M

+MFO [in ALEA.Qmeasure]
+mfun2_mon [in ALEA.Ccpo]
+Mlet_mon2 [in ALEA.Qmeasure]
+MO [in ALEA.Qmeasure]
+monotonic_lub_comp [in ALEA.Ccpo]
+monotonic_comp_mon [in ALEA.Ccpo]
+monotonic_comp [in ALEA.Ccpo]
+monotonic_stable [in ALEA.Ccpo]
+monotonic2_stable2 [in ALEA.Ccpo]
+mon_fun_lub [in ALEA.Ccpo]
+mon_app2_mon [in ALEA.Ccpo]
+mon_app2 [in ALEA.Ccpo]
+mon_diag [in ALEA.Ccpo]
+mon_cte [in ALEA.Ccpo]
+mon_id [in ALEA.Ccpo]
+mon_seq_lift_right [in ALEA.Ccpo]
+mon_seq_lift_left [in ALEA.Ccpo]
+mon_fun_mon [in ALEA.Ccpo]
+mon2_intro [in ALEA.Ccpo]
+mshift_continuous [in ALEA.Ccpo]
+

N

+natO [in ALEA.Ccpo]
+

O

+Odistr [in ALEA.Qmeasure]
+OrderEqRefl [in ALEA.Ccpo]
+OrderEqSym [in ALEA.Ccpo]
+OrderEqTrans [in ALEA.Ccpo]
+OrderEquiv [in ALEA.Ccpo]
+

R

+ratO [in ALEA.Qmeasure]
+

S

+shift_mon2 [in ALEA.Ccpo]
+shift_fun_mon [in ALEA.Ccpo]
+shift_mon_fun [in ALEA.Ccpo]
+

+

Abbreviation Index

+

B

+Basis [in Combi.MPoly.homogsym]
+box [in Combi.Combi.skewpart]
+box [in Combi.Combi.skewpart]
+box [in Combi.Combi.skewpart]
+boxex [in Combi.Combi.skewpart]
+box_in [in Combi.Combi.partition]
+braidred [in Combi.SymGroup.presentSn]
+Builders_1.x [in Combi.Basic.ordtype]
+Builders_6.f_mono [in Combi.Combi.composition]
+Builders_6.f'_can [in Combi.Combi.composition]
+Builders_6.f_can [in Combi.Combi.composition]
+Builders_6.f' [in Combi.Combi.composition]
+Builders_6.f [in Combi.Combi.composition]
+Builders_6.T' [in Combi.Combi.composition]
+Builders_6.disp' [in Combi.Combi.composition]
+Builders_1.f_mono [in Combi.Combi.composition]
+Builders_1.f'_can [in Combi.Combi.composition]
+Builders_1.f_can [in Combi.Combi.composition]
+Builders_1.f' [in Combi.Combi.composition]
+Builders_1.f [in Combi.Combi.composition]
+Builders_1.T' [in Combi.Combi.composition]
+Builders_1.disp' [in Combi.Combi.composition]
+bump [in Combi.LRrule.Schensted]
+bumpRow [in Combi.LRrule.Schensted]
+

C

+cfdot [in Combi.SymGroup.Frobenius_char]
+classX [in Combi.SymGroup.towerSn]
+conj [in Combi.HookFormula.hook]
+coord [in Combi.MPoly.homogsym]
+ct [in Combi.SymGroup.towerSn]
+ct [in Combi.SymGroup.towerSn]
+ct [in Combi.SymGroup.towerSn]
+

D

+Delta [in Combi.MPoly.antisym]
+

E

+E [in Combi.MPoly.homogsym]
+E [in Combi.MPoly.sympoly]
+E [in Combi.MPoly.sympoly]
+E [in Combi.MPoly.sympoly]
+enum_box_in [in Combi.Combi.partition]
+

F

+forest [in Combi.Combi.ordtree]
+

G

+G [in Combi.MPoly.sympoly]
+geL [in Combi.LRrule.plactic]
+gtL [in Combi.LRrule.plactic]
+

H

+HLF [in Combi.HookFormula.hook]
+homlang [in Combi.LRrule.freeSchur]
+HS [in Combi.SymGroup.Frobenius_char]
+HS [in Combi.SymGroup.Frobenius_char]
+HSC [in Combi.MPoly.Cauchy]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.homogsym]
+HSF [in Combi.MPoly.MurnaghanNakayama]
+HSF [in Combi.MPoly.MurnaghanNakayama]
+HSFR [in Combi.MPoly.homogsym]
+HSFS [in Combi.MPoly.homogsym]
+

I

+Ik [in Combi.LRrule.Greene]
+Inhabited [in Combi.Basic.ordtype]
+Inhabited.clone [in Combi.Basic.ordtype]
+Inhabited.copy [in Combi.Basic.ordtype]
+Inhabited.Exports.inhType [in Combi.Basic.ordtype]
+Inhabited.on [in Combi.Basic.ordtype]
+Inhabited.on_ [in Combi.Basic.ordtype]
+InhFinite [in Combi.Basic.ordtype]
+InhFinite.clone [in Combi.Basic.ordtype]
+InhFinite.copy [in Combi.Basic.ordtype]
+InhFinite.Exports.inhFinType [in Combi.Basic.ordtype]
+InhFinite.on [in Combi.Basic.ordtype]
+InhFinite.on_ [in Combi.Basic.ordtype]
+InhFinLattice [in Combi.Basic.ordtype]
+InhFinLattice.clone [in Combi.Basic.ordtype]
+InhFinLattice.copy [in Combi.Basic.ordtype]
+InhFinLattice.Exports.inhFinLatticeType [in Combi.Basic.ordtype]
+InhFinLattice.on [in Combi.Basic.ordtype]
+InhFinLattice.on_ [in Combi.Basic.ordtype]
+InhFinOrder [in Combi.Basic.ordtype]
+InhFinOrder.clone [in Combi.Basic.ordtype]
+InhFinOrder.copy [in Combi.Basic.ordtype]
+InhFinOrder.Exports.inhFinOrderType [in Combi.Basic.ordtype]
+InhFinOrder.on [in Combi.Basic.ordtype]
+InhFinOrder.on_ [in Combi.Basic.ordtype]
+InhFinPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.clone [in Combi.Basic.ordtype]
+InhFinPOrder.copy [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.inhFinPOrderType [in Combi.Basic.ordtype]
+InhFinPOrder.on [in Combi.Basic.ordtype]
+InhFinPOrder.on_ [in Combi.Basic.ordtype]
+InhLattice [in Combi.Basic.ordtype]
+InhLattice.clone [in Combi.Basic.ordtype]
+InhLattice.copy [in Combi.Basic.ordtype]
+InhLattice.Exports.inhLatticeType [in Combi.Basic.ordtype]
+InhLattice.on [in Combi.Basic.ordtype]
+InhLattice.on_ [in Combi.Basic.ordtype]
+InhOrder [in Combi.Basic.ordtype]
+InhOrder.clone [in Combi.Basic.ordtype]
+InhOrder.copy [in Combi.Basic.ordtype]
+InhOrder.Exports.inhOrderType [in Combi.Basic.ordtype]
+InhOrder.on [in Combi.Basic.ordtype]
+InhOrder.on_ [in Combi.Basic.ordtype]
+InhPOrder [in Combi.Basic.ordtype]
+InhPOrder.clone [in Combi.Basic.ordtype]
+InhPOrder.copy [in Combi.Basic.ordtype]
+InhPOrder.Exports.inhPOrderType [in Combi.Basic.ordtype]
+InhPOrder.on [in Combi.Basic.ordtype]
+InhPOrder.on_ [in Combi.Basic.ordtype]
+InhTBLattice [in Combi.Basic.ordtype]
+InhTBLattice.clone [in Combi.Basic.ordtype]
+InhTBLattice.copy [in Combi.Basic.ordtype]
+InhTBLattice.Exports.inhTBLatticeType [in Combi.Basic.ordtype]
+InhTBLattice.on [in Combi.Basic.ordtype]
+InhTBLattice.on_ [in Combi.Basic.ordtype]
+ins [in Combi.LRrule.Schensted]
+inspos [in Combi.LRrule.Schensted]
+insRow [in Combi.LRrule.Schensted]
+isInhabited [in Combi.Basic.ordtype]
+isInhabitedType [in Combi.Basic.ordtype]
+isInhabitedType.axioms [in Combi.Basic.ordtype]
+isInhabitedType.Build [in Combi.Basic.ordtype]
+isInhabited.axioms [in Combi.Basic.ordtype]
+isInhabited.Build [in Combi.Basic.ordtype]
+IsoBottom [in Combi.Combi.composition]
+IsoBottom.axioms [in Combi.Combi.composition]
+IsoBottom.Build [in Combi.Combi.composition]
+IsoTop [in Combi.Combi.composition]
+IsoTop.axioms [in Combi.Combi.composition]
+IsoTop.Build [in Combi.Combi.composition]
+is_homsym [in Combi.MPoly.homogsym]
+is_row [in Combi.Combi.tableau]
+is_row [in Combi.Combi.tableau]
+

L

+leL [in Combi.LRrule.plactic]
+lsh [in Combi.LRrule.Greene]
+ltL [in Combi.LRrule.plactic]
+

M

+m [in Combi.MPoly.Cauchy]
+m [in Combi.MPoly.homogsym]
+m [in Combi.MPoly.sympoly]
+mxvec_index [in Combi.MPoly.Cauchy]
+

N

+n [in Combi.MPoly.Cauchy]
+n [in Combi.MPoly.Cauchy]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.homogsym]
+n [in Combi.MPoly.Schur_mpoly]
+n [in Combi.MPoly.Schur_mpoly]
+n [in Combi.Basic.unitriginv]
+n [in Combi.MPoly.Schur_altdef]
+n [in Combi.MPoly.Schur_altdef]
+n [in Combi.MPoly.Schur_altdef]
+n [in Combi.MPoly.Schur_altdef]
+n [in Combi.LRrule.freeSchur]
+n [in Combi.LRrule.therule]
+n [in Combi.LRrule.therule]
+n [in Combi.SymGroup.weak_order]
+n [in Combi.SymGroup.weak_order]
+n [in Combi.MPoly.MurnaghanNakayama]
+n [in Combi.MPoly.MurnaghanNakayama]
+n [in Combi.MPoly.MurnaghanNakayama]
+n [in Combi.MPoly.MurnaghanNakayama]
+n [in Combi.MPoly.MurnaghanNakayama]
+n [in Combi.SymGroup.presentSn]
+n [in Combi.SymGroup.presentSn]
+n [in Combi.SymGroup.presentSn]
+n [in Combi.SymGroup.presentSn]
+n [in Combi.LRrule.implem]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+n [in Combi.MPoly.sympoly]
+neig4 [in Combi.Combi.skewpart]
+neig4 [in Combi.Combi.skewpart]
+NoSetContainingBoth.posa [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.posb [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swapX [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_setX [in Combi.LRrule.Greene_inv]
+nvar [in Combi.SymGroup.Frobenius_char]
+nvar [in Combi.MPoly.sympoly]
+

P

+P [in Combi.MPoly.Schur_altdef]
+pact [in Combi.Combi.permuted]
+Pair [in Combi.LRrule.Schensted]
+PermLattice.Exports.n [in Combi.SymGroup.weak_order]
+PermLattice.n [in Combi.SymGroup.weak_order]
+Pla [in Combi.MPoly.homogsym]
+Plla [in Combi.MPoly.homogsym]
+Pm [in Combi.MPoly.Schur_altdef]
+pol [in Combi.MPoly.Cauchy]
+Pol [in Combi.MPoly.homogsym]
+polX [in Combi.MPoly.Cauchy]
+polXY [in Combi.MPoly.Cauchy]
+polY [in Combi.MPoly.Cauchy]
+polZ [in Combi.MPoly.Cauchy]
+pos [in Combi.LRrule.Schensted]
+

R

+reduced [in Combi.SymGroup.presentSn]
+reduced [in Combi.SymGroup.presentSn]
+RefinementOrder.Exports.intcompnref [in Combi.Combi.composition]
+RefinementOrder.SetIn [in Combi.Combi.composition]
+RefinmentOrder.Exports.join_finer_eq [in Combi.Combi.setpartition]
+reprS [in Combi.SymGroup.reprSn]
+res [in Combi.Combi.skewpart]
+rho [in Combi.MPoly.antisym]
+rho [in Combi.MPoly.antisym]
+rho [in Combi.MPoly.Schur_altdef]
+rho [in Combi.MPoly.Schur_altdef]
+rho [in Combi.MPoly.Schur_altdef]
+rho [in Combi.MPoly.Schur_altdef]
+rho [in Combi.MPoly.MurnaghanNakayama]
+rsh [in Combi.LRrule.Greene]
+

S

+S [in Combi.MPoly.sympoly]
+Schur [in Combi.LRrule.freeSchur]
+Schur [in Combi.LRrule.therule]
+Schur [in Combi.LRrule.therule]
+Schur [in Combi.LRrule.implem]
+scover [in Combi.LRrule.Greene]
+SetContainingBothLeft.posa [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.posb [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.swap [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.swap_set [in Combi.LRrule.Greene_inv]
+SF [in Combi.MPoly.homogsym]
+SF [in Combi.MPoly.MurnaghanNakayama]
+SF [in Combi.MPoly.MurnaghanNakayama]
+SF [in Combi.MPoly.MurnaghanNakayama]
+SF [in Combi.MPoly.MurnaghanNakayama]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SF [in Combi.MPoly.sympoly]
+SnXm [in Combi.SymGroup.towerSn]
+SnXm [in Combi.SymGroup.towerSn]
+sorted [in Combi.SSRcomplements.sorted]
+SP [in Combi.MPoly.sympoly]
+start [in Combi.Combi.skewpart]
+stop [in Combi.Combi.skewpart]
+stpn [in Combi.HookFormula.Frobenius_ident]
+sz [in Combi.MPoly.Schur_altdef]
+

T

+T [in Combi.MPoly.Schur_altdef]
+TabPair [in Combi.LRrule.Schensted]
+Tm [in Combi.MPoly.Schur_altdef]
+

W

+WeakOrder.Exports.n [in Combi.SymGroup.weak_order]
+WeakOrder.n [in Combi.SymGroup.weak_order]
+word [in Combi.Basic.congr]
+word [in Combi.Basic.congr]
+word [in Combi.Basic.congr]
+word [in Combi.Basic.congr]
+wp [in Combi.Combi.permuted]
+

Y

+YoungLattice.Exports.intpartYoung [in Combi.Combi.partition]
+

+

Definition Index

+

A

+add [in Combi.LRrule.implem]
+add_mesym [in Combi.MPoly.Schur_altdef]
+add_ribbon_intpartn [in Combi.Combi.skewpart]
+add_ribbon [in Combi.Combi.skewpart]
+add_ribbon_on [in Combi.Combi.skewpart]
+admissible [in ALEA.Ccpo]
+admissible2 [in ALEA.Ccpo]
+allLeq [in Combi.Basic.ordtype]
+allLtn [in Combi.Basic.ordtype]
+alternpol [in Combi.MPoly.antisym]
+antim [in Combi.MPoly.antisym]
+antisym [in Combi.MPoly.antisym]
+antisym_keyed [in Combi.MPoly.antisym]
+append_nth [in Combi.Combi.stdtab]
+app2 [in ALEA.Ccpo]
+arm_length [in Combi.HookFormula.hook]
+auxbij [in Combi.LRrule.Yam_plact]
+auxbij_inv [in Combi.LRrule.Yam_plact]
+

B

+bal_of_Dyck [in Combi.Combi.Dyckword]
+bal_hsz [in Combi.Combi.Dyckword]
+bijLR [in Combi.LRrule.therule]
+bijLRyam [in Combi.LRrule.therule]
+bij_LRsupport [in Combi.LRrule.freeSchur]
+bintree_bintreesz__canonical__fintype_SubFinite [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__fintype_Finite [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_SubCountable [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_Countable [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_SubChoice [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__choice_Choice [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_SubEquality [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_Equality [in Combi.Combi.bintree]
+bintree_bintreesz__canonical__eqtype_SubType [in Combi.Combi.bintree]
+bintree_bintree__canonical__choice_Countable [in Combi.Combi.bintree]
+bintree_bintree__canonical__choice_Choice [in Combi.Combi.bintree]
+bintree_bintree__canonical__eqtype_Equality [in Combi.Combi.bintree]
+bintree_sind [in Combi.Combi.bintree]
+bintree_rec [in Combi.Combi.bintree]
+bintree_ind [in Combi.Combi.bintree]
+bintree_rect [in Combi.Combi.bintree]
+bintree_of_Dyck [in Combi.Combi.Dyckword]
+bin_to_ordtree [in Combi.Combi.ordtree]
+bin_to_forest [in Combi.Combi.ordtree]
+bool_of_brace [in Combi.Combi.Dyckword]
+BoxIn [in Combi.Combi.partition]
+brace_of_bool [in Combi.Combi.Dyckword]
+brace_sind [in Combi.Combi.Dyckword]
+brace_rec [in Combi.Combi.Dyckword]
+brace_ind [in Combi.Combi.Dyckword]
+brace_rect [in Combi.Combi.Dyckword]
+braidC [in Combi.SymGroup.presentSn]
+braidclass [in Combi.SymGroup.presentSn]
+braidcongr [in Combi.SymGroup.presentSn]
+braidrule [in Combi.SymGroup.presentSn]
+braid_reduces [in Combi.SymGroup.presentSn]
+braid_cat [in Combi.SymGroup.presentSn]
+braid_catr [in Combi.SymGroup.presentSn]
+braid_catl [in Combi.SymGroup.presentSn]
+braid_rcons [in Combi.SymGroup.presentSn]
+braid_cons [in Combi.SymGroup.presentSn]
+braid_aba [in Combi.SymGroup.presentSn]
+Builders_1.HB_unnamed_factory_3 [in Combi.Basic.ordtype]
+Builders_6.Builders_6_T__canonical__Order_TPOrder [in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_TPreorder [in Combi.Combi.composition]
+Builders_6.HB_unnamed_factory_8 [in Combi.Combi.composition]
+Builders_6.top [in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_POrder [in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__Order_Preorder [in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__choice_Choice [in Combi.Combi.composition]
+Builders_6.Builders_6_T__canonical__eqtype_Equality [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_BPOrder [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_BPreorder [in Combi.Combi.composition]
+Builders_1.HB_unnamed_factory_3 [in Combi.Combi.composition]
+Builders_1.bottom [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_POrder [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__Order_Preorder [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__choice_Choice [in Combi.Combi.composition]
+Builders_1.Builders_1_T__canonical__eqtype_Equality [in Combi.Combi.composition]
+bump [in Combi.LRrule.Schensted]
+bumped [in Combi.LRrule.Schensted]
+bumprow [in Combi.LRrule.Schensted]
+

C

+canporbit [in Combi.SymGroup.cycletype]
+canword [in Combi.SymGroup.presentSn]
+cast_intpartn [in Combi.Combi.partition]
+cast_set [in Combi.SSRcomplements.ordcast]
+cast_cons [in Combi.LRrule.Greene]
+Catalan [in Combi.Combi.Dyckword]
+Catalan_bin [in Combi.Combi.bintree]
+Catalan_bin_leq [in Combi.Combi.bintree]
+catlang [in Combi.LRrule.freeSchur]
+cat_left [in Combi.Combi.bintree]
+cat_Dyck [in Combi.Combi.Dyckword]
+Cauchy_co_hpXY__canonical__Algebra_Additive [in Combi.MPoly.Cauchy]
+Cauchy_co_hp__canonical__GRing_Linear [in Combi.MPoly.Cauchy]
+Cauchy_co_hp__canonical__Algebra_Additive [in Combi.MPoly.Cauchy]
+Cauchy_kernel [in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_LRMorphism [in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_Linear [in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__GRing_RMorphism [in Combi.MPoly.Cauchy]
+Cauchy_evalXY__canonical__Algebra_Additive [in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_LRMorphism [in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_Linear [in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__GRing_RMorphism [in Combi.MPoly.Cauchy]
+Cauchy_polY_XY__canonical__Algebra_Additive [in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_LRMorphism [in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_Linear [in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__GRing_RMorphism [in Combi.MPoly.Cauchy]
+Cauchy_polX_XY__canonical__Algebra_Additive [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzSemiAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzSemiAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzLalgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzLSemiAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzLalgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzLSemiAlgebra [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_Lmodule [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_LSemiModule [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzRing [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_NzSemiRing [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzRing [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__GRing_PzSemiRing [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_Zmodule [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_Nmodule [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddSemigroup [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddUMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_AddMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseZmodule [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_ChoiceBaseAddUMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseAddUMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_ChoiceBaseAddMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__Algebra_BaseAddMagma [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__choice_Choice [in Combi.MPoly.Cauchy]
+Cauchy_polXY__canonical__eqtype_Equality [in Combi.MPoly.Cauchy]
+cfextprod [in Combi.SymGroup.towerSn]
+cfuniCT [in Combi.SymGroup.cycletype]
+changeUT [in Combi.LRrule.freeSchur]
+charfun [in Combi.HookFormula.hook]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.std]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.std]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.std]
+choice_Choice__to__eqtype_hasDecEq [in Combi.MPoly.homogsym]
+choice_Choice__to__choice_hasChoice [in Combi.MPoly.homogsym]
+choice_Countable__to__choice_Choice_isCountable__78 [in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq__76 [in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice__74 [in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable__21 [in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq__19 [in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice__17 [in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.partition]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.partition]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.partition]
+choice_Countable__to__choice_Choice_isCountable__18 [in Combi.Combi.bintree]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.bintree]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.bintree]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.bintree]
+choice_Choice__to__choice_hasChoice [in Combi.Combi.bintree]
+choice_Choice__to__eqtype_hasDecEq [in Combi.LRrule.Schensted]
+choice_Choice__to__choice_hasChoice__14 [in Combi.LRrule.Schensted]
+choice_Choice__to__choice_hasChoice [in Combi.LRrule.Schensted]
+choice_Countable__to__choice_Choice_isCountable__25 [in Combi.Combi.Dyckword]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.Dyckword]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.Dyckword]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.Dyckword]
+choice_Choice__to__choice_hasChoice [in Combi.Combi.Dyckword]
+choice_Countable__to__choice_Choice_isCountable__20 [in Combi.Combi.stdtab]
+choice_Countable__to__eqtype_hasDecEq__18 [in Combi.Combi.stdtab]
+choice_Countable__to__choice_hasChoice__16 [in Combi.Combi.stdtab]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.stdtab]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.stdtab]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.stdtab]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.tableau]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.tableau]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.tableau]
+choice_Countable__to__choice_Choice_isCountable__18 [in Combi.Combi.ordtree]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.ordtree]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.ordtree]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.ordtree]
+choice_Choice__to__choice_hasChoice [in Combi.Combi.ordtree]
+choice_Countable__to__choice_Choice_isCountable__24 [in Combi.Combi.Yamanouchi]
+choice_Countable__to__eqtype_hasDecEq__22 [in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_hasChoice__20 [in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.Yamanouchi]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.Yamanouchi]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.permuted]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.permuted]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.permuted]
+choice_Countable__to__choice_Choice_isCountable [in Combi.SymGroup.presentSn]
+choice_Countable__to__eqtype_hasDecEq [in Combi.SymGroup.presentSn]
+choice_Countable__to__choice_hasChoice [in Combi.SymGroup.presentSn]
+choice_Choice__to__eqtype_hasDecEq [in Combi.MPoly.sympoly]
+choice_Choice__to__choice_hasChoice [in Combi.MPoly.sympoly]
+choice_Countable__to__choice_Choice_isCountable__28 [in Combi.Combi.composition]
+choice_Countable__to__eqtype_hasDecEq__26 [in Combi.Combi.composition]
+choice_Countable__to__choice_hasChoice__24 [in Combi.Combi.composition]
+choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.composition]
+choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.composition]
+choice_Countable__to__choice_hasChoice [in Combi.Combi.composition]
+choose_corner [in Combi.HookFormula.hook]
+class [in ALEA.Misc]
+classCT [in Combi.SymGroup.cycletype]
+classfun_cfdot__canonical__sesquilinear_Dot [in Combi.SymGroup.Frobenius_char]
+classfun_cfdot__canonical__sesquilinear_Hermitian [in Combi.SymGroup.Frobenius_char]
+classfun_cfdot__canonical__sesquilinear_Bilinear [in Combi.SymGroup.Frobenius_char]
+CMbij [in Combi.SymGroup.cycletype]
+cnvarhomsym [in Combi.MPoly.homogsym]
+cnvarsym [in Combi.MPoly.sympoly]
+cocode [in Combi.SymGroup.presentSn]
+cocode_rec [in Combi.SymGroup.presentSn]
+code [in Combi.SymGroup.presentSn]
+coeff_prodgen [in Combi.MPoly.sympoly]
+coeff_prodgen_seq [in Combi.MPoly.sympoly]
+colcomp [in Combi.Combi.composition]
+colcompn [in Combi.Combi.composition]
+colpartn [in Combi.Combi.partition]
+commword [in Combi.LRrule.freeSchur]
+comp [in ALEA.Ccpo]
+compn [in ALEA.Misc]
+composition_intcompn__canonical__fintype_SubFinite [in Combi.Combi.composition]
+composition_intcompn__canonical__fintype_Finite [in Combi.Combi.composition]
+composition_intcompn__canonical__choice_SubCountable [in Combi.Combi.composition]
+composition_intcompn__canonical__choice_Countable [in Combi.Combi.composition]
+composition_intcompn__canonical__choice_SubChoice [in Combi.Combi.composition]
+composition_intcompn__canonical__choice_Choice [in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_SubEquality [in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_Equality [in Combi.Combi.composition]
+composition_intcompn__canonical__eqtype_SubType [in Combi.Combi.composition]
+composition_intcomp__canonical__choice_SubCountable [in Combi.Combi.composition]
+composition_intcomp__canonical__choice_Countable [in Combi.Combi.composition]
+composition_intcomp__canonical__choice_SubChoice [in Combi.Combi.composition]
+composition_intcomp__canonical__choice_Choice [in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_SubEquality [in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_Equality [in Combi.Combi.composition]
+composition_intcomp__canonical__eqtype_SubType [in Combi.Combi.composition]
+congrrule [in Combi.Basic.congr]
+congruence_rule [in Combi.Basic.congr]
+congruence_rel [in Combi.Basic.congr]
+conjbij [in Combi.SymGroup.cycletype]
+conj_intpartn [in Combi.Combi.partition]
+conj_intpart [in Combi.Combi.partition]
+conj_part [in Combi.Combi.partition]
+conj_stdtabsh [in Combi.Combi.stdtab]
+conj_stdtabn [in Combi.Combi.stdtab]
+conj_tab_expl2 [in Combi.Combi.stdtab]
+conj_tab_expl1 [in Combi.Combi.stdtab]
+conj_tab [in Combi.Combi.stdtab]
+conn4_skew [in Combi.Combi.skewpart]
+cont_app [in ALEA.Ccpo]
+cont2 [in ALEA.Ccpo]
+corner_box [in Combi.HookFormula.hook]
+covers [in Combi.Basic.ordtype]
+co_hpXY [in Combi.MPoly.Cauchy]
+co_hp [in Combi.MPoly.Cauchy]
+cpo_ord_equiv [in ALEA.Ccpo]
+cshift [in ALEA.Ccpo]
+cte [in ALEA.Ccpo]
+CTpartn [in Combi.SymGroup.cycletype]
+cut_k [in Combi.Combi.vectNK]
+cut3 [in Combi.Combi.vectNK]
+cycle_typeSn [in Combi.SymGroup.cycletype]
+cycle_type [in Combi.SymGroup.cycletype]
+cycle_dec [in Combi.SymGroup.cycles]
+cyclic [in Combi.SymGroup.cycles]
+cymap [in Combi.SymGroup.cycletype]
+cymapcan [in Combi.SymGroup.cycletype]
+

D

+Datatypes_nat__canonical__ordtype_InhOrder [in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_InhLattice [in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_InhPOrder [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhTBLattice [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinOrder [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinLattice [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhFinite [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhOrder [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhLattice [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_InhPOrder [in Combi.Basic.ordtype]
+Datatypes_prod__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+Datatypes_option__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+Datatypes_nat__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+Datatypes_bool__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+Datatypes_unit__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+decr_nth_intpart [in Combi.Combi.partition]
+decr_nth [in Combi.Combi.partition]
+decr_yam [in Combi.Combi.Yamanouchi]
+Delta [in Combi.SymGroup.presentSn]
+depth_ordtree [in Combi.Combi.ordtree]
+descset [in Combi.Combi.composition]
+dhomog_of_homogsym [in Combi.MPoly.homogsym]
+diag [in ALEA.Ccpo]
+diff_shape [in Combi.Combi.partition]
+disjoint_psupports [in Combi.SymGroup.cycles]
+disp [in Combi.LRrule.extract]
+dominant [in Combi.MPoly.antisym]
+dominate [in Combi.Combi.tableau]
+dominate_rev_trans [in Combi.Combi.tableau]
+dominate_rec [in Combi.Combi.tableau]
+double_app [in ALEA.Ccpo]
+dyck [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_SubCountable [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_Countable [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_SubChoice [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__choice_Choice [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_SubEquality [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_Equality [in Combi.Combi.Dyckword]
+Dyckword_Dyck__canonical__eqtype_SubType [in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__fintype_Finite [in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__choice_Countable [in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__choice_Choice [in Combi.Combi.Dyckword]
+Dyckword_brace__canonical__eqtype_Equality [in Combi.Combi.Dyckword]
+Dyck_of_bal [in Combi.Combi.Dyckword]
+Dyck_hsz [in Combi.Combi.Dyckword]
+Dyck_of_bintree [in Combi.Combi.Dyckword]
+Dyck_word [in Combi.Combi.Dyckword]
+Dyck_prefix [in Combi.Combi.Dyckword]
+

E

+eltr [in Combi.SymGroup.presentSn]
+eltrD [in Combi.SymGroup.presentSn]
+eltrL [in Combi.SymGroup.presentSn]
+eltrp [in Combi.MPoly.antisym]
+eltrR [in Combi.SymGroup.presentSn]
+empty_intpart [in Combi.Combi.partition]
+ends_at [in Combi.HookFormula.hook]
+enum_stdwordn [in Combi.Combi.std]
+enum_box_skew [in Combi.Combi.partition]
+enum_partn [in Combi.Combi.partition]
+enum_partns [in Combi.Combi.partition]
+enum_partnsk [in Combi.Combi.partition]
+enum_bintreesz [in Combi.Combi.bintree]
+enum_bintreesz_leq [in Combi.Combi.bintree]
+enum_trace [in Combi.HookFormula.hook]
+enum_stdtabn [in Combi.Combi.stdtab]
+enum_stdtabsh [in Combi.Combi.stdtab]
+enum_ordtreesz [in Combi.Combi.ordtree]
+enum_yameval [in Combi.Combi.Yamanouchi]
+enum_yamevaln [in Combi.Combi.Yamanouchi]
+enum_codesz [in Combi.SymGroup.presentSn]
+enum_union [in Combi.Basic.combclass]
+enum_compn [in Combi.Combi.composition]
+enum_compn_rec [in Combi.Combi.composition]
+eqeval [in Combi.MPoly.Schur_altdef]
+eqtype_Equality__to__eqtype_hasDecEq [in Combi.LRrule.Schensted]
+eqtype_Equality__to__eqtype_hasDecEq [in Combi.Combi.Dyckword]
+eq_inv [in Combi.Combi.std]
+eq_bintree [in Combi.Combi.bintree]
+eq_ord [in ALEA.Ccpo]
+eq_rel [in ALEA.Ccpo]
+eq_ordtree [in Combi.Combi.ordtree]
+eq_forest [in Combi.Combi.ordtree]
+eq_nat2_dec [in ALEA.Misc]
+eval [in Combi.MPoly.Schur_altdef]
+evalseq [in Combi.Combi.Yamanouchi]
+evalseq_count [in Combi.Combi.Yamanouchi]
+evalXY [in Combi.MPoly.Cauchy]
+Example1.choice_Countable__to__choice_Choice_isCountable [in Combi.Basic.combclass]
+Example1.choice_Countable__to__eqtype_hasDecEq [in Combi.Basic.combclass]
+Example1.choice_Countable__to__choice_hasChoice [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__fintype_SubFinite [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__fintype_Finite [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_SubCountable [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_Countable [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_SubChoice [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__choice_Choice [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_SubEquality [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_Equality [in Combi.Basic.combclass]
+Example1.Example1_isOne__canonical__eqtype_SubType [in Combi.Basic.combclass]
+Example1.fintype_Finite__to__fintype_isFinite [in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_15 [in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_10 [in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_9 [in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_8 [in Combi.Basic.combclass]
+Example1.HB_unnamed_mixin_7 [in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_3 [in Combi.Basic.combclass]
+Example1.HB_unnamed_factory_1 [in Combi.Basic.combclass]
+Example1.is_one [in Combi.Basic.combclass]
+Example2.choice_Countable__to__choice_Choice_isCountable [in Combi.Basic.combclass]
+Example2.choice_Countable__to__eqtype_hasDecEq [in Combi.Basic.combclass]
+Example2.choice_Countable__to__choice_hasChoice [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__fintype_SubFinite [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__fintype_Finite [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_SubCountable [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_Countable [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_SubChoice [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__choice_Choice [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_SubEquality [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_Equality [in Combi.Basic.combclass]
+Example2.Example2_isOne__canonical__eqtype_SubType [in Combi.Basic.combclass]
+Example2.fintype_Finite__to__fintype_isFinite [in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_30 [in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_25 [in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_24 [in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_23 [in Combi.Basic.combclass]
+Example2.HB_unnamed_mixin_22 [in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_18 [in Combi.Basic.combclass]
+Example2.HB_unnamed_factory_16 [in Combi.Basic.combclass]
+Example2.is_one [in Combi.Basic.combclass]
+Example3.choice_Countable__to__choice_Choice_isCountable [in Combi.Basic.combclass]
+Example3.choice_Countable__to__eqtype_hasDecEq [in Combi.Basic.combclass]
+Example3.choice_Countable__to__choice_hasChoice [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__fintype_SubFinite [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__fintype_Finite [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_SubCountable [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_Countable [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_SubChoice [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__choice_Choice [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_SubEquality [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_Equality [in Combi.Basic.combclass]
+Example3.Example3_isOne__canonical__eqtype_SubType [in Combi.Basic.combclass]
+Example3.fintype_Finite__to__fintype_isFinite [in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_45 [in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_40 [in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_39 [in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_38 [in Combi.Basic.combclass]
+Example3.HB_unnamed_mixin_37 [in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_33 [in Combi.Basic.combclass]
+Example3.HB_unnamed_factory_31 [in Combi.Basic.combclass]
+Example3.is_one [in Combi.Basic.combclass]
+exc [in ALEA.Misc]
+extprod_mx_repr [in Combi.SymGroup.towerSn]
+extract [in Combi.LRrule.Greene]
+extractpred [in Combi.LRrule.Greene]
+ext_tab [in Combi.MPoly.Schur_altdef]
+ext_tab_fun [in Combi.MPoly.Schur_altdef]
+

F

+famY [in Combi.MPoly.Cauchy]
+famYinv [in Combi.MPoly.Cauchy]
+fbbij [in Combi.Combi.fibered_set]
+Fchar [in Combi.SymGroup.Frobenius_char]
+Fchar_inv [in Combi.SymGroup.Frobenius_char]
+fcomp [in ALEA.Ccpo]
+fcomp2 [in ALEA.Ccpo]
+Fcontm [in ALEA.Ccpo]
+fcont_compn [in ALEA.Ccpo]
+fcont_SEQ [in ALEA.Ccpo]
+fcont_COMP [in ALEA.Ccpo]
+fcont_Comp [in ALEA.Ccpo]
+fcont_comp [in ALEA.Ccpo]
+fcont_mshift [in ALEA.Ccpo]
+fcont_ishift [in ALEA.Ccpo]
+fcont_app [in ALEA.Ccpo]
+fcont_lub [in ALEA.Ccpo]
+fcont_fun [in ALEA.Ccpo]
+fcont0 [in ALEA.Ccpo]
+fcont2_comp [in ALEA.Ccpo]
+fcont2_COMP [in ALEA.Ccpo]
+feq [in ALEA.Misc]
+fiber [in Combi.Combi.fibered_set]
+Fif [in ALEA.Ccpo]
+fif [in ALEA.Ccpo]
+filter_le_tab [in Combi.Combi.skewtab]
+filter_gt_tab [in Combi.Combi.tableau]
+Finite [in ALEA.Qmeasure]
+finset_set_of__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.std]
+fintype_Finite__to__fintype_isFinite__87 [in Combi.Combi.partition]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.partition]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.bintree]
+fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.bintree]
+fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.bintree]
+fintype_Finite__to__choice_hasChoice [in Combi.Combi.bintree]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.Dyckword]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.subseq]
+fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.subseq]
+fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.subseq]
+fintype_Finite__to__choice_hasChoice [in Combi.Combi.subseq]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.ordtree]
+fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.ordtree]
+fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.ordtree]
+fintype_Finite__to__choice_hasChoice [in Combi.Combi.ordtree]
+fintype_Finite__to__fintype_isFinite__33 [in Combi.Combi.Yamanouchi]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.Yamanouchi]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.setpartition]
+fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.setpartition]
+fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.setpartition]
+fintype_Finite__to__choice_hasChoice [in Combi.Combi.setpartition]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.permuted]
+fintype_ordinal__canonical__Order_FinTBTotal [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBTotal [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BTotal [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBDistrLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBDistrLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BDistrLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderTBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBJoinSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BJoinSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinBMeetSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBMeetSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BMeetSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTBPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinBPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TBPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_BPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTJoinSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_FinTPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhFinite [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TTotal [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TDistrLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_InhPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubTLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_JoinSubTLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_MeetSubTLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_SubPOrderTLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TLattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TJoinSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TMeetSemilattice [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__Order_TPOrder [in Combi.Basic.ordtype]
+fintype_ordinal__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+fintype_Finite__to__fintype_isFinite [in Combi.SymGroup.presentSn]
+fintype_Finite__to__fintype_isFinite [in Combi.Combi.composition]
+FIXP [in ALEA.Ccpo]
+Fixp [in ALEA.Ccpo]
+fixp [in ALEA.Ccpo]
+Fixp_cont [in ALEA.Ccpo]
+fixp_cte [in ALEA.Ccpo]
+Flip [in ALEA.Qmeasure]
+flip [in ALEA.Qmeasure]
+flip [in Combi.Combi.bintree]
+flipsz [in Combi.Combi.bintree]
+fnatO_intro [in ALEA.Ccpo]
+forest_to_bintree [in Combi.Combi.ordtree]
+fprob [in ALEA.Qmeasure]
+freeSchur [in Combi.LRrule.freeSchur]
+Frobenius_char_Fchar_inv__canonical__GRing_Linear [in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar_inv__canonical__Algebra_Additive [in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar__canonical__GRing_Linear [in Combi.SymGroup.Frobenius_char]
+Frobenius_char_Fchar__canonical__Algebra_Additive [in Combi.SymGroup.Frobenius_char]
+from_vct [in Combi.Combi.bintree]
+from_vct_acc [in Combi.Combi.bintree]
+from_vct_rec [in Combi.Combi.bintree]
+from_left [in Combi.Combi.bintree]
+from_revdual [in Combi.LRrule.plactic]
+from_descset [in Combi.Combi.composition]
+fun_ext [in ALEA.Ccpo]
+fun2 [in ALEA.Ccpo]
+

G

+genclass [in Combi.Basic.congr]
+genclass_hom [in Combi.Basic.congr]
+genclass_multhom [in Combi.Basic.congr]
+gencongr [in Combi.Basic.congr]
+gencongr_hom [in Combi.Basic.congr]
+gencongr_multhom [in Combi.Basic.congr]
+get_tab [in Combi.Combi.tableau]
+Greene_col [in Combi.LRrule.Greene]
+Greene_row [in Combi.LRrule.Greene]
+Greene_rel [in Combi.LRrule.Greene]
+Greene_rel_t [in Combi.LRrule.Greene]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__76 [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__Algebra_isNmodMorphism [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__70 [in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__65 [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__59 [in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__53 [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable__48 [in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism__42 [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__GRing_isScalable [in Combi.MPoly.Cauchy]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [in Combi.MPoly.Cauchy]
+GRing_Zmodule_isLmodule__to__GRing_Nmodule_isLSemiModule [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__GRing_PzSemiRing_isNonZero [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__GRing_Nmodule_isPzSemiRing [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_AddMagma_isAddSemigroup [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseAddMagma_isAddMagma [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseAddUMagma_isAddUMagma [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_BaseZmoduleNmodule_isZmodule [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasAdd [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasZero [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__Algebra_hasOpp [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__eqtype_hasDecEq [in Combi.MPoly.Cauchy]
+GRing_NzRing__to__choice_hasChoice [in Combi.MPoly.Cauchy]
+GRing_isLinear__to__Algebra_isNmodMorphism__77 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__75 [in Combi.MPoly.homogsym]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__65 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__63 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__58 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__56 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism__45 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable__43 [in Combi.MPoly.homogsym]
+GRing_isLinear__to__Algebra_isNmodMorphism [in Combi.MPoly.homogsym]
+GRing_isLinear__to__GRing_isScalable [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseAddMagma_isAddMagma [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_isSubBaseAddUMagma [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasAdd [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_AddMagma_isAddSemigroup [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasZero [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseZmoduleNmodule_isZmodule [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_hasOpp [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__Algebra_BaseAddUMagma_isAddUMagma [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__GRing_isSubLSemiModule [in Combi.MPoly.homogsym]
+GRing_SubChoice_isSubLmodule__to__GRing_Nmodule_isLSemiModule [in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__Algebra_isAddClosed [in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__Algebra_isOppClosed [in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__GRing_isScaleClosed [in Combi.MPoly.homogsym]
+GRing_isSubmodClosed__to__GRing_isScaleClosed [in Combi.MPoly.antisym]
+GRing_isZmodClosed__to__Algebra_isOppClosed [in Combi.MPoly.antisym]
+GRing_isZmodClosed__to__Algebra_isAddClosed [in Combi.MPoly.antisym]
+GRing_isLinear__to__Algebra_isNmodMorphism__17 [in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__GRing_isScalable__15 [in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__Algebra_isNmodMorphism [in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__GRing_isScalable [in Combi.SymGroup.Frobenius_char]
+GRing_isLinear__to__Algebra_isNmodMorphism__108 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__106 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__99 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__97 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__90 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__88 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism__81 [in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable__79 [in Combi.MPoly.sympoly]
+GRing_isZmodMorphism__to__Algebra_isNmodMorphism [in Combi.MPoly.sympoly]
+GRing_SubComUnitRing_isSubIntegralDomain__to__GRing_ComUnitRing_isIntegral [in Combi.MPoly.sympoly]
+GRing_SubNzRing_isSubUnitRing__to__GRing_NzRing_hasMulInverse [in Combi.MPoly.sympoly]
+GRing_SubSemiRing_isSubComSemiRing__to__GRing_SemiRing_hasCommutativeMul [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_AddMagma_isAddSemigroup [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasZero [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseZmoduleNmodule_isZmodule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasOpp [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseAddUMagma_isAddUMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_isSubPzSemiRing [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_Nmodule_isPzSemiRing [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_BaseAddMagma_isAddMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_isSubLSemiModule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_Nmodule_isLSemiModule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_LSemiModule_isLSemiAlgebra [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_isSubBaseAddUMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_LSemiAlgebra_isSemiAlgebra [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__Algebra_hasAdd [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzAlgebra__to__GRing_PzSemiRing_isNonZero [in Combi.MPoly.sympoly]
+GRing_isLinear__to__Algebra_isNmodMorphism [in Combi.MPoly.sympoly]
+GRing_isLinear__to__GRing_isScalable [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasAdd [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_AddMagma_isAddSemigroup [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasZero [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseZmoduleNmodule_isZmodule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_hasOpp [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseAddUMagma_isAddUMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_isSubPzSemiRing [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_Nmodule_isPzSemiRing [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_BaseAddMagma_isAddMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_isSubLSemiModule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_Nmodule_isLSemiModule [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_LSemiModule_isLSemiAlgebra [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__Algebra_isSubBaseAddUMagma [in Combi.MPoly.sympoly]
+GRing_SubChoice_isSubNzLalgebra__to__GRing_PzSemiRing_isNonZero [in Combi.MPoly.sympoly]
+

H

+hasincr [in Combi.MPoly.Schur_altdef]
+has_no_square [in Combi.Combi.skewpart]
+HB_unnamed_mixin_77 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_74 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_72 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_68 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_66 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_63 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_61 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_57 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_55 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_54 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_51 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_50 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_46 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_44 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_43 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_40 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_39 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_36 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_34 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_33 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_31 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_29 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_27 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_26 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_24 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_23 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_22 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_21 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_20 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_19 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_18 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_17 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_16 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_15 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_14 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_13 [in Combi.MPoly.Cauchy]
+HB_unnamed_factory_1 [in Combi.MPoly.Cauchy]
+HB_unnamed_mixin_15 [in Combi.Combi.std]
+HB_unnamed_factory_10 [in Combi.Combi.std]
+HB_unnamed_mixin_9 [in Combi.Combi.std]
+HB_unnamed_mixin_8 [in Combi.Combi.std]
+HB_unnamed_mixin_7 [in Combi.Combi.std]
+HB_unnamed_factory_3 [in Combi.Combi.std]
+HB_unnamed_factory_1 [in Combi.Combi.std]
+HB_unnamed_factory_86 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_84 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_83 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_80 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_79 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_78 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_73 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_71 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_70 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_68 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_67 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_66 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_61 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_60 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_59 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_54 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_53 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_51 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_50 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_48 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_47 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_46 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_41 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_40 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_39 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_36 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_35 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_34 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_33 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_32 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_31 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_30 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_29 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_28 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_27 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_26 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_15 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_14 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_13 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_12 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_8 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_7 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_6 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_3 [in Combi.MPoly.homogsym]
+HB_unnamed_factory_1 [in Combi.MPoly.homogsym]
+HB_unnamed_mixin_88 [in Combi.Combi.partition]
+HB_unnamed_factory_82 [in Combi.Combi.partition]
+HB_unnamed_mixin_81 [in Combi.Combi.partition]
+HB_unnamed_mixin_80 [in Combi.Combi.partition]
+HB_unnamed_mixin_79 [in Combi.Combi.partition]
+HB_unnamed_factory_72 [in Combi.Combi.partition]
+HB_unnamed_factory_70 [in Combi.Combi.partition]
+HB_unnamed_mixin_35 [in Combi.Combi.partition]
+HB_unnamed_factory_32 [in Combi.Combi.partition]
+HB_unnamed_mixin_30 [in Combi.Combi.partition]
+HB_unnamed_factory_25 [in Combi.Combi.partition]
+HB_unnamed_mixin_24 [in Combi.Combi.partition]
+HB_unnamed_mixin_23 [in Combi.Combi.partition]
+HB_unnamed_mixin_22 [in Combi.Combi.partition]
+HB_unnamed_factory_15 [in Combi.Combi.partition]
+HB_unnamed_factory_13 [in Combi.Combi.partition]
+HB_unnamed_mixin_12 [in Combi.Combi.partition]
+HB_unnamed_factory_10 [in Combi.Combi.partition]
+HB_unnamed_mixin_9 [in Combi.Combi.partition]
+HB_unnamed_mixin_8 [in Combi.Combi.partition]
+HB_unnamed_mixin_7 [in Combi.Combi.partition]
+HB_unnamed_factory_3 [in Combi.Combi.partition]
+HB_unnamed_factory_1 [in Combi.Combi.partition]
+HB_unnamed_mixin_28 [in Combi.Combi.bintree]
+HB_unnamed_factory_23 [in Combi.Combi.bintree]
+HB_unnamed_mixin_21 [in Combi.Combi.bintree]
+HB_unnamed_mixin_20 [in Combi.Combi.bintree]
+HB_unnamed_mixin_19 [in Combi.Combi.bintree]
+HB_unnamed_factory_14 [in Combi.Combi.bintree]
+HB_unnamed_factory_12 [in Combi.Combi.bintree]
+HB_unnamed_mixin_11 [in Combi.Combi.bintree]
+HB_unnamed_factory_7 [in Combi.Combi.bintree]
+HB_unnamed_mixin_6 [in Combi.Combi.bintree]
+HB_unnamed_factory_3 [in Combi.Combi.bintree]
+HB_unnamed_factory_1 [in Combi.Combi.bintree]
+HB_unnamed_mixin_3 [in Combi.SymGroup.towerSn]
+HB_unnamed_factory_1 [in Combi.SymGroup.towerSn]
+HB_unnamed_mixin_10 [in Combi.MPoly.antisym]
+HB_unnamed_factory_6 [in Combi.MPoly.antisym]
+HB_unnamed_mixin_5 [in Combi.MPoly.antisym]
+HB_unnamed_mixin_4 [in Combi.MPoly.antisym]
+HB_unnamed_factory_1 [in Combi.MPoly.antisym]
+HB_unnamed_mixin_17 [in Combi.LRrule.Schensted]
+HB_unnamed_mixin_16 [in Combi.LRrule.Schensted]
+HB_unnamed_factory_12 [in Combi.LRrule.Schensted]
+HB_unnamed_factory_10 [in Combi.LRrule.Schensted]
+HB_unnamed_mixin_9 [in Combi.LRrule.Schensted]
+HB_unnamed_factory_6 [in Combi.LRrule.Schensted]
+HB_unnamed_mixin_5 [in Combi.LRrule.Schensted]
+HB_unnamed_factory_3 [in Combi.LRrule.Schensted]
+HB_unnamed_factory_1 [in Combi.LRrule.Schensted]
+HB_unnamed_mixin_19 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_18 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_13 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_12 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_11 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_8 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_6 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_4 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_3 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_factory_1 [in Combi.SymGroup.Frobenius_char]
+HB_unnamed_mixin_28 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_27 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_26 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_21 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_19 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_18 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_13 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_12 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_8 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_7 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_4 [in Combi.Combi.Dyckword]
+HB_unnamed_mixin_3 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_1 [in Combi.Combi.Dyckword]
+HB_unnamed_factory_24 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_23 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_22 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_21 [in Combi.Combi.stdtab]
+HB_unnamed_factory_14 [in Combi.Combi.stdtab]
+HB_unnamed_factory_12 [in Combi.Combi.stdtab]
+HB_unnamed_factory_10 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_9 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_8 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_7 [in Combi.Combi.stdtab]
+HB_unnamed_factory_3 [in Combi.Combi.stdtab]
+HB_unnamed_factory_1 [in Combi.Combi.stdtab]
+HB_unnamed_mixin_11 [in Combi.Combi.subseq]
+HB_unnamed_mixin_10 [in Combi.Combi.subseq]
+HB_unnamed_mixin_9 [in Combi.Combi.subseq]
+HB_unnamed_mixin_8 [in Combi.Combi.subseq]
+HB_unnamed_factory_3 [in Combi.Combi.subseq]
+HB_unnamed_factory_1 [in Combi.Combi.subseq]
+HB_unnamed_factory_10 [in Combi.Combi.tableau]
+HB_unnamed_mixin_9 [in Combi.Combi.tableau]
+HB_unnamed_mixin_8 [in Combi.Combi.tableau]
+HB_unnamed_mixin_7 [in Combi.Combi.tableau]
+HB_unnamed_factory_3 [in Combi.Combi.tableau]
+HB_unnamed_factory_1 [in Combi.Combi.tableau]
+HB_unnamed_mixin_28 [in Combi.Combi.ordtree]
+HB_unnamed_factory_23 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_21 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_20 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_19 [in Combi.Combi.ordtree]
+HB_unnamed_factory_14 [in Combi.Combi.ordtree]
+HB_unnamed_factory_12 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_11 [in Combi.Combi.ordtree]
+HB_unnamed_factory_7 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_6 [in Combi.Combi.ordtree]
+HB_unnamed_factory_3 [in Combi.Combi.ordtree]
+HB_unnamed_factory_1 [in Combi.Combi.ordtree]
+HB_unnamed_mixin_34 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_28 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_27 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_26 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_25 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_18 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_16 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_15 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_10 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_9 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_8 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_7 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_3 [in Combi.Combi.Yamanouchi]
+HB_unnamed_factory_1 [in Combi.Combi.Yamanouchi]
+HB_unnamed_mixin_14 [in Combi.Combi.setpartition]
+HB_unnamed_factory_12 [in Combi.Combi.setpartition]
+HB_unnamed_mixin_11 [in Combi.Combi.setpartition]
+HB_unnamed_mixin_10 [in Combi.Combi.setpartition]
+HB_unnamed_mixin_9 [in Combi.Combi.setpartition]
+HB_unnamed_mixin_8 [in Combi.Combi.setpartition]
+HB_unnamed_factory_3 [in Combi.Combi.setpartition]
+HB_unnamed_factory_1 [in Combi.Combi.setpartition]
+hb_strip [in Combi.Combi.skewpart]
+HB_unnamed_mixin_15 [in Combi.Combi.permuted]
+HB_unnamed_factory_10 [in Combi.Combi.permuted]
+HB_unnamed_mixin_9 [in Combi.Combi.permuted]
+HB_unnamed_mixin_8 [in Combi.Combi.permuted]
+HB_unnamed_mixin_7 [in Combi.Combi.permuted]
+HB_unnamed_factory_3 [in Combi.Combi.permuted]
+HB_unnamed_factory_1 [in Combi.Combi.permuted]
+HB_unnamed_factory_112 [in Combi.Basic.ordtype]
+HB_unnamed_factory_101 [in Combi.Basic.ordtype]
+HB_unnamed_factory_92 [in Combi.Basic.ordtype]
+HB_unnamed_factory_81 [in Combi.Basic.ordtype]
+HB_unnamed_factory_70 [in Combi.Basic.ordtype]
+HB_unnamed_factory_61 [in Combi.Basic.ordtype]
+HB_unnamed_factory_54 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_52 [in Combi.Basic.ordtype]
+HB_unnamed_factory_49 [in Combi.Basic.ordtype]
+HB_unnamed_factory_44 [in Combi.Basic.ordtype]
+HB_unnamed_factory_39 [in Combi.Basic.ordtype]
+HB_unnamed_factory_35 [in Combi.Basic.ordtype]
+HB_unnamed_factory_33 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_31 [in Combi.Basic.ordtype]
+HB_unnamed_factory_28 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_26 [in Combi.Basic.ordtype]
+HB_unnamed_factory_23 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_21 [in Combi.Basic.ordtype]
+HB_unnamed_factory_18 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_16 [in Combi.Basic.ordtype]
+HB_unnamed_factory_13 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_12 [in Combi.Basic.ordtype]
+HB_unnamed_factory_9 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_8 [in Combi.Basic.ordtype]
+HB_unnamed_factory_6 [in Combi.Basic.ordtype]
+HB_unnamed_mixin_15 [in Combi.SymGroup.presentSn]
+HB_unnamed_factory_10 [in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_9 [in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_8 [in Combi.SymGroup.presentSn]
+HB_unnamed_mixin_7 [in Combi.SymGroup.presentSn]
+HB_unnamed_factory_3 [in Combi.SymGroup.presentSn]
+HB_unnamed_factory_1 [in Combi.SymGroup.presentSn]
+HB_unnamed_factory_111 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_110 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_109 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_104 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_102 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_101 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_100 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_95 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_93 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_92 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_91 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_86 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_84 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_83 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_82 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_77 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_75 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_73 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_72 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_70 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_69 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_67 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_66 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_64 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_63 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_61 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_60 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_44 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_42 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_41 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_40 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_37 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_36 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_35 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_34 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_33 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_32 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_31 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_30 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_29 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_28 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_27 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_26 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_25 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_24 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_23 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_8 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_7 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_6 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_3 [in Combi.MPoly.sympoly]
+HB_unnamed_factory_1 [in Combi.MPoly.sympoly]
+HB_unnamed_mixin_37 [in Combi.Combi.composition]
+HB_unnamed_factory_32 [in Combi.Combi.composition]
+HB_unnamed_mixin_31 [in Combi.Combi.composition]
+HB_unnamed_mixin_30 [in Combi.Combi.composition]
+HB_unnamed_mixin_29 [in Combi.Combi.composition]
+HB_unnamed_factory_22 [in Combi.Combi.composition]
+HB_unnamed_factory_20 [in Combi.Combi.composition]
+HB_unnamed_mixin_19 [in Combi.Combi.composition]
+HB_unnamed_mixin_18 [in Combi.Combi.composition]
+HB_unnamed_mixin_17 [in Combi.Combi.composition]
+HB_unnamed_factory_13 [in Combi.Combi.composition]
+HB_unnamed_factory_11 [in Combi.Combi.composition]
+head_leq_last_row [in Combi.Combi.tableau]
+height [in Combi.Combi.Dyckword]
+height_simpl [in Combi.Combi.Dyckword]
+Hla [in Combi.MPoly.Schur_altdef]
+Hlamu [in Combi.MPoly.Schur_altdef]
+homlang [in Combi.LRrule.freeSchur]
+homogsym_homsymdot__canonical__sesquilinear_Dot [in Combi.MPoly.homogsym]
+homogsym_homsymdot__canonical__sesquilinear_Hermitian [in Combi.MPoly.homogsym]
+homogsym_homsymdot__canonical__sesquilinear_Bilinear [in Combi.MPoly.homogsym]
+homogsym_cnvarhomsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_cnvarhomsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_map_homsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_map_homsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_omegahomsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_omegahomsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_in_homsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_in_homsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_homsymprod__canonical__sesquilinear_Bilinear [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__vector_Vector [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__vector_SemiVector [in Combi.MPoly.homogsym]
+homogsym_dhomog_of_homogsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_dhomog_of_homogsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_homsym__canonical__GRing_Linear [in Combi.MPoly.homogsym]
+homogsym_homsym__canonical__Algebra_Additive [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_SubLmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_Lmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_SubLSemiModule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__GRing_LSemiModule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubZmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubNmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_Zmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_Nmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddSemigroup [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubAddUMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddUMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_AddMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_SubBaseAddUMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseZmodule [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_ChoiceBaseAddUMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseAddUMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_ChoiceBaseAddMagma [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__Algebra_BaseAddMagma [in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__GRing_SubmodClosed [in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_ZmodClosed [in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_AddClosed [in Combi.MPoly.homogsym]
+homogsym_is_homsym__canonical__Algebra_OppClosed [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__choice_SubChoice [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__choice_Choice [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_SubEquality [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_Equality [in Combi.MPoly.homogsym]
+homogsym_homogsym__canonical__eqtype_SubType [in Combi.MPoly.homogsym]
+homsymdot [in Combi.MPoly.homogsym]
+homsyme [in Combi.MPoly.homogsym]
+homsymh [in Combi.MPoly.homogsym]
+homsymm [in Combi.MPoly.homogsym]
+homsymp [in Combi.MPoly.homogsym]
+homsymprod [in Combi.MPoly.homogsym]
+homsyms [in Combi.MPoly.homogsym]
+hookpart [in Combi.Combi.partition]
+hookpartn [in Combi.Combi.partition]
+hook_boxes [in Combi.HookFormula.hook]
+hook_box [in Combi.HookFormula.hook]
+hook_box_indices [in Combi.HookFormula.hook]
+hook_length_prod [in Combi.HookFormula.hook]
+hook_length [in Combi.HookFormula.hook]
+hyper_stdtabn [in Combi.LRrule.freeSchur]
+hyper_stdtab [in Combi.LRrule.freeSchur]
+hyper_stdtabsh [in Combi.Combi.stdtab]
+hyper_yameval [in Combi.Combi.Yamanouchi]
+hyper_yam [in Combi.Combi.Yamanouchi]
+hyper_yam_rev [in Combi.Combi.Yamanouchi]
+

I

+Id [in ALEA.Ccpo]
+id [in ALEA.Ccpo]
+ifte [in ALEA.Misc]
+Imon [in ALEA.Ccpo]
+imon [in ALEA.Ccpo]
+imon2 [in ALEA.Ccpo]
+Imon2 [in ALEA.Ccpo]
+included [in Combi.Combi.partition]
+incr_first_n [in Combi.Combi.partition]
+indporbit [in Combi.SymGroup.cycletype]
+indtree [in Combi.Combi.ordtree]
+indtreeforest [in Combi.Combi.ordtree]
+inh [in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited__to__choice_Choice [in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited__to__eqtype_Equality [in Combi.Basic.ordtype]
+Inhabited.Exports.ordtype_Inhabited_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+Inhabited.pack_ [in Combi.Basic.ordtype]
+Inhabited.phant_on_ [in Combi.Basic.ordtype]
+Inhabited.phant_clone [in Combi.Basic.ordtype]
+InhFinite.Exports.join_ordtype_InhFinite_between_fintype_Finite_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinite.Exports.join_ordtype_InhFinite_between_choice_Countable_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__fintype_Finite [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__fintype_Finite_class [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__choice_Countable [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__choice_Countable_class [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__choice_Choice [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhFinite.Exports.ordtype_InhFinite_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhFinite.pack_ [in Combi.Basic.ordtype]
+InhFinite.phant_on_ [in Combi.Basic.ordtype]
+InhFinite.phant_clone [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinLattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinJoinSemilattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinMeetSemilattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_Lattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinPOrder_and_Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinPOrder_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_Order_FinPreorder_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_Lattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_ordtype_InhFinite_and_Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_fintype_Finite_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.join_ordtype_InhFinLattice_between_choice_Countable_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinLattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinJoinSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinJoinSemilattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinMeetSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinMeetSemilattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhFinPOrder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinPOrder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_FinPreorder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_FinPreorder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhFinite_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__fintype_Finite [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__fintype_Finite_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__choice_Countable [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__choice_Countable_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhLattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_Lattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_Lattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_JoinSemilattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_MeetSemilattice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_POrder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__Order_Preorder [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__choice_Choice [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhFinLattice.Exports.ordtype_InhFinLattice_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhFinLattice.pack_ [in Combi.Basic.ordtype]
+InhFinLattice.phant_on_ [in Combi.Basic.ordtype]
+InhFinLattice.phant_clone [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinTotal_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinDistrLattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinLattice_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinLattice_and_Order_Total [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinLattice_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinJoinSemilattice_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinMeetSemilattice_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinPOrder_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinPOrder_and_Order_Total [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinPOrder_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_FinPreorder_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinite_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_ordtype_InhFinite_and_Order_Total [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_fintype_Finite_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_choice_Countable_and_ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.join_ordtype_InhFinOrder_between_Order_DistrLattice_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinTotal [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinTotal_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinDistrLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinDistrLattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinLattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinLattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinJoinSemilattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinJoinSemilattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinMeetSemilattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinMeetSemilattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinPOrder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinPOrder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_FinPreorder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_FinPreorder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhFinite_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__fintype_Finite [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__fintype_Finite_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__choice_Countable [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__choice_Countable_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhOrder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Total [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Total_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_DistrLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_DistrLattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhLattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhLattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Lattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Lattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_JoinSemilattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_MeetSemilattice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_POrder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__Order_Preorder [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__choice_Choice [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhFinOrder.Exports.ordtype_InhFinOrder_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhFinOrder.pack_ [in Combi.Basic.ordtype]
+InhFinOrder.phant_on_ [in Combi.Basic.ordtype]
+InhFinOrder.phant_clone [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPOrder_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_Order_FinPreorder_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_Order_POrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_ordtype_InhFinite_and_Order_Preorder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_fintype_Finite_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.join_ordtype_InhFinPOrder_between_choice_Countable_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_FinPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_FinPOrder_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_FinPreorder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_FinPreorder_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_InhFinite [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_InhFinite_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__fintype_Finite [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__fintype_Finite_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__choice_Countable [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__choice_Countable_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_POrder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__Order_Preorder [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__choice_Choice [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhFinPOrder.Exports.ordtype_InhFinPOrder_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhFinPOrder.pack_ [in Combi.Basic.ordtype]
+InhFinPOrder.phant_on_ [in Combi.Basic.ordtype]
+InhFinPOrder.phant_clone [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_Lattice [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_InhPOrder_and_Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_Lattice [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.join_ordtype_InhLattice_between_ordtype_Inhabited_and_Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_Lattice [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_Lattice_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_JoinSemilattice_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_MeetSemilattice_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_POrder [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__Order_Preorder [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__choice_Choice [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhLattice.Exports.ordtype_InhLattice_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhLattice.pack_ [in Combi.Basic.ordtype]
+InhLattice.phant_on_ [in Combi.Basic.ordtype]
+InhLattice.phant_clone [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_Order_DistrLattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_InhLattice_and_Order_Total [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_InhPOrder_and_Order_Total [in Combi.Basic.ordtype]
+InhOrder.Exports.join_ordtype_InhOrder_between_ordtype_Inhabited_and_Order_Total [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Total [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Total_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_DistrLattice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_DistrLattice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_InhLattice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_InhLattice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Lattice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Lattice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_JoinSemilattice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_MeetSemilattice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_POrder [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__Order_Preorder [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__choice_Choice [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhOrder.Exports.ordtype_InhOrder_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhOrder.pack_ [in Combi.Basic.ordtype]
+InhOrder.phant_on_ [in Combi.Basic.ordtype]
+InhOrder.phant_clone [in Combi.Basic.ordtype]
+InhPOrder.Exports.join_ordtype_InhPOrder_between_ordtype_Inhabited_and_Order_POrder [in Combi.Basic.ordtype]
+InhPOrder.Exports.join_ordtype_InhPOrder_between_ordtype_Inhabited_and_Order_Preorder [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__Order_POrder [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__Order_Preorder [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__choice_Choice [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhPOrder.Exports.ordtype_InhPOrder_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhPOrder.pack_ [in Combi.Basic.ordtype]
+InhPOrder.phant_on_ [in Combi.Basic.ordtype]
+InhPOrder.phant_clone [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BLattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BJoinSemilattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BMeetSemilattice_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPOrder_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_Order_BPreorder_and_ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TBPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhLattice_and_Order_TPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TBPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_InhPOrder_and_Order_TPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TBPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.join_ordtype_InhTBLattice_between_ordtype_Inhabited_and_Order_TPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBLattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BLattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBJoinSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BJoinSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBMeetSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BMeetSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBPOrder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BPOrder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TBPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TBPreorder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_BPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_BPreorder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_InhLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_InhLattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_InhPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_InhPOrder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__ordtype_Inhabited [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__ordtype_Inhabited_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TLattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TLattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_Lattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_Lattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TJoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TJoinSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_JoinSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_JoinSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TMeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TMeetSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_MeetSemilattice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_MeetSemilattice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TPOrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TPOrder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_POrder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_POrder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_TPreorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_TPreorder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__Order_Preorder [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__Order_Preorder_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__choice_Choice [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__choice_Choice_class [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice__to__eqtype_Equality [in Combi.Basic.ordtype]
+InhTBLattice.Exports.ordtype_InhTBLattice_class__to__eqtype_Equality_class [in Combi.Basic.ordtype]
+InhTBLattice.pack_ [in Combi.Basic.ordtype]
+InhTBLattice.phant_on_ [in Combi.Basic.ordtype]
+InhTBLattice.phant_clone [in Combi.Basic.ordtype]
+inh_ex [in Combi.Basic.ordtype]
+inporbits [in Combi.SymGroup.permcent]
+ins [in Combi.LRrule.Schensted]
+inscode [in Combi.SymGroup.presentSn]
+insmin [in Combi.LRrule.Schensted]
+inspos [in Combi.LRrule.Schensted]
+inspred [in Combi.LRrule.Schensted]
+insrow [in Combi.LRrule.Schensted]
+instab [in Combi.LRrule.Schensted]
+instabnrow [in Combi.LRrule.Schensted]
+intcompn_behead [in Combi.MPoly.sympoly]
+intcompn_cons [in Combi.MPoly.sympoly]
+intcompn_cast [in Combi.Combi.composition]
+intcomp_of_intcompn [in Combi.Combi.composition]
+IntPartNDom.Exports.botEintpartndom [in Combi.Combi.partition]
+IntPartNDom.Exports.join_intpartnE [in Combi.Combi.partition]
+IntPartNDom.Exports.leEpartdom [in Combi.Combi.partition]
+IntPartNDom.Exports.partdom_conj_intpartn [in Combi.Combi.partition]
+IntPartNDom.Exports.sumn_take_pardom_meet [in Combi.Combi.partition]
+IntPartNDom.Exports.topEintpartndom [in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__fintype_isFinite [in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.partition]
+IntPartNDom.fintype_Finite__to__choice_hasChoice [in Combi.Combi.partition]
+IntPartNDom.from_parttuple [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_142 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_140 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_139 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_138 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_135 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_134 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_133 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_130 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_129 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_125 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_124 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_123 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_122 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_mixin_121 [in Combi.Combi.partition]
+IntPartNDom.HB_unnamed_factory_116 [in Combi.Combi.partition]
+IntPartNDom.intpartndom [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTBPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinTPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhTBLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TBPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_TPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinBPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_BPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhLattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_Lattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_MeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_JoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_FinPreorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhPOrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_POrder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__Order_Preorder [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_InhFinite [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__ordtype_Inhabited [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__fintype_Finite [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__choice_Countable [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__choice_Choice [in Combi.Combi.partition]
+IntPartNDom.IntPartNDom_intpartndom__canonical__eqtype_Equality [in Combi.Combi.partition]
+IntPartNDom.is_parttuple [in Combi.Combi.partition]
+IntPartNDom.join_intpartn [in Combi.Combi.partition]
+IntPartNDom.meet_intpartn [in Combi.Combi.partition]
+IntPartNDom.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [in Combi.Combi.partition]
+IntPartNDom.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [in Combi.Combi.partition]
+IntPartNDom.Order_Le_isPOrder__to__Order_isDuallyPreorder [in Combi.Combi.partition]
+IntPartNDom.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.partition]
+IntPartNDom.ordtype_Inhabited__to__ordtype_isInhabited [in Combi.Combi.partition]
+IntPartNDom.parttuple [in Combi.Combi.partition]
+IntPartNDom.parttuple_minn [in Combi.Combi.partition]
+IntPartNDom.part_fromtuple [in Combi.Combi.partition]
+IntPartNLexi.eqtype_SubType__to__eqtype_isSub [in Combi.Combi.partition]
+IntPartNLexi.Exports.botEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.Exports.leEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.Exports.ltEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.Exports.topEintpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__fintype_isFinite [in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.partition]
+IntPartNLexi.fintype_Finite__to__choice_hasChoice [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_69 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_67 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_65 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_63 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_62 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_61 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_60 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_59 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_58 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_57 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_48 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_47 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_46 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_45 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_44 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_39 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_mixin_38 [in Combi.Combi.partition]
+IntPartNLexi.HB_unnamed_factory_36 [in Combi.Combi.partition]
+IntPartNLexi.intpartnlexi [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhFinite [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhTBLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_InhPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__ordtype_Inhabited [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBTotal [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBDistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTBPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBTotal [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBDistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TBPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TTotal [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TDistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_TPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BTotal [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BDistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinBPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_BPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinTotal [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Total [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinDistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_DistrLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinLattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Lattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_JoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_MeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinPOrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_POrder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_FinPreorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__Order_Preorder [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__fintype_SubFinite [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__fintype_Finite [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_SubCountable [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_Countable [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_SubChoice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__choice_Choice [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_SubEquality [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_Equality [in Combi.Combi.partition]
+IntPartNLexi.IntPartNLexi_intpartnlexi__canonical__eqtype_SubType [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_DistrLattice_isTotal [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_Lattice_isDistributive [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_POrder_isJoinSemilattice [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_POrder_isMeetSemilattice [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.partition]
+IntPartNLexi.Order_Total__to__Order_isDuallyPreorder [in Combi.Combi.partition]
+IntPartNLexi.ordtype_isInhabitedType__to__ordtype_isInhabited [in Combi.Combi.partition]
+intpartnsk_nb [in Combi.Combi.partition]
+intpartns_nb [in Combi.Combi.partition]
+intpartn_of_mon [in Combi.MPoly.homogsym]
+intpartn_nb [in Combi.Combi.partition]
+intpart_of_mon [in Combi.MPoly.homogsym]
+intpart_of_intpartn [in Combi.Combi.partition]
+invbump [in Combi.LRrule.Schensted]
+invbumped [in Combi.LRrule.Schensted]
+invbumprow [in Combi.LRrule.Schensted]
+invcont_size [in Combi.Basic.congr]
+invcont_perm [in Combi.Basic.congr]
+invcont_congr [in Combi.Basic.congr]
+invins [in Combi.LRrule.Schensted]
+invinstabnrow [in Combi.LRrule.Schensted]
+invseq [in Combi.Combi.std]
+invset [in Combi.SymGroup.presentSn]
+invstd [in Combi.Combi.std]
+in_homsym [in Combi.MPoly.homogsym]
+in_skew [in Combi.Combi.partition]
+in_shape [in Combi.Combi.partition]
+in_hook [in Combi.HookFormula.hook]
+Iord [in ALEA.Ccpo]
+irrSG [in Combi.SymGroup.Frobenius_char]
+isglb [in ALEA.Ccpo]
+ishift [in ALEA.Ccpo]
+ishomogsym1 [in Combi.MPoly.homogsym]
+isInhabitedType.phant_axioms [in Combi.Basic.ordtype]
+isInhabitedType.phant_Build [in Combi.Basic.ordtype]
+isInhabited.identity_builder [in Combi.Basic.ordtype]
+isInhabited.phant_axioms [in Combi.Basic.ordtype]
+isInhabited.phant_Build [in Combi.Basic.ordtype]
+IsoBottom.IsoBottom_T__canonical__Order_POrder [in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__Order_Preorder [in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__choice_Choice [in Combi.Combi.composition]
+IsoBottom.IsoBottom_T__canonical__eqtype_Equality [in Combi.Combi.composition]
+IsoBottom.phant_axioms [in Combi.Combi.composition]
+IsoBottom.phant_Build [in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__Order_POrder [in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__Order_Preorder [in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__choice_Choice [in Combi.Combi.composition]
+IsoTop.IsoTop_T__canonical__eqtype_Equality [in Combi.Combi.composition]
+IsoTop.phant_axioms [in Combi.Combi.composition]
+IsoTop.phant_Build [in Combi.Combi.composition]
+is_std_of_n [in Combi.Combi.std]
+is_std [in Combi.Combi.std]
+is_homsym [in Combi.MPoly.homogsym]
+is_part_of_nsk [in Combi.Combi.partition]
+is_part_of_ns [in Combi.Combi.partition]
+is_part_of_n [in Combi.Combi.partition]
+is_add_corner [in Combi.Combi.partition]
+is_rem_corner [in Combi.Combi.partition]
+is_part [in Combi.Combi.partition]
+is_Tamari [in Combi.Combi.bintree]
+is_skew_tableau [in Combi.Combi.skewtab]
+is_skew_yam [in Combi.Combi.skewtab]
+is_RStabpair [in Combi.LRrule.Schensted]
+is_RSpair [in Combi.LRrule.Schensted]
+is_trace [in Combi.HookFormula.hook]
+is_stdtab_of_n [in Combi.Combi.stdtab]
+is_stdtab_of_shape [in Combi.Combi.stdtab]
+is_stdtab [in Combi.Combi.stdtab]
+is_tab_of_shape [in Combi.Combi.tableau]
+is_tableau [in Combi.Combi.tableau]
+is_row_cat2 [in Combi.Combi.tableau]
+is_row_drop [in Combi.Combi.tableau]
+is_row_take [in Combi.Combi.tableau]
+is_row_last [in Combi.Combi.tableau]
+is_row_rconsK [in Combi.Combi.tableau]
+is_row_rcons [in Combi.Combi.tableau]
+is_row_consK [in Combi.Combi.tableau]
+is_row_cons [in Combi.Combi.tableau]
+is_rowP [in Combi.Combi.tableau]
+is_row1P [in Combi.Combi.tableau]
+is_skew_reshape_tableau [in Combi.LRrule.therule]
+is_yam_of_size [in Combi.Combi.Yamanouchi]
+is_yam_of_eval [in Combi.Combi.Yamanouchi]
+is_yam [in Combi.Combi.Yamanouchi]
+is_code_of_size [in Combi.SymGroup.presentSn]
+is_comp_of_n [in Combi.Combi.composition]
+is_comp [in Combi.Combi.composition]
+Iter [in ALEA.Ccpo]
+iter [in ALEA.Ccpo]
+iter_ [in ALEA.Ccpo]
+

J

+join_tab [in Combi.Combi.skewtab]
+join_Dyck [in Combi.Combi.Dyckword]
+

K

+Kostka [in Combi.MPoly.Schur_altdef]
+KostkaInv [in Combi.MPoly.Schur_altdef]
+KostkaMon [in Combi.MPoly.Schur_altdef]
+KostkaTab [in Combi.MPoly.Schur_altdef]
+Kostka_expl3 [in Combi.MPoly.Schur_altdef]
+Kostka_expl2 [in Combi.MPoly.Schur_altdef]
+Kostka_expl1 [in Combi.MPoly.Schur_altdef]
+Kostka_rec [in Combi.MPoly.Schur_altdef]
+ksupp [in Combi.LRrule.Greene]
+ksupp_inj [in Combi.LRrule.Greene]
+

L

+langQ [in Combi.LRrule.shuffle]
+last_big [in Combi.Combi.stdtab]
+leftcomb [in Combi.Combi.bintree]
+leftcombsz [in Combi.Combi.bintree]
+left_branch [in Combi.Combi.bintree]
+leg_length [in Combi.HookFormula.hook]
+length [in Combi.SymGroup.presentSn]
+LeqGeqOrder.anti_geq [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_trans [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_total [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.geq_refl [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.gtn_irr [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.gtn_trans [in Combi.SSRcomplements.sorted]
+LeqGeqOrder.ltn_irr [in Combi.SSRcomplements.sorted]
+linvseq [in Combi.Combi.std]
+LRcoeff [in Combi.LRrule.implem]
+LRsupport [in Combi.LRrule.freeSchur]
+LRtab_coeff [in Combi.LRrule.freeSchur]
+LRtab_set [in Combi.LRrule.freeSchur]
+LRtriple_sind [in Combi.LRrule.shuffle]
+LRtriple_ind [in Combi.LRrule.shuffle]
+LRyamtab_list [in Combi.LRrule.implem]
+LRyamtab_count_rec [in Combi.LRrule.implem]
+LRyamtab_list_rec [in Combi.LRrule.implem]
+LRyam_compute [in Combi.LRrule.therule]
+LRyam_enum [in Combi.LRrule.therule]
+LRyam_coeff [in Combi.LRrule.therule]
+LRyam_set [in Combi.LRrule.therule]
+LRyam_list [in Combi.LRrule.implem]
+lsh_rec [in Combi.LRrule.Greene]
+lsplit_rec [in Combi.LRrule.Greene]
+Lub [in ALEA.Ccpo]
+lub_fun [in ALEA.Ccpo]
+

M

+M [in ALEA.Qmeasure]
+map_homsym [in Combi.MPoly.homogsym]
+map_finer [in Combi.Combi.setpartition]
+map_sympoly [in Combi.MPoly.sympoly]
+Mat [in Combi.Basic.unitriginv]
+maxL [in Combi.Basic.ordtype]
+maxperm [in Combi.SymGroup.presentSn]
+MF [in ALEA.Qmeasure]
+mfinite [in ALEA.Qmeasure]
+mfun2 [in ALEA.Ccpo]
+mindropeq [in Combi.Combi.skewpart]
+minh [in Combi.Combi.Dyckword]
+mininspred [in Combi.LRrule.Schensted]
+Minv [in Combi.Basic.unitriginv]
+MLet [in ALEA.Qmeasure]
+Mlet [in ALEA.Qmeasure]
+MN_coeff_fast [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff_rec [in Combi.MPoly.MurnaghanNakayama]
+MN_coeff [in Combi.MPoly.MurnaghanNakayama]
+monsY [in Combi.MPoly.Cauchy]
+monX [in Combi.MPoly.Cauchy]
+mon_ord_equiv [in ALEA.Ccpo]
+mon_comp [in ALEA.Ccpo]
+mon_fun_subst [in ALEA.Ccpo]
+mon2 [in ALEA.Ccpo]
+morph_of_tinj [in Combi.SymGroup.towerSn]
+mpart [in Combi.MPoly.antisym]
+mseq_cte [in ALEA.Ccpo]
+mseq_lift_right [in ALEA.Ccpo]
+mseq_lift_left [in ALEA.Ccpo]
+mshift [in ALEA.Ccpo]
+mstar [in ALEA.Qmeasure]
+multinomial [in Combi.Combi.multinomial]
+multinomial_rec [in Combi.Combi.multinomial]
+Munit [in ALEA.Qmeasure]
+munit [in ALEA.Qmeasure]
+

N

+nat_repr [in Combi.SymGroup.reprSn]
+nat_mx [in Combi.SymGroup.reprSn]
+ncfuniCT [in Combi.SymGroup.towerSn]
+neig4 [in Combi.Combi.skewpart]
+neig4box [in Combi.Combi.skewpart]
+nil_Dyck [in Combi.Combi.Dyckword]
+NoSetContainingBoth.Q [in Combi.LRrule.Greene_inv]
+NoSetContainingBoth.swap_set [in Combi.LRrule.Greene_inv]
+

O

+Oge [in ALEA.Ccpo]
+Olt [in ALEA.Ccpo]
+omegahomsym [in Combi.MPoly.homogsym]
+omegasf [in Combi.MPoly.sympoly]
+one_letter_choices [in Combi.LRrule.implem]
+orc [in ALEA.Misc]
+Order_dual__canonical__ordtype_InhFinOrder [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinLattice [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinPOrder [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhFinite [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhOrder [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhTBLattice [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhLattice [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_InhPOrder [in Combi.Basic.ordtype]
+Order_dual__canonical__ordtype_Inhabited [in Combi.Basic.ordtype]
+ordtree_ordtreesz__canonical__fintype_SubFinite [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__fintype_Finite [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_SubCountable [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_Countable [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_SubChoice [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__choice_Choice [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_SubEquality [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_Equality [in Combi.Combi.ordtree]
+ordtree_ordtreesz__canonical__eqtype_SubType [in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__choice_Countable [in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__choice_Choice [in Combi.Combi.ordtree]
+ordtree_ordtree__canonical__eqtype_Equality [in Combi.Combi.ordtree]
+ordtype_isInhabitedType__to__ordtype_isInhabited__34 [in Combi.Combi.partition]
+ordtype_isInhabitedType__to__ordtype_isInhabited [in Combi.Combi.partition]
+ordtype_isInhabitedType__to__ordtype_isInhabited [in Combi.Combi.setpartition]
+ordtype_InhFinOrder__to__Order_DistrLattice_isTotal [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_Lattice_isDistributive [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_POrder_isJoinSemilattice [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_POrder_isMeetSemilattice [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__fintype_isFinite [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__choice_Choice_isCountable [in Combi.Basic.ordtype]
+ordtype_InhFinOrder__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_POrder_isJoinSemilattice [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_POrder_isMeetSemilattice [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__fintype_isFinite [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__choice_Choice_isCountable [in Combi.Basic.ordtype]
+ordtype_InhFinLattice__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__fintype_isFinite [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__choice_Choice_isCountable [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhFinPOrder__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_DistrLattice_isTotal [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_Lattice_isDistributive [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_POrder_isJoinSemilattice [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_POrder_isMeetSemilattice [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhOrder__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_POrder_isJoinSemilattice [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_POrder_isMeetSemilattice [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_hasBottom [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_hasTop [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhTBLattice__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_POrder_isJoinSemilattice [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_POrder_isMeetSemilattice [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhLattice__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__Order_isDuallyPreorder [in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_InhPOrder__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__51 [in Combi.Basic.ordtype]
+ordtype_Inhabited__to__eqtype_hasDecEq [in Combi.Basic.ordtype]
+ordtype_Inhabited__to__choice_hasChoice [in Combi.Basic.ordtype]
+ordtype_Inhabited__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__30 [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__25 [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__20 [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__15 [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited__11 [in Combi.Basic.ordtype]
+ordtype_isInhabitedType__to__ordtype_isInhabited [in Combi.Basic.ordtype]
+ord_to_bintree [in Combi.Combi.ordtree]
+outer_shape [in Combi.Combi.partition]
+

P

+pad [in Combi.Combi.partition]
+partdom [in Combi.Combi.partition]
+partdomsh [in Combi.Combi.partition]
+partition_box_skew__canonical__fintype_SubFinite [in Combi.Combi.partition]
+partition_box_skew__canonical__fintype_Finite [in Combi.Combi.partition]
+partition_box_skew__canonical__choice_SubCountable [in Combi.Combi.partition]
+partition_box_skew__canonical__choice_Countable [in Combi.Combi.partition]
+partition_box_skew__canonical__choice_SubChoice [in Combi.Combi.partition]
+partition_box_skew__canonical__choice_Choice [in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_SubEquality [in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_Equality [in Combi.Combi.partition]
+partition_box_skew__canonical__eqtype_SubType [in Combi.Combi.partition]
+partition_intpartn__canonical__ordtype_InhFinite [in Combi.Combi.partition]
+partition_intpartn__canonical__ordtype_Inhabited [in Combi.Combi.partition]
+partition_intpartn__canonical__fintype_SubFinite [in Combi.Combi.partition]
+partition_intpartn__canonical__fintype_Finite [in Combi.Combi.partition]
+partition_intpartn__canonical__choice_SubCountable [in Combi.Combi.partition]
+partition_intpartn__canonical__choice_Countable [in Combi.Combi.partition]
+partition_intpartn__canonical__choice_SubChoice [in Combi.Combi.partition]
+partition_intpartn__canonical__choice_Choice [in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_SubEquality [in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_Equality [in Combi.Combi.partition]
+partition_intpartn__canonical__eqtype_SubType [in Combi.Combi.partition]
+partition_intpart__canonical__ordtype_Inhabited [in Combi.Combi.partition]
+partition_intpart__canonical__choice_SubCountable [in Combi.Combi.partition]
+partition_intpart__canonical__choice_Countable [in Combi.Combi.partition]
+partition_intpart__canonical__choice_SubChoice [in Combi.Combi.partition]
+partition_intpart__canonical__choice_Choice [in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_SubEquality [in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_Equality [in Combi.Combi.partition]
+partition_intpart__canonical__eqtype_SubType [in Combi.Combi.partition]
+partm [in Combi.MPoly.antisym]
+partnCT [in Combi.SymGroup.cycletype]
+partn_of_compn [in Combi.Combi.composition]
+partsums [in Combi.Combi.composition]
+permCT [in Combi.SymGroup.cycletype]
+permcycles [in Combi.SymGroup.permcent]
+permcycles_morphism [in Combi.SymGroup.permcent]
+PermLattice.HB_unnamed_factory_22 [in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_factory_20 [in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_mixin_19 [in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_mixin_18 [in Combi.SymGroup.weak_order]
+PermLattice.HB_unnamed_factory_15 [in Combi.SymGroup.weak_order]
+PermLattice.infperm [in Combi.SymGroup.weak_order]
+PermLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBLattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTBPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinTPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBLattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TBPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TLattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_TPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BLattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BPOrder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinBPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_BPreorder [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinLattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinMeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_Lattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_MeetSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_FinJoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.perm_perm_of__canonical__Order_JoinSemilattice [in Combi.SymGroup.weak_order]
+PermLattice.supperm [in Combi.SymGroup.weak_order]
+permutedact [in Combi.Combi.permuted]
+permuted_action [in Combi.Combi.permuted]
+permuted_permuted__canonical__fintype_SubFinite [in Combi.Combi.permuted]
+permuted_permuted__canonical__fintype_Finite [in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_SubCountable [in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_Countable [in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_SubChoice [in Combi.Combi.permuted]
+permuted_permuted__canonical__choice_Choice [in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_SubEquality [in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_Equality [in Combi.Combi.permuted]
+permuted_permuted__canonical__eqtype_SubType [in Combi.Combi.permuted]
+permuted_seq [in Combi.Combi.permuted]
+permuted_tuple [in Combi.Combi.permuted]
+perm_of_setpart [in Combi.SymGroup.cycletype]
+perm_of_porbit [in Combi.SymGroup.cycletype]
+perm_dec [in Combi.SymGroup.cycles]
+perm_of_invset [in Combi.SymGroup.presentSn]
+perm_partn [in Combi.MPoly.sympoly]
+pfminh [in Combi.Combi.Dyckword]
+plactcongr [in Combi.LRrule.plactic]
+plactrule [in Combi.LRrule.plactic]
+plact_cat [in Combi.LRrule.plactic]
+plact_catr [in Combi.LRrule.plactic]
+plact_catl [in Combi.LRrule.plactic]
+plact_rcons [in Combi.LRrule.plactic]
+plact_cons [in Combi.LRrule.plactic]
+plact1 [in Combi.LRrule.plactic]
+plact1i [in Combi.LRrule.plactic]
+plact2 [in Combi.LRrule.plactic]
+plact2i [in Combi.LRrule.plactic]
+polXY [in Combi.MPoly.Cauchy]
+polXY_scale [in Combi.MPoly.Cauchy]
+polX_XY [in Combi.MPoly.Cauchy]
+polylang [in Combi.LRrule.freeSchur]
+polY_XY [in Combi.MPoly.Cauchy]
+porbit_set [in Combi.SymGroup.cycles]
+posbig [in Combi.Basic.ordtype]
+predi [in Combi.MPoly.antisym]
+pred_LRtriple_fast [in Combi.LRrule.shuffle]
+pred_LRtriple [in Combi.LRrule.shuffle]
+prefixes [in Combi.Combi.Dyckword]
+presentSn_codesz__canonical__fintype_SubFinite [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__fintype_Finite [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_SubCountable [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_Countable [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_SubChoice [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__choice_Choice [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_SubEquality [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_Equality [in Combi.SymGroup.presentSn]
+presentSn_codesz__canonical__eqtype_SubType [in Combi.SymGroup.presentSn]
+prods_codesz [in Combi.SymGroup.presentSn]
+prod_partsum [in Combi.MPoly.sympoly]
+prod_symp_homog [in Combi.MPoly.sympoly]
+prod_symp [in Combi.MPoly.sympoly]
+prod_symh_homog [in Combi.MPoly.sympoly]
+prod_symh [in Combi.MPoly.sympoly]
+prod_syme_homog [in Combi.MPoly.sympoly]
+prod_syme [in Combi.MPoly.sympoly]
+prod_gen [in Combi.MPoly.sympoly]
+psupport [in Combi.SymGroup.cycles]
+

R

+Random [in ALEA.Qmeasure]
+rclass [in Combi.Basic.congr]
+recforest [in Combi.Combi.ordtree]
+rectree [in Combi.Combi.ordtree]
+reduced_word [in Combi.SymGroup.presentSn]
+reduces [in Combi.SymGroup.presentSn]
+RefinementOrder.composition_IsoTop__to__Order_hasTop [in Combi.Combi.composition]
+RefinementOrder.composition_IsoBottom__to__Order_hasBottom [in Combi.Combi.composition]
+RefinementOrder.eqtype_SubType__to__eqtype_isSub [in Combi.Combi.composition]
+RefinementOrder.Exports.botEcompnref [in Combi.Combi.composition]
+RefinementOrder.Exports.compnref_rev [in Combi.Combi.composition]
+RefinementOrder.Exports.descset_join [in Combi.Combi.composition]
+RefinementOrder.Exports.descset_meet [in Combi.Combi.composition]
+RefinementOrder.Exports.descset_mono [in Combi.Combi.composition]
+RefinementOrder.Exports.leEcompnref [in Combi.Combi.composition]
+RefinementOrder.Exports.topEcompnref [in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__fintype_isFinite [in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__eqtype_hasDecEq [in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__choice_Choice_isCountable [in Combi.Combi.composition]
+RefinementOrder.fintype_Finite__to__choice_hasChoice [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_73 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_71 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_70 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_68 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_67 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_63 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_62 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_61 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_58 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_57 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_55 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_54 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_53 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_50 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_49 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_48 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_47 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_46 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_41 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_mixin_40 [in Combi.Combi.composition]
+RefinementOrder.HB_unnamed_factory_38 [in Combi.Combi.composition]
+RefinementOrder.Order_IsoDistrLattice__to__Order_Lattice_isDistributive [in Combi.Combi.composition]
+RefinementOrder.Order_IsoLattice__to__Order_POrder_isMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.Order_IsoLattice__to__Order_POrder_isJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.Order_isPOrder__to__Order_isDuallyPreorder [in Combi.Combi.composition]
+RefinementOrder.Order_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.composition]
+RefinementOrder.ordtype_isInhabitedType__to__ordtype_isInhabited [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBDistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTBPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinTPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhTBLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBDistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TBPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TDistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_TPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BDistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinBPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_BPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinDistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_DistrLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinMeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhLattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_Lattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_MeetSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinJoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_JoinSemilattice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhFinite [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_InhPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__ordtype_Inhabited [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinPOrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_FinPreorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_POrder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__Order_Preorder [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__fintype_SubFinite [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__fintype_Finite [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_SubCountable [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_Countable [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_SubChoice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__choice_Choice [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_SubEquality [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_Equality [in Combi.Combi.composition]
+RefinementOrder.RefinementOrder_type__canonical__eqtype_SubType [in Combi.Combi.composition]
+RefinementOrder.type [in Combi.Combi.composition]
+RefinmentOrder.Exports.is_finer_pblockP [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.is_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.Exports.mem_meet_finerP [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_28 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_27 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_24 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_22 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_20 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_19 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_mixin_18 [in Combi.Combi.setpartition]
+RefinmentOrder.HB_unnamed_factory_15 [in Combi.Combi.setpartition]
+RefinmentOrder.is_finer [in Combi.Combi.setpartition]
+RefinmentOrder.join_finer [in Combi.Combi.setpartition]
+RefinmentOrder.join_finer_eq [in Combi.Combi.setpartition]
+RefinmentOrder.meet_finer [in Combi.Combi.setpartition]
+RefinmentOrder.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.Order_Le_isPOrder__to__Order_isDuallyPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhTBLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhFinLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TLattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_Lattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TMeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_MeetSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TJoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_JoinSemilattice [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTBPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinTPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TBPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_TPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinBPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_BPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhFinPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_FinPreorder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__ordtype_InhPOrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_POrder [in Combi.Combi.setpartition]
+RefinmentOrder.setpartition_setpart__canonical__Order_Preorder [in Combi.Combi.setpartition]
+rembig [in Combi.Basic.ordtype]
+remn [in Combi.Combi.stdtab]
+remn_rec [in Combi.Combi.stdtab]
+rem_corners [in Combi.Combi.partition]
+rem_trail0 [in Combi.Combi.partition]
+res_tab [in Combi.MPoly.Schur_altdef]
+revdual [in Combi.LRrule.plactic]
+revset [in Combi.LRrule.Greene]
+rev_ksupp_inv [in Combi.LRrule.Greene_inv]
+rev_ksupp [in Combi.LRrule.Greene_inv]
+rev_set [in Combi.LRrule.Greene_inv]
+rev_ord_cast [in Combi.LRrule.Greene_inv]
+rev_intcompn [in Combi.Combi.composition]
+rho [in Combi.MPoly.antisym]
+RHSL3 [in Combi.HookFormula.hook]
+RHSL3_trace [in Combi.HookFormula.hook]
+ribbon [in Combi.Combi.skewpart]
+ribbontb_start [in Combi.Combi.skewpart]
+ribbon_on_box_ex [in Combi.Combi.skewpart]
+ribbon_box_ex [in Combi.Combi.skewpart]
+ribbon_textbook [in Combi.Combi.skewpart]
+ribbon_height [in Combi.Combi.skewpart]
+ribbon_on [in Combi.Combi.skewpart]
+ribbon_from [in Combi.Combi.skewpart]
+rightcomb [in Combi.Combi.bintree]
+rightcombsz [in Combi.Combi.bintree]
+right_sizes [in Combi.Combi.bintree]
+rotations [in Combi.Combi.bintree]
+rowcomp [in Combi.Combi.composition]
+rowcompn [in Combi.Combi.composition]
+rowpartn [in Combi.Combi.partition]
+RS [in Combi.LRrule.extract]
+RS [in Combi.LRrule.Schensted]
+RSbij [in Combi.LRrule.Schensted]
+RSbijinv [in Combi.LRrule.Schensted]
+RSbijinvnat [in Combi.LRrule.extract]
+RSbijnat [in Combi.LRrule.extract]
+RSclass [in Combi.LRrule.Schensted]
+rsh_rec [in Combi.LRrule.Greene]
+RSmap [in Combi.LRrule.Schensted]
+RSmapinv [in Combi.LRrule.Schensted]
+RSmapinv2 [in Combi.LRrule.Schensted]
+RSmap_rev [in Combi.LRrule.Schensted]
+rsplit_rec [in Combi.LRrule.Greene]
+RStab [in Combi.LRrule.Schensted]
+RStabinv [in Combi.LRrule.Schensted]
+RStabinvnat [in Combi.LRrule.extract]
+RStabmap [in Combi.LRrule.Schensted]
+RStabnat [in Combi.LRrule.extract]
+rsymrel [in Combi.SymGroup.presentSn]
+RS_rev [in Combi.LRrule.Schensted]
+rtrans [in Combi.Basic.congr]
+

S

+Sch [in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__choice_SubChoice [in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__choice_Choice [in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_SubEquality [in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_Equality [in Combi.LRrule.Schensted]
+Schensted_rstabpair__canonical__eqtype_SubType [in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__choice_SubChoice [in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__choice_Choice [in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_SubEquality [in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_Equality [in Combi.LRrule.Schensted]
+Schensted_rspair__canonical__eqtype_SubType [in Combi.LRrule.Schensted]
+Schur [in Combi.MPoly.Schur_mpoly]
+Sch_rev [in Combi.LRrule.Schensted]
+seq_lift_right [in ALEA.Ccpo]
+seq_lift_left [in ALEA.Ccpo]
+seq_finType [in Combi.Basic.combclass]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear__82 [in Combi.MPoly.homogsym]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [in Combi.MPoly.homogsym]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [in Combi.SymGroup.towerSn]
+sesquilinear_bilinear_isBilinear__to__sesquilinear_isBilinear [in Combi.SymGroup.Frobenius_char]
+SetContainingBothLeft.posc [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.Qbnotin [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.S1 [in Combi.LRrule.Greene_inv]
+SetContainingBothLeft.T1 [in Combi.LRrule.Greene_inv]
+setdiff [in Combi.MPoly.Schur_altdef]
+setpartition_setpart__canonical__ordtype_InhFinite [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__ordtype_Inhabited [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__fintype_SubFinite [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__fintype_Finite [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_SubCountable [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_Countable [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_SubChoice [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__choice_Choice [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_SubEquality [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_Equality [in Combi.Combi.setpartition]
+setpartition_setpart__canonical__eqtype_SubType [in Combi.Combi.setpartition]
+setpart_shape [in Combi.Combi.partition]
+setpart_set1 [in Combi.Combi.setpartition]
+setpart_set0 [in Combi.Combi.setpartition]
+setpart1 [in Combi.Combi.setpartition]
+sfiltergtn [in Combi.LRrule.shuffle]
+sfilterleq [in Combi.LRrule.shuffle]
+shape_res_tab [in Combi.MPoly.Schur_altdef]
+shape_deg [in Combi.Combi.stdtab]
+shcols [in Combi.LRrule.Greene]
+shift [in ALEA.Ccpo]
+shiftinv_pos [in Combi.Basic.ordtype]
+shiftn [in Combi.LRrule.shuffle]
+shiftset [in Combi.LRrule.Greene]
+shift_pos [in Combi.Basic.ordtype]
+shrows [in Combi.LRrule.Greene]
+shsh [in Combi.LRrule.shuffle]
+shuffle [in Combi.LRrule.shuffle]
+shuffle_from_rec [in Combi.LRrule.shuffle]
+signed_repr [in Combi.SymGroup.reprSn]
+signed_mx [in Combi.SymGroup.reprSn]
+sign_char [in Combi.SymGroup.reprSn]
+sign_repr [in Combi.SymGroup.reprSn]
+sign_mx [in Combi.SymGroup.reprSn]
+simplexp [in Combi.MPoly.antisym]
+size_tree [in Combi.Combi.bintree]
+size_tab [in Combi.Combi.tableau]
+size_ordtree [in Combi.Combi.ordtree]
+skew_reshape [in Combi.Combi.skewtab]
+skew_dominate [in Combi.Combi.skewtab]
+slporbits [in Combi.SymGroup.cycletype]
+srel [in Combi.SymGroup.presentSn]
+ssrbool_has_quality__canonical__GRing_SubmodClosed [in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_ZmodClosed [in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_OppClosed [in Combi.MPoly.antisym]
+ssrbool_has_quality__canonical__Algebra_AddClosed [in Combi.MPoly.antisym]
+stable_opp [in ALEA.Qmeasure]
+stable_mull [in ALEA.Qmeasure]
+stable_sub [in ALEA.Qmeasure]
+stable_add [in ALEA.Qmeasure]
+stab_iporbits_map [in Combi.SymGroup.permcent]
+stab_iporbits_porbitmap [in Combi.SymGroup.permcent]
+stab_iporbits [in Combi.SymGroup.permcent]
+startrem [in Combi.Combi.skewpart]
+starts_at [in Combi.HookFormula.hook]
+std [in Combi.Combi.std]
+stdtabn_of_sh [in Combi.Combi.stdtab]
+stdtabshcast [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__fintype_SubFinite [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__fintype_Finite [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_SubCountable [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_Countable [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_SubChoice [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__choice_Choice [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_SubEquality [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_Equality [in Combi.Combi.stdtab]
+stdtab_stdtabn__canonical__eqtype_SubType [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__fintype_SubFinite [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__fintype_Finite [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_SubCountable [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_Countable [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_SubChoice [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__choice_Choice [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_SubEquality [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_Equality [in Combi.Combi.stdtab]
+stdtab_stdtabsh__canonical__eqtype_SubType [in Combi.Combi.stdtab]
+stdtab_of_yam [in Combi.Combi.stdtab]
+std_rec [in Combi.Combi.std]
+std_stdwordn__canonical__fintype_SubFinite [in Combi.Combi.std]
+std_stdwordn__canonical__fintype_Finite [in Combi.Combi.std]
+std_stdwordn__canonical__choice_SubCountable [in Combi.Combi.std]
+std_stdwordn__canonical__choice_Countable [in Combi.Combi.std]
+std_stdwordn__canonical__choice_SubChoice [in Combi.Combi.std]
+std_stdwordn__canonical__choice_Choice [in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_SubEquality [in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_Equality [in Combi.Combi.std]
+std_stdwordn__canonical__eqtype_SubType [in Combi.Combi.std]
+step [in Combi.Basic.congr]
+straighten [in Combi.SymGroup.presentSn]
+straighten_rev [in Combi.SymGroup.presentSn]
+subseqrow [in Combi.LRrule.Schensted]
+subseqrow_n [in Combi.LRrule.Schensted]
+subseq_subseqs__canonical__fintype_SubFinite [in Combi.Combi.subseq]
+subseq_subseqs__canonical__fintype_Finite [in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_SubCountable [in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_Countable [in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_SubChoice [in Combi.Combi.subseq]
+subseq_subseqs__canonical__choice_Choice [in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_SubEquality [in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_Equality [in Combi.Combi.subseq]
+subseq_subseqs__canonical__eqtype_SubType [in Combi.Combi.subseq]
+subType_seq [in Combi.Basic.combclass]
+sub_full [in Combi.Combi.subseq]
+sub_nil [in Combi.Combi.subseq]
+Swap.pos0 [in Combi.LRrule.Greene_inv]
+Swap.pos1 [in Combi.LRrule.Greene_inv]
+Swap.swap [in Combi.LRrule.Greene_inv]
+Swap.swap_set [in Combi.LRrule.Greene_inv]
+symbe [in Combi.MPoly.homogsym]
+symbh [in Combi.MPoly.homogsym]
+symbm [in Combi.MPoly.homogsym]
+symbp [in Combi.MPoly.homogsym]
+symbs [in Combi.MPoly.homogsym]
+syme [in Combi.MPoly.sympoly]
+symh [in Combi.MPoly.sympoly]
+symh_pol [in Combi.MPoly.sympoly]
+symh_pol_bound [in Combi.MPoly.sympoly]
+symm [in Combi.MPoly.sympoly]
+symm_pol [in Combi.MPoly.sympoly]
+symp [in Combi.MPoly.sympoly]
+sympolyf [in Combi.MPoly.sympoly]
+sympolyf_eval [in Combi.MPoly.sympoly]
+SymPolyHomogKey.homogsym1_keyed [in Combi.MPoly.homogsym]
+sympoly_cnvarsym__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_cnvarsym__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_omegasf__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_sympolyf_eval__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_sympolyf__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_map_sympoly__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubIntegralDomain [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_IntegralDomain [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComUnitAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComUnitRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComUnitRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubUnitRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_UnitAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_UnitRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComNzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComNzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComNzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComPzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubComPzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_ComPzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_LRMorphism [in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_RMorphism [in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__GRing_Linear [in Combi.MPoly.sympoly]
+sympoly_sympol__canonical__Algebra_Additive [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzLalgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzLalgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzLalgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzLalgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubLmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_Lmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzLSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzLSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzLSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzLSemiAlgebra [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubLSemiModule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_LSemiModule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubNzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_NzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_SubPzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__GRing_PzSemiRing [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubZmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubNmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_Zmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_Nmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddSemigroup [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubAddUMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddUMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_AddMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseZmodule [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_SubBaseAddUMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_ChoiceBaseAddUMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseAddUMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_ChoiceBaseAddMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__Algebra_BaseAddMagma [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__choice_SubChoice [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__choice_Choice [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_SubEquality [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_Equality [in Combi.MPoly.sympoly]
+sympoly_sympoly__canonical__eqtype_SubType [in Combi.MPoly.sympoly]
+symp_pol [in Combi.MPoly.sympoly]
+syms [in Combi.MPoly.sympoly]
+

T

+tabcols [in Combi.LRrule.Greene]
+tabcolsk [in Combi.LRrule.Greene]
+tableau_tabsh__canonical__fintype_SubFinite [in Combi.Combi.tableau]
+tableau_tabsh__canonical__fintype_Finite [in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_SubCountable [in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_Countable [in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_SubChoice [in Combi.Combi.tableau]
+tableau_tabsh__canonical__choice_Choice [in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_SubEquality [in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_Equality [in Combi.Combi.tableau]
+tableau_tabsh__canonical__eqtype_SubType [in Combi.Combi.tableau]
+tabnat_of_ord [in Combi.MPoly.Schur_altdef]
+tabnat_of_ord_fun [in Combi.MPoly.Schur_altdef]
+tabord_of_nat [in Combi.MPoly.Schur_altdef]
+tabord_of_nat_fun [in Combi.MPoly.Schur_altdef]
+tabrowconst [in Combi.Combi.tableau]
+tabrows [in Combi.LRrule.Greene]
+tabrowsk [in Combi.LRrule.Greene]
+tabsh_reading_RS [in Combi.LRrule.freeSchur]
+tabsh_reading [in Combi.Combi.tableau]
+tabwordshape [in Combi.LRrule.freeSchur]
+tabword_of_tuple [in Combi.LRrule.freeSchur]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBLattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTBPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinTPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBLattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TBPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TLattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_TPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BLattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinBPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_BPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinLattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_Lattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_MeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_JoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinPOrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_FinPreorder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_POrder [in Combi.Combi.bintree]
+TamariLattice.bintree_bintreesz__canonical__Order_Preorder [in Combi.Combi.bintree]
+TamariLattice.Exports.botETamari [in Combi.Combi.bintree]
+TamariLattice.Exports.flipsz_join [in Combi.Combi.bintree]
+TamariLattice.Exports.flipsz_meet [in Combi.Combi.bintree]
+TamariLattice.Exports.right_sizes_meet [in Combi.Combi.bintree]
+TamariLattice.Exports.rotations_Tamari [in Combi.Combi.bintree]
+TamariLattice.Exports.TamariE [in Combi.Combi.bintree]
+TamariLattice.Exports.Tamari_vctleq [in Combi.Combi.bintree]
+TamariLattice.Exports.Tamari_flip [in Combi.Combi.bintree]
+TamariLattice.Exports.topETamari [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_41 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_39 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_38 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_37 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_34 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_33 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_mixin_32 [in Combi.Combi.bintree]
+TamariLattice.HB_unnamed_factory_29 [in Combi.Combi.bintree]
+TamariLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [in Combi.Combi.bintree]
+TamariLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [in Combi.Combi.bintree]
+TamariLattice.Order_Le_isPOrder__to__Order_isDuallyPreorder [in Combi.Combi.bintree]
+TamariLattice.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.bintree]
+TamariLattice.Tamari [in Combi.Combi.bintree]
+TamariLattice.Tjoin [in Combi.Combi.bintree]
+TamariLattice.Tmeet [in Combi.Combi.bintree]
+TamariVector [in Combi.Combi.bintree]
+tclosure [in Combi.SymGroup.presentSn]
+tinj [in Combi.SymGroup.towerSn]
+tinjval [in Combi.SymGroup.towerSn]
+towerSn_cfextprod__canonical__sesquilinear_Bilinear [in Combi.SymGroup.towerSn]
+to_mask [in Combi.Combi.subseq]
+to_word [in Combi.Combi.tableau]
+trace_seq [in Combi.HookFormula.hook]
+tree_decode [in Combi.Combi.bintree]
+tree_encode [in Combi.Combi.bintree]
+trivIseq [in Combi.LRrule.Greene]
+trivsetpart [in Combi.Combi.setpartition]
+trivSimpl [in Combi.SymGroup.presentSn]
+triv_repr [in Combi.SymGroup.reprSn]
+triv_mx [in Combi.SymGroup.reprSn]
+

U

+UDn [in Combi.Combi.Dyckword]
+UnDn [in Combi.Combi.Dyckword]
+undup_finType [in Combi.Basic.combclass]
+unifnat [in ALEA.Qmeasure]
+Uniform [in ALEA.Qmeasure]
+unif_def [in ALEA.Qmeasure]
+unif2fin [in ALEA.Qmeasure]
+union_intpartn [in Combi.Combi.partition]
+union_intpart [in Combi.Combi.partition]
+union_finType [in Combi.Basic.combclass]
+uniq_finType [in Combi.Basic.combclass]
+unitrig [in Combi.Basic.unitriginv]
+Unnamed_thm0 [in Combi.Combi.std]
+Unnamed_thm [in Combi.Combi.std]
+Unnamed_thm10 [in Combi.Combi.bintree]
+Unnamed_thm9 [in Combi.Combi.bintree]
+Unnamed_thm8 [in Combi.Combi.bintree]
+Unnamed_thm7 [in Combi.Combi.bintree]
+Unnamed_thm6 [in Combi.Combi.bintree]
+Unnamed_thm5 [in Combi.Combi.bintree]
+Unnamed_thm4 [in Combi.Combi.bintree]
+Unnamed_thm3 [in Combi.Combi.bintree]
+Unnamed_thm2 [in Combi.Combi.bintree]
+Unnamed_thm1 [in Combi.Combi.bintree]
+Unnamed_thm0 [in Combi.Combi.bintree]
+Unnamed_thm [in Combi.Combi.bintree]
+Unnamed_thm13 [in Combi.LRrule.Schensted]
+Unnamed_thm12 [in Combi.LRrule.Schensted]
+Unnamed_thm11 [in Combi.LRrule.Schensted]
+Unnamed_thm10 [in Combi.LRrule.Schensted]
+Unnamed_thm9 [in Combi.LRrule.Schensted]
+Unnamed_thm8 [in Combi.LRrule.Schensted]
+Unnamed_thm7 [in Combi.LRrule.Schensted]
+Unnamed_thm6 [in Combi.LRrule.Schensted]
+Unnamed_thm5 [in Combi.LRrule.Schensted]
+Unnamed_thm4 [in Combi.LRrule.Schensted]
+Unnamed_thm3 [in Combi.LRrule.Schensted]
+Unnamed_thm2 [in Combi.LRrule.Schensted]
+Unnamed_thm1 [in Combi.LRrule.Schensted]
+Unnamed_thm0 [in Combi.LRrule.Schensted]
+Unnamed_thm [in Combi.LRrule.Schensted]
+Unnamed_thm40 [in Combi.Combi.skewpart]
+Unnamed_thm39 [in Combi.Combi.skewpart]
+Unnamed_thm38 [in Combi.Combi.skewpart]
+Unnamed_thm37 [in Combi.Combi.skewpart]
+Unnamed_thm36 [in Combi.Combi.skewpart]
+Unnamed_thm35 [in Combi.Combi.skewpart]
+Unnamed_thm34 [in Combi.Combi.skewpart]
+Unnamed_thm33 [in Combi.Combi.skewpart]
+Unnamed_thm32 [in Combi.Combi.skewpart]
+Unnamed_thm31 [in Combi.Combi.skewpart]
+Unnamed_thm30 [in Combi.Combi.skewpart]
+Unnamed_thm29 [in Combi.Combi.skewpart]
+Unnamed_thm28 [in Combi.Combi.skewpart]
+Unnamed_thm27 [in Combi.Combi.skewpart]
+Unnamed_thm26 [in Combi.Combi.skewpart]
+Unnamed_thm25 [in Combi.Combi.skewpart]
+Unnamed_thm24 [in Combi.Combi.skewpart]
+Unnamed_thm23 [in Combi.Combi.skewpart]
+Unnamed_thm22 [in Combi.Combi.skewpart]
+Unnamed_thm21 [in Combi.Combi.skewpart]
+Unnamed_thm20 [in Combi.Combi.skewpart]
+Unnamed_thm19 [in Combi.Combi.skewpart]
+Unnamed_thm18 [in Combi.Combi.skewpart]
+Unnamed_thm17 [in Combi.Combi.skewpart]
+Unnamed_thm16 [in Combi.Combi.skewpart]
+Unnamed_thm15 [in Combi.Combi.skewpart]
+Unnamed_thm14 [in Combi.Combi.skewpart]
+Unnamed_thm13 [in Combi.Combi.skewpart]
+Unnamed_thm12 [in Combi.Combi.skewpart]
+Unnamed_thm11 [in Combi.Combi.skewpart]
+Unnamed_thm10 [in Combi.Combi.skewpart]
+Unnamed_thm9 [in Combi.Combi.skewpart]
+Unnamed_thm8 [in Combi.Combi.skewpart]
+Unnamed_thm7 [in Combi.Combi.skewpart]
+Unnamed_thm6 [in Combi.Combi.skewpart]
+Unnamed_thm5 [in Combi.Combi.skewpart]
+Unnamed_thm4 [in Combi.Combi.skewpart]
+Unnamed_thm3 [in Combi.Combi.skewpart]
+Unnamed_thm2 [in Combi.Combi.skewpart]
+Unnamed_thm1 [in Combi.Combi.skewpart]
+Unnamed_thm0 [in Combi.Combi.skewpart]
+Unnamed_thm [in Combi.Combi.skewpart]
+Unnamed_thm8 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm7 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm6 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm5 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm4 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm3 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm2 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm1 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm0 [in Combi.MPoly.MurnaghanNakayama]
+Unnamed_thm [in Combi.MPoly.MurnaghanNakayama]
+usize [in ALEA.Qmeasure]
+

V

+val2pos [in Combi.LRrule.stdplact]
+Vandet [in Combi.MPoly.antisym]
+Vanmx [in Combi.MPoly.antisym]
+Vanprod [in Combi.MPoly.antisym]
+vb_strip [in Combi.Combi.skewpart]
+vctleq [in Combi.Combi.bintree]
+vctmin [in Combi.Combi.bintree]
+vector_Lmodule_hasFinDim__to__vector_LSemiModule_hasFinDim [in Combi.MPoly.homogsym]
+vect_n_k [in Combi.Combi.vectNK]
+versions [in Combi.Combi.std]
+

W

+walk_to_corner [in Combi.HookFormula.hook]
+walk_to_corner_rec [in Combi.HookFormula.hook]
+wcord [in Combi.SymGroup.presentSn]
+WeakOrder.Exports.lepermP [in Combi.SymGroup.weak_order]
+WeakOrder.Exports.leperm_lengthE [in Combi.SymGroup.weak_order]
+WeakOrder.Exports.leperm_length [in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__fintype_isFinite [in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__eqtype_hasDecEq [in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__choice_Choice_isCountable [in Combi.SymGroup.weak_order]
+WeakOrder.fintype_Finite__to__choice_hasChoice [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_14 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_13 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_factory_10 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_9 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_8 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_7 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_mixin_6 [in Combi.SymGroup.weak_order]
+WeakOrder.HB_unnamed_factory_1 [in Combi.SymGroup.weak_order]
+WeakOrder.leperm [in Combi.SymGroup.weak_order]
+WeakOrder.Order_Le_isPOrder__to__Order_isDuallyPreorder [in Combi.SymGroup.weak_order]
+WeakOrder.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_FinPOrder [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_FinPreorder [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_POrder [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__Order_Preorder [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__fintype_Finite [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__choice_Countable [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__choice_Choice [in Combi.SymGroup.weak_order]
+WeakOrder.perm_perm_of__canonical__eqtype_Equality [in Combi.SymGroup.weak_order]
+weight [in ALEA.Qmeasure]
+Wikipedia_Murnaghan_Nakayama [in Combi.SymGroup.Frobenius_char]
+wordcd [in Combi.SymGroup.presentSn]
+wordperm [in Combi.Combi.std]
+

Y

+Yamanouchi_yamn__canonical__fintype_SubFinite [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__fintype_Finite [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_SubCountable [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_Countable [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_SubChoice [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__choice_Choice [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_SubEquality [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_Equality [in Combi.Combi.Yamanouchi]
+Yamanouchi_yamn__canonical__eqtype_SubType [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__fintype_SubFinite [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__fintype_Finite [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_SubCountable [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_Countable [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_SubChoice [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__choice_Choice [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_SubEquality [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_Equality [in Combi.Combi.Yamanouchi]
+Yamanouchi_yameval__canonical__eqtype_SubType [in Combi.Combi.Yamanouchi]
+yamrow [in Combi.LRrule.therule]
+yamtab [in Combi.LRrule.Yam_plact]
+yamtab_rec [in Combi.LRrule.Yam_plact]
+yamtab_shift [in Combi.LRrule.implem]
+yamtab_rows [in Combi.LRrule.implem]
+yam_of_stdtab [in Combi.Combi.stdtab]
+yam_of_stdtab_rec [in Combi.Combi.stdtab]
+Ymon [in Combi.MPoly.Cauchy]
+YoungLattice.choice_Countable__to__choice_Choice_isCountable [in Combi.Combi.partition]
+YoungLattice.choice_Countable__to__eqtype_hasDecEq [in Combi.Combi.partition]
+YoungLattice.choice_Countable__to__choice_hasChoice [in Combi.Combi.partition]
+YoungLattice.Exports.bottom_YoungE [in Combi.Combi.partition]
+YoungLattice.Exports.le_Young_sumn [in Combi.Combi.partition]
+YoungLattice.Exports.le_YoungP [in Combi.Combi.partition]
+YoungLattice.Exports.le_YoungE [in Combi.Combi.partition]
+YoungLattice.Exports.lt_Young_sumn [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_115 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_113 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_111 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_110 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_109 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_106 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_105 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_104 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_101 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_100 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_96 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_95 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_94 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_mixin_93 [in Combi.Combi.partition]
+YoungLattice.HB_unnamed_factory_89 [in Combi.Combi.partition]
+YoungLattice.intpartYoung [in Combi.Combi.partition]
+YoungLattice.join_Young [in Combi.Combi.partition]
+YoungLattice.join_Young_fun [in Combi.Combi.partition]
+YoungLattice.le_Young [in Combi.Combi.partition]
+YoungLattice.meet_Young [in Combi.Combi.partition]
+YoungLattice.meet_Young_fun [in Combi.Combi.partition]
+YoungLattice.Order_Lattice_Meet_isDistrLattice__to__Order_Lattice_isDistributive [in Combi.Combi.partition]
+YoungLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isMeetSemilattice [in Combi.Combi.partition]
+YoungLattice.Order_POrder_MeetJoin_isLattice__to__Order_POrder_isJoinSemilattice [in Combi.Combi.partition]
+YoungLattice.Order_Le_isPOrder__to__Order_isDuallyPreorder [in Combi.Combi.partition]
+YoungLattice.Order_Le_isPOrder__to__Order_Preorder_isDuallyPOrder [in Combi.Combi.partition]
+YoungLattice.ordtype_Inhabited__to__ordtype_isInhabited [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BDistrLattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_DistrLattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BLattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BJoinSemilattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BMeetSemilattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BPOrder [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_BPreorder [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_InhLattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_Lattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_MeetSemilattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_JoinSemilattice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_InhPOrder [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_POrder [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__Order_Preorder [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__ordtype_Inhabited [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__choice_Countable [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__choice_Choice [in Combi.Combi.partition]
+YoungLattice.YoungLattice_intpartYoung__canonical__eqtype_Equality [in Combi.Combi.partition]
+YoungSG [in Combi.SymGroup.Frobenius_char]
+

Z

+zcard [in Combi.SymGroup.permcent]
+zcoeff [in Combi.SymGroup.towerSn]
+

+

Record Index

+

B

+bintreesz [in Combi.Combi.bintree]
+box_skew [in Combi.Combi.partition]
+

C

+codesz [in Combi.SymGroup.presentSn]
+continuous [in ALEA.Ccpo]
+continuous2 [in ALEA.Ccpo]
+cpo [in ALEA.Ccpo]
+

D

+distr [in ALEA.Qmeasure]
+Dyck [in Combi.Combi.Dyckword]
+

E

+Example1.isOne [in Combi.Basic.combclass]
+Example2.isOne [in Combi.Basic.combclass]
+Example3.isOne [in Combi.Basic.combclass]
+

F

+fcont [in ALEA.Ccpo]
+fibered_set [in Combi.Combi.fibered_set]
+fin [in ALEA.Qmeasure]
+fmon [in ALEA.Ccpo]
+

H

+homogsym [in Combi.MPoly.homogsym]
+

I

+Inhabited.axioms_ [in Combi.Basic.ordtype]
+Inhabited.type [in Combi.Basic.ordtype]
+InhFinite.axioms_ [in Combi.Basic.ordtype]
+InhFinite.type [in Combi.Basic.ordtype]
+InhFinLattice.axioms_ [in Combi.Basic.ordtype]
+InhFinLattice.type [in Combi.Basic.ordtype]
+InhFinOrder.axioms_ [in Combi.Basic.ordtype]
+InhFinOrder.type [in Combi.Basic.ordtype]
+InhFinPOrder.axioms_ [in Combi.Basic.ordtype]
+InhFinPOrder.type [in Combi.Basic.ordtype]
+InhLattice.axioms_ [in Combi.Basic.ordtype]
+InhLattice.type [in Combi.Basic.ordtype]
+InhOrder.axioms_ [in Combi.Basic.ordtype]
+InhOrder.type [in Combi.Basic.ordtype]
+InhPOrder.axioms_ [in Combi.Basic.ordtype]
+InhPOrder.type [in Combi.Basic.ordtype]
+InhTBLattice.axioms_ [in Combi.Basic.ordtype]
+InhTBLattice.type [in Combi.Basic.ordtype]
+inputSpec [in Combi.LRrule.implem]
+intcomp [in Combi.Combi.composition]
+intcompn [in Combi.Combi.composition]
+intpart [in Combi.Combi.partition]
+intpartn [in Combi.Combi.partition]
+invariant_context [in Combi.Basic.congr]
+isInhabitedType.axioms_ [in Combi.Basic.ordtype]
+isInhabited.axioms_ [in Combi.Basic.ordtype]
+islub [in ALEA.Ccpo]
+IsoBottom.axioms_ [in Combi.Combi.composition]
+IsoTop.axioms_ [in Combi.Combi.composition]
+

M

+monotonic [in ALEA.Ccpo]
+monotonic2 [in ALEA.Ccpo]
+

O

+ord [in ALEA.Ccpo]
+Order [in ALEA.Ccpo]
+ordtreesz [in Combi.Combi.ordtree]
+outputSpec [in Combi.LRrule.implem]
+

P

+permuted [in Combi.Combi.permuted]
+porbits_map [in Combi.SymGroup.cycletype]
+

R

+rspair [in Combi.LRrule.Schensted]
+rstabpair [in Combi.LRrule.Schensted]
+

S

+SetContainingBothLeft.hypRabc [in Combi.LRrule.Greene_inv]
+setpart [in Combi.Combi.setpartition]
+stable [in ALEA.Ccpo]
+stable2 [in ALEA.Ccpo]
+stdtabn [in Combi.Combi.stdtab]
+stdtabsh [in Combi.Combi.stdtab]
+stdwordn [in Combi.Combi.std]
+subseqs [in Combi.Combi.subseq]
+sympoly [in Combi.MPoly.sympoly]
+

T

+tabsh [in Combi.Combi.tableau]
+

U

+unif [in ALEA.Qmeasure]
+

Y

+yameval [in Combi.Combi.Yamanouchi]
+yamn [in Combi.Combi.Yamanouchi]
+

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(8361 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(178 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(68 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(910 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(57 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3352 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(181 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(17 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(363 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(67 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(261 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(2820 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(68 entries)
+
This page has been generated by coqdoc +
+ +
+ + + \ No newline at end of file diff --git a/combi/1.1.0/toc.html b/combi/1.1.0/toc.html new file mode 100644 index 00000000..c2d69d14 --- /dev/null +++ b/combi/1.1.0/toc.html @@ -0,0 +1,1661 @@ + + + + + +Table of contents + + + + +
+ + + +
+ +
+

Library Combi.Combi.bintree: Binary Trees

+ +

Library Combi.Combi.composition: Integer Composition

+ +

Library Combi.Combi.Dyckword: Dyck Words

+ +

Library Combi.Combi.fibered_set: Bijection beetween fibered sets

+ +

Library Combi.Combi.partition: Integer Partitions

+ +

Library Combi.Combi.permuted: Listing the Permutations of a tuple or seq

+ +

Library Combi.Combi.multinomial: Multinomial Coefficients

+ +

Library Combi.Combi.ordtree

+ +

Library Combi.Combi.setpartition: Set Partitions

+ +

Library Combi.Combi.skewpart: Skew Partitions

+ +

Library Combi.Combi.skewtab: Skew Tableaux

+ +

Library Combi.Combi.std: Standard Words, i.e. Permutation as Words

+ +

Library Combi.Combi.stdtab: Standard Tableaux

+ +

Library Combi.Combi.subseq: Subsequence of a sequence as a fintype

+ +

Library Combi.Combi.tableau: Young Tableaux

+ +

Library Combi.Combi.vectNK: Integer Vector of Given Sums and Sizes

+ +

Library Combi.Combi.Yamanouchi: Yamanouchi Words

+ +

Library Combi.Erdos_Szekeres.Erdos_Szekeres: The Erdös-Szekeres theorem

+ +

Library Combi.LRrule.extract: Extracting the implementation to OCaml

+ +

Library Combi.LRrule.freeSchur: Free Schur functions

+ +

Library Combi.LRrule.Greene: Greene monotone subsequence numbers

+ +

Library Combi.LRrule.Greene_inv: Greene subsequence theorem

+ +

Library Combi.LRrule.implem: A Coq implementation of the Littlewood-Richarson rule

+ +

Library Combi.LRrule.plactic: The plactic monoid

+ +

Library Combi.LRrule.Schensted: The Robinson-Schensted correspondence

+ +

Library Combi.LRrule.shuffle: Shuffle and shifted shuffle

+ +

Library Combi.LRrule.stdplact: Plactic congruences and standardization

+ +

Library Combi.LRrule.therule: The Littlewood-Richardson rule

+ +

Library Combi.LRrule.Yam_plact: Plactic classes and Yamanouchi words

+ +

Library Combi.SSRcomplements.ordcast: Cast between ordinals

+ +

Library Combi.SSRcomplements.sorted: [path] and [sorted] complements

+ +

Library Combi.SSRcomplements.permcomp: Complement on permutations

+ +

Library Combi.SSRcomplements.tools: Missing SSReflect sequence and set lemmas

+ +

Library Combi.HookFormula.Frobenius_ident

+ +

Library Combi.HookFormula.hook: A proof of the Hook-Length formula

+ +

Library Combi.Basic.combclass: Fintypes for Combinatorics

+ +

Library Combi.Basic.congr: Rewriting rule and congruencies of words

+ +

Library Combi.Basic.ordtype: Ordered Types

+ +

Library Combi.Basic.unitriginv: Uni-triangular Matrices

+ +

Library Combi.SymGroup.cycles: The Cycle Decomposition of a Permutation

+ +

Library Combi.SymGroup.cycletype: The Cycle Type of a Permutation

+ +

Library Combi.SymGroup.Frobenius_char: Frobenius characteristic

+ +

Library Combi.SymGroup.permcent: The Centralizer of a Permutation

+ +

Library Combi.SymGroup.presentSn: The Coxeter Presentation of the Symmetric Group

+ +

Library Combi.SymGroup.reprSn: Basic representations of the Symmetric Groups

+ +

Library Combi.SymGroup.towerSn: The Tower of the Symmetric Groups

+ +

Library Combi.SymGroup.weak_order: The weak order on the Symmetric Group

+ +

Library Combi.MPoly.antisym: Antisymmetric polynomials and Vandermonde product

+ +

Library Combi.MPoly.Cauchy: Cauchy formula for symmetric polynomials

+ +

Library Combi.MPoly.homogsym: Homogenous Symmetric Polynomials

+ +

Library Combi.MPoly.sympoly: Symmetric Polynomials

+ +

Library Combi.MPoly.Schur_mpoly: Schur symmetric polynomials

+ +

Library Combi.MPoly.Schur_altdef: Alternants definition of Schur polynomials

+ +

Library Combi.MPoly.MurnaghanNakayama: Murnaghan-Nakayama rule

+ +

Library ALEA.Ccpo

+ +

Library ALEA.Misc

+ +

Library ALEA.Qmeasure: Finite probabilities

+ +
+
This page has been generated by coqdoc +
+ +
+ + + \ No newline at end of file diff --git a/combi/README.html b/combi/README.html index 12bfa875..1ae779d5 100644 --- a/combi/README.html +++ b/combi/README.html @@ -1,23 +1,30 @@ -

Coq-Combi

- -

Formalisation of (algebraic) combinatorics in Coq/MathComp.

- -

Authors

- -

Florent Hivert Florent.Hivert@lisn.fr

- +

Rocq-Combi

+

Formalisation of algebraic combinatorics in Rocq/MathComp.

+

+

Authors

+

Florent Hivert Florent.Hivert@lisn.fr

Contributors:

-
    -
  • Thibaut Benjamin (representation theory of the symmetric groups)
    • +
  • Thibaut Benjamin (representation theory of the symmetric +groups)
  • Jean Christophe Filliâtre (Why3 implementation)
    • -
  • Christine Paulin (SSreflect binding for ALEA + hook length formula)
    • +
  • Christine Paulin (SSreflect binding for ALEA + hook length +formula)
  • Olivier Stietel (hook length formula)
  • Cyril Cohen (MathComp compatibility + nix)
    • +
  • Pierre Roux (MathComp compatibility + nix)
-

This library was supported by additional discussions with:

-
  • Kazuhiko Sakaguchi (port on MathComp2 / Hierarchy Builder)
  • Georges Gonthier
    • @@ -25,70 +32,75 @@

    Authors

  • Pierre Yves Strub
  • the SSReflect mailing list
-

The project was transferred to mathcomp on 2021-10-20.

- -

Contents

- +

Contents

    -

  • basic theory of symmetric functions including

    - +

  • basic theory of symmetric functions +including

      -

    • Schur function and Kostka numbers and the equivalence of the -combinatorial and algebraic (Jacobi) definition of Schur polynomials

      • -

    • the classical bases, Newton formulas and various basis changes

      • +

    • Schur function and Kostka numbers and the +equivalence of the combinatorial and algebraic (Jacobi) definition of +Schur polynomials

      • +

    • the classical bases, Newton formulas and various basis +changes

    • the scalar product and the Cauchy formula

    • -

  • the Littlewood-Richardson rule using Schützenberger approach, it includes

    - +

  • the Littlewood-Richardson rule using +Schützenberger approach, it includes

    • the Robinson-Schensted correspondence

      • -

    • the construction of the plactic monoïd using Greene invariants

      • -

    • the Littlewood-Richardson and Pieri rules using the combinatorial -(tableau) definition of Schur polynomials.

      • +

    • the construction of the plactic monoïd using Greene +invariants

      • +

    • the Littlewood-Richardson and Pieri rules using +the combinatorial (tableau) definition of Schur polynomials.

    - -

    After A. Lascoux, B. Leclerc and J.-Y. Thibon, "The Plactic Monoid" in -Lothaire, M. (2011), Algebraic combinatorics on words, Cambridge University -Press With variant described in G. Duchamp, F. Hivert, and J.-Y. Thibon, -Noncommutative symmetric functions VI. Free quasi-symmetric functions and -related algebras. Internat. J. Algebra Comput. 12 (2002), 671–717.

    • -

  • the Murnaghan-Nakayama rule for converting power sum to Schur function, -it includes

    - +

    After A. Lascoux, B. Leclerc and J.-Y. Thibon, “The Plactic Monoid” +in Lothaire, M. (2011), Algebraic combinatorics on words, Cambridge +University Press With variant described in G. Duchamp, F. Hivert, and +J.-Y. Thibon, Noncommutative symmetric functions VI. Free +quasi-symmetric functions and related algebras. Internat. J. Algebra +Comput. 12 (2002), 671–717.

    • +

  • the Murnaghan-Nakayama rule for converting power +sum to Schur function, it includes

      -

    • two recursive implementations building the tableau upward or downward

      • -

    • a skew version multiplying a Schur function by a power sum expanding the -result on Schur functions.

      • +

    • two recursive implementations building the tableau upward or +downward

      • +

    • a skew version multiplying a Schur function by a power sum +expanding the result on Schur functions.

    • -

  • the character theory of the symmetric Groups. We do not use -representations but rather goes as fast as possible to Frobenius -isomorphism and then uses computations with symmetric polynomials. It includes

    - +

  • the character theory of the symmetric Groups. We +do not use representations but rather goes as fast as possible to +Frobenius isomorphism and then uses computations with symmetric +polynomials. It includes

      -

    • cycle types for permutations (together with Thibaut Benjamin)

      • -

    • The tower structure and the restriction and induction formulas for class -indicator (together with Thibaut Benjamin)

      • -

    • the structure of the centralizer of a permutation

      • +

    • cycle types for permutations (together with Thibaut +Benjamin)

      • +

    • The tower structure and the restriction and induction +formulas for class indicator (together with Thibaut +Benjamin)

      • +

    • the structure of the centralizer of a +permutation

    • Young character and Young Rule

      • -

    • the theory of Frobenius characteristic and Frobenius character formula

      • -

    • the Murnaghan-Nakayama rule for evaluating irreducible characters

      • -

    • the Littlewood-Richardson rule for inducing irreducible characters

      • +

    • the theory of Frobenius characteristic and Frobenius +character formula

      • +

    • the Murnaghan-Nakayama rule for evaluating irreducible +characters

      • +

    • the Littlewood-Richardson rule for inducing irreducible +characters

    • -

  • the Hook-Length Formula for standard Young tableaux -(together with Christine Paulin and Olivier Stietel). We follow closely

    - -

    Greene, C., Nijenhuis, A. and Wilf, H. S. (1979) A probabilistic proof of a -formula for the number of Young tableaux of a given shape. Adv. in -Math. 31, 104–109.

    • -

  • the Erdös Szekeres theorem about increassing and decreassing subsequences

    - -

    from Greene's invariants theorem.

    • +

  • the Hook-Length Formula for standard Young +tableaux (together with Christine Paulin and Olivier Stietel). We follow +closely

    +

    Greene, C., Nijenhuis, A. and Wilf, H. S. (1979) A probabilistic +proof of a formula for the number of Young tableaux of a given shape. +Adv. in Math. 31, 104–109.

    • +

  • the Erdös Szekeres theorem about increassing and +decreassing subsequences

    +

    from Greene’s invariants theorem.

  • various Combinatorial objects including

    -
      -
    • integer partitions and compositions, together with Young's and dominance -lattices
      • +
    • integer partitions and compositions, together with Young’s and +dominance lattices
    • skew partition, horizontal, vertical and ribbon border strip
    • tableaux, standard tableaux, skew tableaux
    • subsequences, integer vectors
      • @@ -97,77 +109,89 @@

      Contents

    • binary trees, Dyck words and Catalan numbers
    • set partition and refinement order
    • -

  • the Coxeter presentation of the symmetric group.

    - +

  • the Coxeter presentation of the symmetric +group.

    We formalize:

    -
    • presentation of the symmetric group generated by elementary transpositions
      • -
    • Matsumoto theorem saying that two reduced words give the same permutation -iff they are equivalent under braid relations
      • +
    • Matsumoto theorem saying that two reduced words give the same +permutation iff they are equivalent under braid relations
    • the Coxeter length and the inversion set
    • the dual Lehmer code of a permutation
    • the weak permutohedron lattice
    • -

  • the factorization of the Vandermonde determinant as the product -of differences.

    • +

  • the factorization of the Vandermonde determinant +as the product of differences.

  • the Tamari lattice on binary trees.

    • -

  • the formula for Catalan numbers counting binary trees and Dyck words.

    - -

    I use a bijective proof using rotations. There is a generating function -proof available in https://github.com/hivert/FormalPowerSeries which I plan -to merge here at some points.

    • +

  • the formula for Catalan numbers counting binary +trees and Dyck words.

    +

    I use a bijective proof using rotations. There is a generating +function proof available in https://github.com/hivert/FormalPowerSeries +which I plan to merge here at some points.

- -

Various unstable/unfinished experiments:

- +

Documentation

- -

Documentation

- - - -

Installation

- +

Various +unstable/unfinished experiments:

+ +

Installation

This library is based on

- - -

Here are the Opam packages I'm using - -coq-hierarchy-builder 1.6.0 -coq-mathcomp-ssreflect 2.1.0 -coq-mathcomp-algebra 2.1.0 -coq-mathcomp-field 2.1.0 -coq-mathcomp-fingroup 2.1.0 -coq-mathcomp-character 2.1.0 -coq-mathcomp-multinomials 2.1.0 -

+

Here are the Opam packages I’m using

+
rocq-hierarchy-builder        1.9.1
+rocq-mathcomp-ssreflect       2.5.0
+rocq-mathcomp-algebra         2.5.0
+rocq-mathcomp-field           2.5.0
+rocq-mathcomp-fingroup        2.5.0
+rocq-mathcomp-character       2.5.0
+coq-mathcomp-multinomials     2.4.0