shorten proofs

Co-authored-by: Holger Thies <thies.holger.5c@kyoto-u.ac.jp>

Author: affeldt-aist

CHANGELOG_UNRELEASED.md

@@ -110,6 +110,9 @@
     `differentiable_row_mx`
   + instance `is_diff_row_mx`
 
+- in `unstable.v`:
+  + lemma `big_split_ord_idem`
+
 ### Changed
 
 - moved from `measurable_structure.v` to `classical_sets.v`:

classical/unstable.v

@@ -651,3 +651,21 @@ End Theory.
 Module Import Exports. HB.reexport. End Exports.
 End Norm.
 Export Norm.Exports.
+
+(* NB: big_split_org in bigop.v requires Monoid.law idx,
+   big_split_ord_idem will be made available from MathComp
+   via PR https://github.com/math-comp/math-comp/pull/1608
+   (MathComp 2.7.0) *)
+Lemma big_split_ord_idem [R : Type] [idx : R] (op : SemiGroup.com_law R) m n
+    (P : pred 'I_(m + n)) F :
+  op idx idx = idx ->
+  \big[op/idx]_(i | P i) F i =
+        op (\big[op/idx]_(i | P (lshift n i)) F (lshift n i))
+        (\big[op/idx]_(i | P (rshift m i)) F (rshift m i)).
+Proof.
+move=> opxx.
+rewrite -(big_map (lshift n) P F) -(big_map (@rshift m _) P F).
+rewrite -big_cat_idem//; congr bigop; apply: (inj_map val_inj).
+rewrite map_cat -!map_comp (map_comp (addn m)) /=.
+by rewrite ![index_enum _]unlock unlock !val_ord_enum -iotaDl addn0 iotaD.
+Qed.

theories/normedtype_theory/matrix_normedtype.v

@@ -2,6 +2,8 @@
 From HB Require Import structures.
 From mathcomp Require Import all_ssreflect_compat finmap ssralg ssrnum matrix.
 From mathcomp Require Import interval interval_inference.
+#[warning="-warn-library-file-internal-analysis"]
+From mathcomp Require Import unstable.
 From mathcomp Require Import boolp contra classical_sets reals topology.
 From mathcomp Require Import prodnormedzmodule tvs pseudometric_normed_Zmodule.
 From mathcomp Require Import normed_module.
@@ -281,34 +283,24 @@ Implicit Types (M : 'M[K]_(m, n1)) (N : 'M[K]_(m, n2)).
 
 Lemma norm_row_mx0r M : `|row_mx M (0 : 'M_(m, n2))| = `|M|.
 Proof.
-apply/eqP; rewrite eq_le; apply/andP; split.
-- rewrite [leLHS]/Num.norm/= !mx_normrE/=; apply/bigmax_le => // -[i j]/= _.
-  rewrite mxE; case: splitP => k kE; last by rewrite mxE normr0.
-  by rewrite [leRHS]/Num.norm/= !mx_normrE/= (le_bigmax _ _ (i, k)).
-- rewrite [leLHS]/Num.norm/= !mx_normrE/=; apply/bigmax_le => // -[i j]/= _.
-  rewrite [leRHS]/Num.norm/= !mx_normrE/=.
-  rewrite (le_trans _(le_bigmax _ _ (i, lshift n2 j)))// mxE/=.
-  case: splitP => // k kE.
-    rewrite (_ : j = k)//; apply: val_inj => /=.
-    by apply/eqP; rewrite -(eqn_add2l n1) -kE.
-  absurd: kE.
-  by rewrite ltn_eqF// (leq_trans (ltn_ord j))// /lshift/= leq_addr.
+rewrite /Num.norm/= !mx_normrE.
+rewrite -!(pair_bigA_idem _ (fun i j => `|_ i j|))/= ?maxxx//.
+apply: eq_bigr => i _.
+have ? := maxxx (0 : K).
+rewrite big_split_ord_idem//= [X in maxr _ X = _]big1_idem//=.
+  by move=> ? _; rewrite row_mxEr mxE normr0.
+by rewrite -bigmax_idr; apply: eq_bigr => j _; rewrite row_mxEl.
 Qed.
 
 Lemma norm_row_mx0l N : `|row_mx (0 : 'M_(m, n1)) N| = `|N|.
 Proof.
-apply/eqP; rewrite eq_le; apply/andP; split.
-- rewrite [leLHS]/Num.norm/= !mx_normrE/=; apply/bigmax_le => // -[i j]/= _.
-  rewrite mxE; case: splitP => k kE; first by rewrite mxE normr0.
-  by rewrite [leRHS]/Num.norm/= !mx_normrE/= (le_bigmax _ _ (i, k)).
-- rewrite [leLHS]/Num.norm/= !mx_normrE/=; apply/bigmax_le => // -[i j]/= _.
-  rewrite [leRHS]/Num.norm/= !mx_normrE/=.
-  rewrite (le_trans _ (le_bigmax _ _ (i, rshift n1 j)))// mxE/=.
-  case: splitP => // k kE.
-    absurd: kE.
-    by rewrite gtn_eqF// (leq_trans (ltn_ord k))// /rshift/= leq_addr.
-  rewrite (_ : j = k)//; apply: val_inj => /=.
-  by apply/eqP; rewrite -(eqn_add2l n1) -kE.
+rewrite /Num.norm/= !mx_normrE.
+rewrite -!(pair_bigA_idem _ (fun i j => `|_ i j|))/= ?maxxx//.
+apply: eq_bigr => i _.
+have ? := maxxx (0 : K).
+rewrite big_split_ord_idem//= [X in maxr X _ = _]big1_idem//=.
+  by move=> ? _; rewrite row_mxEl mxE normr0.
+by rewrite -bigmax_idl; apply: eq_bigr => j _; rewrite row_mxEr.
 Qed.
 
 Lemma norm_row_mx M N : `|row_mx M N| = Num.max `|M| `|N|.