feat: stationary sets (#39727)

We define stationary sets as sets intersecting all club sets, and prove basic theorems about them.

Author: vihdzp

Mathlib/Order/Cofinal.lean

@@ -52,6 +52,11 @@ theorem IsCofinal.mono {s t : Set α} (h : s ⊆ t) (hs : IsCofinal s) : IsCofin
   obtain ⟨b, hb, hb'⟩ := hs a
   exact ⟨b, h hb, hb'⟩
 
+theorem IsCofinal.nonempty [Nonempty α] {s : Set α} (h : IsCofinal s) : s.Nonempty := by
+  inhabit α
+  obtain ⟨x, hx, _⟩ := h default
+  exact ⟨x, hx⟩
+
 end LE
 
 section Preorder

Mathlib/SetTheory/Cardinal/Cofinality/Club.lean

@@ -10,11 +10,13 @@ public import Mathlib.Order.IsNormal
 public import Mathlib.SetTheory.Cardinal.Cofinality.Basic
 
 /-!
-# Club sets
+# Club sets and stationary sets
 
-A subset of a well-ordered type `α` is called a club set when it is closed in the order topology and
-cofinal. If `α` has no maximum, then an equivalent condition is that `α` is closed and unbounded;
-hence the name.
+A subset of a well-ordered type `α` is called a **club set** when it is closed in the order topology
+and cofinal. If `α` has no maximum, then an equivalent condition is that `α` is closed and
+unbounded; hence the name.
+
+A **stationary set** is a set which intersects all club sets.
 
 ## Implementation notes
 
@@ -27,7 +29,9 @@ public section
 
 universe u v
 
-open Cardinal Order
+open Cardinal Order Set
+
+variable {α : Type v} {s t : Set α} {x : α} [LinearOrder α]
 
 /-- A club set is closed under suprema and cofinal. -/
 structure IsClub {α : Type*} [LinearOrder α] (s : Set α) where
@@ -40,8 +44,6 @@ structure IsClub {α : Type*} [LinearOrder α] (s : Set α) where
 
 namespace IsClub
 
-variable {α : Type v} {s t : Set α} {x : α} [LinearOrder α]
-
 @[simp]
 theorem of_isEmpty [IsEmpty α] {s : Set α} : IsClub s :=
   ⟨.of_isEmpty, .of_isEmpty⟩
@@ -50,6 +52,12 @@ theorem of_isEmpty [IsEmpty α] {s : Set α} : IsClub s :=
 protected theorem univ : IsClub (α := α) .univ :=
   ⟨.univ, .univ⟩
 
+protected theorem nonempty [Nonempty α] (hs : IsClub s) : s.Nonempty :=
+  hs.isCofinal.nonempty
+
+theorem _root_.isClub_empty_iff : IsClub (α := α) ∅ ↔ IsEmpty α :=
+  ⟨fun h ↦ isCofinal_empty_iff.1 h.isCofinal, fun _ ↦ .of_isEmpty⟩
+
 protected theorem union (hs : IsClub s) (ht : IsClub t) : IsClub (s ∪ t) :=
   ⟨hs.dirSupClosed.union ht.dirSupClosed, hs.isCofinal.mono Set.subset_union_left⟩
 
@@ -63,12 +71,12 @@ theorem csSup_mem {α} [ConditionallyCompleteLinearOrder α] {s t : Set α}
 theorem sInter_of_orderTop {s : Set (Set α)} [OrderTop α] (hs : ∀ x ∈ s, IsClub x) :
     IsClub (⋂₀ s) := by
   refine ⟨.sInter fun x hx ↦ (hs x hx).dirSupClosed, ?_⟩
-  rw [isCofinal_iff_top_mem, Set.mem_sInter]
+  rw [isCofinal_iff_top_mem, mem_sInter]
   exact fun x hx ↦ (hs x hx).isCofinal.top_mem
 
 theorem iInter_of_orderTop {ι : Type*} {f : ι → Set α} [OrderTop α] (hs : ∀ i, IsClub (f i)) :
     IsClub (⋂ i, f i) := by
-  rw [← Set.sInter_range]
+  rw [← sInter_range]
   exact .sInter_of_orderTop (by simpa)
 
 theorem sInter_of_cof_le_one {s : Set (Set α)} (hα : cof α ≤ 1) (hs : ∀ x ∈ s, IsClub x) :
@@ -80,11 +88,10 @@ theorem sInter_of_cof_le_one {s : Set (Set α)} (hα : cof α ≤ 1) (hs : ∀ x
 
 theorem iInter_of_cof_le_one {ι : Type*} {f : ι → Set α} (hα : cof α ≤ 1) (hs : ∀ i, IsClub (f i)) :
     IsClub (⋂ i, f i) := by
-  rw [← Set.sInter_range]
+  rw [← sInter_range]
   exact .sInter_of_cof_le_one hα (by simpa)
 
 section WellFoundedLT
-
 variable [WellFoundedLT α]
 
 attribute [local instance]
@@ -102,7 +109,7 @@ protected theorem sInter {s : Set (Set α)} (hα : cof α ≠ ℵ₀) (hsα : #s
     refine .of_not_isCofinal fun hg ↦ (cof_le hg).not_gt (hα.trans_le' ?_)
     simpa using mk_range_le_lift (f := g)
   refine ⟨_, fun t ht ↦ ?_, le_csSup hg ⟨0, rfl⟩⟩
-  apply (hs t ht).isLUB_mem (t := .range fun n ↦ f ⟨t, ht⟩ (g n)) _ (Set.range_nonempty _)
+  apply (hs t ht).isLUB_mem (t := .range fun n ↦ f ⟨t, ht⟩ (g n)) _ (range_nonempty _)
   · refine ⟨?_, fun b hb ↦ csSup_le' ?_⟩ <;> rintro _ ⟨n, rfl⟩
     · apply (le_csSup (.of_not_isCofinal _) _).trans (le_csSup hg ⟨n + 1, rfl⟩)
       · exact fun hg' ↦ (cof_le hg').not_gt (mk_range_le.trans_lt hsα)
@@ -113,19 +120,27 @@ protected theorem sInter {s : Set (Set α)} (hα : cof α ≠ ℵ₀) (hsα : #s
 protected theorem iInter {ι : Type u} {f : ι → Set α} (hα : cof α ≠ ℵ₀)
     (hι : Cardinal.lift.{v} #ι < Cardinal.lift.{u} (cof α)) (hf : ∀ i, IsClub (f i)) :
     IsClub (⋂ i, f i) := by
-  rw [← Set.sInter_range]
+  rw [← sInter_range]
   refine IsClub.sInter hα ?_ (by simpa)
   rw [← Cardinal.lift_lt]
   exact mk_range_le_lift.trans_lt hι
 
-protected theorem inter {s t : Set α} (hα : cof α ≠ ℵ₀) (hs : IsClub s) (ht : IsClub t) :
-    IsClub (s ∩ t) := by
-  rw [← Set.sInter_pair]
-  have H : ∀ x ∈ ({s, t} : Set _), IsClub x := by simpa [hs]
-  obtain hα | hα' := hα.lt_or_gt
-  · rw [cof_lt_aleph0_iff] at hα
-    exact .sInter_of_cof_le_one hα H
-  · exact .sInter hα (hα'.trans_le' <| by simp) H
+theorem sInter_of_countable {s : Set (Set α)} (hα : cof α ≠ ℵ₀) (hsα : s.Countable)
+    (hs : ∀ x ∈ s, IsClub x) : IsClub (⋂₀ s) := by
+  obtain hα | hα := hα.lt_or_gt
+  · apply IsClub.sInter_of_cof_le_one _ hs
+    rwa [← cof_lt_aleph0_iff]
+  · apply IsClub.sInter hα.ne' (hα.trans_le' _) hs
+    rwa [le_aleph0_iff_set_countable]
+
+theorem iInter_of_countable {ι : Sort*} {f : ι → Set α} [Countable ι] (hα : cof α ≠ ℵ₀)
+    (hf : ∀ i, IsClub (f i)) : IsClub (⋂ i, f i) := by
+  rw [← sInter_range]
+  apply IsClub.sInter_of_countable hα (countable_range f)
+  simpa
+
+protected theorem inter (hα : cof α ≠ ℵ₀) (hs : IsClub s) (ht : IsClub t) : IsClub (s ∩ t) := by
+  simpa [hs, ht] using IsClub.sInter_of_countable (s := {s, t}) hα
 
 theorem _root_.Order.IsNormal.isClub_range {f : α → α} (hf : IsNormal f) : IsClub (.range f) :=
   ⟨hf.dirSupClosed_range, fun x ↦ ⟨_, ⟨x, rfl⟩, hf.strictMono.le_apply⟩⟩
@@ -134,7 +149,7 @@ theorem _root_.Order.IsNormal.isClub_fixedPoints {f : α → α} (hα : cof α 
     IsClub f.fixedPoints := by
   cases isEmpty_or_nonempty α; · simp
   refine ⟨fun s hs hs₀ _ a ha ↦ (hf.map_isLUB ha hs₀).unique ?_, fun a ↦ ?_⟩
-  · rwa [Set.image_congr hs, Set.image_id']
+  · rwa [image_congr hs, image_id']
   · cases topOrderOrNoTopOrder α with
     | inl => use ⊤; simpa using! hf.strictMono.id_le ⊤
     | inr h =>
@@ -147,3 +162,115 @@ theorem _root_.Order.IsNormal.isClub_fixedPoints {f : α → α} (hα : cof α 
 
 end WellFoundedLT
 end IsClub
+
+/-! ### Stationary sets -/
+
+/-- A set is called stationary when it intersects all club sets. -/
+@[expose]
+def IsStationary (s : Set α) : Prop :=
+  ∀ ⦃t⦄, IsClub t → (s ∩ t).Nonempty
+
+theorem not_isStationary_iff : ¬ IsStationary s ↔ ∃ t, IsClub t ∧ Disjoint s t := by
+  simp [IsStationary, disjoint_iff, not_nonempty_iff_eq_empty]
+
+@[gcongr]
+theorem IsStationary.mono (hs : IsStationary s) (h : s ⊆ t) : IsStationary t :=
+  fun _u hu ↦ (hs hu).mono (inter_subset_inter_left _ h)
+
+theorem IsStationary.nonempty (hs : IsStationary s) : s.Nonempty := by
+  simpa using hs .univ
+
+theorem isStationary_univ_iff : IsStationary (.univ (α := α)) ↔ Nonempty α := by
+  simp [IsStationary, ← not_imp_not (b := IsClub _), not_nonempty_iff_eq_empty,
+    isClub_empty_iff]
+
+@[simp]
+protected theorem IsStationary.univ [Nonempty α] : IsStationary (.univ (α := α)) :=
+  isStationary_univ_iff.2 ‹_›
+
+@[simp]
+theorem not_isStationary_empty : ¬ IsStationary (∅ : Set α) := by
+  intro h
+  simpa using h .univ
+
+@[simp]
+theorem not_isStationary_of_isEmpty [IsEmpty α] : ¬ IsStationary s :=
+  s.eq_empty_of_isEmpty ▸ not_isStationary_empty
+
+theorem IsStationary.of_not_isCofinal_compl (hs : ¬ IsCofinal sᶜ) : IsStationary s := by
+  intro t ht
+  obtain ⟨a, ha⟩ := not_isCofinal_iff.1 hs
+  obtain ⟨b, hb, hb'⟩ := ht.isCofinal a
+  refine ⟨b, ?_, hb⟩
+  contrapose! ha
+  exact ⟨b, ha, hb'⟩
+
+theorem isStationary_sUnion_iff_of_cof_le_one {s : Set (Set α)} (hα : cof α ≤ 1) :
+    IsStationary (⋃₀ s) ↔ ∃ x ∈ s, IsStationary x where
+  mp h := by
+    contrapose! h
+    simp_rw [not_isStationary_iff] at h ⊢
+    choose f hf hxf using h
+    refine ⟨⋂ x : s, f _ x.2, ?_, ?_⟩
+    · apply IsClub.iInter_of_cof_le_one hα
+      simpa
+    · rw [disjoint_sUnion_left]
+      exact fun x hx ↦ (hxf _ hx).mono_right (iInter_subset _ ⟨x, hx⟩)
+  mpr := fun ⟨x, hxs, hx⟩ ↦ hx.mono (subset_sUnion_of_mem hxs)
+
+theorem isStationary_iUnion_iff_of_cof_le_one {ι : Sort*} {f : ι → Set α} (hα : cof α ≤ 1) :
+    IsStationary (⋃ i, f i) ↔ ∃ i, IsStationary (f i) := by
+  rw [← sUnion_range, isStationary_sUnion_iff_of_cof_le_one hα]
+  simp
+
+theorem isStationary_sUnion_iff_of_orderTop [OrderTop α] {s : Set (Set α)} :
+    IsStationary (⋃₀ s) ↔ ∃ x ∈ s, IsStationary x :=
+  isStationary_sUnion_iff_of_cof_le_one (by simp)
+
+theorem isStationary_iUnion_iff_of_orderTop [OrderTop α] {ι : Sort*} {f : ι → Set α} :
+    IsStationary (⋃ i, f i) ↔ ∃ i, IsStationary (f i) :=
+  isStationary_iUnion_iff_of_cof_le_one (by simp)
+
+section WellFoundedLT
+variable [WellFoundedLT α]
+
+theorem IsClub.isStationary [Nonempty α] (hα : cof α ≠ ℵ₀) (hs : IsClub s) : IsStationary s :=
+  fun _ ht ↦ (hs.inter hα ht).nonempty
+
+theorem isStationary_sUnion_iff {s : Set (Set α)} (hα : cof α ≠ ℵ₀) (hsα : #s < cof α) :
+    IsStationary (⋃₀ s) ↔ ∃ x ∈ s, IsStationary x where
+  mp h := by
+    contrapose! h
+    simp_rw [not_isStationary_iff] at h ⊢
+    choose f hf hxf using h
+    refine ⟨⋂ x : s, f _ x.2, ?_, ?_⟩
+    · apply IsClub.iInter hα <;> simpa
+    · rw [disjoint_sUnion_left]
+      exact fun x hx ↦ (hxf _ hx).mono_right (iInter_subset _ ⟨x, hx⟩)
+  mpr := fun ⟨x, hxs, hx⟩ ↦ hx.mono (subset_sUnion_of_mem hxs)
+
+theorem isStationary_iUnion_iff {ι : Type u} {f : ι → Set α} (hα : cof α ≠ ℵ₀)
+    (hι : lift.{v} #ι < lift.{u} (cof α)) : IsStationary (⋃ i, f i) ↔ ∃ i, IsStationary (f i) := by
+  rw [← sUnion_range, isStationary_sUnion_iff hα]
+  · simp
+  · rw [← Cardinal.lift_lt]
+    exact mk_range_le_lift.trans_lt hι
+
+theorem isStationary_sUnion_iff_of_countable {s : Set (Set α)} (hα : cof α ≠ ℵ₀)
+    (hsα : s.Countable) : IsStationary (⋃₀ s) ↔ ∃ x ∈ s, IsStationary x := by
+  obtain hα | hα := hα.lt_or_gt
+  · apply isStationary_sUnion_iff_of_cof_le_one
+    rwa [← cof_lt_aleph0_iff]
+  · apply isStationary_sUnion_iff hα.ne' (hα.trans_le' _)
+    rwa [le_aleph0_iff_set_countable]
+
+theorem isStationary_iUnion_iff_of_countable {ι : Sort*} {f : ι → Set α} [Countable ι]
+    (hα : cof α ≠ ℵ₀) : IsStationary (⋃ i, f i) ↔ ∃ i, IsStationary (f i) := by
+  rw [← sUnion_range, isStationary_sUnion_iff_of_countable hα (countable_range f)]
+  simp
+
+theorem isStationary_union_iff (hα : cof α ≠ ℵ₀) :
+    IsStationary (s ∪ t) ↔ IsStationary s ∨ IsStationary t := by
+  simpa using isStationary_sUnion_iff_of_countable (s := {s, t}) hα
+
+end WellFoundedLT