union

Author: vihdzp

Mathlib/SetTheory/Cardinal/Cofinality/Club.lean

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+/-
+Copyright (c) 2026 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+-/
+module
+
+public import Mathlib.SetTheory.Cardinal.Cofinality.Basic
+public import Mathlib.Order.DirSupClosed
+
+/-!
+# Club 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.
+
+## Implementation notes
+
+To avoid importing topology in the ordinals, we spell out the closure property using `DirSupClosed`.
+For any type equipped with the Scott-Hausdorff topology (which includes well-orders with the order
+topology), `DirSupClosed s` and `IsClosed s` are equivalent predicates.
+-/
+
+@[expose] public section
+
+universe u v
+
+open Cardinal Order
+
+/-- A club set is closed under suprema and cofinal. -/
+structure IsClub {α : Type*} [LinearOrder α] (s : Set α) where
+  /-- Club sets are closed under suprema. If `α` is a well-order with the order topology, this
+  condition is equivalent to `IsClosed s`. -/
+  dirSupClosed : DirSupClosed s
+  /-- Club sets are cofinal. If `α` has no maximum, this condition is equivalent to `¬ BddAbove s`.
+  See `not_bddAbove_iff_isCofinal`. -/
+  isCofinal : IsCofinal s
+
+variable {α : Type v} {s t : Set α} {x : α} [LinearOrder α]
+
+@[simp]
+theorem IsClub.of_isEmpty [IsEmpty α] (s : Set α) : IsClub s :=
+  ⟨.of_isEmpty s, .of_isEmpty s⟩
+
+@[simp]
+theorem IsClub.univ : IsClub (.univ (α := α)) :=
+  ⟨.univ, .univ⟩
+
+theorem IsClub.union (hs : IsClub s) (ht : IsClub t) : IsClub (s ∪ t) :=
+  ⟨hs.dirSupClosed.union ht.dirSupClosed, hs.isCofinal.mono Set.subset_union_left⟩
+
+theorem IsClub.isLUB_mem (hs : IsClub s) (ht : t ⊆ s) (ht₀ : t.Nonempty) (hx : IsLUB t x) : x ∈ s :=
+  hs.dirSupClosed ht ht₀ (Std.Total.directedOn _) hx
+
+theorem IsClub.csSup_mem {α} [ConditionallyCompleteLinearOrder α] {s t : Set α}
+    (hs : IsClub s) (ht : t ⊆ s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) : sSup t ∈ s :=
+  hs.isLUB_mem ht ht₀ (isLUB_csSup ht₀ ht₁)
+
+theorem IsClub.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]
+  exact fun x hx ↦ (hs x hx).isCofinal.top_mem
+
+theorem IsClub.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]
+  exact .sInter_of_orderTop (by simpa)
+
+theorem IsClub.sInter_of_cof_le_one {s : Set (Set α)} (hα : cof α ≤ 1)
+    (hs : ∀ x ∈ s, IsClub x) : IsClub (⋂₀ s) := by
+  cases isEmpty_or_nonempty α; · simp
+  cases topOrderOrNoTopOrder α
+  · exact .sInter_of_orderTop hs
+  · cases one_lt_cof.not_ge hα
+
+theorem IsClub.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]
+  exact .sInter_of_cof_le_one hα (by simpa)
+
+section WellFoundedLT
+
+variable [WellFoundedLT α]
+
+attribute [local instance]
+  WellFoundedLT.toOrderBot WellFoundedLT.conditionallyCompleteLinearOrderBot
+
+theorem IsClub.sInter {s : Set (Set α)} (hα : cof α ≠ ℵ₀) (hsα : #s < cof α)
+    (hs : ∀ x ∈ s, IsClub x) : IsClub (⋂₀ s) := by
+  cases isEmpty_or_nonempty α; · simp
+  obtain hα | hα := hα.lt_or_gt
+  · exact .sInter_of_cof_le_one (cof_lt_aleph0_iff.1 hα) hs
+  refine ⟨.sInter fun x hx ↦ (hs x hx).dirSupClosed, fun a ↦ ?_⟩
+  choose f hf using fun x : s ↦ (hs _ x.2).isCofinal
+  let g : ℕ → α := Nat.rec a fun _ IH ↦ sSup (.range (f · IH))
+  have hg : BddAbove (.range g) := by
+    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 _)
+  · 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)
+      · use ⟨t, ht⟩
+    · exact (hf ⟨t, ht⟩ _).2.trans <| hb ⟨_, rfl⟩
+  · grind
+
+theorem IsClub.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]
+  refine IsClub.sInter hα ?_ (by simpa)
+  rw [← Cardinal.lift_lt]
+  exact mk_range_le_lift.trans_lt hι
+
+theorem IsClub.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 Order.IsNormal.isClub_fixedPoints {f : α → α} (hα : cof α ≠ ℵ₀) (hf : IsNormal f) :
+    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']
+  · cases topOrderOrNoTopOrder α with
+    | inl => use ⊤; simpa using hf.strictMono.id_le ⊤
+    | inr h =>
+      rw [noTopOrder_iff_noMaxOrder] at h
+      suffices BddAbove (.range fun n ↦ f^[n] a) from
+        ⟨_, hf.iSup_iterate_mem_fixedPoints a this, le_csSup this ⟨0, rfl⟩⟩
+      refine .of_not_isCofinal fun h ↦ (cof_le h).not_gt
+        ((aleph0_le_cof.lt_of_ne' hα).trans_le' ?_)
+      simpa using mk_range_le_lift (f := fun n : ℕ ↦ f^[n] a)
+
+end WellFoundedLT