[Merged by Bors] - feat: a cofinal set has a cofinal subset of order type (cof α).ord

Mathlib/SetTheory/Cardinal/Aleph.lean

@@ -303,7 +303,7 @@ theorem ord_preAleph (o : Ordinal) : (preAleph o).ord = preOmega o := by
 
 @[simp]
 theorem _root_.Ordinal.type_lt_cardinal : typeLT Cardinal = Ordinal.univ.{u, u + 1} := by
-  simpa using preAleph.symm.toRelIsoLT.ordinal_type_eq
+  simpa using preAleph.symm.ordinalType_congr
 
 @[deprecated (since := "2026-03-20")] alias type_cardinal := type_lt_cardinal
 

Mathlib/SetTheory/Cardinal/Cofinality/Basic.lean

@@ -48,6 +48,7 @@ theorem le_cof_iff {c : Cardinal} : c ≤ cof α ↔ ∀ s : Set α, IsCofinal s
 @[deprecated (since := "2026-02-18")] alias le_cof := le_cof_iff
 
 variable (α) in
+/-- Every well-order has a cofinal subset of cardinal `cof α`. -/
 theorem exists_cof_eq : ∃ s : Set α, IsCofinal s ∧ #s = cof α := by
   obtain ⟨s, hs⟩ := ciInf_mem fun s : {s : Set α // IsCofinal s} ↦ #s
   exact ⟨s.1, s.2, hs⟩
@@ -133,6 +134,9 @@ theorem cof_congr_of_strictMono {f : α → γ} (hf : StrictMono f) (hf' : IsCof
     cof α = cof γ := by
   simpa using lift_cof_congr_of_strictMono hf hf'
 
+theorem cof_eq_of_isCofinal {s : Set α} (hs : IsCofinal s) : cof s = cof α :=
+  cof_congr_of_strictMono (Subtype.strictMono_coe _) (by simpa)
+
 @[simp]
 theorem cof_lt_aleph0_iff : cof α < ℵ₀ ↔ cof α ≤ 1 := by
   refine ⟨fun h ↦ ?_, (lt_of_le_of_lt · one_lt_aleph0)⟩

Mathlib/SetTheory/Cardinal/Cofinality/Ordinal.lean

@@ -151,7 +151,9 @@ theorem cof_omega0 : cof ω = ℵ₀ :=
 
 @[deprecated (since := "2026-02-18")] alias cof_eq_one_iff_is_succ := cof_eq_one_iff
 
-theorem exists_ord_cof_eq (α : Type*) [LinearOrder α] [WellFoundedLT α] :
+variable (α) in
+/-- Every well-order has a cofinal subset of order type `(cof α).ord`. -/
+theorem exists_ord_cof_eq [LinearOrder α] [WellFoundedLT α] :
     ∃ s : Set α, IsCofinal s ∧ typeLT s = (Order.cof α).ord := by
   obtain ⟨s, hs, hs'⟩ := exists_cof_eq α
   obtain ⟨r, hr, hr'⟩ := exists_ord_eq s
@@ -168,16 +170,21 @@ theorem exists_ord_cof_eq (α : Type*) [LinearOrder α] [WellFoundedLT α] :
 
 @[deprecated (since := "2026-05-25")] alias ord_cof_eq := exists_ord_cof_eq
 
+/-- Every cofinal set has a cofinal subset of order type `(cof α).ord`. -/
+theorem exists_ord_cof_eq_of_isCofinal [LinearOrder α] [WellFoundedLT α]
+    {s : Set α} (hs : IsCofinal s) : ∃ t ⊆ s, IsCofinal t ∧ typeLT t = (Order.cof α).ord := by
+  obtain ⟨t, ht, ht'⟩ := exists_ord_cof_eq s
+  rw [cof_eq_of_isCofinal hs] at ht'
+  refine ⟨t, ?_, hs.trans ht, ?_⟩
+  · simp
+  · rw [← ht']
+    exact ((Subtype.strictMono_coe _).strictMonoOn _).orderIso.ordinalType_congr.symm
+
 @[simp]
 theorem _root_.Order.cof_ord_cof (α : Type*) [LinearOrder α] [WellFoundedLT α] :
     (Order.cof α).ord.cof = Order.cof α := by
   obtain ⟨s, hs, hs'⟩ := exists_ord_cof_eq α
-  rw [← hs', cof_type]
-  apply le_antisymm
-  · rw [← card_ord (Order.cof α), ← hs', card_type]
-    exact cof_le_cardinalMk s
-  · rw [le_cof_iff]
-    exact fun t ht ↦ (cof_le (hs.trans ht)).trans_eq (mk_image_eq Subtype.val_injective)
+  rw [← hs', cof_type, cof_eq_of_isCofinal hs]
 
 @[simp]
 theorem cof_ord_cof (o : Ordinal) : o.cof.ord.cof = o.cof := by

Mathlib/SetTheory/Ordinal/Basic.lean

@@ -165,12 +165,19 @@ theorem type_eq {α β} {r : α → α → Prop} {s : β → β → Prop} [IsWel
     type r = type s ↔ Nonempty (r ≃r s) :=
   Quotient.eq'
 
-theorem _root_.RelIso.ordinal_type_eq {α β} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r]
-    [IsWellOrder β s] (h : r ≃r s) : type r = type s :=
+theorem _root_.RelIso.ordinalType_congr {α β} {r : α → α → Prop} {s : β → β → Prop}
+    [IsWellOrder α r] [IsWellOrder β s] (h : r ≃r s) : type r = type s :=
   type_eq.2 ⟨h⟩
 
+@[deprecated (since := "2026-05-25")]
+alias _root_.RelIso.ordinal_type_eq := RelIso.ordinalType_congr
+
+theorem _root_.OrderIso.ordinalType_congr {α β} [LinearOrder α] [LinearOrder β]
+    [WellFoundedLT α] [WellFoundedLT β] (h : α ≃o β) : typeLT α = typeLT β :=
+  h.toRelIsoLT.ordinalType_congr
+
 theorem type_eq_zero_of_empty (r) [IsWellOrder α r] [IsEmpty α] : type r = 0 :=
-  (RelIso.relIsoOfIsEmpty r _).ordinal_type_eq
+  (RelIso.relIsoOfIsEmpty r _).ordinalType_congr
 
 @[simp]
 theorem type_eq_zero_iff_isEmpty [IsWellOrder α r] : type r = 0 ↔ IsEmpty α := by
@@ -191,7 +198,7 @@ theorem type_empty : type (@emptyRelation Empty) = 0 :=
 
 theorem type_eq_one_of_unique (r) [IsWellOrder α r] [Nonempty α] [Subsingleton α] : type r = 1 := by
   cases nonempty_unique α
-  exact (RelIso.ofUniqueOfIrrefl r _).ordinal_type_eq
+  exact (RelIso.ofUniqueOfIrrefl r _).ordinalType_congr
 
 @[simp]
 theorem type_eq_one_iff_unique [IsWellOrder α r] : type r = 1 ↔ Nonempty (Unique α) :=
@@ -410,7 +417,7 @@ def typein (r : α → α → Prop) [IsWellOrder α r] : @PrincipalSeg α Ordina
     exact (PrincipalSeg.ofElement _ _).ordinal_type_lt
   · refine inductionOn a ?_
     rintro β s wo ⟨g⟩
-    exact ⟨_, g.subrelIso.ordinal_type_eq⟩
+    exact ⟨_, g.subrelIso.ordinalType_congr⟩
 
 @[simp]
 theorem type_subrel (r : α → α → Prop) [IsWellOrder α r] (a : α) :
@@ -431,7 +438,7 @@ theorem typein_lt_self {o : Ordinal} (i : o.ToType) : typein (α := o.ToType) (
 @[simp]
 theorem typein_top {α β} {r : α → α → Prop} {s : β → β → Prop}
     [IsWellOrder α r] [IsWellOrder β s] (f : r ≺i s) : typein s f.top = type r :=
-  f.subrelIso.ordinal_type_eq
+  f.subrelIso.ordinalType_congr
 
 @[simp]
 theorem typein_lt_typein (r : α → α → Prop) [IsWellOrder α r] {a b : α} :
@@ -659,12 +666,12 @@ theorem type_lt_ulift [LinearOrder α] [WellFoundedLT α] :
 theorem _root_.RelIso.ordinal_lift_type_eq {r : α → α → Prop} {s : β → β → Prop}
     [IsWellOrder α r] [IsWellOrder β s] (f : r ≃r s) : lift.{v} (type r) = lift.{u} (type s) :=
   ((RelIso.preimage Equiv.ulift r).trans <|
-      f.trans (RelIso.preimage Equiv.ulift s).symm).ordinal_type_eq
+      f.trans (RelIso.preimage Equiv.ulift s).symm).ordinalType_congr
 
 @[simp]
 theorem type_preimage {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β ≃ α) :
     type (f ⁻¹'o r) = type r :=
-  (RelIso.preimage f r).ordinal_type_eq
+  (RelIso.preimage f r).ordinalType_congr
 
 @[simp]
 theorem type_lift_preimage (r : α → α → Prop) [IsWellOrder α r]
@@ -830,11 +837,11 @@ the addition, together with properties of the other operations, are proved in
 every element of `o₁` is smaller than every element of `o₂`. -/
 instance add : Add Ordinal.{u} :=
   ⟨fun o₁ o₂ => Quotient.liftOn₂ o₁ o₂ (fun ⟨_, r, _⟩ ⟨_, s, _⟩ => type (Sum.Lex r s))
-    fun _ _ _ _ ⟨f⟩ ⟨g⟩ => (RelIso.sumLexCongr f g).ordinal_type_eq⟩
+    fun _ _ _ _ ⟨f⟩ ⟨g⟩ => (RelIso.sumLexCongr f g).ordinalType_congr⟩
 
 instance addMonoidWithOne : AddMonoidWithOne Ordinal.{u} where
-  zero_add o := inductionOn o fun α _ _ => (RelIso.emptySumLex _ _).ordinal_type_eq
-  add_zero o := inductionOn o fun α _ _ => (RelIso.sumLexEmpty _ _).ordinal_type_eq
+  zero_add o := inductionOn o fun α _ _ => (RelIso.emptySumLex _ _).ordinalType_congr
+  add_zero o := inductionOn o fun α _ _ => (RelIso.sumLexEmpty _ _).ordinalType_congr
   add_assoc o₁ o₂ o₃ :=
     Quotient.inductionOn₃ o₁ o₂ o₃ fun _ _ _ ↦ Quot.sound ⟨⟨sumAssoc .., by simp⟩⟩
   nsmul := nsmulRec
@@ -876,7 +883,7 @@ instance existsAddOfLE : ExistsAddOfLE Ordinal where
   exists_add_of_le {a b} := by
     refine inductionOn₂ a b fun α r _ β s _ ⟨f⟩ ↦ ?_
     obtain ⟨γ, t, _, ⟨g⟩⟩ := f.exists_sum_relIso
-    exact ⟨type t, g.ordinal_type_eq.symm⟩
+    exact ⟨type t, g.ordinalType_congr.symm⟩
 
 instance canonicallyOrderedAdd : CanonicallyOrderedAdd Ordinal where
   le_add_self a b := by simpa using add_le_add_left bot_le a