chore: dualize ConditionallyCompleteLattice/Indexed

Author: lua-vr

Mathlib/Order/ConditionallyCompleteLattice/Basic.lean

@@ -46,23 +46,18 @@ Extension of `sSup` and `sInf` from a preorder `α` to `WithTop α` and `WithBot
 variable [Preorder α]
 
 open Classical in
+@[to_dual]
 noncomputable instance WithTop.instSupSet [SupSet α] :
     SupSet (WithTop α) :=
   ⟨fun S =>
     if ⊤ ∈ S then ⊤ else if BddAbove ((fun (a : α) ↦ ↑a) ⁻¹' S : Set α) then
       ↑(sSup ((fun (a : α) ↦ (a : WithTop α)) ⁻¹' S : Set α)) else ⊤⟩
 
 open Classical in
+@[to_dual]
 noncomputable instance WithTop.instInfSet [InfSet α] : InfSet (WithTop α) :=
   ⟨fun S => if S ⊆ {⊤} ∨ ¬BddBelow S then ⊤ else ↑(sInf ((fun (a : α) ↦ ↑a) ⁻¹' S : Set α))⟩
 
-noncomputable instance WithBot.instSupSet [SupSet α] : SupSet (WithBot α) :=
-  ⟨(WithTop.instInfSet (α := αᵒᵈ)).sInf⟩
-
-noncomputable instance WithBot.instInfSet [InfSet α] :
-    InfSet (WithBot α) :=
-  ⟨(WithTop.instSupSet (α := αᵒᵈ)).sSup⟩
-
 theorem WithTop.sSup_eq [SupSet α] {s : Set (WithTop α)} (hs : ⊤ ∉ s)
     (hs' : BddAbove ((↑) ⁻¹' s : Set α)) : sSup s = ↑(sSup ((↑) ⁻¹' s) : α) :=
   (if_neg hs).trans <| if_pos hs'
@@ -79,10 +74,11 @@ theorem WithBot.sSup_eq [SupSet α] {s : Set (WithBot α)} (hs : ¬s ⊆ {⊥})
     sSup s = ↑(sSup ((↑) ⁻¹' s) : α) :=
   WithTop.sInf_eq (α := αᵒᵈ) hs h's
 
-@[simp]
+@[to_dual (attr := simp)]
 theorem WithTop.sInf_empty [InfSet α] : sInf (∅ : Set (WithTop α)) = ⊤ :=
   if_pos <| by simp
 
+@[to_dual (attr := norm_cast)]
 theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
     ↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by
   classical
@@ -95,6 +91,7 @@ theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddB
   · rw [preimage_image_eq]
     exact Option.some_injective _
 
+@[to_dual (attr := norm_cast)]
 theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) :
     ↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by
   classical
@@ -103,24 +100,10 @@ theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) :
   · exact Option.some_injective _
   · rintro ⟨x, _, ⟨⟩⟩
 
-@[simp]
-theorem WithBot.sSup_empty [SupSet α] : sSup (∅ : Set (WithBot α)) = ⊥ :=
-  WithTop.sInf_empty (α := αᵒᵈ)
-
 theorem WithBot.sInf_empty (α : Type*) [CompleteLattice α] : (sInf ∅ : WithBot α) = ⊤ := by
   rw [WithBot.sInf_eq (by simp) (OrderBot.bddBelow _), Set.preimage_empty, _root_.sInf_empty,
     WithBot.coe_top]
 
-@[norm_cast]
-theorem WithBot.coe_sSup' [SupSet α] {s : Set α} (hs : s.Nonempty) (h's : BddAbove s) :
-    ↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithBot α) :=
-  WithTop.coe_sInf' (α := αᵒᵈ) hs h's
-
-@[norm_cast]
-theorem WithBot.coe_sInf' [InfSet α] {s : Set α} (hs : BddBelow s) :
-    ↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithBot α) :=
-  WithTop.coe_sSup' (α := αᵒᵈ) hs
-
 end
 
 instance ConditionallyCompleteLinearOrder.toLinearOrder [h : ConditionallyCompleteLinearOrder α] :

Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean

@@ -33,29 +33,22 @@ Extension of `iSup` and `iInf` from a preorder `α` to `WithTop α` and `WithBot
 
 variable [Preorder α]
 
-@[simp]
+@[to_dual (attr := simp)]
 theorem WithTop.iInf_empty [IsEmpty ι] [InfSet α] (f : ι → WithTop α) :
     ⨅ i, f i = ⊤ := by rw [iInf, range_eq_empty, WithTop.sInf_empty]
 
-@[norm_cast]
+@[to_dual (attr := norm_cast)]
 theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) :
     ↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by
   rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp, Function.comp_def]
 
-@[norm_cast]
+set_option pp.explicit true in
+@[to_dual (attr := norm_cast)]
 theorem WithTop.coe_iSup [SupSet α] (f : ι → α) (h : BddAbove (Set.range f)) :
     ↑(⨆ i, f i) = (⨆ i, f i : WithTop α) := by
   rw [iSup, iSup, WithTop.coe_sSup' h, ← range_comp, Function.comp_def]
 
-@[simp]
-theorem WithBot.ciSup_empty [IsEmpty ι] [SupSet α] (f : ι → WithBot α) :
-    ⨆ i, f i = ⊥ :=
-  WithTop.iInf_empty (α := αᵒᵈ) _
-
-@[norm_cast]
-theorem WithBot.coe_iSup [Nonempty ι] [SupSet α] {f : ι → α} (hf : BddAbove (range f)) :
-    ↑(⨆ i, f i) = (⨆ i, f i : WithBot α) :=
-  WithTop.coe_iInf (α := αᵒᵈ) hf
+@[deprecated (since := "2026-05-04")] alias WithBot.ciSup_empty := WithBot.iSup_empty
 
 theorem WithBot.coe_biSup {ι : Type*} {s : Set ι} (hs : s.Nonempty)
     {α : Type*} [CompleteLattice α] (f : ι → α) :
@@ -69,10 +62,11 @@ theorem WithBot.coe_biSup {ι : Type*} {s : Set ι} (hs : s.Nonempty)
   · simpa only [iSup_pos h] using by apply le_biSup _ h
   · simpa only [iSup_neg h] using le_trans (by simp) (le_biSup _ hj)
 
-@[norm_cast]
-theorem WithBot.coe_iInf [InfSet α] (f : ι → α) (h : BddBelow (Set.range f)) :
-    ↑(⨅ i, f i) = (⨅ i, f i : WithBot α) :=
-  WithTop.coe_iSup (α := αᵒᵈ) _ h
+@[to_dual existing]
+theorem WithTop.coe_biInf {ι : Type*} {s : Set ι} (hs : s.Nonempty)
+    {α : Type*} [CompleteLattice α] (f : ι → α) :
+    ⨅ i ∈ s, f i = ⨅ i ∈ s, (f i : WithTop α) :=
+  WithBot.coe_biSup (α := αᵒᵈ) hs f
 
 theorem WithBot.coe_biInf {ι : Type*} {s : Set ι} {α : Type*} [CompleteLattice α] (f : ι → α) :
     ⨅ i ∈ s, f i = ⨅ i ∈ s, (f i : WithBot α) := by
@@ -82,59 +76,47 @@ theorem WithBot.coe_biInf {ι : Type*} {s : Set ι} {α : Type*} [CompleteLattic
   · simpa only [iInf_pos h] using by apply biInf_le _ h
   · simp [iInf_neg h]
 
+@[to_dual existing]
+theorem WithTop.coe_biSup {ι : Type*} {s : Set ι} {α : Type*} [CompleteLattice α] (f : ι → α) :
+    ⨆ i ∈ s, f i = ⨆ i ∈ s, (f i : WithTop α) :=
+  WithBot.coe_biInf (α := αᵒᵈ) f
+
 end
 
 section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α] {a b : α}
 
+@[to_dual]
 theorem isLUB_ciSup [Nonempty ι] {f : ι → α} (H : BddAbove (range f)) :
     IsLUB (range f) (⨆ i, f i) :=
   isLUB_csSup (range_nonempty f) H
 
+@[to_dual]
 theorem isLUB_ciSup_set {f : β → α} {s : Set β} (H : BddAbove (f '' s)) (Hne : s.Nonempty) :
     IsLUB (f '' s) (⨆ i : s, f i) := by
   rw [← sSup_image']
   exact isLUB_csSup (Hne.image _) H
 
-theorem isGLB_ciInf [Nonempty ι] {f : ι → α} (H : BddBelow (range f)) :
-    IsGLB (range f) (⨅ i, f i) :=
-  isGLB_csInf (range_nonempty f) H
-
-theorem isGLB_ciInf_set {f : β → α} {s : Set β} (H : BddBelow (f '' s)) (Hne : s.Nonempty) :
-    IsGLB (f '' s) (⨅ i : s, f i) :=
-  isLUB_ciSup_set (α := αᵒᵈ) H Hne
-
+@[to_dual le_ciInf_iff]
 theorem ciSup_le_iff [Nonempty ι] {f : ι → α} {a : α} (hf : BddAbove (range f)) :
     iSup f ≤ a ↔ ∀ i, f i ≤ a :=
   (isLUB_le_iff <| isLUB_ciSup hf).trans forall_mem_range
 
-theorem le_ciInf_iff [Nonempty ι] {f : ι → α} {a : α} (hf : BddBelow (range f)) :
-    a ≤ iInf f ↔ ∀ i, a ≤ f i :=
-  (le_isGLB_iff <| isGLB_ciInf hf).trans forall_mem_range
-
+@[to_dual le_ciInf_set_iff]
 theorem ciSup_set_le_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)
     (hf : BddAbove (f '' s)) : ⨆ i : s, f i ≤ a ↔ ∀ i ∈ s, f i ≤ a :=
   (isLUB_le_iff <| isLUB_ciSup_set hf hs).trans forall_mem_image
 
-theorem le_ciInf_set_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)
-    (hf : BddBelow (f '' s)) : (a ≤ ⨅ i : s, f i) ↔ ∀ i ∈ s, a ≤ f i :=
-  (le_isGLB_iff <| isGLB_ciInf_set hf hs).trans forall_mem_image
-
+@[to_dual]
 theorem IsLUB.ciSup_eq [Nonempty ι] {f : ι → α} (H : IsLUB (range f) a) : ⨆ i, f i = a :=
   H.csSup_eq (range_nonempty f)
 
+@[to_dual]
 theorem IsLUB.ciSup_set_eq {s : Set β} {f : β → α} (H : IsLUB (f '' s) a) (Hne : s.Nonempty) :
     ⨆ i : s, f i = a :=
   IsLUB.csSup_eq (image_eq_range f s ▸ H) (image_eq_range f s ▸ Hne.image f)
 
-theorem IsGLB.ciInf_eq [Nonempty ι] {f : ι → α} (H : IsGLB (range f) a) : ⨅ i, f i = a :=
-  H.csInf_eq (range_nonempty f)
-
-theorem IsGLB.ciInf_set_eq {s : Set β} {f : β → α} (H : IsGLB (f '' s) a) (Hne : s.Nonempty) :
-    ⨅ i : s, f i = a :=
-  IsGLB.csInf_eq (image_eq_range f s ▸ H) (image_eq_range f s ▸ Hne.image f)
-
 /-- The indexed supremum of a function is bounded above by a uniform bound -/
 @[to_dual le_ciInf /-- The indexed minimum of a function is bounded below by a uniform
 lower bound -/]
@@ -147,9 +129,12 @@ theorem ciSup₂_le [Nonempty ι] [∀ i, Nonempty (κ i)] {f : ∀ i, κ i →
   ciSup_le fun i ↦ ciSup_le <| h i
 
 /-- The indexed supremum of a function is bounded below by the value taken at one point -/
+@[to_dual ciInf_le /-- The indexed infimum of a function is bounded above by the value taken at one
+point -/]
 theorem le_ciSup {f : ι → α} (H : BddAbove (range f)) (c : ι) : f c ≤ iSup f :=
   le_csSup H (mem_range_self _)
 
+@[to_dual ciInf_le_of_le]
 theorem le_ciSup_of_le {f : ι → α} (H : BddAbove (range f)) (c : ι) (h : a ≤ f c) : a ≤ iSup f :=
   le_trans h (le_ciSup H c)