topology.algebra.order.compact | Mathlib porting status

(last sync)

feat(topology/algebra/order/compact): remove conditional completeness assumption in is_compact.exists_forall_le (#18991)

Diff

@@ -136,80 +136,39 @@ is_compact_iff_compact_space.mp is_compact_Icc
 end
 
 /-!
-### Min and max elements of a compact set
+### Extreme value theorem
 -/
 
-variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
-  [order_topology α] [topological_space β] [topological_space γ]
-
-lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  Inf s ∈ s :=
-hs.is_closed.cInf_mem ne_s hs.bdd_below
-
-lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s :=
-@is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s
-
-lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_glb s (Inf s) :=
-is_glb_cInf ne_s hs.bdd_below
-
-lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_lub s (Sup s) :=
-@is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s
-
-lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_least s (Inf s) :=
-⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
+section linear_order
 
-lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_greatest s (Sup s) :=
-@is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s
+variables {α β γ : Type*} [linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
 
 lemma is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x, is_least s x :=
-⟨_, hs.is_least_Inf ne_s⟩
+begin
+  haveI : nonempty s := ne_s.to_subtype,
+  suffices : (s ∩ ⋂ x ∈ s, Iic x).nonempty,
+    from ⟨this.some, this.some_spec.1, mem_Inter₂.mp this.some_spec.2⟩,
+  rw bInter_eq_Inter,
+  by_contra H,
+  rw not_nonempty_iff_eq_empty at H,
+  rcases hs.elim_directed_family_closed (λ x : s, Iic ↑x) (λ x, is_closed_Iic) H
+    ((is_total.directed coe).mono_comp _ (λ _ _, Iic_subset_Iic.mpr)) with ⟨x, hx⟩,
+  exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
+end
 
 lemma is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x, is_greatest s x :=
-⟨_, hs.is_greatest_Sup ne_s⟩
+@is_compact.exists_is_least αᵒᵈ _ _ _ _ hs ne_s
 
 lemma is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x ∈ s, is_glb s x :=
-⟨_, hs.Inf_mem ne_s, hs.is_glb_Inf ne_s⟩
+exists_imp_exists (λ x (hx : is_least s x), ⟨hx.1, hx.is_glb⟩) (hs.exists_is_least ne_s)
 
 lemma is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x ∈ s, is_lub s x :=
-⟨_, hs.Sup_mem ne_s, hs.is_lub_Sup ne_s⟩
-
-lemma is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
-in ⟨x, hxs, hx.symm, λ y hy,
-  hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩
-
-lemma is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-@is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-
-lemma is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s,  Inf (f '' s) = f x :=
-let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩
-
-lemma is_compact.exists_Sup_image_eq :
-  ∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
-  ∃ x ∈ s, Sup (f '' s) = f x :=
-@is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _
-
-lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
-  s = Icc (Inf s) (Sup s) :=
-eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
-
-/-!
-### Extreme value theorem
--/
+@is_compact.exists_is_glb αᵒᵈ _ _ _ _ hs ne_s
 
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 lemma is_compact.exists_forall_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
@@ -277,6 +236,68 @@ lemma continuous.exists_forall_ge [nonempty β] {f : β → α}
   ∃ x, ∀ y, f y ≤ f x :=
 @continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 
+/-- A continuous function with compact support has a global minimum. -/
+@[to_additive "A continuous function with compact support has a global minimum."]
+lemma continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  ∃ (x : β), ∀ (y : β), f x ≤ f y :=
+begin
+  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
+  rw [mem_lower_bounds, forall_range_iff] at hx,
+  exact ⟨x, hx⟩,
+end
+
+/-- A continuous function with compact support has a global maximum. -/
+@[to_additive "A continuous function with compact support has a global maximum."]
+lemma continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  ∃ (x : β), ∀ (y : β), f y ≤ f x :=
+@continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
+
+/-- A compact set is bounded below -/
+lemma is_compact.bdd_below [nonempty α] {s : set α} (hs : is_compact s) : bdd_below s :=
+begin
+  cases s.eq_empty_or_nonempty,
+  { rw h,
+    exact bdd_below_empty },
+  { obtain ⟨a, ha, has⟩ := hs.exists_is_least h,
+    exact ⟨a, has⟩ },
+end
+
+/-- A compact set is bounded above -/
+lemma is_compact.bdd_above [nonempty α] {s : set α} (hs : is_compact s) : bdd_above s :=
+@is_compact.bdd_below αᵒᵈ _ _ _ _ _ hs
+
+/-- A continuous function is bounded below on a compact set. -/
+lemma is_compact.bdd_below_image [nonempty α] {f : β → α} {K : set β}
+  (hK : is_compact K) (hf : continuous_on f K) : bdd_below (f '' K) :=
+(hK.image_of_continuous_on hf).bdd_below
+
+/-- A continuous function is bounded above on a compact set. -/
+lemma is_compact.bdd_above_image [nonempty α] {f : β → α} {K : set β}
+  (hK : is_compact K) (hf : continuous_on f K) : bdd_above (f '' K) :=
+@is_compact.bdd_below_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
+
+/-- A continuous function with compact support is bounded below. -/
+@[to_additive /-" A continuous function with compact support is bounded below. "-/]
+lemma continuous.bdd_below_range_of_has_compact_mul_support [has_one α] {f : β → α}
+  (hf : continuous f) (h : has_compact_mul_support f) : bdd_below (range f) :=
+(h.is_compact_range hf).bdd_below
+
+/-- A continuous function with compact support is bounded above. -/
+@[to_additive /-" A continuous function with compact support is bounded above. "-/]
+lemma continuous.bdd_above_range_of_has_compact_mul_support [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  bdd_above (range f) :=
+@continuous.bdd_below_range_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ hf h
+
+end linear_order
+
+section conditionally_complete_linear_order
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
+
 lemma is_compact.Sup_lt_iff_of_continuous {f : β → α}
   {K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
   Sup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
@@ -294,23 +315,71 @@ lemma is_compact.lt_Inf_iff_of_continuous {α β : Type*}
   y < Inf (f '' K) ↔ ∀ x ∈ K, y < f x :=
 @is_compact.Sup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 
-/-- A continuous function with compact support has a global minimum. -/
-@[to_additive "A continuous function with compact support has a global minimum."]
-lemma continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
-  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
-  ∃ (x : β), ∀ (y : β), f x ≤ f y :=
-begin
-  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
-  rw [mem_lower_bounds, forall_range_iff] at hx,
-  exact ⟨x, hx⟩,
-end
+end conditionally_complete_linear_order
 
-/-- A continuous function with compact support has a global maximum. -/
-@[to_additive "A continuous function with compact support has a global maximum."]
-lemma continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
-  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
-  ∃ (x : β), ∀ (y : β), f y ≤ f x :=
-@continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
+/-!
+### Min and max elements of a compact set
+-/
+
+section order_closed_topology
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
+
+lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  Inf s ∈ s :=
+let ⟨a, ha⟩ := hs.exists_is_least ne_s in
+ha.Inf_mem
+
+lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s :=
+@is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_glb s (Inf s) :=
+is_glb_cInf ne_s hs.bdd_below
+
+lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_lub s (Sup s) :=
+@is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_least s (Inf s) :=
+⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
+
+lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_greatest s (Sup s) :=
+@is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
+in ⟨x, hxs, hx.symm, λ y hy,
+  hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩
+
+lemma is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+@is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+
+lemma is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s,  Inf (f '' s) = f x :=
+let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩
+
+lemma is_compact.exists_Sup_image_eq :
+  ∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
+  ∃ x ∈ s, Sup (f '' s) = f x :=
+@is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _
+
+end order_closed_topology
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_topology α] [topological_space β] [topological_space γ]
+
+lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
+  s = Icc (Inf s) (Sup s) :=
+eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
 
 lemma is_compact.continuous_Sup {f : γ → β → α}
   {K : set β} (hK : is_compact K) (hf : continuous ↿f) :

(first ported)

Changes in mathlib3port

mathlib3

mathlib3port

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
 -/
-import Topology.Algebra.Order.IntermediateValue
-import Topology.LocalExtr
+import Topology.Order.IntermediateValue
+import Topology.Order.LocalExtr
 
 #align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"

bump to nightly-2024-03-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

22f01ce4

Diff

@@ -173,7 +173,7 @@ theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonem
   exact ⟨this.some, this.some_spec.1, mem_Inter₂.mp this.some_spec.2⟩
   rw [bInter_eq_Inter]
   by_contra H
-  rw [not_nonempty_iff_eq_empty] at H 
+  rw [not_nonempty_iff_eq_empty] at H
   rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H
       ((IsTotal.directed coe).mono_comp _ fun _ _ => Iic_subset_Iic.mpr) with
     ⟨x, hx⟩
@@ -294,7 +294,7 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
     (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y :=
   by
   obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_isLeast (range_nonempty _)
-  rw [mem_lowerBounds, forall_range_iff] at hx 
+  rw [mem_lowerBounds, forall_range_iff] at hx
   exact ⟨x, hx⟩
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
 -/
-import Mathbin.Topology.Algebra.Order.IntermediateValue
-import Mathbin.Topology.LocalExtr
+import Topology.Algebra.Order.IntermediateValue
+import Topology.LocalExtr
 
 #align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.algebra.order.compact
-! leanprover-community/mathlib commit 3efd324a3a31eaa40c9d5bfc669c4fafee5f9423
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Algebra.Order.IntermediateValue
 import Mathbin.Topology.LocalExtr
 
+#align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"
+
 /-!
 # Compactness of a closed interval

bump to nightly-2023-06-23-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

f333abb9

Diff

@@ -616,13 +616,13 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β}
 #align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
 -/
 
-#print IsCompact.exists_local_min_mem_open /-
-theorem IsCompact.exists_local_min_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
+#print IsCompact.exists_isLocalMin_mem_open /-
+theorem IsCompact.exists_isLocalMin_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
     (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
     (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
   by
   obtain ⟨x, hx, hfx⟩ := ht.exists_local_min_on_mem_subset hf hz hfz
   exact ⟨x, hx, hfx.is_local_min (Filter.mem_of_superset (hs.mem_nhds hx) hst)⟩
-#align is_compact.exists_local_min_mem_open IsCompact.exists_local_min_mem_open
+#align is_compact.exists_local_min_mem_open IsCompact.exists_isLocalMin_mem_open
 -/

bump to nightly-2023-06-22-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

2424c6fa

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.algebra.order.compact
-! leanprover-community/mathlib commit 50832daea47b195a48b5b33b1c8b2162c48c3afc
+! leanprover-community/mathlib commit 3efd324a3a31eaa40c9d5bfc669c4fafee5f9423
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -159,125 +159,52 @@ instance compactSpace_Icc (a b : α) : CompactSpace (Icc a b) :=
 end
 
 /-!
-### Min and max elements of a compact set
--/
-
-
-variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
-  [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
-
-#print IsCompact.sInf_mem /-
-theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
-  hs.IsClosed.csInf_mem ne_s hs.BddBelow
-#align is_compact.Inf_mem IsCompact.sInf_mem
--/
-
-#print IsCompact.sSup_mem /-
-theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
-  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.Sup_mem IsCompact.sSup_mem
--/
-
-#print IsCompact.isGLB_sInf /-
-theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGLB s (sInf s) :=
-  isGLB_csInf ne_s hs.BddBelow
-#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
+### Extreme value theorem
 -/
 
-#print IsCompact.isLUB_sSup /-
-theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLUB s (sSup s) :=
-  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
--/
 
-#print IsCompact.isLeast_sInf /-
-theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLeast s (sInf s) :=
-  ⟨hs.csInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
-#align is_compact.is_least_Inf IsCompact.isLeast_sInf
--/
+section LinearOrder
 
-#print IsCompact.isGreatest_sSup /-
-theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGreatest s (sSup s) :=
-  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
--/
+variable {α β γ : Type _} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α]
+  [TopologicalSpace β] [TopologicalSpace γ]
 
 #print IsCompact.exists_isLeast /-
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x, IsLeast s x :=
-  ⟨_, hs.isLeast_csInf ne_s⟩
+    ∃ x, IsLeast s x := by
+  haveI : Nonempty s := ne_s.to_subtype
+  suffices : (s ∩ ⋂ x ∈ s, Iic x).Nonempty
+  exact ⟨this.some, this.some_spec.1, mem_Inter₂.mp this.some_spec.2⟩
+  rw [bInter_eq_Inter]
+  by_contra H
+  rw [not_nonempty_iff_eq_empty] at H 
+  rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H
+      ((IsTotal.directed coe).mono_comp _ fun _ _ => Iic_subset_Iic.mpr) with
+    ⟨x, hx⟩
+  exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast
 -/
 
 #print IsCompact.exists_isGreatest /-
 theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsGreatest s x :=
-  ⟨_, hs.isGreatest_sSup ne_s⟩
+  @IsCompact.exists_isLeast αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 -/
 
 #print IsCompact.exists_isGLB /-
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
-  ⟨_, hs.csInf_mem ne_s, hs.isGLB_sInf ne_s⟩
+  Exists.imp (fun x (hx : IsLeast s x) => ⟨hx.1, hx.IsGLB⟩) (hs.exists_isLeast ne_s)
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 -/
 
 #print IsCompact.exists_isLUB /-
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
-  ⟨_, hs.csSup_mem ne_s, hs.isLUB_sSup ne_s⟩
+  @IsCompact.exists_isGLB αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 -/
 
-#print IsCompact.exists_sInf_image_eq_and_le /-
-theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).csInf_mem (ne_s.image f)
-  ⟨x, hxs, hx.symm, fun y hy =>
-    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
-#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
--/
-
-#print IsCompact.exists_sSup_image_eq_and_ge /-
-theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
--/
-
-#print IsCompact.exists_sInf_image_eq /-
-theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
-  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
-  ⟨x, hxs, hx⟩
-#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
--/
-
-#print IsCompact.exists_sSup_image_eq /-
-theorem IsCompact.exists_sSup_image_eq :
-    ∀ {s : Set β},
-      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
-  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
-#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
--/
-
-#print eq_Icc_of_connected_compact /-
-theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
-    s = Icc (sInf s) (sSup s) :=
-  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
-#align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
--/
-
-/-!
-### Extreme value theorem
--/
-
-
 #print IsCompact.exists_forall_le /-
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
@@ -363,6 +290,91 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 -/
 
+#print Continuous.exists_forall_le_of_hasCompactMulSupport /-
+/-- A continuous function with compact support has a global minimum. -/
+@[to_additive "A continuous function with compact support has a global minimum."]
+theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
+    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y :=
+  by
+  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_isLeast (range_nonempty _)
+  rw [mem_lowerBounds, forall_range_iff] at hx 
+  exact ⟨x, hx⟩
+#align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
+#align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
+-/
+
+#print Continuous.exists_forall_ge_of_hasCompactMulSupport /-
+/-- A continuous function with compact support has a global maximum. -/
+@[to_additive "A continuous function with compact support has a global maximum."]
+theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
+    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f y ≤ f x :=
+  @Continuous.exists_forall_le_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ _ hf h
+#align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
+#align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
+-/
+
+#print IsCompact.bddBelow /-
+/-- A compact set is bounded below -/
+theorem IsCompact.bddBelow [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBelow s :=
+  by
+  cases s.eq_empty_or_nonempty
+  · rw [h]
+    exact bddBelow_empty
+  · obtain ⟨a, ha, has⟩ := hs.exists_is_least h
+    exact ⟨a, has⟩
+#align is_compact.bdd_below IsCompact.bddBelow
+-/
+
+#print IsCompact.bddAbove /-
+/-- A compact set is bounded above -/
+theorem IsCompact.bddAbove [Nonempty α] {s : Set α} (hs : IsCompact s) : BddAbove s :=
+  @IsCompact.bddBelow αᵒᵈ _ _ _ _ _ hs
+#align is_compact.bdd_above IsCompact.bddAbove
+-/
+
+#print IsCompact.bddBelow_image /-
+/-- A continuous function is bounded below on a compact set. -/
+theorem IsCompact.bddBelow_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
+    (hf : ContinuousOn f K) : BddBelow (f '' K) :=
+  (hK.image_of_continuousOn hf).BddBelow
+#align is_compact.bdd_below_image IsCompact.bddBelow_image
+-/
+
+#print IsCompact.bddAbove_image /-
+/-- A continuous function is bounded above on a compact set. -/
+theorem IsCompact.bddAbove_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
+    (hf : ContinuousOn f K) : BddAbove (f '' K) :=
+  @IsCompact.bddBelow_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
+#align is_compact.bdd_above_image IsCompact.bddAbove_image
+-/
+
+#print Continuous.bddBelow_range_of_hasCompactMulSupport /-
+/-- A continuous function with compact support is bounded below. -/
+@[to_additive " A continuous function with compact support is bounded below. "]
+theorem Continuous.bddBelow_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
+    (h : HasCompactMulSupport f) : BddBelow (range f) :=
+  (h.isCompact_range hf).BddBelow
+#align continuous.bdd_below_range_of_has_compact_mul_support Continuous.bddBelow_range_of_hasCompactMulSupport
+#align continuous.bdd_below_range_of_has_compact_support Continuous.bddBelow_range_of_hasCompactSupport
+-/
+
+#print Continuous.bddAbove_range_of_hasCompactMulSupport /-
+/-- A continuous function with compact support is bounded above. -/
+@[to_additive " A continuous function with compact support is bounded above. "]
+theorem Continuous.bddAbove_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
+    (h : HasCompactMulSupport f) : BddAbove (range f) :=
+  @Continuous.bddBelow_range_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ hf h
+#align continuous.bdd_above_range_of_has_compact_mul_support Continuous.bddAbove_range_of_hasCompactMulSupport
+#align continuous.bdd_above_range_of_has_compact_support Continuous.bddAbove_range_of_hasCompactSupport
+-/
+
+end LinearOrder
+
+section ConditionallyCompleteLinearOrder
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
 #print IsCompact.sSup_lt_iff_of_continuous /-
 theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
     (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
@@ -385,27 +397,101 @@ theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompl
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 -/
 
-#print Continuous.exists_forall_le_of_hasCompactMulSupport /-
-/-- A continuous function with compact support has a global minimum. -/
-@[to_additive "A continuous function with compact support has a global minimum."]
-theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
-    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y :=
-  by
-  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_isLeast (range_nonempty _)
-  rw [mem_lowerBounds, forall_range_iff] at hx 
-  exact ⟨x, hx⟩
-#align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
-#align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
+end ConditionallyCompleteLinearOrder
+
+/-!
+### Min and max elements of a compact set
 -/
 
-#print Continuous.exists_forall_ge_of_hasCompactMulSupport /-
-/-- A continuous function with compact support has a global maximum. -/
-@[to_additive "A continuous function with compact support has a global maximum."]
-theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
-    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f y ≤ f x :=
-  @Continuous.exists_forall_le_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ _ hf h
-#align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
-#align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
+
+section OrderClosedTopology
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
+#print IsCompact.sInf_mem /-
+theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
+  let ⟨a, ha⟩ := hs.exists_isLeast ne_s
+  ha.csInf_mem
+#align is_compact.Inf_mem IsCompact.sInf_mem
+-/
+
+#print IsCompact.sSup_mem /-
+theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
+  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.Sup_mem IsCompact.sSup_mem
+-/
+
+#print IsCompact.isGLB_sInf /-
+theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGLB s (sInf s) :=
+  isGLB_csInf ne_s hs.BddBelow
+#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
+-/
+
+#print IsCompact.isLUB_sSup /-
+theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLUB s (sSup s) :=
+  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
+-/
+
+#print IsCompact.isLeast_sInf /-
+theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLeast s (sInf s) :=
+  ⟨hs.csInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
+#align is_compact.is_least_Inf IsCompact.isLeast_sInf
+-/
+
+#print IsCompact.isGreatest_sSup /-
+theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGreatest s (sSup s) :=
+  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
+-/
+
+#print IsCompact.exists_sInf_image_eq_and_le /-
+theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).csInf_mem (ne_s.image f)
+  ⟨x, hxs, hx.symm, fun y hy =>
+    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
+#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
+-/
+
+#print IsCompact.exists_sSup_image_eq_and_ge /-
+theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
+-/
+
+#print IsCompact.exists_sInf_image_eq /-
+theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
+  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
+  ⟨x, hxs, hx⟩
+#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
+-/
+
+#print IsCompact.exists_sSup_image_eq /-
+theorem IsCompact.exists_sSup_image_eq :
+    ∀ {s : Set β},
+      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
+  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
+#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
+-/
+
+end OrderClosedTopology
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
+#print eq_Icc_of_connected_compact /-
+theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
+    s = Icc (sInf s) (sSup s) :=
+  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
+#align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -166,33 +166,45 @@ end
 variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
+#print IsCompact.sInf_mem /-
 theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
   hs.IsClosed.csInf_mem ne_s hs.BddBelow
 #align is_compact.Inf_mem IsCompact.sInf_mem
+-/
 
+#print IsCompact.sSup_mem /-
 theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
   @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.Sup_mem IsCompact.sSup_mem
+-/
 
+#print IsCompact.isGLB_sInf /-
 theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsGLB s (sInf s) :=
   isGLB_csInf ne_s hs.BddBelow
 #align is_compact.is_glb_Inf IsCompact.isGLB_sInf
+-/
 
+#print IsCompact.isLUB_sSup /-
 theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsLUB s (sSup s) :=
   @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.is_lub_Sup IsCompact.isLUB_sSup
+-/
 
+#print IsCompact.isLeast_sInf /-
 theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsLeast s (sInf s) :=
   ⟨hs.csInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
 #align is_compact.is_least_Inf IsCompact.isLeast_sInf
+-/
 
+#print IsCompact.isGreatest_sSup /-
 theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsGreatest s (sSup s) :=
   @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
+-/
 
 #print IsCompact.exists_isLeast /-
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
@@ -208,50 +220,65 @@ theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.No
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 -/
 
+#print IsCompact.exists_isGLB /-
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
   ⟨_, hs.csInf_mem ne_s, hs.isGLB_sInf ne_s⟩
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
+-/
 
+#print IsCompact.exists_isLUB /-
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
   ⟨_, hs.csSup_mem ne_s, hs.isLUB_sSup ne_s⟩
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
+-/
 
+#print IsCompact.exists_sInf_image_eq_and_le /-
 theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
   let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).csInf_mem (ne_s.image f)
   ⟨x, hxs, hx.symm, fun y hy =>
     hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
 #align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
+-/
 
+#print IsCompact.exists_sSup_image_eq_and_ge /-
 theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
   @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
 #align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
+-/
 
+#print IsCompact.exists_sInf_image_eq /-
 theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
   let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
   ⟨x, hxs, hx⟩
 #align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
+-/
 
+#print IsCompact.exists_sSup_image_eq /-
 theorem IsCompact.exists_sSup_image_eq :
     ∀ {s : Set β},
       IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
   @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
 #align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
+-/
 
+#print eq_Icc_of_connected_compact /-
 theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
     s = Icc (sInf s) (sSup s) :=
   eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
 #align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
+-/
 
 /-!
 ### Extreme value theorem
 -/
 
 
+#print IsCompact.exists_forall_le /-
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
     (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
@@ -259,14 +286,18 @@ theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Non
   rcases(hs.image_of_continuous_on hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
   exact ⟨x, hxs, ball_image_iff.1 hx⟩
 #align is_compact.exists_forall_le IsCompact.exists_forall_le
+-/
 
+#print IsCompact.exists_forall_ge /-
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 theorem IsCompact.exists_forall_ge :
     ∀ {s : Set β},
       IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
   @IsCompact.exists_forall_le αᵒᵈ _ _ _ _ _
 #align is_compact.exists_forall_ge IsCompact.exists_forall_ge
+-/
 
+#print ContinuousOn.exists_forall_le' /-
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/
 theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
@@ -281,7 +312,9 @@ theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : Contin
   by_cases hyK : y ∈ K
   exacts [hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
 #align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'
+-/
 
+#print ContinuousOn.exists_forall_ge' /-
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
 theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
@@ -289,6 +322,7 @@ theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : Contin
     ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
   @ContinuousOn.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
 #align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'
+-/
 
 #print Continuous.exists_forall_le' /-
 /-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
@@ -329,6 +363,7 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 -/
 
+#print IsCompact.sSup_lt_iff_of_continuous /-
 theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
     (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
   by
@@ -339,13 +374,16 @@ theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : I
   refine' (csSup_le (h0K.image f) _).trans_lt (h x hx)
   rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
 #align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
+-/
 
+#print IsCompact.lt_sInf_iff_of_continuous /-
 theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
     y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
   @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
+-/
 
 #print Continuous.exists_forall_le_of_hasCompactMulSupport /-
 /-- A continuous function with compact support has a global minimum. -/
@@ -371,6 +409,7 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print IsCompact.continuous_sSup /-
 theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) :=
   by
@@ -400,11 +439,14 @@ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCo
         (show Continuous (f x') from hf.comp <| Continuous.Prod.mk x').ContinuousOn]
     exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy'))
 #align is_compact.continuous_Sup IsCompact.continuous_sSup
+-/
 
+#print IsCompact.continuous_sInf /-
 theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
   @IsCompact.continuous_sSup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
 #align is_compact.continuous_Inf IsCompact.continuous_sInf
+-/
 
 namespace ContinuousOn
 
@@ -418,12 +460,15 @@ variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopolog
 
 open scoped Interval
 
+#print ContinuousOn.image_Icc /-
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
     f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
   eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
     (isCompact_Icc.image_of_continuousOn h)
 #align continuous_on.image_Icc ContinuousOn.image_Icc
+-/
 
+#print ContinuousOn.image_uIcc_eq_Icc /-
 theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = Icc (sInf (f '' [a, b])) (sSup (f '' [a, b])) :=
   by
@@ -431,7 +476,9 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
   · simp_rw [uIcc_of_le h2] at h ⊢; exact h.image_Icc h2
   · simp_rw [uIcc_of_ge h2] at h ⊢; exact h.image_Icc h2
 #align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Icc
+-/
 
+#print ContinuousOn.image_uIcc /-
 theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = [sInf (f '' [a, b]), sSup (f '' [a, b])] :=
   by
@@ -439,7 +486,9 @@ theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
   refine' csInf_le_csSup _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
   exacts [bddBelow_Icc, bddAbove_Icc]
 #align continuous_on.image_uIcc ContinuousOn.image_uIcc
+-/
 
+#print ContinuousOn.sInf_image_Icc_le /-
 theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
     sInf (f '' Icc a b) ≤ f c :=
   by
@@ -450,7 +499,9 @@ theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
         ⟨csInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
           le_csSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
 #align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_le
+-/
 
+#print ContinuousOn.le_sSup_image_Icc /-
 theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
     f c ≤ sSup (f '' Icc a b) :=
   by
@@ -461,9 +512,11 @@ theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
         ⟨csInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
           le_csSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
 #align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Icc
+-/
 
 end ContinuousOn
 
+#print IsCompact.exists_isLocalMinOn_mem_subset /-
 theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β} {z : β}
     (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
     ∃ x ∈ s, IsLocalMinOn f t x :=
@@ -475,7 +528,9 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β}
   have h2 : x ∈ s := key hx h1
   refine' ⟨x, h2, eventually_nhdsWithin_of_forall hfx⟩
 #align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
+-/
 
+#print IsCompact.exists_local_min_mem_open /-
 theorem IsCompact.exists_local_min_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
     (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
     (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
@@ -483,4 +538,5 @@ theorem IsCompact.exists_local_min_mem_open {f : β → α} {s t : Set β} {z :
   obtain ⟨x, hx, hfx⟩ := ht.exists_local_min_on_mem_subset hf hz hfz
   exact ⟨x, hx, hfx.is_local_min (Filter.mem_of_superset (hs.mem_nhds hx) hst)⟩
 #align is_compact.exists_local_min_mem_open IsCompact.exists_local_min_mem_open
+-/

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -72,7 +72,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   rw [le_principal_iff]
   have hpt : ∀ x ∈ Icc a b, {x} ∉ f := fun x hx hxf =>
     hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x))
-  set s := { x ∈ Icc a b | Icc a x ∉ f }
+  set s := {x ∈ Icc a b | Icc a x ∉ f}
   have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
   have sbd : BddAbove s := ⟨b, hsb⟩
   have ha : a ∈ s := by simp [hpt, hab]

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -279,7 +279,7 @@ theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : Contin
     ((hK.inter_right hsc).insert x₀).exists_forall_le (insert_nonempty _ _) (hf.mono hsub)
   refine' ⟨x, hsub hx, fun y hy => _⟩
   by_cases hyK : y ∈ K
-  exacts[hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
+  exacts [hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
 #align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
@@ -354,7 +354,7 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
     (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y :=
   by
   obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_isLeast (range_nonempty _)
-  rw [mem_lowerBounds, forall_range_iff] at hx
+  rw [mem_lowerBounds, forall_range_iff] at hx 
   exact ⟨x, hx⟩
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
@@ -428,8 +428,8 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = Icc (sInf (f '' [a, b])) (sSup (f '' [a, b])) :=
   by
   cases' le_total a b with h2 h2
-  · simp_rw [uIcc_of_le h2] at h⊢; exact h.image_Icc h2
-  · simp_rw [uIcc_of_ge h2] at h⊢; exact h.image_Icc h2
+  · simp_rw [uIcc_of_le h2] at h ⊢; exact h.image_Icc h2
+  · simp_rw [uIcc_of_ge h2] at h ⊢; exact h.image_Icc h2
 #align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Icc
 
 theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
@@ -437,7 +437,7 @@ theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
   by
   refine' h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm
   refine' csInf_le_csSup _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
-  exacts[bddBelow_Icc, bddAbove_Icc]
+  exacts [bddBelow_Icc, bddAbove_Icc]
 #align continuous_on.image_uIcc ContinuousOn.image_uIcc
 
 theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -35,7 +35,7 @@ compact, extreme value theorem
 
 open Filter OrderDual TopologicalSpace Function Set
 
-open Filter Topology
+open scoped Filter Topology
 
 /-!
 ### Compactness of a closed interval
@@ -290,6 +290,7 @@ theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : Contin
   @ContinuousOn.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
 #align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'
 
+#print Continuous.exists_forall_le' /-
 /-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
 away from compact sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ : β)
@@ -299,27 +300,34 @@ theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ :
       (by rwa [principal_univ, inf_top_eq])
   ⟨x, fun y => hx y (mem_univ y)⟩
 #align continuous.exists_forall_le' Continuous.exists_forall_le'
+-/
 
+#print Continuous.exists_forall_ge' /-
 /-- The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
 away from compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge' {f : β → α} (hf : Continuous f) (x₀ : β)
     (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x :=
   @Continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h
 #align continuous.exists_forall_ge' Continuous.exists_forall_ge'
+-/
 
+#print Continuous.exists_forall_le /-
 /-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
 sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuous f)
     (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by inhabit β;
   exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_at_top _)
 #align continuous.exists_forall_le Continuous.exists_forall_le
+-/
 
+#print Continuous.exists_forall_ge /-
 /-- The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
 compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuous f)
     (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x :=
   @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
+-/
 
 theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
     (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
@@ -339,6 +347,7 @@ theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompl
   @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
+#print Continuous.exists_forall_le_of_hasCompactMulSupport /-
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
 theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -349,7 +358,9 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
   exact ⟨x, hx⟩
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
+-/
 
+#print Continuous.exists_forall_ge_of_hasCompactMulSupport /-
 /-- A continuous function with compact support has a global maximum. -/
 @[to_additive "A continuous function with compact support has a global maximum."]
 theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -357,6 +368,7 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
   @Continuous.exists_forall_le_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ _ hf h
 #align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
@@ -404,7 +416,7 @@ namespace ContinuousOn
 variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β}
   {a b c : α}
 
-open Interval
+open scoped Interval
 
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
     f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=

bump to nightly-2023-05-28-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

ba5d8b97

Diff

@@ -166,65 +166,29 @@ end
 variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
-/- warning: is_compact.Inf_mem -> IsCompact.sInf_mem is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
-Case conversion may be inaccurate. Consider using '#align is_compact.Inf_mem IsCompact.sInf_memₓ'. -/
 theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
   hs.IsClosed.csInf_mem ne_s hs.BddBelow
 #align is_compact.Inf_mem IsCompact.sInf_mem
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
-Case conversion may be inaccurate. Consider using '#align is_compact.Sup_mem IsCompact.sSup_memₓ'. -/
 theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
   @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.Sup_mem IsCompact.sSup_mem
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGLB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_glb_Inf IsCompact.isGLB_sInfₓ'. -/
 theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsGLB s (sInf s) :=
   isGLB_csInf ne_s hs.BddBelow
 #align is_compact.is_glb_Inf IsCompact.isGLB_sInf
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_lub_Sup IsCompact.isLUB_sSupₓ'. -/
 theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsLUB s (sSup s) :=
   @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
 #align is_compact.is_lub_Sup IsCompact.isLUB_sSup
 
-/- warning: is_compact.is_least_Inf -> IsCompact.isLeast_sInf is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_least_Inf IsCompact.isLeast_sInfₓ'. -/
 theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsLeast s (sInf s) :=
   ⟨hs.csInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
 #align is_compact.is_least_Inf IsCompact.isLeast_sInf
 
-/- warning: is_compact.is_greatest_Sup -> IsCompact.isGreatest_sSup is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSupₓ'. -/
 theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     IsGreatest s (sSup s) :=
   @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
@@ -244,34 +208,16 @@ theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.No
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 -/
 
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-Case conversion may be inaccurate. Consider using '#align is_compact.exists_is_glb IsCompact.exists_isGLBₓ'. -/
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
   ⟨_, hs.csInf_mem ne_s, hs.isGLB_sInf ne_s⟩
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 
-/- warning: is_compact.exists_is_lub -> IsCompact.exists_isLUB is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Exists.{succ u1} α (fun (x : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) => IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s x)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s x)))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_is_lub IsCompact.exists_isLUBₓ'. -/
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
   ⟨_, hs.csSup_mem ne_s, hs.isLUB_sSup ne_s⟩
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
-/- warning: is_compact.exists_Inf_image_eq_and_le -> IsCompact.exists_sInf_image_eq_and_le is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_leₓ'. -/
 theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
   let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).csInf_mem (ne_s.image f)
@@ -279,47 +225,23 @@ theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (n
     hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
 #align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
 
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_geₓ'. -/
 theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
   @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
 #align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
 
-/- warning: is_compact.exists_Inf_image_eq -> IsCompact.exists_sInf_image_eq is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eqₓ'. -/
 theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
     {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
   let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
   ⟨x, hxs, hx⟩
 #align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
 
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-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eqₓ'. -/
 theorem IsCompact.exists_sSup_image_eq :
     ∀ {s : Set β},
       IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
   @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
 #align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
 
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-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
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-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
-Case conversion may be inaccurate. Consider using '#align eq_Icc_of_connected_compact eq_Icc_of_connected_compactₓ'. -/
 theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
     s = Icc (sInf s) (sSup s) :=
   eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
@@ -330,12 +252,6 @@ theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ :
 -/
 
 
-/- warning: is_compact.exists_forall_le -> IsCompact.exists_forall_le is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_forall_le IsCompact.exists_forall_leₓ'. -/
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
     (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
@@ -344,12 +260,6 @@ theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Non
   exact ⟨x, hxs, ball_image_iff.1 hx⟩
 #align is_compact.exists_forall_le IsCompact.exists_forall_le
 
-/- warning: is_compact.exists_forall_ge -> IsCompact.exists_forall_ge is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_forall_ge IsCompact.exists_forall_geₓ'. -/
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 theorem IsCompact.exists_forall_ge :
     ∀ {s : Set β},
@@ -357,12 +267,6 @@ theorem IsCompact.exists_forall_ge :
   @IsCompact.exists_forall_le αᵒᵈ _ _ _ _ _
 #align is_compact.exists_forall_ge IsCompact.exists_forall_ge
 
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'ₓ'. -/
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/
 theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
@@ -378,12 +282,6 @@ theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : Contin
   exacts[hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
 #align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'
 
-/- warning: continuous_on.exists_forall_ge' -> ContinuousOn.exists_forall_ge' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'ₓ'. -/
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
 theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
@@ -392,12 +290,6 @@ theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : Contin
   @ContinuousOn.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
 #align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'
 
-/- warning: continuous.exists_forall_le' -> Continuous.exists_forall_le' is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))
-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le' Continuous.exists_forall_le'ₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
 away from compact sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ : β)
@@ -408,12 +300,6 @@ theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ :
   ⟨x, fun y => hx y (mem_univ y)⟩
 #align continuous.exists_forall_le' Continuous.exists_forall_le'
 
-/- warning: continuous.exists_forall_ge' -> Continuous.exists_forall_ge' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))
-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge' Continuous.exists_forall_ge'ₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
 away from compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge' {f : β → α} (hf : Continuous f) (x₀ : β)
@@ -421,12 +307,6 @@ theorem Continuous.exists_forall_ge' {f : β → α} (hf : Continuous f) (x₀ :
   @Continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h
 #align continuous.exists_forall_ge' Continuous.exists_forall_ge'
 
-/- warning: continuous.exists_forall_le -> Continuous.exists_forall_le is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (Filter.Tendsto.{u2, u1} β α f (Filter.cocompact.{u2} β _inst_4) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))
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-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le Continuous.exists_forall_leₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
 sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuous f)
@@ -434,12 +314,6 @@ theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuo
   exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_at_top _)
 #align continuous.exists_forall_le Continuous.exists_forall_le
 
-/- warning: continuous.exists_forall_ge -> Continuous.exists_forall_ge is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge Continuous.exists_forall_geₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
 compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuous f)
@@ -447,12 +321,6 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
   @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 
-/- warning: is_compact.Sup_lt_iff_of_continuous -> IsCompact.sSup_lt_iff_of_continuous is a dubious translation:
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
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-Case conversion may be inaccurate. Consider using '#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuousₓ'. -/
 theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
     (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
   by
@@ -464,12 +332,6 @@ theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : I
   rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
 #align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
 
-/- warning: is_compact.lt_Inf_iff_of_continuous -> IsCompact.lt_sInf_iff_of_continuous is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuousₓ'. -/
 theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
@@ -477,12 +339,6 @@ theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompl
   @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
-/- warning: continuous.exists_forall_le_of_has_compact_mul_support -> Continuous.exists_forall_le_of_hasCompactMulSupport is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupportₓ'. -/
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
 theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -494,12 +350,6 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
 
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-Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupportₓ'. -/
 /-- A continuous function with compact support has a global maximum. -/
 @[to_additive "A continuous function with compact support has a global maximum."]
 theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -508,12 +358,6 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
 #align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
 
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-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.{u3} γ] {f : γ -> β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} γ β) α (Prod.topologicalSpace.{u3, u2} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max u3 u2 u1, max u3 u2, u1} (γ -> β -> α) (Prod.{u3, u2} γ β) α (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} γ (β -> α) β α (Function.hasUncurryBase.{u2, u1} β α)) f)) -> (Continuous.{u3, u1} γ α _inst_5 _inst_2 (fun (x : γ) => SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α (f x) K)))
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-Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Sup IsCompact.continuous_sSupₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) :=
@@ -545,12 +389,6 @@ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCo
     exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy'))
 #align is_compact.continuous_Sup IsCompact.continuous_sSup
 
-/- warning: is_compact.continuous_Inf -> IsCompact.continuous_sInf is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Inf IsCompact.continuous_sInfₓ'. -/
 theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
   @IsCompact.continuous_sSup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
@@ -568,24 +406,12 @@ variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopolog
 
 open Interval
 
-/- warning: continuous_on.image_Icc -> ContinuousOn.image_Icc is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) a b) -> (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))) a b) -> (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.image_Icc ContinuousOn.image_Iccₓ'. -/
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
     f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
   eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
     (isCompact_Icc.image_of_continuousOn h)
 #align continuous_on.image_Icc ContinuousOn.image_Icc
 
-/- warning: continuous_on.image_uIcc_eq_Icc -> ContinuousOn.image_uIcc_eq_Icc is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Iccₓ'. -/
 theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = Icc (sInf (f '' [a, b])) (sSup (f '' [a, b])) :=
   by
@@ -594,12 +420,6 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
   · simp_rw [uIcc_of_ge h2] at h⊢; exact h.image_Icc h2
 #align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Icc
 
-/- warning: continuous_on.image_uIcc -> ContinuousOn.image_uIcc is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.uIcc.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.uIcc.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc ContinuousOn.image_uIccₓ'. -/
 theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = [sInf (f '' [a, b]), sSup (f '' [a, b])] :=
   by
@@ -608,12 +428,6 @@ theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
   exacts[bddBelow_Icc, bddAbove_Icc]
 #align continuous_on.image_uIcc ContinuousOn.image_uIcc
 
-/- warning: continuous_on.Inf_image_Icc_le -> ContinuousOn.sInf_image_Icc_le is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (f c))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (f c))
-Case conversion may be inaccurate. Consider using '#align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_leₓ'. -/
 theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
     sInf (f '' Icc a b) ≤ f c :=
   by
@@ -625,12 +439,6 @@ theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
           le_csSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
 #align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_le
 
-/- warning: continuous_on.le_Sup_image_Icc -> ContinuousOn.le_sSup_image_Icc is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (f c) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (f c) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Iccₓ'. -/
 theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
     f c ≤ sSup (f '' Icc a b) :=
   by
@@ -644,12 +452,6 @@ theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
 
 end ContinuousOn
 
-/- warning: is_compact.exists_local_min_on_mem_subset -> IsCompact.exists_isLocalMinOn_mem_subset is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMinOn.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f t x)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z t) -> (forall (z' : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (Set.instSDiffSet.{u2} β) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (IsLocalMinOn.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f t x)))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subsetₓ'. -/
 theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β} {z : β}
     (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
     ∃ x ∈ s, IsLocalMinOn f t x :=
@@ -662,12 +464,6 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β}
   refine' ⟨x, h2, eventually_nhdsWithin_of_forall hfx⟩
 #align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
 
-/- warning: is_compact.exists_local_min_mem_open -> IsCompact.exists_local_min_mem_open is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) s t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (IsOpen.{u2} β _inst_4 s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMin.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f x)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) s t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z t) -> (forall (z' : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (Set.instSDiffSet.{u2} β) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (IsOpen.{u2} β _inst_4 s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (IsLocalMin.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f x)))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_local_min_mem_open IsCompact.exists_local_min_mem_openₓ'. -/
 theorem IsCompact.exists_local_min_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
     (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
     (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -66,9 +66,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
     CompactIccSpace α := by
   refine' ⟨fun a b => _⟩
-  cases' le_or_lt a b with hab hab
-  swap
-  · simp [hab]
+  cases' le_or_lt a b with hab hab; swap; · simp [hab]
   refine' isCompact_iff_ultrafilter_le_nhds.2 fun f hf => _
   contrapose! hf
   rw [le_principal_iff]
@@ -78,15 +76,13 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
   have sbd : BddAbove s := ⟨b, hsb⟩
   have ha : a ∈ s := by simp [hpt, hab]
-  rcases hab.eq_or_lt with (rfl | hlt)
-  · exact ha.2
+  rcases hab.eq_or_lt with (rfl | hlt); · exact ha.2
   set c := Sup s
   have hsc : IsLUB s c := isLUB_csSup ⟨a, ha⟩ sbd
   have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
   specialize hf c hc
   have hcs : c ∈ s := by
-    cases' hc.1.eq_or_lt with heq hlt
-    · rwa [← HEq]
+    cases' hc.1.eq_or_lt with heq hlt; · rwa [← HEq]
     refine' ⟨hc, fun hcf => hf fun U hU => _⟩
     rcases(mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with
       ⟨x, hxc, hxU⟩
@@ -98,16 +94,13 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
     exact
       subset.trans Icc_subset_Icc_union_Ioc
         (union_subset_union subset.rfl <| Ioc_subset_Ioc_left hy.1.le)
-  cases' hc.2.eq_or_lt with heq hlt
-  · rw [← HEq]
-    exact hcs.2
+  cases' hc.2.eq_or_lt with heq hlt; · rw [← HEq]; exact hcs.2
   contrapose! hf
   intro U hU
   rcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
       (mem_nhdsWithin_of_mem_nhds hU) with
     ⟨y, hxy, hyU⟩
-  refine' mem_of_superset _ hyU
-  clear! U
+  refine' mem_of_superset _ hyU; clear! U
   have hy : y ∈ Icc a b := ⟨hc.1.trans hxy.1.le, hxy.2⟩
   by_cases hay : Icc a y ∈ f
   · refine' mem_of_superset (f.diff_mem_iff.2 ⟨f.diff_mem_iff.2 ⟨hay, hcs.2⟩, hpt y hy⟩) _
@@ -437,9 +430,7 @@ Case conversion may be inaccurate. Consider using '#align continuous.exists_fora
 /-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
 sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuous f)
-    (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y :=
-  by
-  inhabit β
+    (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by inhabit β;
   exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_at_top _)
 #align continuous.exists_forall_le Continuous.exists_forall_le
 
@@ -528,8 +519,7 @@ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCo
     (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) :=
   by
   rcases eq_empty_or_nonempty K with (rfl | h0K)
-  · simp_rw [image_empty]
-    exact continuous_const
+  · simp_rw [image_empty]; exact continuous_const
   rw [continuous_iff_continuousAt]
   intro x
   obtain ⟨y, hyK, h2y, hy⟩ :=
@@ -545,10 +535,7 @@ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCo
   · refine'
       (this.1 z hz).mono fun x' hx' => hx'.trans_le <| le_csSup _ <| mem_image_of_mem (f x') hyK
     exact hK.bdd_above_image (hf.comp <| Continuous.Prod.mk x').ContinuousOn
-  · have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z :=
-      by
-      rintro ⟨x', y'⟩ ⟨hx', hy'⟩
-      cases hx'
+  · have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z := by rintro ⟨x', y'⟩ ⟨hx', hy'⟩; cases hx';
       exact (hy y' hy').trans_lt hz
     obtain ⟨u, v, hu, hv, hxu, hKv, huv⟩ :=
       generalized_tube_lemma isCompact_singleton hK (is_open_Iio.preimage hf) h
@@ -603,10 +590,8 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
     f '' [a, b] = Icc (sInf (f '' [a, b])) (sSup (f '' [a, b])) :=
   by
   cases' le_total a b with h2 h2
-  · simp_rw [uIcc_of_le h2] at h⊢
-    exact h.image_Icc h2
-  · simp_rw [uIcc_of_ge h2] at h⊢
-    exact h.image_Icc h2
+  · simp_rw [uIcc_of_le h2] at h⊢; exact h.image_Icc h2
+  · simp_rw [uIcc_of_ge h2] at h⊢; exact h.image_Icc h2
 #align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Icc
 
 /- warning: continuous_on.image_uIcc -> ContinuousOn.image_uIcc is a dubious translation:

bump to nightly-2023-05-16-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

9cb92e8c

Diff

@@ -275,7 +275,7 @@ theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempt
 
 /- warning: is_compact.exists_Inf_image_eq_and_le -> IsCompact.exists_sInf_image_eq_and_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_leₓ'. -/
@@ -288,7 +288,7 @@ theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (n
 
 /- warning: is_compact.exists_Sup_image_eq_and_ge -> IsCompact.exists_sSup_image_eq_and_ge is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_geₓ'. -/
@@ -339,7 +339,7 @@ theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ :
 
 /- warning: is_compact.exists_forall_le -> IsCompact.exists_forall_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_forall_le IsCompact.exists_forall_leₓ'. -/
@@ -353,7 +353,7 @@ theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Non
 
 /- warning: is_compact.exists_forall_ge -> IsCompact.exists_forall_ge is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_forall_ge IsCompact.exists_forall_geₓ'. -/
@@ -366,7 +366,7 @@ theorem IsCompact.exists_forall_ge :
 
 /- warning: continuous_on.exists_forall_le' -> ContinuousOn.exists_forall_le' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'ₓ'. -/
@@ -387,7 +387,7 @@ theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : Contin
 
 /- warning: continuous_on.exists_forall_ge' -> ContinuousOn.exists_forall_ge' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'ₓ'. -/
@@ -399,7 +399,12 @@ theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : Contin
   @ContinuousOn.exists_forall_le' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
 #align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'
 
-#print Continuous.exists_forall_le' /-
+/- warning: continuous.exists_forall_le' -> Continuous.exists_forall_le' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le' Continuous.exists_forall_le'ₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
 away from compact sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ : β)
@@ -409,18 +414,26 @@ theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ :
       (by rwa [principal_univ, inf_top_eq])
   ⟨x, fun y => hx y (mem_univ y)⟩
 #align continuous.exists_forall_le' Continuous.exists_forall_le'
--/
 
-#print Continuous.exists_forall_ge' /-
+/- warning: continuous.exists_forall_ge' -> Continuous.exists_forall_ge' is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (forall (x₀ : β), (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Filter.cocompact.{u2} β _inst_4)) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge' Continuous.exists_forall_ge'ₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
 away from compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge' {f : β → α} (hf : Continuous f) (x₀ : β)
     (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x :=
   @Continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h
 #align continuous.exists_forall_ge' Continuous.exists_forall_ge'
--/
 
-#print Continuous.exists_forall_le /-
+/- warning: continuous.exists_forall_le -> Continuous.exists_forall_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (Filter.Tendsto.{u2, u1} β α f (Filter.cocompact.{u2} β _inst_4) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (Filter.Tendsto.{u2, u1} β α f (Filter.cocompact.{u2} β _inst_4) (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le Continuous.exists_forall_leₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
 sets, then it has a global minimum. -/
 theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuous f)
@@ -429,20 +442,23 @@ theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuo
   inhabit β
   exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_at_top _)
 #align continuous.exists_forall_le Continuous.exists_forall_le
--/
 
-#print Continuous.exists_forall_ge /-
+/- warning: continuous.exists_forall_ge -> Continuous.exists_forall_ge is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (Filter.Tendsto.{u2, u1} β α f (Filter.cocompact.{u2} β _inst_4) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (Filter.Tendsto.{u2, u1} β α f (Filter.cocompact.{u2} β _inst_4) (Filter.atBot.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge Continuous.exists_forall_geₓ'. -/
 /-- The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
 compact sets, then it has a global maximum. -/
 theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuous f)
     (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x :=
   @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
--/
 
 /- warning: is_compact.Sup_lt_iff_of_continuous -> IsCompact.sSup_lt_iff_of_continuous is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
 Case conversion may be inaccurate. Consider using '#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuousₓ'. -/
@@ -459,7 +475,7 @@ theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : I
 
 /- warning: is_compact.lt_Inf_iff_of_continuous -> IsCompact.lt_sInf_iff_of_continuous is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_6 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_7 : TopologicalSpace.{u1} α] [_inst_8 : OrderTopology.{u1} α _inst_7 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_9 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)) (Set.image.{u2, u1} β α f K))) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (f x))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_6 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_7 : TopologicalSpace.{u1} α] [_inst_8 : OrderTopology.{u1} α _inst_7 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_9 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)) (Set.image.{u2, u1} β α f K))) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (f x))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_6 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_7 : TopologicalSpace.{u2} α] [_inst_8 : OrderTopology.{u2} α _inst_7 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u1} β] {f : β -> α} {K : Set.{u1} β}, (IsCompact.{u1} β _inst_9 K) -> (Set.Nonempty.{u1} β K) -> (ContinuousOn.{u1, u2} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (InfSet.sInf.{u2} α (ConditionallyCompleteLattice.toInfSet.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)) (Set.image.{u1, u2} β α f K))) (forall (x : β), (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x K) -> (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (f x))))
 Case conversion may be inaccurate. Consider using '#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuousₓ'. -/
@@ -470,7 +486,12 @@ theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompl
   @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
-#print Continuous.exists_forall_le_of_hasCompactMulSupport /-
+/- warning: continuous.exists_forall_le_of_has_compact_mul_support -> Continuous.exists_forall_le_of_hasCompactMulSupport is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] [_inst_7 : One.{u1} α] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (HasCompactMulSupport.{u2, u1} β α _inst_4 _inst_7 f) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] [_inst_7 : One.{u1} α] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (HasCompactMulSupport.{u2, u1} β α _inst_4 _inst_7 f) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupportₓ'. -/
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
 theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -481,9 +502,13 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
   exact ⟨x, hx⟩
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport
--/
 
-#print Continuous.exists_forall_ge_of_hasCompactMulSupport /-
+/- warning: continuous.exists_forall_ge_of_has_compact_mul_support -> Continuous.exists_forall_ge_of_hasCompactMulSupport is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] [_inst_7 : One.{u1} α] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (HasCompactMulSupport.{u2, u1} β α _inst_4 _inst_7 f) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : Nonempty.{succ u2} β] [_inst_7 : One.{u1} α] {f : β -> α}, (Continuous.{u2, u1} β α _inst_4 _inst_2 f) -> (HasCompactMulSupport.{u2, u1} β α _inst_4 _inst_7 f) -> (Exists.{succ u2} β (fun (x : β) => forall (y : β), LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))
+Case conversion may be inaccurate. Consider using '#align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupportₓ'. -/
 /-- A continuous function with compact support has a global maximum. -/
 @[to_additive "A continuous function with compact support has a global maximum."]
 theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -491,7 +516,6 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
   @Continuous.exists_forall_le_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ _ hf h
 #align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
--/
 
 /- warning: is_compact.continuous_Sup -> IsCompact.continuous_sSup is a dubious translation:
 lean 3 declaration is
@@ -559,7 +583,7 @@ open Interval
 
 /- warning: continuous_on.image_Icc -> ContinuousOn.image_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) a b) -> (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) a b) -> (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))) a b) -> (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_Icc ContinuousOn.image_Iccₓ'. -/
@@ -571,7 +595,7 @@ theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
 
 /- warning: continuous_on.image_uIcc_eq_Icc -> ContinuousOn.image_uIcc_eq_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Iccₓ'. -/
@@ -587,7 +611,7 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
 
 /- warning: continuous_on.image_uIcc -> ContinuousOn.image_uIcc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.uIcc.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.uIcc.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.uIcc.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc ContinuousOn.image_uIccₓ'. -/
@@ -601,7 +625,7 @@ theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
 
 /- warning: continuous_on.Inf_image_Icc_le -> ContinuousOn.sInf_image_Icc_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (f c))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (f c))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (f c))
 Case conversion may be inaccurate. Consider using '#align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_leₓ'. -/
@@ -618,7 +642,7 @@ theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
 
 /- warning: continuous_on.le_Sup_image_Icc -> ContinuousOn.le_sSup_image_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (f c) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (f c) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (f c) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Iccₓ'. -/
@@ -637,7 +661,7 @@ end ContinuousOn
 
 /- warning: is_compact.exists_local_min_on_mem_subset -> IsCompact.exists_isLocalMinOn_mem_subset is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMinOn.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f t x)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMinOn.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f t x)))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z t) -> (forall (z' : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (Set.instSDiffSet.{u2} β) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (IsLocalMinOn.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f t x)))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subsetₓ'. -/
@@ -655,7 +679,7 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β}
 
 /- warning: is_compact.exists_local_min_mem_open -> IsCompact.exists_local_min_mem_open is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) s t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (IsOpen.{u2} β _inst_4 s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMin.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f x)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) s t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z t) -> (forall (z' : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (BooleanAlgebra.toHasSdiff.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β)) t s)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (IsOpen.{u2} β _inst_4 s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => IsLocalMin.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f x)))
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {s : Set.{u2} β} {t : Set.{u2} β} {z : β}, (IsCompact.{u2} β _inst_4 t) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) s t) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f t) -> (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z t) -> (forall (z' : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) z' (SDiff.sdiff.{u2} (Set.{u2} β) (Set.instSDiffSet.{u2} β) t s)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f z) (f z'))) -> (IsOpen.{u2} β _inst_4 s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (IsLocalMin.{u2, u1} β α _inst_4 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) f x)))
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_local_min_mem_open IsCompact.exists_local_min_mem_openₓ'. -/

bump to nightly-2023-05-14-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

d31da313

Diff

@@ -81,7 +81,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   rcases hab.eq_or_lt with (rfl | hlt)
   · exact ha.2
   set c := Sup s
-  have hsc : IsLUB s c := isLUB_csupₛ ⟨a, ha⟩ sbd
+  have hsc : IsLUB s c := isLUB_csSup ⟨a, ha⟩ sbd
   have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
   specialize hf c hc
   have hcs : c ∈ s := by
@@ -173,81 +173,81 @@ end
 variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
-/- warning: is_compact.Inf_mem -> IsCompact.infₛ_mem is a dubious translation:
+/- warning: is_compact.Inf_mem -> IsCompact.sInf_mem is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
-Case conversion may be inaccurate. Consider using '#align is_compact.Inf_mem IsCompact.infₛ_memₓ'. -/
-theorem IsCompact.infₛ_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : infₛ s ∈ s :=
-  hs.IsClosed.cinfₛ_mem ne_s hs.BddBelow
-#align is_compact.Inf_mem IsCompact.infₛ_mem
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
+Case conversion may be inaccurate. Consider using '#align is_compact.Inf_mem IsCompact.sInf_memₓ'. -/
+theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
+  hs.IsClosed.csInf_mem ne_s hs.BddBelow
+#align is_compact.Inf_mem IsCompact.sInf_mem
 
-/- warning: is_compact.Sup_mem -> IsCompact.supₛ_mem is a dubious translation:
+/- warning: is_compact.Sup_mem -> IsCompact.sSup_mem is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
-Case conversion may be inaccurate. Consider using '#align is_compact.Sup_mem IsCompact.supₛ_memₓ'. -/
-theorem IsCompact.supₛ_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : supₛ s ∈ s :=
-  @IsCompact.infₛ_mem αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.Sup_mem IsCompact.supₛ_mem
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) s)
+Case conversion may be inaccurate. Consider using '#align is_compact.Sup_mem IsCompact.sSup_memₓ'. -/
+theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
+  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.Sup_mem IsCompact.sSup_mem
 
-/- warning: is_compact.is_glb_Inf -> IsCompact.isGLB_infₛ is a dubious translation:
+/- warning: is_compact.is_glb_Inf -> IsCompact.isGLB_sInf is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGLB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGLB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGLB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_glb_Inf IsCompact.isGLB_infₛₓ'. -/
-theorem IsCompact.isGLB_infₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGLB s (infₛ s) :=
-  isGLB_cinfₛ ne_s hs.BddBelow
-#align is_compact.is_glb_Inf IsCompact.isGLB_infₛ
-
-/- warning: is_compact.is_lub_Sup -> IsCompact.isLUB_supₛ is a dubious translation:
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGLB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+Case conversion may be inaccurate. Consider using '#align is_compact.is_glb_Inf IsCompact.isGLB_sInfₓ'. -/
+theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGLB s (sInf s) :=
+  isGLB_csInf ne_s hs.BddBelow
+#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
+
+/- warning: is_compact.is_lub_Sup -> IsCompact.isLUB_sSup is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_lub_Sup IsCompact.isLUB_supₛₓ'. -/
-theorem IsCompact.isLUB_supₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLUB s (supₛ s) :=
-  @IsCompact.isGLB_infₛ αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_lub_Sup IsCompact.isLUB_supₛ
-
-/- warning: is_compact.is_least_Inf -> IsCompact.isLeast_infₛ is a dubious translation:
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLUB.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+Case conversion may be inaccurate. Consider using '#align is_compact.is_lub_Sup IsCompact.isLUB_sSupₓ'. -/
+theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLUB s (sSup s) :=
+  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
+
+/- warning: is_compact.is_least_Inf -> IsCompact.isLeast_sInf is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_least_Inf IsCompact.isLeast_infₛₓ'. -/
-theorem IsCompact.isLeast_infₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLeast s (infₛ s) :=
-  ⟨hs.cinfₛ_mem ne_s, (hs.isGLB_infₛ ne_s).1⟩
-#align is_compact.is_least_Inf IsCompact.isLeast_infₛ
-
-/- warning: is_compact.is_greatest_Sup -> IsCompact.isGreatest_supₛ is a dubious translation:
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+Case conversion may be inaccurate. Consider using '#align is_compact.is_least_Inf IsCompact.isLeast_sInfₓ'. -/
+theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLeast s (sInf s) :=
+  ⟨hs.csInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
+#align is_compact.is_least_Inf IsCompact.isLeast_sInf
+
+/- warning: is_compact.is_greatest_Sup -> IsCompact.isGreatest_sSup is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
-Case conversion may be inaccurate. Consider using '#align is_compact.is_greatest_Sup IsCompact.isGreatest_supₛₓ'. -/
-theorem IsCompact.isGreatest_supₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGreatest s (supₛ s) :=
-  @IsCompact.isLeast_infₛ αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_greatest_Sup IsCompact.isGreatest_supₛ
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_2 s) -> (Set.Nonempty.{u1} α s) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) s (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s))
+Case conversion may be inaccurate. Consider using '#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSupₓ'. -/
+theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGreatest s (sSup s) :=
+  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
 
 #print IsCompact.exists_isLeast /-
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsLeast s x :=
-  ⟨_, hs.isLeast_cinfₛ ne_s⟩
+  ⟨_, hs.isLeast_csInf ne_s⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast
 -/
 
 #print IsCompact.exists_isGreatest /-
 theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsGreatest s x :=
-  ⟨_, hs.isGreatest_supₛ ne_s⟩
+  ⟨_, hs.isGreatest_sSup ne_s⟩
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 -/
 
@@ -259,7 +259,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_is_glb IsCompact.exists_isGLBₓ'. -/
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
-  ⟨_, hs.cinfₛ_mem ne_s, hs.isGLB_infₛ ne_s⟩
+  ⟨_, hs.csInf_mem ne_s, hs.isGLB_sInf ne_s⟩
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 
 /- warning: is_compact.exists_is_lub -> IsCompact.exists_isLUB is a dubious translation:
@@ -270,66 +270,66 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_compact.exists_is_lub IsCompact.exists_isLUBₓ'. -/
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
-  ⟨_, hs.csupₛ_mem ne_s, hs.isLUB_supₛ ne_s⟩
+  ⟨_, hs.csSup_mem ne_s, hs.isLUB_sSup ne_s⟩
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
-/- warning: is_compact.exists_Inf_image_eq_and_le -> IsCompact.exists_infₛ_image_eq_and_le is a dubious translation:
+/- warning: is_compact.exists_Inf_image_eq_and_le -> IsCompact.exists_sInf_image_eq_and_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_infₛ_image_eq_and_leₓ'. -/
-theorem IsCompact.exists_infₛ_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, infₛ (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).cinfₛ_mem (ne_s.image f)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y)))))))
+Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_leₓ'. -/
+theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).csInf_mem (ne_s.image f)
   ⟨x, hxs, hx.symm, fun y hy =>
-    hx.trans_le <| cinfₛ_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
-#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_infₛ_image_eq_and_le
+    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).BddBelow <| mem_image_of_mem f hy⟩
+#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
 
-/- warning: is_compact.exists_Sup_image_eq_and_ge -> IsCompact.exists_supₛ_image_eq_and_ge is a dubious translation:
+/- warning: is_compact.exists_Sup_image_eq_and_ge -> IsCompact.exists_sSup_image_eq_and_ge is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_supₛ_image_eq_and_geₓ'. -/
-theorem IsCompact.exists_supₛ_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, supₛ (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-  @IsCompact.exists_infₛ_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_supₛ_image_eq_and_ge
-
-/- warning: is_compact.exists_Inf_image_eq -> IsCompact.exists_infₛ_image_eq is a dubious translation:
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x)))))))
+Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_geₓ'. -/
+theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
+
+/- warning: is_compact.exists_Inf_image_eq -> IsCompact.exists_sInf_image_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq IsCompact.exists_infₛ_image_eqₓ'. -/
-theorem IsCompact.exists_infₛ_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, infₛ (f '' s) = f x :=
-  let ⟨x, hxs, hx, _⟩ := hs.exists_infₛ_image_eq_and_le ne_s hf
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
+Case conversion may be inaccurate. Consider using '#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eqₓ'. -/
+theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
+  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
   ⟨x, hxs, hx⟩
-#align is_compact.exists_Inf_image_eq IsCompact.exists_infₛ_image_eq
+#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
 
-/- warning: is_compact.exists_Sup_image_eq -> IsCompact.exists_supₛ_image_eq is a dubious translation:
+/- warning: is_compact.exists_Sup_image_eq -> IsCompact.exists_sSup_image_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
-Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq IsCompact.exists_supₛ_image_eqₓ'. -/
-theorem IsCompact.exists_supₛ_image_eq :
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β}, (IsCompact.{u2} β _inst_4 s) -> (Set.Nonempty.{u2} β s) -> (forall {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f s)) (f x)))))
+Case conversion may be inaccurate. Consider using '#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eqₓ'. -/
+theorem IsCompact.exists_sSup_image_eq :
     ∀ {s : Set β},
-      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, supₛ (f '' s) = f x :=
-  @IsCompact.exists_infₛ_image_eq αᵒᵈ _ _ _ _ _
-#align is_compact.exists_Sup_image_eq IsCompact.exists_supₛ_image_eq
+      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
+  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
+#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
 
 /- warning: eq_Icc_of_connected_compact -> eq_Icc_of_connected_compact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] {s : Set.{u1} α}, (IsConnected.{u1} α _inst_2 s) -> (IsCompact.{u1} α _inst_2 s) -> (Eq.{succ u1} (Set.{u1} α) s (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) s)))
 Case conversion may be inaccurate. Consider using '#align eq_Icc_of_connected_compact eq_Icc_of_connected_compactₓ'. -/
 theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
-    s = Icc (infₛ s) (supₛ s) :=
-  eq_Icc_cinfₛ_csupₛ_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
+    s = Icc (sInf s) (sSup s) :=
+  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.BddBelow h₂.BddAbove h₂.IsClosed
 #align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
 
 /-!
@@ -440,35 +440,35 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 -/
 
-/- warning: is_compact.Sup_lt_iff_of_continuous -> IsCompact.supₛ_lt_iff_of_continuous is a dubious translation:
+/- warning: is_compact.Sup_lt_iff_of_continuous -> IsCompact.sSup_lt_iff_of_continuous is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
-Case conversion may be inaccurate. Consider using '#align is_compact.Sup_lt_iff_of_continuous IsCompact.supₛ_lt_iff_of_continuousₓ'. -/
-theorem IsCompact.supₛ_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
-    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : supₛ (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α f K)) y) (forall (x : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) y)))
+Case conversion may be inaccurate. Consider using '#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuousₓ'. -/
+theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
+    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
   by
   refine'
-    ⟨fun h x hx => (le_csupₛ (hK.bdd_above_image hf) <| mem_image_of_mem f hx).trans_lt h, fun h =>
+    ⟨fun h x hx => (le_csSup (hK.bdd_above_image hf) <| mem_image_of_mem f hx).trans_lt h, fun h =>
       _⟩
   obtain ⟨x, hx, h2x⟩ := hK.exists_forall_ge h0K hf
-  refine' (csupₛ_le (h0K.image f) _).trans_lt (h x hx)
+  refine' (csSup_le (h0K.image f) _).trans_lt (h x hx)
   rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
-#align is_compact.Sup_lt_iff_of_continuous IsCompact.supₛ_lt_iff_of_continuous
+#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
 
-/- warning: is_compact.lt_Inf_iff_of_continuous -> IsCompact.lt_infₛ_iff_of_continuous is a dubious translation:
+/- warning: is_compact.lt_Inf_iff_of_continuous -> IsCompact.lt_sInf_iff_of_continuous is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_6 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_7 : TopologicalSpace.{u1} α] [_inst_8 : OrderTopology.{u1} α _inst_7 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_9 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)) (Set.image.{u2, u1} β α f K))) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (f x))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_6 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_7 : TopologicalSpace.{u1} α] [_inst_8 : OrderTopology.{u1} α _inst_7 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u2} β] {f : β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_9 K) -> (Set.Nonempty.{u2} β K) -> (ContinuousOn.{u2, u1} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)) (Set.image.{u2, u1} β α f K))) (forall (x : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x K) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_6)))))) y (f x))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_6 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_7 : TopologicalSpace.{u2} α] [_inst_8 : OrderTopology.{u2} α _inst_7 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u1} β] {f : β -> α} {K : Set.{u1} β}, (IsCompact.{u1} β _inst_9 K) -> (Set.Nonempty.{u1} β K) -> (ContinuousOn.{u1, u2} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (InfSet.infₛ.{u2} α (ConditionallyCompleteLattice.toInfSet.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)) (Set.image.{u1, u2} β α f K))) (forall (x : β), (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x K) -> (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (f x))))
-Case conversion may be inaccurate. Consider using '#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_infₛ_iff_of_continuousₓ'. -/
-theorem IsCompact.lt_infₛ_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_6 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_7 : TopologicalSpace.{u2} α] [_inst_8 : OrderTopology.{u2} α _inst_7 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))] [_inst_9 : TopologicalSpace.{u1} β] {f : β -> α} {K : Set.{u1} β}, (IsCompact.{u1} β _inst_9 K) -> (Set.Nonempty.{u1} β K) -> (ContinuousOn.{u1, u2} β α _inst_9 _inst_7 f K) -> (forall (y : α), Iff (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (InfSet.sInf.{u2} α (ConditionallyCompleteLattice.toInfSet.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)) (Set.image.{u1, u2} β α f K))) (forall (x : β), (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) x K) -> (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_6)))))) y (f x))))
+Case conversion may be inaccurate. Consider using '#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuousₓ'. -/
+theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
-    y < infₛ (f '' K) ↔ ∀ x ∈ K, y < f x :=
-  @IsCompact.supₛ_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
-#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_infₛ_iff_of_continuous
+    y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
+  @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
+#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
 #print Continuous.exists_forall_le_of_hasCompactMulSupport /-
 /-- A continuous function with compact support has a global minimum. -/
@@ -493,15 +493,15 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
 -/
 
-/- warning: is_compact.continuous_Sup -> IsCompact.continuous_supₛ is a dubious translation:
+/- warning: is_compact.continuous_Sup -> IsCompact.continuous_sSup is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.{u3} γ] {f : γ -> β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} γ β) α (Prod.topologicalSpace.{u3, u2} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max u3 u2 u1, max u3 u2, u1} (γ -> β -> α) (Prod.{u3, u2} γ β) α (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} γ (β -> α) β α (Function.hasUncurryBase.{u2, u1} β α)) f)) -> (Continuous.{u3, u1} γ α _inst_5 _inst_2 (fun (x : γ) => SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α (f x) K)))
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.{u3} γ] {f : γ -> β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} γ β) α (Prod.topologicalSpace.{u3, u2} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max u3 u2 u1, max u3 u2, u1} (γ -> β -> α) (Prod.{u3, u2} γ β) α (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} γ (β -> α) β α (Function.hasUncurryBase.{u2, u1} β α)) f)) -> (Continuous.{u3, u1} γ α _inst_5 _inst_2 (fun (x : γ) => SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α (f x) K)))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u3} β] [_inst_5 : TopologicalSpace.{u2} γ] {f : γ -> β -> α} {K : Set.{u3} β}, (IsCompact.{u3} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u2, u3} γ β) α (instTopologicalSpaceProd.{u2, u3} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max (max u1 u3) u2, max u3 u2, u1} (γ -> β -> α) (Prod.{u2, u3} γ β) α (Function.hasUncurryInduction.{u2, max u1 u3, u3, u1} γ (β -> α) β α (Function.hasUncurryBase.{u3, u1} β α)) f)) -> (Continuous.{u2, u1} γ α _inst_5 _inst_2 (fun (x : γ) => SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u3, u1} β α (f x) K)))
-Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Sup IsCompact.continuous_supₛₓ'. -/
+  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u3} β] [_inst_5 : TopologicalSpace.{u2} γ] {f : γ -> β -> α} {K : Set.{u3} β}, (IsCompact.{u3} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u2, u3} γ β) α (instTopologicalSpaceProd.{u2, u3} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max (max u1 u3) u2, max u3 u2, u1} (γ -> β -> α) (Prod.{u2, u3} γ β) α (Function.hasUncurryInduction.{u2, max u1 u3, u3, u1} γ (β -> α) β α (Function.hasUncurryBase.{u3, u1} β α)) f)) -> (Continuous.{u2, u1} γ α _inst_5 _inst_2 (fun (x : γ) => SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u3, u1} β α (f x) K)))
+Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Sup IsCompact.continuous_sSupₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-theorem IsCompact.continuous_supₛ {f : γ → β → α} {K : Set β} (hK : IsCompact K)
-    (hf : Continuous ↿f) : Continuous fun x => supₛ (f x '' K) :=
+theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
+    (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) :=
   by
   rcases eq_empty_or_nonempty K with (rfl | h0K)
   · simp_rw [image_empty]
@@ -519,7 +519,7 @@ theorem IsCompact.continuous_supₛ {f : γ → β → α} {K : Set β} (hK : Is
         x)
   refine' ⟨fun z hz => _, fun z hz => _⟩
   · refine'
-      (this.1 z hz).mono fun x' hx' => hx'.trans_le <| le_csupₛ _ <| mem_image_of_mem (f x') hyK
+      (this.1 z hz).mono fun x' hx' => hx'.trans_le <| le_csSup _ <| mem_image_of_mem (f x') hyK
     exact hK.bdd_above_image (hf.comp <| Continuous.Prod.mk x').ContinuousOn
   · have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z :=
       by
@@ -532,18 +532,18 @@ theorem IsCompact.continuous_supₛ {f : γ → β → α} {K : Set β} (hK : Is
     rw [hK.Sup_lt_iff_of_continuous h0K
         (show Continuous (f x') from hf.comp <| Continuous.Prod.mk x').ContinuousOn]
     exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy'))
-#align is_compact.continuous_Sup IsCompact.continuous_supₛ
+#align is_compact.continuous_Sup IsCompact.continuous_sSup
 
-/- warning: is_compact.continuous_Inf -> IsCompact.continuous_infₛ is a dubious translation:
+/- warning: is_compact.continuous_Inf -> IsCompact.continuous_sInf is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.{u3} γ] {f : γ -> β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} γ β) α (Prod.topologicalSpace.{u3, u2} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max u3 u2 u1, max u3 u2, u1} (γ -> β -> α) (Prod.{u3, u2} γ β) α (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} γ (β -> α) β α (Function.hasUncurryBase.{u2, u1} β α)) f)) -> (Continuous.{u3, u1} γ α _inst_5 _inst_2 (fun (x : γ) => InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α (f x) K)))
+  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_5 : TopologicalSpace.{u3} γ] {f : γ -> β -> α} {K : Set.{u2} β}, (IsCompact.{u2} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u3, u2} γ β) α (Prod.topologicalSpace.{u3, u2} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max u3 u2 u1, max u3 u2, u1} (γ -> β -> α) (Prod.{u3, u2} γ β) α (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} γ (β -> α) β α (Function.hasUncurryBase.{u2, u1} β α)) f)) -> (Continuous.{u3, u1} γ α _inst_5 _inst_2 (fun (x : γ) => InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toHasInf.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u2, u1} β α (f x) K)))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u3} β] [_inst_5 : TopologicalSpace.{u2} γ] {f : γ -> β -> α} {K : Set.{u3} β}, (IsCompact.{u3} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u2, u3} γ β) α (instTopologicalSpaceProd.{u2, u3} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max (max u1 u3) u2, max u3 u2, u1} (γ -> β -> α) (Prod.{u2, u3} γ β) α (Function.hasUncurryInduction.{u2, max u1 u3, u3, u1} γ (β -> α) β α (Function.hasUncurryBase.{u3, u1} β α)) f)) -> (Continuous.{u2, u1} γ α _inst_5 _inst_2 (fun (x : γ) => InfSet.infₛ.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u3, u1} β α (f x) K)))
-Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Inf IsCompact.continuous_infₛₓ'. -/
-theorem IsCompact.continuous_infₛ {f : γ → β → α} {K : Set β} (hK : IsCompact K)
-    (hf : Continuous ↿f) : Continuous fun x => infₛ (f x '' K) :=
-  @IsCompact.continuous_supₛ αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
-#align is_compact.continuous_Inf IsCompact.continuous_infₛ
+  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u3} β] [_inst_5 : TopologicalSpace.{u2} γ] {f : γ -> β -> α} {K : Set.{u3} β}, (IsCompact.{u3} β _inst_4 K) -> (Continuous.{max u3 u2, u1} (Prod.{u2, u3} γ β) α (instTopologicalSpaceProd.{u2, u3} γ β _inst_5 _inst_4) _inst_2 (Function.HasUncurry.uncurry.{max (max u1 u3) u2, max u3 u2, u1} (γ -> β -> α) (Prod.{u2, u3} γ β) α (Function.hasUncurryInduction.{u2, max u1 u3, u3, u1} γ (β -> α) β α (Function.hasUncurryBase.{u3, u1} β α)) f)) -> (Continuous.{u2, u1} γ α _inst_5 _inst_2 (fun (x : γ) => InfSet.sInf.{u1} α (ConditionallyCompleteLattice.toInfSet.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) (Set.image.{u3, u1} β α (f x) K)))
+Case conversion may be inaccurate. Consider using '#align is_compact.continuous_Inf IsCompact.continuous_sInfₓ'. -/
+theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
+    (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
+  @IsCompact.continuous_sSup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
+#align is_compact.continuous_Inf IsCompact.continuous_sInf
 
 namespace ContinuousOn
 
@@ -559,24 +559,24 @@ open Interval
 
 /- warning: continuous_on.image_Icc -> ContinuousOn.image_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) a b) -> (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.infₛ.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (SupSet.supₛ.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) a b) -> (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))) a b) -> (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.infₛ.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (SupSet.supₛ.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)))))
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))) a b) -> (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_Icc ContinuousOn.image_Iccₓ'. -/
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
-    f '' Icc a b = Icc (infₛ <| f '' Icc a b) (supₛ <| f '' Icc a b) :=
+    f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
   eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
     (isCompact_Icc.image_of_continuousOn h)
 #align continuous_on.image_Icc ContinuousOn.image_Icc
 
 /- warning: continuous_on.image_uIcc_eq_Icc -> ContinuousOn.image_uIcc_eq_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.infₛ.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.supₛ.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.Icc.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.infₛ.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.supₛ.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.Icc.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Iccₓ'. -/
 theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
-    f '' [a, b] = Icc (infₛ (f '' [a, b])) (supₛ (f '' [a, b])) :=
+    f '' [a, b] = Icc (sInf (f '' [a, b])) (sSup (f '' [a, b])) :=
   by
   cases' le_total a b with h2 h2
   · simp_rw [uIcc_of_le h2] at h⊢
@@ -587,51 +587,51 @@ theorem image_uIcc_eq_Icc (h : ContinuousOn f <| [a, b]) :
 
 /- warning: continuous_on.image_uIcc -> ContinuousOn.image_uIcc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.uIcc.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (InfSet.infₛ.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.supₛ.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) -> (Eq.{succ u2} (Set.{u2} β) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)) (Set.uIcc.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b))) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.uIcc.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)) a b)))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.uIcc.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (InfSet.infₛ.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.supₛ.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) -> (Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)) (Set.uIcc.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b))) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.uIcc.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)) a b)))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.image_uIcc ContinuousOn.image_uIccₓ'. -/
 theorem image_uIcc (h : ContinuousOn f <| [a, b]) :
-    f '' [a, b] = [infₛ (f '' [a, b]), supₛ (f '' [a, b])] :=
+    f '' [a, b] = [sInf (f '' [a, b]), sSup (f '' [a, b])] :=
   by
   refine' h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm
-  refine' cinfₛ_le_csupₛ _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
+  refine' csInf_le_csSup _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
   exacts[bddBelow_Icc, bddAbove_Icc]
 #align continuous_on.image_uIcc ContinuousOn.image_uIcc
 
-/- warning: continuous_on.Inf_image_Icc_le -> ContinuousOn.infₛ_image_Icc_le is a dubious translation:
+/- warning: continuous_on.Inf_image_Icc_le -> ContinuousOn.sInf_image_Icc_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (InfSet.infₛ.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (f c))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (InfSet.sInf.{u2} β (ConditionallyCompleteLattice.toHasInf.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))) (f c))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (InfSet.infₛ.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (f c))
-Case conversion may be inaccurate. Consider using '#align continuous_on.Inf_image_Icc_le ContinuousOn.infₛ_image_Icc_leₓ'. -/
-theorem infₛ_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
-    infₛ (f '' Icc a b) ≤ f c :=
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (InfSet.sInf.{u1} β (ConditionallyCompleteLattice.toInfSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))) (f c))
+Case conversion may be inaccurate. Consider using '#align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_leₓ'. -/
+theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
+    sInf (f '' Icc a b) ≤ f c :=
   by
   rw [h.image_Icc (nonempty_Icc.mp (Set.nonempty_of_mem hc))]
   exact
-    cinfₛ_le bddBelow_Icc
+    csInf_le bddBelow_Icc
       (mem_Icc.mpr
-        ⟨cinfₛ_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
-          le_csupₛ (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
-#align continuous_on.Inf_image_Icc_le ContinuousOn.infₛ_image_Icc_le
+        ⟨csInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
+          le_csSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
+#align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_le
 
-/- warning: continuous_on.le_Sup_image_Icc -> ContinuousOn.le_supₛ_image_Icc is a dubious translation:
+/- warning: continuous_on.le_Sup_image_Icc -> ContinuousOn.le_sSup_image_Icc is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (f c) (SupSet.supₛ.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] [_inst_6 : DenselyOrdered.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u2} β] [_inst_8 : OrderTopology.{u2} β _inst_4 (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u1, u2} α β _inst_2 _inst_4 f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) c (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b)) -> (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (ConditionallyCompleteLattice.toLattice.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)))))) (f c) (SupSet.sSup.{u2} β (ConditionallyCompleteLattice.toHasSup.{u2} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} β _inst_7)) (Set.image.{u1, u2} α β f (Set.Icc.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1))))) a b))))
 but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (f c) (SupSet.supₛ.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))))
-Case conversion may be inaccurate. Consider using '#align continuous_on.le_Sup_image_Icc ContinuousOn.le_supₛ_image_Iccₓ'. -/
-theorem le_supₛ_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
-    f c ≤ supₛ (f '' Icc a b) :=
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : ConditionallyCompleteLinearOrder.{u2} α] [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : OrderTopology.{u2} α _inst_2 (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u1} β] [_inst_6 : DenselyOrdered.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))))] [_inst_7 : ConditionallyCompleteLinearOrder.{u1} β] [_inst_8 : OrderTopology.{u1} β _inst_4 (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))] {f : α -> β} {a : α} {b : α} {c : α}, (ContinuousOn.{u2, u1} α β _inst_2 _inst_4 f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) c (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b)) -> (LE.le.{u1} β (Preorder.toLE.{u1} β (PartialOrder.toPreorder.{u1} β (SemilatticeInf.toPartialOrder.{u1} β (Lattice.toSemilatticeInf.{u1} β (ConditionallyCompleteLattice.toLattice.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)))))) (f c) (SupSet.sSup.{u1} β (ConditionallyCompleteLattice.toSupSet.{u1} β (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} β _inst_7)) (Set.image.{u2, u1} α β f (Set.Icc.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u2} α _inst_1))))) a b))))
+Case conversion may be inaccurate. Consider using '#align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Iccₓ'. -/
+theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
+    f c ≤ sSup (f '' Icc a b) :=
   by
   rw [h.image_Icc (nonempty_Icc.mp (Set.nonempty_of_mem hc))]
   exact
-    le_csupₛ bddAbove_Icc
+    le_csSup bddAbove_Icc
       (mem_Icc.mpr
-        ⟨cinfₛ_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
-          le_csupₛ (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
-#align continuous_on.le_Sup_image_Icc ContinuousOn.le_supₛ_image_Icc
+        ⟨csInf_le (is_compact_Icc.bdd_below_image h) ⟨c, hc, rfl⟩,
+          le_csSup (is_compact_Icc.bdd_above_image h) ⟨c, hc, rfl⟩⟩)
+#align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Icc
 
 end ContinuousOn

bump to nightly-2023-02-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

2097f8af

Diff

@@ -366,9 +366,9 @@ theorem IsCompact.exists_forall_ge :
 
 /- warning: continuous_on.exists_forall_le' -> ContinuousOn.exists_forall_le' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (HasInf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (HasInf.inf.{u2} (Filter.{u2} β) (Filter.instHasInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x₀) (f x)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f y))))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'ₓ'. -/
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/
@@ -387,9 +387,9 @@ theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : Contin
 
 /- warning: continuous_on.exists_forall_ge' -> ContinuousOn.exists_forall_ge' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (HasInf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => Exists.{0} (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) (fun (H : Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s) => forall (y : β), (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (HasInf.inf.{u2} (Filter.{u2} β) (Filter.instHasInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : ConditionallyCompleteLinearOrder.{u1} α] [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : OrderTopology.{u1} α _inst_2 (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))] [_inst_4 : TopologicalSpace.{u2} β] {s : Set.{u2} β} {f : β -> α}, (ContinuousOn.{u2, u1} β α _inst_4 _inst_2 f s) -> (IsClosed.{u2} β _inst_4 s) -> (forall {x₀ : β}, (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x₀ s) -> (Filter.Eventually.{u2} β (fun (x : β) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f x) (f x₀)) (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) (Filter.cocompact.{u2} β _inst_4) (Filter.principal.{u2} β s))) -> (Exists.{succ u2} β (fun (x : β) => And (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s) (forall (y : β), (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) y s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (f y) (f x))))))
 Case conversion may be inaccurate. Consider using '#align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'ₓ'. -/
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

chore: replace refine' that already have a ?_ (#12261)

Co-authored-by: Moritz Firsching <firsching@google.com>

94b44d32

Diff

@@ -78,7 +78,7 @@ instance [TopologicalSpace α] [Preorder α] [CompactIccSpace α] : CompactIccSp
 instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (α : Type*)
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
     CompactIccSpace α := by
-  refine' .mk'' fun {a b} hlt => ?_
+  refine .mk'' fun {a b} hlt => ?_
   rcases le_or_lt a b with hab | hab
   swap
   · simp [hab]

doc: convert many comments into doc comments (#11940)

All of these changes appear to be oversights to me.

678a5c79

Diff

@@ -571,19 +571,17 @@ theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ :
   eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed
 #align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
 
-/- If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a
+/-- If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a
 conditionally complete linear order, and `K : Set β` is a compact set, then
-`fun x ↦ sSup (f x '' K)` is a continuous function.
-
-Porting note (#11215): TODO: generalize. The following version seems to be true:
+`fun x ↦ sSup (f x '' K)` is a continuous function. -/
+/- Porting note (#11215): TODO: generalize. The following version seems to be true:
 ```
 theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
     (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y)))
     (hgc : ContinuousOn g K) :
     Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _
 ```
-Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov).
--/
+Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov). -/
 theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) := by
   rcases eq_empty_or_nonempty K with (rfl | h0K)

refactor(Topology/Order/Basic): split up large file (#11992)

This splits up the file Mathlib/Topology/Order/Basic.lean (currently > 2000 lines) into several smaller files.

b8145add

Diff

@@ -6,6 +6,7 @@ Authors: Patrick Massot, Yury Kudryashov
 import Mathlib.Topology.Order.LocalExtr
 import Mathlib.Topology.Order.IntermediateValue
 import Mathlib.Topology.Support
+import Mathlib.Topology.Order.IsLUB
 
 #align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"

style: remove redundant instance arguments (#11581)

I removed some redundant instance arguments throughout Mathlib. To do this, I used VS Code's regex search. See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/repeating.20instances.20from.20variable.20command I closed the previous PR for this and reopened it.

acda20c6

Diff

@@ -202,7 +202,7 @@ theorem IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompa
   IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
-theorem cocompact_le_atBot_atTop [LinearOrder α] [CompactIccSpace α] :
+theorem cocompact_le_atBot_atTop [CompactIccSpace α] :
     cocompact α ≤ atBot ⊔ atTop := by
   refine fun s hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
   · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
@@ -211,7 +211,7 @@ theorem cocompact_le_atBot_atTop [LinearOrder α] [CompactIccSpace α] :
     refine ⟨Icc t u, isCompact_Icc, fun x hx ↦ ?_⟩
     exact (not_and_or.mp hx).casesOn (fun h ↦ ht x (le_of_not_le h)) fun h ↦ hu x (le_of_not_le h)
 
-theorem cocompact_le_atBot [LinearOrder α] [OrderTop α] [CompactIccSpace α] :
+theorem cocompact_le_atBot [OrderTop α] [CompactIccSpace α] :
     cocompact α ≤ atBot := by
   refine fun _ hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
   · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
@@ -219,11 +219,11 @@ theorem cocompact_le_atBot [LinearOrder α] [OrderTop α] [CompactIccSpace α] :
     refine ⟨Icc t ⊤, isCompact_Icc, fun _ hx ↦ ?_⟩
     exact (not_and_or.mp hx).casesOn (fun h ↦ ht _ (le_of_not_le h)) (fun h ↦ (h le_top).elim)
 
-theorem cocompact_le_atTop [LinearOrder α] [OrderBot α] [CompactIccSpace α] :
+theorem cocompact_le_atTop [OrderBot α] [CompactIccSpace α] :
     cocompact α ≤ atTop :=
   cocompact_le_atBot (α := αᵒᵈ)
 
-theorem atBot_le_cocompact [LinearOrder α] [NoMinOrder α] [ClosedIicTopology α] :
+theorem atBot_le_cocompact [NoMinOrder α] [ClosedIicTopology α] :
     atBot ≤ cocompact α := by
   refine fun s hs ↦ ?_
   obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
@@ -236,26 +236,26 @@ theorem atBot_le_cocompact [LinearOrder α] [NoMinOrder α] [ClosedIicTopology 
     exact Filter.mem_atBot_sets.mpr ⟨b, fun b' hb' ↦ hts <| Classical.byContradiction
       fun hc ↦ LT.lt.false <| hb'.trans_lt <| hb.trans_le <| ha.2 (not_not_mem.mp hc)⟩
 
-theorem atTop_le_cocompact [LinearOrder α] [NoMaxOrder α] [ClosedIciTopology α] :
+theorem atTop_le_cocompact [NoMaxOrder α] [ClosedIciTopology α] :
     atTop ≤ cocompact α :=
   atBot_le_cocompact (α := αᵒᵈ)
 
-theorem atBot_atTop_le_cocompact [LinearOrder α] [NoMinOrder α] [NoMaxOrder α]
+theorem atBot_atTop_le_cocompact [NoMinOrder α] [NoMaxOrder α]
     [OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α :=
   sup_le atBot_le_cocompact atTop_le_cocompact
 
 @[simp 900]
-theorem cocompact_eq_atBot_atTop [LinearOrder α] [NoMaxOrder α] [NoMinOrder α]
+theorem cocompact_eq_atBot_atTop [NoMaxOrder α] [NoMinOrder α]
     [OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop :=
   cocompact_le_atBot_atTop.antisymm atBot_atTop_le_cocompact
 
 @[simp]
-theorem cocompact_eq_atBot [LinearOrder α] [NoMinOrder α] [OrderTop α]
+theorem cocompact_eq_atBot [NoMinOrder α] [OrderTop α]
     [ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot :=
   cocompact_le_atBot.antisymm atBot_le_cocompact
 
 @[simp]
-theorem cocompact_eq_atTop [LinearOrder α] [NoMaxOrder α] [OrderBot α]
+theorem cocompact_eq_atTop [NoMaxOrder α] [OrderBot α]
     [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop :=
   cocompact_le_atTop.antisymm atTop_le_cocompact

move(Topology/Order): Move anything that doesn't concern algebra (#11610)

Move files from Topology.Algebra.Order to Topology.Order when they do not contain any algebra. Also move Topology.LocalExtr to Topology.Order.LocalExtr.

According to git, the moves are:

Mathlib/Topology/{Algebra => }/Order/ExtendFrom.lean
Mathlib/Topology/{Algebra => }/Order/ExtrClosure.lean
Mathlib/Topology/{Algebra => }/Order/Filter.lean
Mathlib/Topology/{Algebra => }/Order/IntermediateValue.lean
Mathlib/Topology/{Algebra => }/Order/LeftRight.lean
Mathlib/Topology/{Algebra => }/Order/LeftRightLim.lean
Mathlib/Topology/{Algebra => }/Order/MonotoneContinuity.lean
Mathlib/Topology/{Algebra => }/Order/MonotoneConvergence.lean
Mathlib/Topology/{Algebra => }/Order/ProjIcc.lean
Mathlib/Topology/{Algebra => }/Order/T5.lean
Mathlib/Topology/{ => Order}/LocalExtr.lean

cd9902cc

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
 -/
-import Mathlib.Topology.Algebra.Order.IntermediateValue
-import Mathlib.Topology.LocalExtr
+import Mathlib.Topology.Order.LocalExtr
+import Mathlib.Topology.Order.IntermediateValue
 import Mathlib.Topology.Support
 
 #align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"

chore: avoid some unused variables (#11583)

These will be caught by the linter in a future lean version.

818ec230

Diff

@@ -109,9 +109,10 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
     rw [diff_subset_iff]
     exact Subset.trans Icc_subset_Icc_union_Ioc <| union_subset_union Subset.rfl <|
       Ioc_subset_Ioc_left hy.1.le
-  rcases hc.2.eq_or_lt with (rfl | hlt); · exact hcs.2
-  contrapose! hf
-  intro U hU
+  rcases hc.2.eq_or_lt with (rfl | hlt)
+  · exact hcs.2
+  exfalso
+  refine hf fun U hU => ?_
   rcases (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
       (mem_nhdsWithin_of_mem_nhds hU) with
     ⟨y, hxy, hyU⟩

chore: classify new theorem / theorem porting notes (#11432)

Classifies by adding issue number #10756 to porting notes claiming anything equivalent to:

"added theorem"
"added theorems"
"new theorem"
"new theorems"
"added lemma"
"new lemma"
"new lemmas"

55fe4027

Diff

@@ -55,13 +55,13 @@ export CompactIccSpace (isCompact_Icc)
 
 variable {α : Type*}
 
--- Porting note: new lemma;
+-- Porting note (#10756): new lemma;
 -- Porting note (#11215): TODO: make it the definition
 lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
     (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
   isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty
 
--- Porting note: new lemma;
+-- Porting note (#10756): new lemma;
 -- Porting note (#11215): TODO: drop one `'`
 lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
     (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α :=

chore: classify new lemma porting notes (#11217)

Classifies by adding issue number #10756 to porting notes claiming anything semantically equivalent to:

"new lemma"
"added lemma"

68df7d0d

Diff

@@ -258,7 +258,7 @@ theorem cocompact_eq_atTop [LinearOrder α] [NoMaxOrder α] [OrderBot α]
     [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop :=
   cocompact_le_atTop.antisymm atTop_le_cocompact
 
--- Porting note: new lemma; defeq to the old one but allows us to use dot notation
+-- Porting note (#10756): new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
@@ -283,7 +283,7 @@ theorem IsCompact.exists_forall_le [ClosedIicTopology α] {s : Set β} (hs : IsC
   hs.exists_isMinOn ne_s hf
 #align is_compact.exists_forall_le IsCompact.exists_forall_le
 
--- Porting note: new lemma; defeq to the old one but allows us to use dot notation
+-- Porting note (#10756): new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 theorem IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x :=

chore: classify todo porting notes (#11216)

Classifies by adding issue number #11215 to porting notes claiming "TODO".

86f95a40

Diff

@@ -55,12 +55,14 @@ export CompactIccSpace (isCompact_Icc)
 
 variable {α : Type*}
 
--- Porting note: new lemma; TODO: make it the definition
+-- Porting note: new lemma;
+-- Porting note (#11215): TODO: make it the definition
 lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
     (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
   isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty
 
--- Porting note: new lemma; TODO: drop one `'`
+-- Porting note: new lemma;
+-- Porting note (#11215): TODO: drop one `'`
 lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
     (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α :=
   .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h
@@ -571,7 +573,7 @@ theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ :
 conditionally complete linear order, and `K : Set β` is a compact set, then
 `fun x ↦ sSup (f x '' K)` is a continuous function.
 
-Porting note: todo: generalize. The following version seems to be true:
+Porting note (#11215): TODO: generalize. The following version seems to be true:
 ```
 theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
     (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y)))

chore: Remove ball and bex from lemma names (#10816)

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

c697e040

Diff

@@ -261,7 +261,7 @@ theorem cocompact_eq_atTop [LinearOrder α] [NoMaxOrder α] [OrderBot α]
 theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
   rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
-  exact ⟨x, hxs, ball_image_iff.1 hx⟩
+  exact ⟨x, hxs, forall_mem_image.1 hx⟩
 
 /-- If a continuous function lies strictly above `a` on a compact set,
   it has a lower bound strictly above `a`. -/
@@ -369,7 +369,7 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology 
     [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
     ∃ x : β, ∀ y : β, f x ≤ f y := by
   obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.isCompact_range hf).exists_isLeast (range_nonempty _)
-  rw [mem_lowerBounds, forall_range_iff] at hx
+  rw [mem_lowerBounds, forall_mem_range] at hx
   exact ⟨x, hx⟩
 #align continuous.exists_forall_le_of_has_compact_mul_support Continuous.exists_forall_le_of_hasCompactMulSupport
 #align continuous.exists_forall_le_of_has_compact_support Continuous.exists_forall_le_of_hasCompactSupport

feat: version of the extreme value theorem without a non-emptiness assumption (#9872)

Adapted and generalised from sphere-eversion; I'm just upstreaming it.

39f3b00d

Diff

@@ -263,6 +263,17 @@ theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCom
   rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
   exact ⟨x, hxs, ball_image_iff.1 hx⟩
 
+/-- If a continuous function lies strictly above `a` on a compact set,
+  it has a lower bound strictly above `a`. -/
+theorem IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
+    {s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
+    ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b := by
+  rcases s.eq_empty_or_nonempty with (rfl | hs')
+  · obtain ⟨a', ha'⟩ := exists_gt a
+    exact ⟨a', ha', fun _ a ↦ a.elim⟩
+  · obtain ⟨x, hx, hx'⟩ := hs.exists_isMinOn hs' hf
+    exact ⟨f x, hf' x hx, hx'⟩
+
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 @[deprecated IsCompact.exists_isMinOn]
 theorem IsCompact.exists_forall_le [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -55,12 +55,12 @@ export CompactIccSpace (isCompact_Icc)
 
 variable {α : Type*}
 
--- porting note: new lemma; TODO: make it the definition
+-- Porting note: new lemma; TODO: make it the definition
 lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
     (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
   isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty
 
--- porting note: new lemma; TODO: drop one `'`
+-- Porting note: new lemma; TODO: drop one `'`
 lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
     (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α :=
   .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h
@@ -90,7 +90,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   have ha : a ∈ s := by simp [s, hpt, hab]
   rcases hab.eq_or_lt with (rfl | _hlt)
   · exact ha.2
-  -- porting note: the `obtain` below was instead
+  -- Porting note: the `obtain` below was instead
   -- `set c := Sup s`
   -- `have hsc : IsLUB s c := isLUB_csSup ⟨a, ha⟩ sbd`
   obtain ⟨c, hsc⟩ : ∃ c, IsLUB s c := ⟨sSup s, isLUB_csSup ⟨a, ha⟩ ⟨b, hsb⟩⟩
@@ -256,7 +256,7 @@ theorem cocompact_eq_atTop [LinearOrder α] [NoMaxOrder α] [OrderBot α]
     [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop :=
   cocompact_le_atTop.antisymm atTop_le_cocompact
 
--- porting note: new lemma; defeq to the old one but allows us to use dot notation
+-- Porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
@@ -270,7 +270,7 @@ theorem IsCompact.exists_forall_le [ClosedIicTopology α] {s : Set β} (hs : IsC
   hs.exists_isMinOn ne_s hf
 #align is_compact.exists_forall_le IsCompact.exists_forall_le
 
--- porting note: new lemma; defeq to the old one but allows us to use dot notation
+-- Porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 theorem IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x :=
@@ -531,7 +531,7 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset [ClosedIicTopology α] {f : β
   ⟨x, hxs, h.localize⟩
 #align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
 
--- porting note: rfc: assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and
+-- Porting note: rfc: assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and
 -- `IsOpen s`?
 theorem IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β}
     {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)

chore: more backporting of simp changes from #10995 (#11001)

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

d9589c14

Diff

@@ -87,7 +87,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   set s := { x ∈ Icc a b | Icc a x ∉ f }
   have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
   have sbd : BddAbove s := ⟨b, hsb⟩
-  have ha : a ∈ s := by simp [hpt, hab]
+  have ha : a ∈ s := by simp [s, hpt, hab]
   rcases hab.eq_or_lt with (rfl | _hlt)
   · exact ha.2
   -- porting note: the `obtain` below was instead

chore: remove stream-of-consciousness uses of have, replace and suffices (#10640)

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

a13e8040

Diff

@@ -174,8 +174,8 @@ variable {α β γ : Type*} [LinearOrder α] [TopologicalSpace α]
 theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
     (ne_s : s.Nonempty) : ∃ x, IsLeast s x := by
   haveI : Nonempty s := ne_s.to_subtype
-  suffices : (s ∩ ⋂ x ∈ s, Iic x).Nonempty
-  · exact ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
+  suffices (s ∩ ⋂ x ∈ s, Iic x).Nonempty from
+    ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
   rw [biInter_eq_iInter]
   by_contra H
   rw [not_nonempty_iff_eq_empty] at H

feat: generalize cocompact_eq (#10285)

example use case: cocompact_le with integrable_iff_integrableAtFilter_cocompact from #10248 becomes a way to prove integrability from big-O estimates (e.g. #10258)

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

117789e2

Diff

@@ -65,6 +65,12 @@ lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
     (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α :=
   .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h
 
+instance [TopologicalSpace α] [Preorder α] [CompactIccSpace α] : CompactIccSpace (αᵒᵈ) where
+  isCompact_Icc := by
+    intro a b
+    convert isCompact_Icc (α := α) (a := b) (b := a) using 1
+    exact dual_Icc (α := α)
+
 /-- A closed interval in a conditionally complete linear order is compact. -/
 instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (α : Type*)
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
@@ -193,6 +199,63 @@ theorem IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompa
   IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
+theorem cocompact_le_atBot_atTop [LinearOrder α] [CompactIccSpace α] :
+    cocompact α ≤ atBot ⊔ atTop := by
+  refine fun s hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
+  · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
+  · obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs.1
+    obtain ⟨u, hu⟩ := mem_atTop_sets.mp hs.2
+    refine ⟨Icc t u, isCompact_Icc, fun x hx ↦ ?_⟩
+    exact (not_and_or.mp hx).casesOn (fun h ↦ ht x (le_of_not_le h)) fun h ↦ hu x (le_of_not_le h)
+
+theorem cocompact_le_atBot [LinearOrder α] [OrderTop α] [CompactIccSpace α] :
+    cocompact α ≤ atBot := by
+  refine fun _ hs ↦ mem_cocompact.mpr <| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro
+  · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩
+  · obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs
+    refine ⟨Icc t ⊤, isCompact_Icc, fun _ hx ↦ ?_⟩
+    exact (not_and_or.mp hx).casesOn (fun h ↦ ht _ (le_of_not_le h)) (fun h ↦ (h le_top).elim)
+
+theorem cocompact_le_atTop [LinearOrder α] [OrderBot α] [CompactIccSpace α] :
+    cocompact α ≤ atTop :=
+  cocompact_le_atBot (α := αᵒᵈ)
+
+theorem atBot_le_cocompact [LinearOrder α] [NoMinOrder α] [ClosedIicTopology α] :
+    atBot ≤ cocompact α := by
+  refine fun s hs ↦ ?_
+  obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
+  refine (Set.eq_empty_or_nonempty t).casesOn (fun h_empty ↦ ?_) (fun h_nonempty ↦ ?_)
+  · rewrite [compl_univ_iff.mpr h_empty, univ_subset_iff] at hts
+    convert univ_mem
+  · haveI := h_nonempty.nonempty
+    obtain ⟨a, ha⟩ := ht.exists_isLeast h_nonempty
+    obtain ⟨b, hb⟩ := exists_lt a
+    exact Filter.mem_atBot_sets.mpr ⟨b, fun b' hb' ↦ hts <| Classical.byContradiction
+      fun hc ↦ LT.lt.false <| hb'.trans_lt <| hb.trans_le <| ha.2 (not_not_mem.mp hc)⟩
+
+theorem atTop_le_cocompact [LinearOrder α] [NoMaxOrder α] [ClosedIciTopology α] :
+    atTop ≤ cocompact α :=
+  atBot_le_cocompact (α := αᵒᵈ)
+
+theorem atBot_atTop_le_cocompact [LinearOrder α] [NoMinOrder α] [NoMaxOrder α]
+    [OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α :=
+  sup_le atBot_le_cocompact atTop_le_cocompact
+
+@[simp 900]
+theorem cocompact_eq_atBot_atTop [LinearOrder α] [NoMaxOrder α] [NoMinOrder α]
+    [OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop :=
+  cocompact_le_atBot_atTop.antisymm atBot_atTop_le_cocompact
+
+@[simp]
+theorem cocompact_eq_atBot [LinearOrder α] [NoMinOrder α] [OrderTop α]
+    [ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot :=
+  cocompact_le_atBot.antisymm atBot_le_cocompact
+
+@[simp]
+theorem cocompact_eq_atTop [LinearOrder α] [NoMaxOrder α] [OrderBot α]
+    [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop :=
+  cocompact_le_atTop.antisymm atTop_le_cocompact
+
 -- porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)

chore(Topology): remove autoImplicit from most remaining files (#9865)

d1054e16

Diff

@@ -27,9 +27,6 @@ We also prove that the image of a closed interval under a continuous map is a cl
 compact, extreme value theorem
 -/
 
-set_option autoImplicit true
-
-
 open Filter OrderDual TopologicalSpace Function Set
 
 open scoped Filter Topology
@@ -56,6 +53,8 @@ class CompactIccSpace (α : Type*) [TopologicalSpace α] [Preorder α] : Prop wh
 
 export CompactIccSpace (isCompact_Icc)
 
+variable {α : Type*}
+
 -- porting note: new lemma; TODO: make it the definition
 lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
     (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where

chore: reduce imports (#9830)

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

f4c4ca61

Diff

@@ -5,6 +5,7 @@ Authors: Patrick Massot, Yury Kudryashov
 -/
 import Mathlib.Topology.Algebra.Order.IntermediateValue
 import Mathlib.Topology.LocalExtr
+import Mathlib.Topology.Support
 
 #align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"

chore(*): drop $/<| before fun (#9361)

Subset of #9319

3694e27d

Diff

@@ -58,7 +58,7 @@ export CompactIccSpace (isCompact_Icc)
 -- porting note: new lemma; TODO: make it the definition
 lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
     (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
-  isCompact_Icc {a b} := by_cases h $ fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty
+  isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty
 
 -- porting note: new lemma; TODO: drop one `'`
 lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
@@ -452,13 +452,13 @@ theorem IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → 
     {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
     (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x :=
   let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf
-  ⟨x, by_contra <| fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
+  ⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
 
 theorem IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β}
     {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
     (hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x :=
   let ⟨x, hxt, hfx⟩ := ht.exists_isMaxOn ⟨z, hz⟩ hf
-  ⟨x, by_contra <| fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
+  ⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
 
 @[deprecated IsCompact.exists_isMinOn_mem_subset]
 theorem IsCompact.exists_isLocalMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}

chore: remove uses of cases' (#9171)

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

3fca282c

Diff

@@ -70,7 +70,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
     CompactIccSpace α := by
   refine' .mk'' fun {a b} hlt => ?_
-  cases' le_or_lt a b with hab hab
+  rcases le_or_lt a b with hab | hab
   swap
   · simp [hab]
   refine' isCompact_iff_ultrafilter_le_nhds.2 fun f hf => _

feat: generalize some lemmas to directed types (#7852)

New lemmas / instances

An archimedean ordered semiring is directed upwards.
Filter.hasAntitoneBasis_atTop;
Filter.HasAntitoneBasis.iInf_principal;

Fix typos

Docstrings: "if the agree" -> "if they agree".
ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self -> ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self.

Weaken typeclass assumptions

From a semilattice to a directed type

MeasureTheory.tendsto_measure_iUnion;
MeasureTheory.tendsto_measure_iInter;
Monotone.directed_le, Monotone.directed_ge;
Antitone.directed_le, Antitone.directed_ge;
directed_of_sup, renamed to directed_of_isDirected_le;
directed_of_inf, renamed to directed_of_isDirected_ge;

From a strict ordered semiring to an ordered semiring

tendsto_nat_cast_atTop_atTop;
Filter.Eventually.nat_cast_atTop;
atTop_hasAntitoneBasis_of_archimedean;

65b7d164

Diff

@@ -174,7 +174,7 @@ theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCom
   by_contra H
   rw [not_nonempty_iff_eq_empty] at H
   rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H
-      (directed_of_inf fun _ _ h => Iic_subset_Iic.mpr h) with ⟨x, hx⟩
+      (Monotone.directed_ge fun _ _ h => Iic_subset_Iic.mpr h) with ⟨x, hx⟩
   exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast

chore: missing spaces after rcases, convert and congrm (#7725)

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

c5594244

Diff

@@ -104,7 +104,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   rcases hc.2.eq_or_lt with (rfl | hlt); · exact hcs.2
   contrapose! hf
   intro U hU
-  rcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
+  rcases (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
       (mem_nhdsWithin_of_mem_nhds hU) with
     ⟨y, hxy, hyU⟩
   refine' mem_of_superset _ hyU; clear! U

chore(Topology/../Order): weaken TC assumptions (#7553)

Generalize some lemmas from ConditionallyCompleteLinearOrder to LinearOrder. Also drop an unused variable.

8ec1ec7d

Diff

@@ -381,10 +381,10 @@ end ConditionallyCompleteLinearOrder
 ### Min and max elements of a compact set
 -/
 
-section OrderClosedTopology
+section InfSup
 
-variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
-  [TopologicalSpace β] [TopologicalSpace γ]
+variable {α β : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [TopologicalSpace β]
 
 theorem IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
     (ne_s : s.Nonempty) : sInf s ∈ s :=
@@ -442,6 +442,12 @@ theorem IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs :
   IsCompact.exists_sInf_image_eq (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
 
+end InfSup
+
+section ExistsExtr
+
+variable {α β : Type*} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β]
+
 theorem IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
     {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
     (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x :=
@@ -477,7 +483,7 @@ theorem IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β →
   let ⟨x, hxs, h⟩ := ht.exists_isMaxOn_mem_subset hf hz hfz
   ⟨x, hxs, h.isLocalMax <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩
 
-end OrderClosedTopology
+end ExistsExtr
 
 variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]

feat: weaken assumptions for IsCompact.existsIsLeast and all of its variations (#6345)

As discussed a while ago on Zulip, we introduce classes expressing that a certain topology has closed Icis/Iics, which is sufficient to get boundedness of compacts on the desired side. The main application is that these now apply to types satisfying UpperTopology/LowerTopology, which will allow us to apply these compactness results to semicontinuous functions.

The naming was discussed here

b5ece309

Diff

@@ -162,11 +162,11 @@ end
 
 section LinearOrder
 
-variable {α β γ : Type*} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α]
+variable {α β γ : Type*} [LinearOrder α] [TopologicalSpace α]
   [TopologicalSpace β] [TopologicalSpace γ]
 
-theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x, IsLeast s x := by
+theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : ∃ x, IsLeast s x := by
   haveI : Nonempty s := ne_s.to_subtype
   suffices : (s ∩ ⋂ x ∈ s, Iic x).Nonempty
   · exact ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
@@ -178,53 +178,53 @@ theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonem
   exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast
 
-theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x, IsGreatest s x :=
+theorem IsCompact.exists_isGreatest [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : ∃ x, IsGreatest s x :=
   IsCompact.exists_isLeast (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 
-theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x ∈ s, IsGLB s x :=
+theorem IsCompact.exists_isGLB [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : ∃ x ∈ s, IsGLB s x :=
   (hs.exists_isLeast ne_s).imp (fun x (hx : IsLeast s x) => ⟨hx.1, hx.isGLB⟩)
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 
-theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x ∈ s, IsLUB s x :=
+theorem IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : ∃ x ∈ s, IsLUB s x :=
   IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
 -- porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
-theorem IsCompact.exists_isMinOn {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
-    (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
+theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
   rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
   exact ⟨x, hxs, ball_image_iff.1 hx⟩
 
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 @[deprecated IsCompact.exists_isMinOn]
-theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
-    (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
+theorem IsCompact.exists_forall_le [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
   hs.exists_isMinOn ne_s hf
 #align is_compact.exists_forall_le IsCompact.exists_forall_le
 
 -- porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
-theorem IsCompact.exists_isMaxOn : ∀ {s : Set β}, IsCompact s → s.Nonempty → ∀ {f : β → α},
-    ContinuousOn f s → ∃ x ∈ s, IsMaxOn f s x :=
-  IsCompact.exists_isMinOn (α := αᵒᵈ)
+theorem IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x :=
+  IsCompact.exists_isMinOn (α := αᵒᵈ) hs ne_s hf
 
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 @[deprecated IsCompact.exists_isMaxOn]
-theorem IsCompact.exists_forall_ge : ∀ {s : Set β}, IsCompact s → s.Nonempty → ∀ {f : β → α},
-    ContinuousOn f s → ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
-  IsCompact.exists_isMaxOn
+theorem IsCompact.exists_forall_ge [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
+  IsCompact.exists_isMaxOn hs ne_s hf
 #align is_compact.exists_forall_ge IsCompact.exists_forall_ge
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/
-theorem ContinuousOn.exists_isMinOn' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
-    (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) :
-    ∃ x ∈ s, IsMinOn f s x := by
+theorem ContinuousOn.exists_isMinOn' [ClosedIicTopology α] {s : Set β} {f : β → α}
+    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
+    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, IsMinOn f s x := by
   rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩
   have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset_iff.2 ⟨h₀, inter_subset_right _ _⟩
   obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
@@ -236,32 +236,32 @@ theorem ContinuousOn.exists_isMinOn' {s : Set β} {f : β → α} (hf : Continuo
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/
 @[deprecated ContinuousOn.exists_isMinOn']
-theorem ContinuousOn.exists_forall_le' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
-    (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) :
-    ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
+theorem ContinuousOn.exists_forall_le' [ClosedIicTopology α] {s : Set β} {f : β → α}
+    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
+    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y :=
   hf.exists_isMinOn' hsc h₀ hc
 #align continuous_on.exists_forall_le' ContinuousOn.exists_forall_le'
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
-theorem ContinuousOn.exists_isMaxOn' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
-    (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) :
-    ∃ x ∈ s, IsMaxOn f s x :=
+theorem ContinuousOn.exists_isMaxOn' [ClosedIciTopology α] {s : Set β} {f : β → α}
+    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
+    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, IsMaxOn f s x :=
   ContinuousOn.exists_isMinOn' (α := αᵒᵈ) hf hsc h₀ hc
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
 @[deprecated ContinuousOn.exists_isMaxOn']
-theorem ContinuousOn.exists_forall_ge' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
-    (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) :
-    ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
+theorem ContinuousOn.exists_forall_ge' [ClosedIciTopology α] {s : Set β} {f : β → α}
+    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
+    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x :=
   hf.exists_isMaxOn' hsc h₀ hc
 #align continuous_on.exists_forall_ge' ContinuousOn.exists_forall_ge'
 
 /-- The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
 away from compact sets, then it has a global minimum. -/
-theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ : β)
-    (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y :=
+theorem Continuous.exists_forall_le' [ClosedIicTopology α] {f : β → α} (hf : Continuous f)
+    (x₀ : β) (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y :=
   let ⟨x, _, hx⟩ := hf.continuousOn.exists_forall_le' isClosed_univ (mem_univ x₀)
     (by rwa [principal_univ, inf_top_eq])
   ⟨x, fun y => hx y (mem_univ y)⟩
@@ -269,30 +269,31 @@ theorem Continuous.exists_forall_le' {f : β → α} (hf : Continuous f) (x₀ :
 
 /-- The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
 away from compact sets, then it has a global maximum. -/
-theorem Continuous.exists_forall_ge' {f : β → α} (hf : Continuous f) (x₀ : β)
-    (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x :=
-  @Continuous.exists_forall_le' αᵒᵈ _ _ _ _ _ _ hf x₀ h
+theorem Continuous.exists_forall_ge' [ClosedIciTopology α] {f : β → α} (hf : Continuous f)
+    (x₀ : β) (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x :=
+  Continuous.exists_forall_le' (α := αᵒᵈ) hf x₀ h
 #align continuous.exists_forall_ge' Continuous.exists_forall_ge'
 
 /-- The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
 sets, then it has a global minimum. -/
-theorem Continuous.exists_forall_le [Nonempty β] {f : β → α} (hf : Continuous f)
-    (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by
+theorem Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α}
+    (hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by
   inhabit β
   exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _)
 #align continuous.exists_forall_le Continuous.exists_forall_le
 
 /-- The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
 compact sets, then it has a global maximum. -/
-theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuous f)
-    (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x :=
-  @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
+theorem Continuous.exists_forall_ge [ClosedIciTopology α] [Nonempty β] {f : β → α}
+    (hf : Continuous f) (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x :=
+  Continuous.exists_forall_le (α := αᵒᵈ) hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
-theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
-    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y := by
+theorem Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β]
+    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
+    ∃ x : β, ∀ y : β, f x ≤ f y := by
   obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.isCompact_range hf).exists_isLeast (range_nonempty _)
   rw [mem_lowerBounds, forall_range_iff] at hx
   exact ⟨x, hx⟩
@@ -301,14 +302,16 @@ theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One 
 
 /-- A continuous function with compact support has a global maximum. -/
 @[to_additive "A continuous function with compact support has a global maximum."]
-theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
-    (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f y ≤ f x :=
-  @Continuous.exists_forall_le_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ _ hf h
+theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β]
+    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
+    ∃ x : β, ∀ y : β, f y ≤ f x :=
+  Continuous.exists_forall_le_of_hasCompactMulSupport (α := αᵒᵈ) hf h
 #align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
 
 /-- A compact set is bounded below -/
-theorem IsCompact.bddBelow [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBelow s := by
+theorem IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
+    BddBelow s := by
   rcases s.eq_empty_or_nonempty with rfl | hne
   · exact bddBelow_empty
   · obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
@@ -316,35 +319,36 @@ theorem IsCompact.bddBelow [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBe
 #align is_compact.bdd_below IsCompact.bddBelow
 
 /-- A compact set is bounded above -/
-theorem IsCompact.bddAbove [Nonempty α] {s : Set α} (hs : IsCompact s) : BddAbove s :=
-  @IsCompact.bddBelow αᵒᵈ _ _ _ _ _ hs
+theorem IsCompact.bddAbove [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
+    BddAbove s :=
+  IsCompact.bddBelow (α := αᵒᵈ) hs
 #align is_compact.bdd_above IsCompact.bddAbove
 
 /-- A continuous function is bounded below on a compact set. -/
-theorem IsCompact.bddBelow_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
-    (hf : ContinuousOn f K) : BddBelow (f '' K) :=
+theorem IsCompact.bddBelow_image [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β}
+    (hK : IsCompact K) (hf : ContinuousOn f K) : BddBelow (f '' K) :=
   (hK.image_of_continuousOn hf).bddBelow
 #align is_compact.bdd_below_image IsCompact.bddBelow_image
 
 /-- A continuous function is bounded above on a compact set. -/
-theorem IsCompact.bddAbove_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
-    (hf : ContinuousOn f K) : BddAbove (f '' K) :=
-  @IsCompact.bddBelow_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
+theorem IsCompact.bddAbove_image [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β}
+    (hK : IsCompact K) (hf : ContinuousOn f K) : BddAbove (f '' K) :=
+  IsCompact.bddBelow_image (α := αᵒᵈ) hK hf
 #align is_compact.bdd_above_image IsCompact.bddAbove_image
 
 /-- A continuous function with compact support is bounded below. -/
 @[to_additive " A continuous function with compact support is bounded below. "]
-theorem Continuous.bddBelow_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
-    (h : HasCompactMulSupport f) : BddBelow (range f) :=
+theorem Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α]
+    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) :=
   (h.isCompact_range hf).bddBelow
 #align continuous.bdd_below_range_of_has_compact_mul_support Continuous.bddBelow_range_of_hasCompactMulSupport
 #align continuous.bdd_below_range_of_has_compact_support Continuous.bddBelow_range_of_hasCompactSupport
 
 /-- A continuous function with compact support is bounded above. -/
 @[to_additive " A continuous function with compact support is bounded above. "]
-theorem Continuous.bddAbove_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
-    (h : HasCompactMulSupport f) : BddAbove (range f) :=
-  @Continuous.bddBelow_range_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ hf h
+theorem Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α]
+    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) :=
+  Continuous.bddBelow_range_of_hasCompactMulSupport (α := αᵒᵈ) hf h
 #align continuous.bdd_above_range_of_has_compact_mul_support Continuous.bddAbove_range_of_hasCompactMulSupport
 #align continuous.bdd_above_range_of_has_compact_support Continuous.bddAbove_range_of_hasCompactSupport
 
@@ -353,10 +357,11 @@ end LinearOrder
 section ConditionallyCompleteLinearOrder
 
 variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
-  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+  [TopologicalSpace β] [TopologicalSpace γ]
 
-theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
-    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y := by
+theorem IsCompact.sSup_lt_iff_of_continuous [ClosedIciTopology α] {f : β → α} {K : Set β}
+    (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
+    sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y := by
   refine' ⟨fun h x hx => (le_csSup (hK.bddAbove_image hf) <| mem_image_of_mem f hx).trans_lt h,
     fun h => _⟩
   obtain ⟨x, hx, h2x⟩ := hK.exists_forall_ge h0K hf
@@ -364,11 +369,10 @@ theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : I
   rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
 #align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
 
-theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type*} [ConditionallyCompleteLinearOrder α]
-    [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
+theorem IsCompact.lt_sInf_iff_of_continuous [ClosedIicTopology α] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
     y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
-  @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
+  IsCompact.sSup_lt_iff_of_continuous (α := αᵒᵈ) hK h0K hf y
 #align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
 end ConditionallyCompleteLinearOrder
@@ -380,61 +384,99 @@ end ConditionallyCompleteLinearOrder
 section OrderClosedTopology
 
 variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
-  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+  [TopologicalSpace β] [TopologicalSpace γ]
 
-theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
+theorem IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : sInf s ∈ s :=
   let ⟨_a, ha⟩ := hs.exists_isLeast ne_s
   ha.csInf_mem
 #align is_compact.Inf_mem IsCompact.sInf_mem
 
-theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
-  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
+theorem IsCompact.sSup_mem [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : sSup s ∈ s :=
+  IsCompact.sInf_mem (α := αᵒᵈ) hs ne_s
 #align is_compact.Sup_mem IsCompact.sSup_mem
 
-theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGLB s (sInf s) :=
+theorem IsCompact.isGLB_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : IsGLB s (sInf s) :=
   isGLB_csInf ne_s hs.bddBelow
 #align is_compact.is_glb_Inf IsCompact.isGLB_sInf
 
-theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLUB s (sSup s) :=
-  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
+theorem IsCompact.isLUB_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : IsLUB s (sSup s) :=
+  IsCompact.isGLB_sInf (α := αᵒᵈ) hs ne_s
 #align is_compact.is_lub_Sup IsCompact.isLUB_sSup
 
-theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLeast s (sInf s) :=
+theorem IsCompact.isLeast_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : IsLeast s (sInf s) :=
   ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
 #align is_compact.is_least_Inf IsCompact.isLeast_sInf
 
-theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGreatest s (sSup s) :=
-  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
+theorem IsCompact.isGreatest_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : IsGreatest s (sSup s) :=
+  IsCompact.isLeast_sInf (α := αᵒᵈ) hs ne_s
 #align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
 
-theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+theorem IsCompact.exists_sInf_image_eq_and_le [ClosedIicTopology α] {s : Set β}
+    (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
+    ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
   let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
   ⟨x, hxs, hx.symm, fun _y hy =>
     hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩
 #align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
 
-theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+theorem IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology α] {s : Set β}
+    (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
+    ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+  IsCompact.exists_sInf_image_eq_and_le (α := αᵒᵈ) hs ne_s hf
 #align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
 
-theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
+theorem IsCompact.exists_sInf_image_eq [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
   let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
   ⟨x, hxs, hx⟩
 #align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
 
-theorem IsCompact.exists_sSup_image_eq :
-    ∀ {s : Set β},
-      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
-  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
+theorem IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
+    (ne_s : s.Nonempty) : ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
+  IsCompact.exists_sInf_image_eq (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
 
+theorem IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
+    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
+    (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x :=
+  let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf
+  ⟨x, by_contra <| fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
+
+theorem IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β}
+    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
+    (hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x :=
+  let ⟨x, hxt, hfx⟩ := ht.exists_isMaxOn ⟨z, hz⟩ hf
+  ⟨x, by_contra <| fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
+
+@[deprecated IsCompact.exists_isMinOn_mem_subset]
+theorem IsCompact.exists_isLocalMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
+    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
+    (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsLocalMinOn f t x :=
+  let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
+  ⟨x, hxs, h.localize⟩
+#align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
+
+-- porting note: rfc: assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and
+-- `IsOpen s`?
+theorem IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β}
+    {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
+    (hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
+  let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
+  ⟨x, hxs, h.isLocalMin <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩
+#align is_compact.exists_local_min_mem_open IsCompact.exists_isLocalMin_mem_open
+
+theorem IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β → α} {s t : Set β}
+    {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
+    (hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s) : ∃ x ∈ s, IsLocalMax f x :=
+  let ⟨x, hxs, h⟩ := ht.exists_isMaxOn_mem_subset hf hz hfz
+  ⟨x, hxs, h.isLocalMax <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩
+
 end OrderClosedTopology
 
 variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
@@ -488,7 +530,7 @@ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCo
 
 theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
     (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
-  @IsCompact.continuous_sSup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
+  IsCompact.continuous_sSup (α := αᵒᵈ) hK hf
 #align is_compact.continuous_Inf IsCompact.continuous_sInf
 
 namespace ContinuousOn
@@ -535,28 +577,3 @@ theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
 #align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Icc
 
 end ContinuousOn
-
--- porting note: todo: add dual versions
-
-theorem IsCompact.exists_isMinOn_mem_subset {f : β → α} {s t : Set β} {z : β}
-    (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
-    ∃ x ∈ s, IsMinOn f t x :=
-  let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf
-  ⟨x, by_contra <| fun hxs => (hfz x ⟨hxt, hxs⟩).not_le (hfx hz), hfx⟩
-
-@[deprecated IsCompact.exists_isMinOn_mem_subset]
-theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β} {z : β}
-    (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
-    ∃ x ∈ s, IsLocalMinOn f t x :=
-  let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
-  ⟨x, hxs, h.localize⟩
-#align is_compact.exists_local_min_on_mem_subset IsCompact.exists_isLocalMinOn_mem_subset
-
--- porting note: rfc: assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and
--- `IsOpen s`?
-theorem IsCompact.exists_isLocalMin_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
-    (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
-    (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
-  let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
-  ⟨x, hxs, h.isLocalMin <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩
-#align is_compact.exists_local_min_mem_open IsCompact.exists_isLocalMin_mem_open

fix: disable autoImplicit globally (#6528)

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

Assuming variables are in scope, but pasting the lemma in the wrong section
Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

0c24f831

Diff

@@ -26,6 +26,8 @@ We also prove that the image of a closed interval under a continuous map is a cl
 compact, extreme value theorem
 -/
 
+set_option autoImplicit true
+
 
 open Filter OrderDual TopologicalSpace Function Set

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -46,7 +46,7 @@ We also prove some simple lemmas about spaces with this property.
 /-- This typeclass says that all closed intervals in `α` are compact. This is true for all
 conditionally complete linear orders with order topology and products (finite or infinite)
 of such spaces. -/
-class CompactIccSpace (α : Type _) [TopologicalSpace α] [Preorder α] : Prop where
+class CompactIccSpace (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
   /-- A closed interval `Set.Icc a b` is a compact set for all `a` and `b`. -/
   isCompact_Icc : ∀ {a b : α}, IsCompact (Icc a b)
 #align compact_Icc_space CompactIccSpace
@@ -64,7 +64,7 @@ lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
   .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h
 
 /-- A closed interval in a conditionally complete linear order is compact. -/
-instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (α : Type _)
+instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (α : Type*)
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
     CompactIccSpace α := by
   refine' .mk'' fun {a b} hlt => ?_
@@ -114,21 +114,21 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   · exact ((hsc.1 ⟨hy, hay⟩).not_lt hxy.1).elim
 #align conditionally_complete_linear_order.to_compact_Icc_space ConditionallyCompleteLinearOrder.toCompactIccSpace
 
-instance {ι : Type _} {α : ι → Type _} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
+instance {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
     [∀ i, CompactIccSpace (α i)] : CompactIccSpace (∀ i, α i) :=
   ⟨fun {a b} => (pi_univ_Icc a b ▸ isCompact_univ_pi) fun _ => isCompact_Icc⟩
 
-instance Pi.compact_Icc_space' {α β : Type _} [Preorder β] [TopologicalSpace β]
+instance Pi.compact_Icc_space' {α β : Type*} [Preorder β] [TopologicalSpace β]
     [CompactIccSpace β] : CompactIccSpace (α → β) :=
   inferInstance
 #align pi.compact_Icc_space' Pi.compact_Icc_space'
 
-instance {α β : Type _} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] [Preorder β]
+instance {α β : Type*} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] [Preorder β]
     [TopologicalSpace β] [CompactIccSpace β] : CompactIccSpace (α × β) :=
   ⟨fun {a b} => (Icc_prod_eq a b).symm ▸ isCompact_Icc.prod isCompact_Icc⟩
 
 /-- An unordered closed interval is compact. -/
-theorem isCompact_uIcc {α : Type _} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α]
+theorem isCompact_uIcc {α : Type*} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α]
     {a b : α} : IsCompact (uIcc a b) :=
   isCompact_Icc
 #align is_compact_uIcc isCompact_uIcc
@@ -139,14 +139,14 @@ theorem isCompact_uIcc {α : Type _} [LinearOrder α] [TopologicalSpace α] [Com
 We do not register an instance for a `[CompactIccSpace α]` because this would only add instances
 for products (indexed or not) of complete linear orders, and we have instances with higher priority
 that cover these cases. -/
-instance (priority := 100) compactSpace_of_completeLinearOrder {α : Type _} [CompleteLinearOrder α]
+instance (priority := 100) compactSpace_of_completeLinearOrder {α : Type*} [CompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] : CompactSpace α :=
   ⟨by simp only [← Icc_bot_top, isCompact_Icc]⟩
 #align compact_space_of_complete_linear_order compactSpace_of_completeLinearOrder
 
 section
 
-variable {α : Type _} [Preorder α] [TopologicalSpace α] [CompactIccSpace α]
+variable {α : Type*} [Preorder α] [TopologicalSpace α] [CompactIccSpace α]
 
 instance compactSpace_Icc (a b : α) : CompactSpace (Icc a b) :=
   isCompact_iff_compactSpace.mp isCompact_Icc
@@ -160,7 +160,7 @@ end
 
 section LinearOrder
 
-variable {α β γ : Type _} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α]
+variable {α β γ : Type*} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α]
   [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
@@ -350,7 +350,7 @@ end LinearOrder
 
 section ConditionallyCompleteLinearOrder
 
-variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
@@ -362,7 +362,7 @@ theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : I
   rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
 #align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
 
-theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
+theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type*} [ConditionallyCompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
     y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
@@ -377,7 +377,7 @@ end ConditionallyCompleteLinearOrder
 
 section OrderClosedTopology
 
-variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
@@ -435,7 +435,7 @@ theorem IsCompact.exists_sSup_image_eq :
 
 end OrderClosedTopology
 
-variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+variable {α β γ : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.algebra.order.compact
-! leanprover-community/mathlib commit 3efd324a3a31eaa40c9d5bfc669c4fafee5f9423
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Algebra.Order.IntermediateValue
 import Mathlib.Topology.LocalExtr
 
+#align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"
+
 /-!
 # Compactness of a closed interval

feat(Data.Set.Basic/Data.Finset.Basic): rename insert_subset (#5450)

Currently, (for both Set and Finset) insert_subset is an iff lemma stating that insert a s ⊆ t if and only if a ∈ t and s ⊆ t. For both types, this PR renames this lemma to insert_subset_iff, and adds an insert_subset lemma that gives the implication just in the reverse direction : namely theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t .

This both aligns the naming with union_subset and union_subset_iff, and removes the need for the awkward insert_subset.mpr ⟨_,_⟩ idiom. It touches a lot of files (too many to list), but in a trivial way.

d8b62864

Diff

@@ -227,7 +227,7 @@ theorem ContinuousOn.exists_isMinOn' {s : Set β} {f : β → α} (hf : Continuo
     (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) :
     ∃ x ∈ s, IsMinOn f s x := by
   rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩
-  have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset.2 ⟨h₀, inter_subset_right _ _⟩
+  have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset_iff.2 ⟨h₀, inter_subset_right _ _⟩
   obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
     ((hK.inter_right hsc).insert x₀).exists_forall_le (insert_nonempty _ _) (hf.mono hsub)
   refine' ⟨x, hsub hx, fun y hy => _⟩

remove conditional completeness assumption in IsCompact.exists_isMinOn (#5388)

Also rename 2 lemmas

IsCompact.exists_local_min_mem_open -> IsCompact.exists_isLocalMin_mem_open;
Metric.exists_local_min_mem_ball -> Metric.exists_isLocal_min_mem_ball.

fcb2f0f3

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.algebra.order.compact
-! leanprover-community/mathlib commit 4c19a16e4b705bf135cf9a80ac18fcc99c438514
+! leanprover-community/mathlib commit 3efd324a3a31eaa40c9d5bfc669c4fafee5f9423
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -32,7 +32,7 @@ compact, extreme value theorem
 
 open Filter OrderDual TopologicalSpace Function Set
 
-open Filter Topology
+open scoped Filter Topology
 
 /-!
 ### Compactness of a closed interval
@@ -92,29 +92,23 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
   specialize hf c hc
   have hcs : c ∈ s := by
-    cases' hc.1.eq_or_lt with heq hlt
-    · rwa [← heq]
+    rcases hc.1.eq_or_lt with (rfl | hlt); · assumption
     refine' ⟨hc, fun hcf => hf fun U hU => _⟩
-    rcases(mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with
-      ⟨x, hxc, hxU⟩
-    rcases((hsc.frequently_mem ⟨a, ha⟩).and_eventually
-          (Ioc_mem_nhdsWithin_Iic ⟨hxc, le_rfl⟩)).exists with
-      ⟨y, ⟨_hyab, hyf⟩, hy⟩
+    rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU)
+      with ⟨x, hxc, hxU⟩
+    rcases ((hsc.frequently_mem ⟨a, ha⟩).and_eventually
+      (Ioc_mem_nhdsWithin_Iic ⟨hxc, le_rfl⟩)).exists with ⟨y, ⟨_hyab, hyf⟩, hy⟩
     refine' mem_of_superset (f.diff_mem_iff.2 ⟨hcf, hyf⟩) (Subset.trans _ hxU)
     rw [diff_subset_iff]
-    exact
-      Subset.trans Icc_subset_Icc_union_Ioc
-        (union_subset_union Subset.rfl <| Ioc_subset_Ioc_left hy.1.le)
-  cases' hc.2.eq_or_lt with heq hlt
-  · rw [← heq]
-    exact hcs.2
+    exact Subset.trans Icc_subset_Icc_union_Ioc <| union_subset_union Subset.rfl <|
+      Ioc_subset_Ioc_left hy.1.le
+  rcases hc.2.eq_or_lt with (rfl | hlt); · exact hcs.2
   contrapose! hf
   intro U hU
   rcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
       (mem_nhdsWithin_of_mem_nhds hU) with
     ⟨y, hxy, hyU⟩
-  refine' mem_of_superset _ hyU
-  clear! U
+  refine' mem_of_superset _ hyU; clear! U
   have hy : y ∈ Icc a b := ⟨hc.1.trans hxy.1.le, hxy.2⟩
   by_cases hay : Icc a y ∈ f
   · refine' mem_of_superset (f.diff_mem_iff.2 ⟨f.diff_mem_iff.2 ⟨hay, hcs.2⟩, hpt y hy⟩) _
@@ -164,100 +158,48 @@ instance compactSpace_Icc (a b : α) : CompactSpace (Icc a b) :=
 end
 
 /-!
-### Min and max elements of a compact set
+### Extreme value theorem
 -/
 
+section LinearOrder
 
-variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
-  [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
-
-theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
-  hs.isClosed.csInf_mem ne_s hs.bddBelow
-#align is_compact.Inf_mem IsCompact.sInf_mem
-
-theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
-  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.Sup_mem IsCompact.sSup_mem
-
-theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGLB s (sInf s) :=
-  isGLB_csInf ne_s hs.bddBelow
-#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
-
-theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLUB s (sSup s) :=
-  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
-
-theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLeast s (sInf s) :=
-  ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
-#align is_compact.is_least_Inf IsCompact.isLeast_sInf
-
-theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGreatest s (sSup s) :=
-  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
+variable {α β γ : Type _} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α]
+  [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    ∃ x, IsLeast s x :=
-  ⟨_, hs.isLeast_sInf ne_s⟩
+    ∃ x, IsLeast s x := by
+  haveI : Nonempty s := ne_s.to_subtype
+  suffices : (s ∩ ⋂ x ∈ s, Iic x).Nonempty
+  · exact ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
+  rw [biInter_eq_iInter]
+  by_contra H
+  rw [not_nonempty_iff_eq_empty] at H
+  rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H
+      (directed_of_inf fun _ _ h => Iic_subset_Iic.mpr h) with ⟨x, hx⟩
+  exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast
 
 theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsGreatest s x :=
-  ⟨_, hs.isGreatest_sSup ne_s⟩
+  IsCompact.exists_isLeast (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
-  ⟨_, hs.sInf_mem ne_s, hs.isGLB_sInf ne_s⟩
+  (hs.exists_isLeast ne_s).imp (fun x (hx : IsLeast s x) => ⟨hx.1, hx.isGLB⟩)
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
-  ⟨_, hs.sSup_mem ne_s, hs.isLUB_sSup ne_s⟩
+  IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
-theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
-  ⟨x, hxs, hx.symm, fun _y hy =>
-    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩
-#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
-
-theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
-
-theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
-  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
-  ⟨x, hxs, hx⟩
-#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
-
-theorem IsCompact.exists_sSup_image_eq :
-    ∀ {s : Set β},
-      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
-  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
-#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
-
-theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
-    s = Icc (sInf s) (sSup s) :=
-  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed
-#align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
-
-/-!
-### Extreme value theorem
--/
-
 -- porting note: new lemma; defeq to the old one but allows us to use dot notation
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_isMinOn {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
     (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
-  rcases hs.exists_sInf_image_eq_and_le ne_s hf with ⟨x, hxs, -, hx⟩
-  exact ⟨x, hxs, hx⟩
+  rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
+  exact ⟨x, hxs, ball_image_iff.1 hx⟩
 
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 @[deprecated IsCompact.exists_isMinOn]
@@ -270,7 +212,7 @@ theorem IsCompact.exists_forall_le {s : Set β} (hs : IsCompact s) (ne_s : s.Non
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 theorem IsCompact.exists_isMaxOn : ∀ {s : Set β}, IsCompact s → s.Nonempty → ∀ {f : β → α},
     ContinuousOn f s → ∃ x ∈ s, IsMaxOn f s x :=
-  @IsCompact.exists_isMinOn αᵒᵈ _ _ _ _ _
+  IsCompact.exists_isMinOn (α := αᵒᵈ)
 
 /-- The **extreme value theorem**: a continuous function realizes its maximum on a compact set. -/
 @[deprecated IsCompact.exists_isMaxOn]
@@ -306,7 +248,7 @@ smaller than a value in its image away from compact sets, then it has a maximum
 theorem ContinuousOn.exists_isMaxOn' {s : Set β} {f : β → α} (hf : ContinuousOn f s)
     (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) :
     ∃ x ∈ s, IsMaxOn f s x :=
-  @ContinuousOn.exists_isMinOn' αᵒᵈ _ _ _ _ _ _ _ hf hsc _ h₀ hc
+  ContinuousOn.exists_isMinOn' (α := αᵒᵈ) hf hsc h₀ hc
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 smaller than a value in its image away from compact sets, then it has a maximum on this set. -/
@@ -348,18 +290,6 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
   @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 
-theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
-    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
-  ((hK.image_of_continuousOn hf).isGreatest_sSup (h0K.image _)).lt_iff.trans ball_image_iff
-#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
-
-theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
-    [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
-    (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
-    y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
-  @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
-#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
-
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
 theorem Continuous.exists_forall_le_of_hasCompactMulSupport [Nonempty β] [One α] {f : β → α}
@@ -378,6 +308,144 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
 #align continuous.exists_forall_ge_of_has_compact_mul_support Continuous.exists_forall_ge_of_hasCompactMulSupport
 #align continuous.exists_forall_ge_of_has_compact_support Continuous.exists_forall_ge_of_hasCompactSupport
 
+/-- A compact set is bounded below -/
+theorem IsCompact.bddBelow [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBelow s := by
+  rcases s.eq_empty_or_nonempty with rfl | hne
+  · exact bddBelow_empty
+  · obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
+    exact ⟨a, has⟩
+#align is_compact.bdd_below IsCompact.bddBelow
+
+/-- A compact set is bounded above -/
+theorem IsCompact.bddAbove [Nonempty α] {s : Set α} (hs : IsCompact s) : BddAbove s :=
+  @IsCompact.bddBelow αᵒᵈ _ _ _ _ _ hs
+#align is_compact.bdd_above IsCompact.bddAbove
+
+/-- A continuous function is bounded below on a compact set. -/
+theorem IsCompact.bddBelow_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
+    (hf : ContinuousOn f K) : BddBelow (f '' K) :=
+  (hK.image_of_continuousOn hf).bddBelow
+#align is_compact.bdd_below_image IsCompact.bddBelow_image
+
+/-- A continuous function is bounded above on a compact set. -/
+theorem IsCompact.bddAbove_image [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K)
+    (hf : ContinuousOn f K) : BddAbove (f '' K) :=
+  @IsCompact.bddBelow_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
+#align is_compact.bdd_above_image IsCompact.bddAbove_image
+
+/-- A continuous function with compact support is bounded below. -/
+@[to_additive " A continuous function with compact support is bounded below. "]
+theorem Continuous.bddBelow_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
+    (h : HasCompactMulSupport f) : BddBelow (range f) :=
+  (h.isCompact_range hf).bddBelow
+#align continuous.bdd_below_range_of_has_compact_mul_support Continuous.bddBelow_range_of_hasCompactMulSupport
+#align continuous.bdd_below_range_of_has_compact_support Continuous.bddBelow_range_of_hasCompactSupport
+
+/-- A continuous function with compact support is bounded above. -/
+@[to_additive " A continuous function with compact support is bounded above. "]
+theorem Continuous.bddAbove_range_of_hasCompactMulSupport [One α] {f : β → α} (hf : Continuous f)
+    (h : HasCompactMulSupport f) : BddAbove (range f) :=
+  @Continuous.bddBelow_range_of_hasCompactMulSupport αᵒᵈ _ _ _ _ _ _ _ hf h
+#align continuous.bdd_above_range_of_has_compact_mul_support Continuous.bddAbove_range_of_hasCompactMulSupport
+#align continuous.bdd_above_range_of_has_compact_support Continuous.bddAbove_range_of_hasCompactSupport
+
+end LinearOrder
+
+section ConditionallyCompleteLinearOrder
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
+theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
+    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y := by
+  refine' ⟨fun h x hx => (le_csSup (hK.bddAbove_image hf) <| mem_image_of_mem f hx).trans_lt h,
+    fun h => _⟩
+  obtain ⟨x, hx, h2x⟩ := hK.exists_forall_ge h0K hf
+  refine' (csSup_le (h0K.image f) _).trans_lt (h x hx)
+  rintro _ ⟨x', hx', rfl⟩; exact h2x x' hx'
+#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
+
+theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
+    [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
+    (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
+    y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
+  @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
+#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
+
+end ConditionallyCompleteLinearOrder
+
+/-!
+### Min and max elements of a compact set
+-/
+
+section OrderClosedTopology
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderClosedTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
+theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
+  let ⟨_a, ha⟩ := hs.exists_isLeast ne_s
+  ha.csInf_mem
+#align is_compact.Inf_mem IsCompact.sInf_mem
+
+theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
+  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.Sup_mem IsCompact.sSup_mem
+
+theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGLB s (sInf s) :=
+  isGLB_csInf ne_s hs.bddBelow
+#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
+
+theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLUB s (sSup s) :=
+  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
+
+theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLeast s (sInf s) :=
+  ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
+#align is_compact.is_least_Inf IsCompact.isLeast_sInf
+
+theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGreatest s (sSup s) :=
+  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
+
+theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
+  ⟨x, hxs, hx.symm, fun _y hy =>
+    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩
+#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
+
+theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
+
+theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
+  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
+  ⟨x, hxs, hx⟩
+#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
+
+theorem IsCompact.exists_sSup_image_eq :
+    ∀ {s : Set β},
+      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
+  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
+#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
+
+end OrderClosedTopology
+
+variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
+  [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
+
+theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
+    s = Icc (sInf s) (sSup s) :=
+  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed
+#align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
+
 /- If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a
 conditionally complete linear order, and `K : Set β` is a compact set, then
 `fun x ↦ sSup (f x '' K)` is a continuous function.
@@ -433,7 +501,7 @@ namespace ContinuousOn
 variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β}
   {a b c : α}
 
-open Interval
+open scoped Interval
 
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
     f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
@@ -487,9 +555,9 @@ theorem IsCompact.exists_isLocalMinOn_mem_subset {f : β → α} {s t : Set β}
 
 -- porting note: rfc: assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and
 -- `IsOpen s`?
-theorem IsCompact.exists_local_min_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
+theorem IsCompact.exists_isLocalMin_mem_open {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t)
     (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
     (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
   let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
   ⟨x, hxs, h.isLocalMin <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩
-#align is_compact.exists_local_min_mem_open IsCompact.exists_local_min_mem_open
+#align is_compact.exists_local_min_mem_open IsCompact.exists_isLocalMin_mem_open

chore(Topology/{Paracompact + PartitionOfUnity + Algebra/Order/Compact}): restore a more faithful mathported proof, using the "newer" push_neg (#5132)

These are just three cases where a mathported proof had to be changed, due to unwanted behaviour with push_neg/contradiction. Since #5082 is supposed to fix some issues with these tactics, some of these workarounds can be removed.

Note: this was not in any way systematic, I simply grepped the line after a contra, then grepped for a not and then selected 3 likely candidates. There are potentially more situations like these.

f2ca44a9

Diff

@@ -71,41 +71,56 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
     [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] :
     CompactIccSpace α := by
   refine' .mk'' fun {a b} hlt => ?_
+  cases' le_or_lt a b with hab hab
+  swap
+  · simp [hab]
   refine' isCompact_iff_ultrafilter_le_nhds.2 fun f hf => _
-  by_contra H
-  simp only [not_exists, not_and] at H -- porting note: `contrapose!` fails
-  rw [le_principal_iff] at hf
+  contrapose! hf
+  rw [le_principal_iff]
   have hpt : ∀ x ∈ Icc a b, {x} ∉ f := fun x hx hxf =>
-    H x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x))
+    hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x))
   set s := { x ∈ Icc a b | Icc a x ∉ f }
   have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
-  have ha : a ∈ s := by simp [hpt, hlt.le]
+  have sbd : BddAbove s := ⟨b, hsb⟩
+  have ha : a ∈ s := by simp [hpt, hab]
+  rcases hab.eq_or_lt with (rfl | _hlt)
+  · exact ha.2
+  -- porting note: the `obtain` below was instead
+  -- `set c := Sup s`
+  -- `have hsc : IsLUB s c := isLUB_csSup ⟨a, ha⟩ sbd`
   obtain ⟨c, hsc⟩ : ∃ c, IsLUB s c := ⟨sSup s, isLUB_csSup ⟨a, ha⟩ ⟨b, hsb⟩⟩
   have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
-  specialize H c hc
+  specialize hf c hc
   have hcs : c ∈ s := by
-    rcases hc.1.eq_or_lt with (rfl | hlt)
-    · assumption
-    refine' ⟨hc, fun hcf => H fun U hU => _⟩
-    rcases exists_Ioc_subset_of_mem_nhds' hU hlt with ⟨x, hxc, hxU⟩
-    rcases ((hsc.frequently_mem ⟨a, ha⟩).and_eventually (Ioc_mem_nhdsWithin_Iic' hxc.2)).exists with
-      ⟨y, ⟨-, hyf⟩, hy⟩
+    cases' hc.1.eq_or_lt with heq hlt
+    · rwa [← heq]
+    refine' ⟨hc, fun hcf => hf fun U hU => _⟩
+    rcases(mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with
+      ⟨x, hxc, hxU⟩
+    rcases((hsc.frequently_mem ⟨a, ha⟩).and_eventually
+          (Ioc_mem_nhdsWithin_Iic ⟨hxc, le_rfl⟩)).exists with
+      ⟨y, ⟨_hyab, hyf⟩, hy⟩
     refine' mem_of_superset (f.diff_mem_iff.2 ⟨hcf, hyf⟩) (Subset.trans _ hxU)
     rw [diff_subset_iff]
-    exact Icc_subset_Icc_union_Ioc.trans (union_subset_union_right _ (Ioc_subset_Ioc_left hy.1.le))
+    exact
+      Subset.trans Icc_subset_Icc_union_Ioc
+        (union_subset_union Subset.rfl <| Ioc_subset_Ioc_left hy.1.le)
   cases' hc.2.eq_or_lt with heq hlt
-  · exact hcs.2 (heq.symm ▸ hf)
-  obtain ⟨y, ⟨hcy, hyb⟩, hyf⟩ : ∃ y ∈ Ioc c b, Ico c y ∉ f
-  · contrapose! H
-    intro U hU
-    rcases exists_Ico_subset_of_mem_nhds' hU hlt with ⟨y, hy, hyU⟩
-    exact mem_of_superset (H _ hy) hyU
-  suffices : y ∈ s
-  · exact hcy.not_le (hsc.1 this)
-  have hy : y ∈ Icc a b := ⟨hc.1.trans hcy.le, hyb⟩
-  refine ⟨hy, fun hay => ?_⟩
-  simp only [← Icc_union_Icc_eq_Icc hc.1 hcy.le, ← Ico_union_right hcy.le,
-    Ultrafilter.union_mem_iff, hyf, hcs.2, hpt _ hy, false_or] at hay
+  · rw [← heq]
+    exact hcs.2
+  contrapose! hf
+  intro U hU
+  rcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
+      (mem_nhdsWithin_of_mem_nhds hU) with
+    ⟨y, hxy, hyU⟩
+  refine' mem_of_superset _ hyU
+  clear! U
+  have hy : y ∈ Icc a b := ⟨hc.1.trans hxy.1.le, hxy.2⟩
+  by_cases hay : Icc a y ∈ f
+  · refine' mem_of_superset (f.diff_mem_iff.2 ⟨f.diff_mem_iff.2 ⟨hay, hcs.2⟩, hpt y hy⟩) _
+    rw [diff_subset_iff, union_comm, Ico_union_right hxy.1.le, diff_subset_iff]
+    exact Icc_subset_Icc_union_Icc
+  · exact ((hsc.1 ⟨hy, hay⟩).not_lt hxy.1).elim
 #align conditionally_complete_linear_order.to_compact_Icc_space ConditionallyCompleteLinearOrder.toCompactIccSpace
 
 instance {ι : Type _} {α : ι → Type _} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]

chore: add space after exacts (#4945)

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

bf799bb9

Diff

@@ -275,7 +275,7 @@ theorem ContinuousOn.exists_isMinOn' {s : Set β} {f : β → α} (hf : Continuo
     ((hK.inter_right hsc).insert x₀).exists_forall_le (insert_nonempty _ _) (hf.mono hsub)
   refine' ⟨x, hsub hx, fun y hy => _⟩
   by_cases hyK : y ∈ K
-  exacts[hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
+  exacts [hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)]
 
 /-- The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
 larger than a value in its image away from compact sets, then it has a minimum on this set. -/

refactor: use the typeclass SProd to implement overloaded notation · ×ˢ · (#4200)

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

6c01dc6e

Diff

@@ -370,7 +370,7 @@ conditionally complete linear order, and `K : Set β` is a compact set, then
 Porting note: todo: generalize. The following version seems to be true:
 ```
 theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
-    (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ᶠ 𝓝[K] y) (𝓝 (g y)))
+    (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y)))
     (hgc : ContinuousOn g K) :
     Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _
 ```

chore: Rename to sSup/iSup (#3938)

As discussed on Zulip

Renames

supₛ → sSup
infₛ → sInf
supᵢ → iSup
infᵢ → iInf
bsupₛ → bsSup
binfₛ → bsInf
bsupᵢ → biSup
binfᵢ → biInf
csupₛ → csSup
cinfₛ → csInf
csupᵢ → ciSup
cinfᵢ → ciInf
unionₛ → sUnion
interₛ → sInter
unionᵢ → iUnion
interᵢ → iInter
bunionₛ → bsUnion
binterₛ → bsInter
bunionᵢ → biUnion
binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

d1eb6264

Diff

@@ -80,7 +80,7 @@ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (
   set s := { x ∈ Icc a b | Icc a x ∉ f }
   have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
   have ha : a ∈ s := by simp [hpt, hlt.le]
-  obtain ⟨c, hsc⟩ : ∃ c, IsLUB s c := ⟨supₛ s, isLUB_csupₛ ⟨a, ha⟩ ⟨b, hsb⟩⟩
+  obtain ⟨c, hsc⟩ : ∃ c, IsLUB s c := ⟨sSup s, isLUB_csSup ⟨a, ha⟩ ⟨b, hsb⟩⟩
   have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
   specialize H c hc
   have hcs : c ∈ s := by
@@ -156,81 +156,81 @@ end
 variable {α β γ : Type _} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α]
   [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ]
 
-theorem IsCompact.infₛ_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : infₛ s ∈ s :=
-  hs.isClosed.cinfₛ_mem ne_s hs.bddBelow
-#align is_compact.Inf_mem IsCompact.infₛ_mem
+theorem IsCompact.sInf_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s :=
+  hs.isClosed.csInf_mem ne_s hs.bddBelow
+#align is_compact.Inf_mem IsCompact.sInf_mem
 
-theorem IsCompact.supₛ_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : supₛ s ∈ s :=
-  @IsCompact.infₛ_mem αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.Sup_mem IsCompact.supₛ_mem
+theorem IsCompact.sSup_mem {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s :=
+  @IsCompact.sInf_mem αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.Sup_mem IsCompact.sSup_mem
 
-theorem IsCompact.isGLB_infₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGLB s (infₛ s) :=
-  isGLB_cinfₛ ne_s hs.bddBelow
-#align is_compact.is_glb_Inf IsCompact.isGLB_infₛ
+theorem IsCompact.isGLB_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGLB s (sInf s) :=
+  isGLB_csInf ne_s hs.bddBelow
+#align is_compact.is_glb_Inf IsCompact.isGLB_sInf
 
-theorem IsCompact.isLUB_supₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLUB s (supₛ s) :=
-  @IsCompact.isGLB_infₛ αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_lub_Sup IsCompact.isLUB_supₛ
+theorem IsCompact.isLUB_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLUB s (sSup s) :=
+  @IsCompact.isGLB_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_lub_Sup IsCompact.isLUB_sSup
 
-theorem IsCompact.isLeast_infₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsLeast s (infₛ s) :=
-  ⟨hs.infₛ_mem ne_s, (hs.isGLB_infₛ ne_s).1⟩
-#align is_compact.is_least_Inf IsCompact.isLeast_infₛ
+theorem IsCompact.isLeast_sInf {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsLeast s (sInf s) :=
+  ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩
+#align is_compact.is_least_Inf IsCompact.isLeast_sInf
 
-theorem IsCompact.isGreatest_supₛ {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
-    IsGreatest s (supₛ s) :=
-  @IsCompact.isLeast_infₛ αᵒᵈ _ _ _ _ hs ne_s
-#align is_compact.is_greatest_Sup IsCompact.isGreatest_supₛ
+theorem IsCompact.isGreatest_sSup {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
+    IsGreatest s (sSup s) :=
+  @IsCompact.isLeast_sInf αᵒᵈ _ _ _ _ hs ne_s
+#align is_compact.is_greatest_Sup IsCompact.isGreatest_sSup
 
 theorem IsCompact.exists_isLeast {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsLeast s x :=
-  ⟨_, hs.isLeast_infₛ ne_s⟩
+  ⟨_, hs.isLeast_sInf ne_s⟩
 #align is_compact.exists_is_least IsCompact.exists_isLeast
 
 theorem IsCompact.exists_isGreatest {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x, IsGreatest s x :=
-  ⟨_, hs.isGreatest_supₛ ne_s⟩
+  ⟨_, hs.isGreatest_sSup ne_s⟩
 #align is_compact.exists_is_greatest IsCompact.exists_isGreatest
 
 theorem IsCompact.exists_isGLB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsGLB s x :=
-  ⟨_, hs.infₛ_mem ne_s, hs.isGLB_infₛ ne_s⟩
+  ⟨_, hs.sInf_mem ne_s, hs.isGLB_sInf ne_s⟩
 #align is_compact.exists_is_glb IsCompact.exists_isGLB
 
 theorem IsCompact.exists_isLUB {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) :
     ∃ x ∈ s, IsLUB s x :=
-  ⟨_, hs.supₛ_mem ne_s, hs.isLUB_supₛ ne_s⟩
+  ⟨_, hs.sSup_mem ne_s, hs.isLUB_sSup ne_s⟩
 #align is_compact.exists_is_lub IsCompact.exists_isLUB
 
-theorem IsCompact.exists_infₛ_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, infₛ (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).infₛ_mem (ne_s.image f)
+theorem IsCompact.exists_sInf_image_eq_and_le {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+  let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
   ⟨x, hxs, hx.symm, fun _y hy =>
-    hx.trans_le <| cinfₛ_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩
-#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_infₛ_image_eq_and_le
+    hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩
+#align is_compact.exists_Inf_image_eq_and_le IsCompact.exists_sInf_image_eq_and_le
 
-theorem IsCompact.exists_supₛ_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, supₛ (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-  @IsCompact.exists_infₛ_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_supₛ_image_eq_and_ge
+theorem IsCompact.exists_sSup_image_eq_and_ge {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+  @IsCompact.exists_sInf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+#align is_compact.exists_Sup_image_eq_and_ge IsCompact.exists_sSup_image_eq_and_ge
 
-theorem IsCompact.exists_infₛ_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
-    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, infₛ (f '' s) = f x :=
-  let ⟨x, hxs, hx, _⟩ := hs.exists_infₛ_image_eq_and_le ne_s hf
+theorem IsCompact.exists_sInf_image_eq {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty)
+    {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
+  let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
   ⟨x, hxs, hx⟩
-#align is_compact.exists_Inf_image_eq IsCompact.exists_infₛ_image_eq
+#align is_compact.exists_Inf_image_eq IsCompact.exists_sInf_image_eq
 
-theorem IsCompact.exists_supₛ_image_eq :
+theorem IsCompact.exists_sSup_image_eq :
     ∀ {s : Set β},
-      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, supₛ (f '' s) = f x :=
-  @IsCompact.exists_infₛ_image_eq αᵒᵈ _ _ _ _ _
-#align is_compact.exists_Sup_image_eq IsCompact.exists_supₛ_image_eq
+      IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
+  @IsCompact.exists_sInf_image_eq αᵒᵈ _ _ _ _ _
+#align is_compact.exists_Sup_image_eq IsCompact.exists_sSup_image_eq
 
 theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
-    s = Icc (infₛ s) (supₛ s) :=
-  eq_Icc_cinfₛ_csupₛ_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed
+    s = Icc (sInf s) (sSup s) :=
+  eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed
 #align eq_Icc_of_connected_compact eq_Icc_of_connected_compact
 
 /-!
@@ -241,7 +241,7 @@ theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ :
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 theorem IsCompact.exists_isMinOn {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
     (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
-  rcases hs.exists_infₛ_image_eq_and_le ne_s hf with ⟨x, hxs, -, hx⟩
+  rcases hs.exists_sInf_image_eq_and_le ne_s hf with ⟨x, hxs, -, hx⟩
   exact ⟨x, hxs, hx⟩
 
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
@@ -333,17 +333,17 @@ theorem Continuous.exists_forall_ge [Nonempty β] {f : β → α} (hf : Continuo
   @Continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 #align continuous.exists_forall_ge Continuous.exists_forall_ge
 
-theorem IsCompact.supₛ_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
-    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : supₛ (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
-  ((hK.image_of_continuousOn hf).isGreatest_supₛ (h0K.image _)).lt_iff.trans ball_image_iff
-#align is_compact.Sup_lt_iff_of_continuous IsCompact.supₛ_lt_iff_of_continuous
+theorem IsCompact.sSup_lt_iff_of_continuous {f : β → α} {K : Set β} (hK : IsCompact K)
+    (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
+  ((hK.image_of_continuousOn hf).isGreatest_sSup (h0K.image _)).lt_iff.trans ball_image_iff
+#align is_compact.Sup_lt_iff_of_continuous IsCompact.sSup_lt_iff_of_continuous
 
-theorem IsCompact.lt_infₛ_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
+theorem IsCompact.lt_sInf_iff_of_continuous {α β : Type _} [ConditionallyCompleteLinearOrder α]
     [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] {f : β → α} {K : Set β}
     (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
-    y < infₛ (f '' K) ↔ ∀ x ∈ K, y < f x :=
-  @IsCompact.supₛ_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
-#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_infₛ_iff_of_continuous
+    y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
+  @IsCompact.sSup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
+#align is_compact.lt_Inf_iff_of_continuous IsCompact.lt_sInf_iff_of_continuous
 
 /-- A continuous function with compact support has a global minimum. -/
 @[to_additive "A continuous function with compact support has a global minimum."]
@@ -365,33 +365,33 @@ theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [Nonempty β] [One 
 
 /- If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a
 conditionally complete linear order, and `K : Set β` is a compact set, then
-`fun x ↦ supₛ (f x '' K)` is a continuous function.
+`fun x ↦ sSup (f x '' K)` is a continuous function.
 
 Porting note: todo: generalize. The following version seems to be true:
 ```
-theorem IsCompact.tendsto_supₛ {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
+theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
     (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ᶠ 𝓝[K] y) (𝓝 (g y)))
     (hgc : ContinuousOn g K) :
-    Tendsto (fun x => supₛ (f x '' K)) l (𝓝 (supₛ (g '' K))) := _
+    Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _
 ```
 Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov).
 -/
-theorem IsCompact.continuous_supₛ {f : γ → β → α} {K : Set β} (hK : IsCompact K)
-    (hf : Continuous ↿f) : Continuous fun x => supₛ (f x '' K) := by
+theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
+    (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) := by
   rcases eq_empty_or_nonempty K with (rfl | h0K)
   · simp_rw [image_empty]
     exact continuous_const
   rw [continuous_iff_continuousAt]
   intro x
   obtain ⟨y, hyK, h2y, hy⟩ :=
-    hK.exists_supₛ_image_eq_and_ge h0K
+    hK.exists_sSup_image_eq_and_ge h0K
       (show Continuous fun y => f x y from hf.comp <| Continuous.Prod.mk x).continuousOn
   rw [ContinuousAt, h2y, tendsto_order]
   have := tendsto_order.mp ((show Continuous fun x => f x y
     from hf.comp <| continuous_id.prod_mk continuous_const).tendsto x)
   refine' ⟨fun z hz => _, fun z hz => _⟩
   · refine' (this.1 z hz).mono fun x' hx' =>
-      hx'.trans_le <| le_csupₛ _ <| mem_image_of_mem (f x') hyK
+      hx'.trans_le <| le_csSup _ <| mem_image_of_mem (f x') hyK
     exact hK.bddAbove_image (hf.comp <| Continuous.Prod.mk x').continuousOn
   · have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z := by
       rintro ⟨x', y'⟩ ⟨(rfl : x' = x), hy'⟩
@@ -399,15 +399,15 @@ theorem IsCompact.continuous_supₛ {f : γ → β → α} {K : Set β} (hK : Is
     obtain ⟨u, v, hu, _, hxu, hKv, huv⟩ :=
       generalized_tube_lemma isCompact_singleton hK (isOpen_Iio.preimage hf) h
     refine' eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) fun x' hx' => _
-    rw [hK.supₛ_lt_iff_of_continuous h0K
+    rw [hK.sSup_lt_iff_of_continuous h0K
         (show Continuous (f x') from hf.comp <| Continuous.Prod.mk x').continuousOn]
     exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy'))
-#align is_compact.continuous_Sup IsCompact.continuous_supₛ
+#align is_compact.continuous_Sup IsCompact.continuous_sSup
 
-theorem IsCompact.continuous_infₛ {f : γ → β → α} {K : Set β} (hK : IsCompact K)
-    (hf : Continuous ↿f) : Continuous fun x => infₛ (f x '' K) :=
-  @IsCompact.continuous_supₛ αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
-#align is_compact.continuous_Inf IsCompact.continuous_infₛ
+theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
+    (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
+  @IsCompact.continuous_sSup αᵒᵈ β γ _ _ _ _ _ _ _ hK hf
+#align is_compact.continuous_Inf IsCompact.continuous_sInf
 
 namespace ContinuousOn
 
@@ -421,36 +421,36 @@ variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopolog
 open Interval
 
 theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
-    f '' Icc a b = Icc (infₛ <| f '' Icc a b) (supₛ <| f '' Icc a b) :=
+    f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
   eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
     (isCompact_Icc.image_of_continuousOn h)
 #align continuous_on.image_Icc ContinuousOn.image_Icc
 
 theorem image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) :
-    f '' [[a, b]] = Icc (infₛ (f '' [[a, b]])) (supₛ (f '' [[a, b]])) :=
+    f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) :=
   image_Icc min_le_max h
 #align continuous_on.image_uIcc_eq_Icc ContinuousOn.image_uIcc_eq_Icc
 
 theorem image_uIcc (h : ContinuousOn f <| [[a, b]]) :
-    f '' [[a, b]] = [[infₛ (f '' [[a, b]]), supₛ (f '' [[a, b]])]] := by
+    f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] := by
   refine' h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm
-  refine' cinfₛ_le_csupₛ _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
+  refine' csInf_le_csSup _ _ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
   exacts [bddBelow_Icc, bddAbove_Icc]
 #align continuous_on.image_uIcc ContinuousOn.image_uIcc
 
-theorem infₛ_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
-    infₛ (f '' Icc a b) ≤ f c := by
+theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
+    sInf (f '' Icc a b) ≤ f c := by
   have := mem_image_of_mem f hc
   rw [h.image_Icc (hc.1.trans hc.2)] at this
   exact this.1
-#align continuous_on.Inf_image_Icc_le ContinuousOn.infₛ_image_Icc_le
+#align continuous_on.Inf_image_Icc_le ContinuousOn.sInf_image_Icc_le
 
-theorem le_supₛ_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
-    f c ≤ supₛ (f '' Icc a b) := by
+theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
+    f c ≤ sSup (f '' Icc a b) := by
   have := mem_image_of_mem f hc
   rw [h.image_Icc (hc.1.trans hc.2)] at this
   exact this.2
-#align continuous_on.le_Sup_image_Icc ContinuousOn.le_supₛ_image_Icc
+#align continuous_on.le_Sup_image_Icc ContinuousOn.le_sSup_image_Icc
 
 end ContinuousOn