topology.noetherian_space ⟷ Mathlib.Topology.NoetherianSpace

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Order.CompactlyGenerated
+import Order.CompactlyGenerated.Basic
 import Topology.Sets.Closeds
 
 #align_import topology.noetherian_space from "leanprover-community/mathlib"@"34ee86e6a59d911a8e4f89b68793ee7577ae79c7"
@@ -156,7 +156,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
 theorem noetherianSpace_set_iff (s : Set α) : NoetherianSpace s ↔ ∀ (t) (_ : t ⊆ s), IsCompact t :=
   by

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

@@ -231,8 +231,8 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
     rintro ⟨k, hk⟩
     cases finset.mem_singleton.mp hk
     exact ⟨h, h₁⟩
-  · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
-    push_neg at h₁ 
+  · rw [isPreirreducible_iff_closed_union_closed] at h₁
+    push_neg at h₁
     obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
     obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
     obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
@@ -258,7 +258,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
       exact z.2
     · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
   obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
-  rw [← e] at hz' 
+  rw [← e] at hz'
   refine' ⟨s, hs, _⟩
   symm
   suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)

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

@@ -217,13 +217,52 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 
 #print TopologicalSpace.NoetherianSpace.exists_finset_irreducible /-
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
-    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by classical
+    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
+  classical
+  have := ((noetherian_space_tfae α).out 0 1).mp inferInstance
+  apply WellFounded.induction this s
+  clear s
+  intro s H
+  by_cases h₁ : IsPreirreducible s.1
+  cases h₂ : s.1.eq_empty_or_nonempty
+  · use∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
+  · use{s}
+    simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
+    rintro ⟨k, hk⟩
+    cases finset.mem_singleton.mp hk
+    exact ⟨h, h₁⟩
+  · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
+    push_neg at h₁ 
+    obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
+    obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
+    obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
+    refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
+    · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
+    · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 -/
 
 #print TopologicalSpace.NoetherianSpace.finite_irreducibleComponents /-
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
-    (irreducibleComponents α).Finite := by classical
+    (irreducibleComponents α).Finite := by
+  classical
+  obtain ⟨S, hS₁, hS₂⟩ := noetherian_space.exists_finset_irreducible (⊤ : closeds α)
+  suffices irreducibleComponents α ⊆ coe '' (S : Set <| closeds α) by
+    exact Set.Finite.subset ((Set.Finite.intro inferInstance).image _) this
+  intro K hK
+  obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α) (H : z ∈ Finset.image coe S), K ⊆ z :=
+    by
+    convert is_irreducible_iff_sUnion_closed.mp hK.1 (S.image coe) _ _
+    · simp only [Finset.mem_image, exists_prop, forall_exists_index, and_imp]
+      rintro _ z hz rfl
+      exact z.2
+    · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
+  obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
+  rw [← e] at hz' 
+  refine' ⟨s, hs, _⟩
+  symm
+  suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)
+  simpa
 #align topological_space.noetherian_space.finite_irreducible_components TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
 -/
 

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

@@ -217,52 +217,13 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 
 #print TopologicalSpace.NoetherianSpace.exists_finset_irreducible /-
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
-    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
-  classical
-  have := ((noetherian_space_tfae α).out 0 1).mp inferInstance
-  apply WellFounded.induction this s
-  clear s
-  intro s H
-  by_cases h₁ : IsPreirreducible s.1
-  cases h₂ : s.1.eq_empty_or_nonempty
-  · use∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
-  · use{s}
-    simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
-    rintro ⟨k, hk⟩
-    cases finset.mem_singleton.mp hk
-    exact ⟨h, h₁⟩
-  · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
-    push_neg at h₁ 
-    obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
-    obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
-    obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
-    refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
-    · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
-    · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
+    ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by classical
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 -/
 
 #print TopologicalSpace.NoetherianSpace.finite_irreducibleComponents /-
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
-    (irreducibleComponents α).Finite := by
-  classical
-  obtain ⟨S, hS₁, hS₂⟩ := noetherian_space.exists_finset_irreducible (⊤ : closeds α)
-  suffices irreducibleComponents α ⊆ coe '' (S : Set <| closeds α) by
-    exact Set.Finite.subset ((Set.Finite.intro inferInstance).image _) this
-  intro K hK
-  obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α) (H : z ∈ Finset.image coe S), K ⊆ z :=
-    by
-    convert is_irreducible_iff_sUnion_closed.mp hK.1 (S.image coe) _ _
-    · simp only [Finset.mem_image, exists_prop, forall_exists_index, and_imp]
-      rintro _ z hz rfl
-      exact z.2
-    · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
-  obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
-  rw [← e] at hz' 
-  refine' ⟨s, hs, _⟩
-  symm
-  suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)
-  simpa
+    (irreducibleComponents α).Finite := by classical
 #align topological_space.noetherian_space.finite_irreducible_components TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
 -/
 

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathbin.Order.CompactlyGenerated
-import Mathbin.Topology.Sets.Closeds
+import Order.CompactlyGenerated
+import Topology.Sets.Closeds
 
 #align_import topology.noetherian_space from "leanprover-community/mathlib"@"34ee86e6a59d911a8e4f89b68793ee7577ae79c7"
 
@@ -156,7 +156,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
 theorem noetherianSpace_set_iff (s : Set α) : NoetherianSpace s ↔ ∀ (t) (_ : t ⊆ s), IsCompact t :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -105,7 +105,8 @@ theorem noetherianSpace_TFAE :
         ∀ s : Opens α, IsCompact (s : Set α)] :=
   by
   tfae_have 1 ↔ 2
-  · refine' (noetherian_space_iff _).trans (Surjective.wellFounded_iff opens.compl_bijective.2 _)
+  · refine'
+      (noetherian_space_iff _).trans (Function.Surjective.wellFounded_iff opens.compl_bijective.2 _)
     exact fun s t => (OrderIso.compl (Set α)).lt_iff_lt.symm
   tfae_have 1 ↔ 4
   · exact noetherian_space_iff_opens α

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -224,8 +224,8 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
   intro s H
   by_cases h₁ : IsPreirreducible s.1
   cases h₂ : s.1.eq_empty_or_nonempty
-  · use ∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
-  · use {s}
+  · use∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
+  · use{s}
     simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
     rintro ⟨k, hk⟩
     cases finset.mem_singleton.mp hk

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module topology.noetherian_space
-! leanprover-community/mathlib commit 34ee86e6a59d911a8e4f89b68793ee7577ae79c7
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Order.CompactlyGenerated
 import Mathbin.Topology.Sets.Closeds
 
+#align_import topology.noetherian_space from "leanprover-community/mathlib"@"34ee86e6a59d911a8e4f89b68793ee7577ae79c7"
+
 /-!
 # Noetherian space
 
@@ -158,7 +155,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
 theorem noetherianSpace_set_iff (s : Set α) : NoetherianSpace s ↔ ∀ (t) (_ : t ⊆ s), IsCompact t :=
   by

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

@@ -84,10 +84,12 @@ protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) :
 #align topological_space.noetherian_space.is_compact TopologicalSpace.NoetherianSpace.isCompact
 -/
 
+#print Inducing.noetherianSpace /-
 protected theorem Inducing.noetherianSpace [NoetherianSpace α] {i : β → α} (hi : Inducing i) :
     NoetherianSpace β :=
   (noetherianSpace_iff_opens _).2 fun s => hi.isCompact_iff.1 (NoetherianSpace.isCompact _)
 #align topological_space.inducing.noetherian_space Inducing.noetherianSpace
+-/
 
 #print TopologicalSpace.NoetherianSpace.set /-
 instance NoetherianSpace.set [h : NoetherianSpace α] (s : Set α) : NoetherianSpace s :=
@@ -99,6 +101,7 @@ variable (α)
 
 example (α : Type _) : Set α ≃o (Set α)ᵒᵈ := by refine' OrderIso.compl (Set α)
 
+#print TopologicalSpace.noetherianSpace_TFAE /-
 theorem noetherianSpace_TFAE :
     TFAE
       [NoetherianSpace α, WellFounded fun s t : Closeds α => s < t, ∀ s : Set α, IsCompact s,
@@ -115,6 +118,7 @@ theorem noetherianSpace_TFAE :
   · exact fun H s => H s
   tfae_finish
 #align topological_space.noetherian_space_tfae TopologicalSpace.noetherianSpace_TFAE
+-/
 
 variable {α β}
 
@@ -128,6 +132,7 @@ instance {α} : NoetherianSpace (CofiniteTopology α) :=
     exact ⟨a, ha, Or.inr hf⟩
   · exact ⟨a, filter.le_principal_iff.mp hs, Or.inl le_rfl⟩
 
+#print TopologicalSpace.noetherianSpace_of_surjective /-
 theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
     (hf' : Function.Surjective f) : NoetherianSpace β :=
   by
@@ -136,17 +141,22 @@ theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf :
   obtain ⟨t, e⟩ := set.image_surjective.mpr hf' s
   exact e ▸ (noetherian_space.is_compact t).image hf
 #align topological_space.noetherian_space_of_surjective TopologicalSpace.noetherianSpace_of_surjective
+-/
 
+#print TopologicalSpace.noetherianSpace_iff_of_homeomorph /-
 theorem noetherianSpace_iff_of_homeomorph (f : α ≃ₜ β) : NoetherianSpace α ↔ NoetherianSpace β :=
   ⟨fun h => @noetherianSpace_of_surjective _ _ h f f.Continuous f.Surjective, fun h =>
     @noetherianSpace_of_surjective _ _ h f.symm f.symm.Continuous f.symm.Surjective⟩
 #align topological_space.noetherian_space_iff_of_homeomorph TopologicalSpace.noetherianSpace_iff_of_homeomorph
+-/
 
+#print TopologicalSpace.NoetherianSpace.range /-
 theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continuous f) :
     NoetherianSpace (Set.range f) :=
   noetherianSpace_of_surjective (Set.codRestrict f _ Set.mem_range_self) (by continuity)
     fun ⟨a, b, h⟩ => ⟨b, Subtype.ext h⟩
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
@@ -207,6 +217,7 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 #align topological_space.finite.to_noetherian_space TopologicalSpace.Finite.to_noetherianSpace
 -/
 
+#print TopologicalSpace.NoetherianSpace.exists_finset_irreducible /-
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
     ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
   classical
@@ -231,6 +242,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
     · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
     · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
+-/
 
 #print TopologicalSpace.NoetherianSpace.finite_irreducibleComponents /-
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -148,7 +148,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
     fun ⟨a, b, h⟩ => ⟨b, Subtype.ext h⟩
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
 theorem noetherianSpace_set_iff (s : Set α) : NoetherianSpace s ↔ ∀ (t) (_ : t ⊆ s), IsCompact t :=
   by

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

@@ -210,49 +210,49 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
     ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
   classical
-    have := ((noetherian_space_tfae α).out 0 1).mp inferInstance
-    apply WellFounded.induction this s
-    clear s
-    intro s H
-    by_cases h₁ : IsPreirreducible s.1
-    cases h₂ : s.1.eq_empty_or_nonempty
-    · use ∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
-    · use {s}
-      simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
-      rintro ⟨k, hk⟩
-      cases finset.mem_singleton.mp hk
-      exact ⟨h, h₁⟩
-    · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
-      push_neg  at h₁ 
-      obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
-      obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
-      obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
-      refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
-      · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
-      · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
+  have := ((noetherian_space_tfae α).out 0 1).mp inferInstance
+  apply WellFounded.induction this s
+  clear s
+  intro s H
+  by_cases h₁ : IsPreirreducible s.1
+  cases h₂ : s.1.eq_empty_or_nonempty
+  · use ∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
+  · use {s}
+    simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
+    rintro ⟨k, hk⟩
+    cases finset.mem_singleton.mp hk
+    exact ⟨h, h₁⟩
+  · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
+    push_neg at h₁ 
+    obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
+    obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
+    obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
+    refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
+    · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
+    · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 
 #print TopologicalSpace.NoetherianSpace.finite_irreducibleComponents /-
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     (irreducibleComponents α).Finite := by
   classical
-    obtain ⟨S, hS₁, hS₂⟩ := noetherian_space.exists_finset_irreducible (⊤ : closeds α)
-    suffices irreducibleComponents α ⊆ coe '' (S : Set <| closeds α) by
-      exact Set.Finite.subset ((Set.Finite.intro inferInstance).image _) this
-    intro K hK
-    obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α) (H : z ∈ Finset.image coe S), K ⊆ z :=
-      by
-      convert is_irreducible_iff_sUnion_closed.mp hK.1 (S.image coe) _ _
-      · simp only [Finset.mem_image, exists_prop, forall_exists_index, and_imp]
-        rintro _ z hz rfl
-        exact z.2
-      · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
-    obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
-    rw [← e] at hz' 
-    refine' ⟨s, hs, _⟩
-    symm
-    suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)
-    simpa
+  obtain ⟨S, hS₁, hS₂⟩ := noetherian_space.exists_finset_irreducible (⊤ : closeds α)
+  suffices irreducibleComponents α ⊆ coe '' (S : Set <| closeds α) by
+    exact Set.Finite.subset ((Set.Finite.intro inferInstance).image _) this
+  intro K hK
+  obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α) (H : z ∈ Finset.image coe S), K ⊆ z :=
+    by
+    convert is_irreducible_iff_sUnion_closed.mp hK.1 (S.image coe) _ _
+    · simp only [Finset.mem_image, exists_prop, forall_exists_index, and_imp]
+      rintro _ z hz rfl
+      exact z.2
+    · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
+  obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
+  rw [← e] at hz' 
+  refine' ⟨s, hs, _⟩
+  symm
+  suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)
+  simpa
 #align topological_space.noetherian_space.finite_irreducible_components TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
 -/
 

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

@@ -175,7 +175,7 @@ theorem NoetherianSpace.iUnion {ι : Type _} (f : ι → Set α) [Finite ι]
     [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) :=
   by
   cases nonempty_fintype ι
-  simp_rw [noetherian_space_set_iff] at hf⊢
+  simp_rw [noetherian_space_set_iff] at hf ⊢
   intro t ht
   rw [← set.inter_eq_left_iff_subset.mpr ht, Set.inter_iUnion]
   exact isCompact_iUnion fun i => hf i _ (Set.inter_subset_right _ _)
@@ -222,13 +222,13 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
       rintro ⟨k, hk⟩
       cases finset.mem_singleton.mp hk
       exact ⟨h, h₁⟩
-    · rw [isPreirreducible_iff_closed_union_closed] at h₁
-      push_neg  at h₁
+    · rw [isPreirreducible_iff_closed_union_closed] at h₁ 
+      push_neg  at h₁ 
       obtain ⟨z₁, z₂, hz₁, hz₂, h, hz₁', hz₂'⟩ := h₁
       obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
       obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
       refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
-      · cases' finset.mem_union.mp k.2 with h' h'; exacts[hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
+      · cases' finset.mem_union.mp k.2 with h' h'; exacts [hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
       · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 
@@ -240,7 +240,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     suffices irreducibleComponents α ⊆ coe '' (S : Set <| closeds α) by
       exact Set.Finite.subset ((Set.Finite.intro inferInstance).image _) this
     intro K hK
-    obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α)(H : z ∈ Finset.image coe S), K ⊆ z :=
+    obtain ⟨z, hz, hz'⟩ : ∃ (z : Set α) (H : z ∈ Finset.image coe S), K ⊆ z :=
       by
       convert is_irreducible_iff_sUnion_closed.mp hK.1 (S.image coe) _ _
       · simp only [Finset.mem_image, exists_prop, forall_exists_index, and_imp]
@@ -248,7 +248,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
         exact z.2
       · exact (Set.subset_univ _).trans ((congr_arg coe hS₂).trans <| by simp).Subset
     obtain ⟨s, hs, e⟩ := finset.mem_image.mp hz
-    rw [← e] at hz'
+    rw [← e] at hz' 
     refine' ⟨s, hs, _⟩
     symm
     suffices K ≤ s by exact this.antisymm (hK.2 (hS₁ ⟨s, hs⟩) this)

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

@@ -84,12 +84,6 @@ protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) :
 #align topological_space.noetherian_space.is_compact TopologicalSpace.NoetherianSpace.isCompact
 -/
 
-/- warning: topological_space.inducing.noetherian_space -> Inducing.noetherianSpace is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] {i : β -> α}, (Inducing.{u2, u1} β α _inst_2 _inst_1 i) -> (TopologicalSpace.NoetherianSpace.{u2} β _inst_2)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align topological_space.inducing.noetherian_space Inducing.noetherianSpaceₓ'. -/
 protected theorem Inducing.noetherianSpace [NoetherianSpace α] {i : β → α} (hi : Inducing i) :
     NoetherianSpace β :=
   (noetherianSpace_iff_opens _).2 fun s => hi.isCompact_iff.1 (NoetherianSpace.isCompact _)
@@ -105,12 +99,6 @@ variable (α)
 
 example (α : Type _) : Set α ≃o (Set α)ᵒᵈ := by refine' OrderIso.compl (Set α)
 
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-Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space_tfae TopologicalSpace.noetherianSpace_TFAEₓ'. -/
 theorem noetherianSpace_TFAE :
     TFAE
       [NoetherianSpace α, WellFounded fun s t : Closeds α => s < t, ∀ s : Set α, IsCompact s,
@@ -140,12 +128,6 @@ instance {α} : NoetherianSpace (CofiniteTopology α) :=
     exact ⟨a, ha, Or.inr hf⟩
   · exact ⟨a, filter.le_principal_iff.mp hs, Or.inl le_rfl⟩
 
-/- warning: topological_space.noetherian_space_of_surjective -> TopologicalSpace.noetherianSpace_of_surjective is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] (f : α -> β), (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Function.Surjective.{succ u1, succ u2} α β f) -> (TopologicalSpace.NoetherianSpace.{u2} β _inst_2)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.NoetherianSpace.{u2} α _inst_1] (f : α -> β), (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Function.Surjective.{succ u2, succ u1} α β f) -> (TopologicalSpace.NoetherianSpace.{u1} β _inst_2)
-Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space_of_surjective TopologicalSpace.noetherianSpace_of_surjectiveₓ'. -/
 theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
     (hf' : Function.Surjective f) : NoetherianSpace β :=
   by
@@ -155,23 +137,11 @@ theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf :
   exact e ▸ (noetherian_space.is_compact t).image hf
 #align topological_space.noetherian_space_of_surjective TopologicalSpace.noetherianSpace_of_surjective
 
-/- warning: topological_space.noetherian_space_iff_of_homeomorph -> TopologicalSpace.noetherianSpace_iff_of_homeomorph is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β], (Homeomorph.{u1, u2} α β _inst_1 _inst_2) -> (Iff (TopologicalSpace.NoetherianSpace.{u1} α _inst_1) (TopologicalSpace.NoetherianSpace.{u2} β _inst_2))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space_iff_of_homeomorph TopologicalSpace.noetherianSpace_iff_of_homeomorphₓ'. -/
 theorem noetherianSpace_iff_of_homeomorph (f : α ≃ₜ β) : NoetherianSpace α ↔ NoetherianSpace β :=
   ⟨fun h => @noetherianSpace_of_surjective _ _ h f f.Continuous f.Surjective, fun h =>
     @noetherianSpace_of_surjective _ _ h f.symm f.symm.Continuous f.symm.Surjective⟩
 #align topological_space.noetherian_space_iff_of_homeomorph TopologicalSpace.noetherianSpace_iff_of_homeomorph
 
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 theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continuous f) :
     NoetherianSpace (Set.range f) :=
   noetherianSpace_of_surjective (Set.codRestrict f _ Set.mem_range_self) (by continuity)
@@ -237,12 +207,6 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 #align topological_space.finite.to_noetherian_space TopologicalSpace.Finite.to_noetherianSpace
 -/
 
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-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] (s : TopologicalSpace.Closeds.{u1} α _inst_1), Exists.{succ u1} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (fun (S : Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) => And (forall (k : Subtype.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.instMembershipFinset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S)), IsIrreducible.{u1} α _inst_1 (SetLike.coe.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.instSetLikeCloseds.{u1} α _inst_1) (Subtype.val.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.instMembershipFinset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S) k))) (Eq.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) s (Finset.sup.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1)))) (BoundedOrder.toOrderBot.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toLE.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SemilatticeSup.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1))))))) (CompleteLattice.toBoundedOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1))) S (id.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducibleₓ'. -/
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
     ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
   classical

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

@@ -252,11 +252,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
     intro s H
     by_cases h₁ : IsPreirreducible s.1
     cases h₂ : s.1.eq_empty_or_nonempty
-    · use ∅
-      refine' ⟨fun k => k.2.elim, _⟩
-      rw [Finset.sup_empty]
-      ext1
-      exact h
+    · use ∅; refine' ⟨fun k => k.2.elim, _⟩; rw [Finset.sup_empty]; ext1; exact h
     · use {s}
       simp only [coe_coe, Finset.sup_singleton, id.def, eq_self_iff_true, and_true_iff]
       rintro ⟨k, hk⟩
@@ -268,8 +264,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
       obtain ⟨S₁, hS₁, hS₁'⟩ := H (s ⊓ ⟨z₁, hz₁⟩) (inf_lt_left.2 hz₁')
       obtain ⟨S₂, hS₂, hS₂'⟩ := H (s ⊓ ⟨z₂, hz₂⟩) (inf_lt_left.2 hz₂')
       refine' ⟨S₁ ∪ S₂, fun k => _, _⟩
-      · cases' finset.mem_union.mp k.2 with h' h'
-        exacts[hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
+      · cases' finset.mem_union.mp k.2 with h' h'; exacts[hS₁ ⟨k, h'⟩, hS₂ ⟨k, h'⟩]
       · rwa [Finset.sup_union, ← hS₁', ← hS₂', ← inf_sup_left, left_eq_inf]
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 

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

@@ -107,7 +107,7 @@ example (α : Type _) : Set α ≃o (Set α)ᵒᵈ := by refine' OrderIso.compl
 
 /- warning: topological_space.noetherian_space_tfae -> TopologicalSpace.noetherianSpace_TFAE is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (TopologicalSpace.NoetherianSpace.{u1} α _inst_1) (List.cons.{0} Prop (WellFounded.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (s : TopologicalSpace.Closeds.{u1} α _inst_1) (t : TopologicalSpace.Closeds.{u1} α _inst_1) => LT.lt.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toLT.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.setLike.{u1} α _inst_1)))) s t)) (List.cons.{0} Prop (forall (s : Set.{u1} α), IsCompact.{u1} α _inst_1 s) (List.cons.{0} Prop (forall (s : TopologicalSpace.Opens.{u1} α _inst_1), IsCompact.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} α _inst_1) α (TopologicalSpace.Opens.setLike.{u1} α _inst_1)))) s)) (List.nil.{0} Prop)))))
+  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (TopologicalSpace.NoetherianSpace.{u1} α _inst_1) (List.cons.{0} Prop (WellFounded.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (s : TopologicalSpace.Closeds.{u1} α _inst_1) (t : TopologicalSpace.Closeds.{u1} α _inst_1) => LT.lt.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toHasLt.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.setLike.{u1} α _inst_1)))) s t)) (List.cons.{0} Prop (forall (s : Set.{u1} α), IsCompact.{u1} α _inst_1 s) (List.cons.{0} Prop (forall (s : TopologicalSpace.Opens.{u1} α _inst_1), IsCompact.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} α _inst_1) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} α _inst_1) α (TopologicalSpace.Opens.setLike.{u1} α _inst_1)))) s)) (List.nil.{0} Prop)))))
 but is expected to have type
   forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (TopologicalSpace.NoetherianSpace.{u1} α _inst_1) (List.cons.{0} Prop (WellFounded.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (s : TopologicalSpace.Closeds.{u1} α _inst_1) (t : TopologicalSpace.Closeds.{u1} α _inst_1) => LT.lt.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toLT.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1))))) s t)) (List.cons.{0} Prop (forall (s : Set.{u1} α), IsCompact.{u1} α _inst_1 s) (List.cons.{0} Prop (forall (s : TopologicalSpace.Opens.{u1} α _inst_1), IsCompact.{u1} α _inst_1 (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} α _inst_1) α (TopologicalSpace.Opens.instSetLikeOpens.{u1} α _inst_1) s)) (List.nil.{0} Prop)))))
 Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space_tfae TopologicalSpace.noetherianSpace_TFAEₓ'. -/
@@ -239,7 +239,7 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 
 /- warning: topological_space.noetherian_space.exists_finset_irreducible -> TopologicalSpace.NoetherianSpace.exists_finset_irreducible is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] (s : TopologicalSpace.Closeds.{u1} α _inst_1), Exists.{succ u1} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (fun (S : Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) => And (forall (k : coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S), IsIrreducible.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (coeTrans.{succ u1, succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (TopologicalSpace.Closeds.{u1} α _inst_1) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.setLike.{u1} α _inst_1)) (coeSubtype.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.Mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.hasMem.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S))))) k)) (Eq.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) s (Finset.sup.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1)))) (BoundedOrder.toOrderBot.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toLE.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SemilatticeSup.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1))))))) (CompleteLattice.toBoundedOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1))) S (id.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1)))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] (s : TopologicalSpace.Closeds.{u1} α _inst_1), Exists.{succ u1} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (fun (S : Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) => And (forall (k : coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S), IsIrreducible.{u1} α _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (Set.{u1} α) (coeTrans.{succ u1, succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) Type.{u1} (Finset.hasCoeToSort.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) S) (TopologicalSpace.Closeds.{u1} α _inst_1) (Set.{u1} α) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.setLike.{u1} α _inst_1)) (coeSubtype.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.Mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.hasMem.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S))))) k)) (Eq.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) s (Finset.sup.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1)))) (BoundedOrder.toOrderBot.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toHasLe.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SemilatticeSup.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1))))))) (CompleteLattice.toBoundedOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.completeLattice.{u1} α _inst_1))) S (id.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1)))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.NoetherianSpace.{u1} α _inst_1] (s : TopologicalSpace.Closeds.{u1} α _inst_1), Exists.{succ u1} (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (fun (S : Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) => And (forall (k : Subtype.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.instMembershipFinset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S)), IsIrreducible.{u1} α _inst_1 (SetLike.coe.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) α (TopologicalSpace.Closeds.instSetLikeCloseds.{u1} α _inst_1) (Subtype.val.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (fun (x : TopologicalSpace.Closeds.{u1} α _inst_1) => Membership.mem.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Finset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) (Finset.instMembershipFinset.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1)) x S) k))) (Eq.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1) s (Finset.sup.{u1, u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1)))) (BoundedOrder.toOrderBot.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Preorder.toLE.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (SemilatticeSup.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (Lattice.toSemilatticeSup.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1))))))) (CompleteLattice.toBoundedOrder.{u1} (TopologicalSpace.Closeds.{u1} α _inst_1) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} α _inst_1))) S (id.{succ u1} (TopologicalSpace.Closeds.{u1} α _inst_1)))))
 Case conversion may be inaccurate. Consider using '#align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducibleₓ'. -/

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

@@ -77,7 +77,7 @@ variable {α β}
 protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) : IsCompact s :=
   by
   refine' isCompact_iff_finite_subcover.2 fun ι U hUo hs => _
-  rcases((noetherian_space_iff_opens α).mp ‹_› ⟨⋃ i, U i, isOpen_unionᵢ hUo⟩).elim_finite_subcover U
+  rcases((noetherian_space_iff_opens α).mp ‹_› ⟨⋃ i, U i, isOpen_iUnion hUo⟩).elim_finite_subcover U
       hUo Set.Subset.rfl with
     ⟨t, ht⟩
   exact ⟨t, hs.trans ht⟩
@@ -200,16 +200,16 @@ theorem noetherian_univ_iff : NoetherianSpace (Set.univ : Set α) ↔ Noetherian
 #align topological_space.noetherian_univ_iff TopologicalSpace.noetherian_univ_iff
 -/
 
-#print TopologicalSpace.NoetherianSpace.unionᵢ /-
-theorem NoetherianSpace.unionᵢ {ι : Type _} (f : ι → Set α) [Finite ι]
+#print TopologicalSpace.NoetherianSpace.iUnion /-
+theorem NoetherianSpace.iUnion {ι : Type _} (f : ι → Set α) [Finite ι]
     [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) :=
   by
   cases nonempty_fintype ι
   simp_rw [noetherian_space_set_iff] at hf⊢
   intro t ht
-  rw [← set.inter_eq_left_iff_subset.mpr ht, Set.inter_unionᵢ]
-  exact isCompact_unionᵢ fun i => hf i _ (Set.inter_subset_right _ _)
-#align topological_space.noetherian_space.Union TopologicalSpace.NoetherianSpace.unionᵢ
+  rw [← set.inter_eq_left_iff_subset.mpr ht, Set.inter_iUnion]
+  exact isCompact_iUnion fun i => hf i _ (Set.inter_subset_right _ _)
+#align topological_space.noetherian_space.Union TopologicalSpace.NoetherianSpace.iUnion
 -/
 
 #print TopologicalSpace.NoetherianSpace.discrete /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -178,7 +178,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
     fun ⟨a, b, h⟩ => ⟨b, Subtype.ext h⟩
 #align topological_space.noetherian_space.range TopologicalSpace.NoetherianSpace.range
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print TopologicalSpace.noetherianSpace_set_iff /-
 theorem noetherianSpace_set_iff (s : Set α) : NoetherianSpace s ↔ ∀ (t) (_ : t ⊆ s), IsCompact t :=
   by

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

Per the style guidelines, λ is disallowed in mathlib. This is close to exhaustive; I left some tactic code alone when it seemed to me that tactic could be upstreamed soon.

Notes

• In lines I was modifying anyway, I also converted => to ↦.
• Also contains some mild in-passing indentation fixes in Mathlib/Order/SupClosed.
• Some doc comments still contained Lean 3 syntax λ x, , which I also replaced.

@@ -231,9 +231,9 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
   have hι' : Finite ι := by rwa [Set.finite_coe_iff]
 
   let U := Z \ ⋃ (x : ι), x
-  have hU0 : U ≠ ∅ := λ r ↦ by
+  have hU0 : U ≠ ∅ := fun r ↦ by
     obtain ⟨Z', hZ'⟩ := isIrreducible_iff_sUnion_closed.mp H.1 hι.toFinset
-      (λ z hz ↦ by
+      (fun z hz ↦ by
         simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hz
         exact isClosed_of_mem_irreducibleComponents _ hz.1)
       (by
@@ -262,6 +262,6 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
       · exact ⟨i, Or.inr i.2, hi⟩
 
   refine ⟨U, hU1 ▸ isOpen_compl_iff.mpr ?_, hU0, sdiff_le⟩
-  exact isClosed_iUnion_of_finite λ i ↦ isClosed_of_mem_irreducibleComponents i.1 i.2.1
+  exact isClosed_iUnion_of_finite fun i ↦ isClosed_of_mem_irreducibleComponents i.1 i.2.1
 
 end TopologicalSpace

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.

@@ -196,7 +196,8 @@ theorem NoetherianSpace.exists_finite_set_isClosed_irreducible [NoetherianSpace
       (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S := by
   lift s to Closeds α using hs
   rcases NoetherianSpace.exists_finite_set_closeds_irreducible s with ⟨S, hSf, hS, rfl⟩
-  refine ⟨(↑) '' S, hSf.image _, Set.ball_image_iff.2 fun S _ => S.2, Set.ball_image_iff.2 hS, ?_⟩
+  refine ⟨(↑) '' S, hSf.image _, Set.forall_mem_image.2 fun S _ ↦ S.2, Set.forall_mem_image.2 hS,
+    ?_⟩
   lift S to Finset (Closeds α) using hSf
   simp [← Finset.sup_id_eq_sSup, Closeds.coe_finset_sup]
 

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".

@@ -70,7 +70,7 @@ protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) :
   exact ⟨t, hs.trans ht⟩
 #align topological_space.noetherian_space.is_compact TopologicalSpace.NoetherianSpace.isCompact
 
--- porting note: fixed NS
+-- Porting note: fixed NS
 protected theorem _root_.Inducing.noetherianSpace [NoetherianSpace α] {i : β → α}
     (hi : Inducing i) : NoetherianSpace β :=
   (noetherianSpace_iff_opens _).2 fun _ => hi.isCompact_iff.2 (NoetherianSpace.isCompact _)

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

@@ -254,7 +254,7 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
     · rw [Set.mem_diff, Decidable.not_and_iff_or_not_not, not_not, Set.mem_iUnion] at h
       rcases h with (h|⟨i, hi⟩)
       · refine ⟨irreducibleComponent a, Or.inr ?_, mem_irreducibleComponent⟩
-        simp only [Set.mem_diff, Set.mem_singleton_iff]
+        simp only [ι, Set.mem_diff, Set.mem_singleton_iff]
         refine ⟨irreducibleComponent_mem_irreducibleComponents _, ?_⟩
         rintro rfl
         exact h mem_irreducibleComponent

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -257,7 +257,7 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
         simp only [Set.mem_diff, Set.mem_singleton_iff]
         refine ⟨irreducibleComponent_mem_irreducibleComponents _, ?_⟩
         rintro rfl
-        refine h mem_irreducibleComponent
+        exact h mem_irreducibleComponent
       · exact ⟨i, Or.inr i.2, hi⟩
 
   refine ⟨U, hU1 ▸ isOpen_compl_iff.mpr ?_, hU0, sdiff_le⟩

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>

@@ -242,8 +242,8 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
     simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hZ'
     exact hZ'.1.2 <| le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2
 
-  have hU1 : U = (⋃ (x : ι), x.1) ᶜ
-  · rw [Set.compl_eq_univ_diff]
+  have hU1 : U = (⋃ (x : ι), x.1) ᶜ := by
+    rw [Set.compl_eq_univ_diff]
     refine le_antisymm (Set.diff_subset_diff le_top <| subset_refl _) ?_
     rw [← Set.compl_eq_univ_diff]
     refine Set.compl_subset_iff_union.mpr (le_antisymm le_top ?_)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -251,7 +251,7 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
     rintro a -
     by_cases h : a ∈ U
     · exact ⟨U, Set.mem_insert _ _, h⟩
-    · rw [Set.mem_diff, Decidable.not_and, not_not, Set.mem_iUnion] at h
+    · rw [Set.mem_diff, Decidable.not_and_iff_or_not_not, not_not, Set.mem_iUnion] at h
       rcases h with (h|⟨i, hi⟩)
       · refine ⟨irreducibleComponent a, Or.inr ?_, mem_irreducibleComponent⟩
         simp only [Set.mem_diff, Set.mem_singleton_iff]

See Zulip thread for the discussion.

@@ -121,7 +121,7 @@ instance {α} : NoetherianSpace (CofiniteTopology α) := by
 
 theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
     (hf' : Function.Surjective f) : NoetherianSpace β :=
-  noetherianSpace_iff_isCompact.2 $ (Set.image_surjective.mpr hf').forall.2 fun s =>
+  noetherianSpace_iff_isCompact.2 <| (Set.image_surjective.mpr hf').forall.2 fun s =>
     (NoetherianSpace.isCompact s).image hf
 #align topological_space.noetherian_space_of_surjective TopologicalSpace.noetherianSpace_of_surjective
 
@@ -240,11 +240,11 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
         rw [Set.diff_eq_empty] at r
         exact r)
     simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hZ'
-    exact hZ'.1.2 $ le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2
+    exact hZ'.1.2 <| le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2
 
   have hU1 : U = (⋃ (x : ι), x.1) ᶜ
   · rw [Set.compl_eq_univ_diff]
-    refine le_antisymm (Set.diff_subset_diff le_top $ subset_refl _) ?_
+    refine le_antisymm (Set.diff_subset_diff le_top <| subset_refl _) ?_
     rw [← Set.compl_eq_univ_diff]
     refine Set.compl_subset_iff_union.mpr (le_antisymm le_top ?_)
     rw [Set.union_comm, ← Set.sUnion_eq_iUnion, ← Set.sUnion_insert]

• @[mk_iff] class MyPred now generates myPred_iff, not MyPred_iff
• add Lean.Name.decapitalize
• fix indentation and a few typos in the docs/comments.

Partially addresses issue #9129

@@ -45,7 +45,7 @@ variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
 namespace TopologicalSpace
 
 /-- Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. -/
-@[mk_iff noetherianSpace_iff]
+@[mk_iff]
 class NoetherianSpace : Prop where
   wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
 #align topological_space.noetherian_space TopologicalSpace.NoetherianSpace

Subset of #9319

@@ -121,7 +121,7 @@ instance {α} : NoetherianSpace (CofiniteTopology α) := by
 
 theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
     (hf' : Function.Surjective f) : NoetherianSpace β :=
-  noetherianSpace_iff_isCompact.2 $ (Set.image_surjective.mpr hf').forall.2 $ fun s =>
+  noetherianSpace_iff_isCompact.2 $ (Set.image_surjective.mpr hf').forall.2 fun s =>
     (NoetherianSpace.isCompact s).image hf
 #align topological_space.noetherian_space_of_surjective TopologicalSpace.noetherianSpace_of_surjective
 

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

@@ -90,7 +90,7 @@ theorem noetherianSpace_TFAE :
       ∀ s : Set α, IsCompact s,
       ∀ s : Opens α, IsCompact (s : Set α)] := by
   tfae_have 1 ↔ 2
-  · refine' (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff  _)
+  · refine' (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff _)
     exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
   tfae_have 1 ↔ 4
   · exact noetherianSpace_iff_opens α

For faster build times and clearer dependencies. No attempt at being exhaustive.

The new import in Clopen.lean had been transitively imported before.

@@ -3,7 +3,6 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathlib.Order.CompactlyGenerated
 import Mathlib.Topology.Sets.Closeds
 
 #align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"

Define sigma-compact subsets of a topological space and show their basic properties.

• compact sets are sigma-compact
• countable unions of (sigma-)compact sets are sigma-compact
• closed subsets of sigma-compact sets are sigma-compact.

Relate them to sigma-compact space: a set is sigma-compact iff it is a sigma-compact space (w.r.t. the subspace topology).

In a later PR, we'll show that sigma-compact measure zero sets are nowhere dense.

• depends on: #7528

Co-authored-by: David Loeffler <d.loeffler.01@cantab.net> Co-authored-by: grunweg <grunweg@posteo.de>

@@ -74,7 +74,7 @@ protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) :
 -- porting note: fixed NS
 protected theorem _root_.Inducing.noetherianSpace [NoetherianSpace α] {i : β → α}
     (hi : Inducing i) : NoetherianSpace β :=
-  (noetherianSpace_iff_opens _).2 fun _ => hi.isCompact_iff.1 (NoetherianSpace.isCompact _)
+  (noetherianSpace_iff_opens _).2 fun _ => hi.isCompact_iff.2 (NoetherianSpace.isCompact _)
 #align topological_space.inducing.noetherian_space Inducing.noetherianSpace
 
 /-- [Stacks: Lemma 0052 (1)](https://stacks.math.columbia.edu/tag/0052)-/
@@ -139,7 +139,7 @@ theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continu
 
 theorem noetherianSpace_set_iff (s : Set α) :
     NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t := by
-  simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff_isCompact_image,
+  simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff,
     Subtype.forall_set_subtype]
 #align topological_space.noetherian_space_set_iff TopologicalSpace.noetherianSpace_set_iff
 

This will improve spaces in the mathlib4 docs.

@@ -33,9 +33,9 @@ of a noetherian scheme (e.g., the spectrum of a noetherian ring) is noetherian.
   is noetherian.
 - `TopologicalSpace.NoetherianSpace.iUnion`: The finite union of noetherian spaces is noetherian.
 - `TopologicalSpace.NoetherianSpace.discrete`: A noetherian and Hausdorff space is discrete.
-- `TopologicalSpace.NoetherianSpace.exists_finset_irreducible` : Every closed subset of a noetherian
+- `TopologicalSpace.NoetherianSpace.exists_finset_irreducible`: Every closed subset of a noetherian
   space is a finite union of irreducible closed subsets.
-- `TopologicalSpace.NoetherianSpace.finite_irreducibleComponents `: The number of irreducible
+- `TopologicalSpace.NoetherianSpace.finite_irreducibleComponents`: The number of irreducible
   components of a noetherian space is finite.
 
 -/

Rename union_eq_left_iff_subset to union_eq_left to match sup_eq_left. Similarly for the right and inter versions.

@@ -152,7 +152,7 @@ theorem NoetherianSpace.iUnion {ι : Type*} (f : ι → Set α) [Finite ι]
     [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) := by
   simp_rw [noetherianSpace_set_iff] at hf ⊢
   intro t ht
-  rw [← Set.inter_eq_left_iff_subset.mpr ht, Set.inter_iUnion]
+  rw [← Set.inter_eq_left.mpr ht, Set.inter_iUnion]
   exact isCompact_iUnion fun i => hf i _ (Set.inter_subset_right _ _)
 #align topological_space.noetherian_space.Union TopologicalSpace.NoetherianSpace.iUnion
 

• Rename Surjective.wellFounded_iff to Function.Surjective.wellFounded_iff.
• Golf the proof.

@@ -91,7 +91,7 @@ theorem noetherianSpace_TFAE :
       ∀ s : Set α, IsCompact s,
       ∀ s : Opens α, IsCompact (s : Set α)] := by
   tfae_have 1 ↔ 2
-  · refine' (noetherianSpace_iff α).trans (Surjective.wellFounded_iff Opens.compl_bijective.2 _)
+  · refine' (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff  _)
     exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
   tfae_have 1 ↔ 4
   · exact noetherianSpace_iff_opens α

@@ -222,7 +222,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
 /-- [Stacks: Lemma 0052 (3)](https://stacks.math.columbia.edu/tag/0052) -/
 theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
     (Z : Set α) (H : Z ∈ irreducibleComponents α) :
-    ∃ (o : Set α) (_ : IsOpen o) (_ : o ≠ ∅), o ≤ Z := by
+    ∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z := by
   classical
 
   let ι : Set (Set α) := irreducibleComponents α \ {Z}
@@ -264,5 +264,4 @@ theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [Noetherian
   refine ⟨U, hU1 ▸ isOpen_compl_iff.mpr ?_, hU0, sdiff_le⟩
   exact isClosed_iUnion_of_finite λ i ↦ isClosed_of_mem_irreducibleComponents i.1 i.2.1
 
-
 end TopologicalSpace

any irreducible components of a noetherian space contains a nonempty open subset

Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -77,6 +77,7 @@ protected theorem _root_.Inducing.noetherianSpace [NoetherianSpace α] {i : β 
   (noetherianSpace_iff_opens _).2 fun _ => hi.isCompact_iff.1 (NoetherianSpace.isCompact _)
 #align topological_space.inducing.noetherian_space Inducing.noetherianSpace
 
+/-- [Stacks: Lemma 0052 (1)](https://stacks.math.columbia.edu/tag/0052)-/
 instance NoetherianSpace.set [NoetherianSpace α] (s : Set α) : NoetherianSpace s :=
   inducing_subtype_val.noetherianSpace
 #align topological_space.noetherian_space.set TopologicalSpace.NoetherianSpace.set
@@ -207,6 +208,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
     using NoetherianSpace.exists_finite_set_closeds_irreducible s
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 
+/-- [Stacks: Lemma 0052 (2)](https://stacks.math.columbia.edu/tag/0052) -/
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     (irreducibleComponents α).Finite := by
   obtain ⟨S : Set (Set α), hSf, hSc, hSi, hSU⟩ :=
@@ -217,4 +219,50 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
   rwa [ht.antisymm (hs.2 (hSi _ htS) ht)]
 #align topological_space.noetherian_space.finite_irreducible_components TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
 
+/-- [Stacks: Lemma 0052 (3)](https://stacks.math.columbia.edu/tag/0052) -/
+theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
+    (Z : Set α) (H : Z ∈ irreducibleComponents α) :
+    ∃ (o : Set α) (_ : IsOpen o) (_ : o ≠ ∅), o ≤ Z := by
+  classical
+
+  let ι : Set (Set α) := irreducibleComponents α \ {Z}
+  have hι : ι.Finite := (NoetherianSpace.finite_irreducibleComponents (α := α)).subset
+    (Set.diff_subset _ _)
+  have hι' : Finite ι := by rwa [Set.finite_coe_iff]
+
+  let U := Z \ ⋃ (x : ι), x
+  have hU0 : U ≠ ∅ := λ r ↦ by
+    obtain ⟨Z', hZ'⟩ := isIrreducible_iff_sUnion_closed.mp H.1 hι.toFinset
+      (λ z hz ↦ by
+        simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hz
+        exact isClosed_of_mem_irreducibleComponents _ hz.1)
+      (by
+        rw [Set.Finite.coe_toFinset, Set.sUnion_eq_iUnion]
+        rw [Set.diff_eq_empty] at r
+        exact r)
+    simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hZ'
+    exact hZ'.1.2 $ le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2
+
+  have hU1 : U = (⋃ (x : ι), x.1) ᶜ
+  · rw [Set.compl_eq_univ_diff]
+    refine le_antisymm (Set.diff_subset_diff le_top $ subset_refl _) ?_
+    rw [← Set.compl_eq_univ_diff]
+    refine Set.compl_subset_iff_union.mpr (le_antisymm le_top ?_)
+    rw [Set.union_comm, ← Set.sUnion_eq_iUnion, ← Set.sUnion_insert]
+    rintro a -
+    by_cases h : a ∈ U
+    · exact ⟨U, Set.mem_insert _ _, h⟩
+    · rw [Set.mem_diff, Decidable.not_and, not_not, Set.mem_iUnion] at h
+      rcases h with (h|⟨i, hi⟩)
+      · refine ⟨irreducibleComponent a, Or.inr ?_, mem_irreducibleComponent⟩
+        simp only [Set.mem_diff, Set.mem_singleton_iff]
+        refine ⟨irreducibleComponent_mem_irreducibleComponents _, ?_⟩
+        rintro rfl
+        refine h mem_irreducibleComponent
+      · exact ⟨i, Or.inr i.2, hi⟩
+
+  refine ⟨U, hU1 ▸ isOpen_compl_iff.mpr ?_, hU0, sdiff_le⟩
+  exact isClosed_iUnion_of_finite λ i ↦ isClosed_of_mem_irreducibleComponents i.1 i.2.1
+
+
 end TopologicalSpace

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

This has nice performance benefits.

@@ -41,7 +41,7 @@ of a noetherian scheme (e.g., the spectrum of a noetherian ring) is noetherian.
 -/
 
 
-variable (α β : Type _) [TopologicalSpace α] [TopologicalSpace β]
+variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
 
 namespace TopologicalSpace
 
@@ -147,7 +147,7 @@ theorem noetherian_univ_iff : NoetherianSpace (Set.univ : Set α) ↔ Noetherian
   noetherianSpace_iff_of_homeomorph (Homeomorph.Set.univ α)
 #align topological_space.noetherian_univ_iff TopologicalSpace.noetherian_univ_iff
 
-theorem NoetherianSpace.iUnion {ι : Type _} (f : ι → Set α) [Finite ι]
+theorem NoetherianSpace.iUnion {ι : Type*} (f : ι → Set α) [Finite ι]
     [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) := by
   simp_rw [noetherianSpace_set_iff] at hf ⊢
   intro t ht

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module topology.noetherian_space
-! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Order.CompactlyGenerated
 import Mathlib.Topology.Sets.Closeds
 
+#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
+
 /-!
 # Noetherian space
 

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -213,7 +213,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     (irreducibleComponents α).Finite := by
   obtain ⟨S : Set (Set α), hSf, hSc, hSi, hSU⟩ :=
-    NoetherianSpace.exists_finite_set_isClosed_irreducible isClosed_univ
+    NoetherianSpace.exists_finite_set_isClosed_irreducible isClosed_univ (α := α)
   refine hSf.subset fun s hs => ?_
   lift S to Finset (Set α) using hSf
   rcases isIrreducible_iff_sUnion_closed.1 hs.1 S hSc (hSU ▸ Set.subset_univ _) with ⟨t, htS, ht⟩

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>

@@ -34,7 +34,7 @@ of a noetherian scheme (e.g., the spectrum of a noetherian ring) is noetherian.
   spaces.
 - `TopologicalSpace.NoetherianSpace.range`: The image of a noetherian space under a continuous map
   is noetherian.
-- `TopologicalSpace.NoetherianSpace.unionᵢ`: The finite union of noetherian spaces is noetherian.
+- `TopologicalSpace.NoetherianSpace.iUnion`: The finite union of noetherian spaces is noetherian.
 - `TopologicalSpace.NoetherianSpace.discrete`: A noetherian and Hausdorff space is discrete.
 - `TopologicalSpace.NoetherianSpace.exists_finset_irreducible` : Every closed subset of a noetherian
   space is a finite union of irreducible closed subsets.
@@ -69,7 +69,7 @@ variable {α β}
 /-- In a Noetherian space, all sets are compact. -/
 protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) : IsCompact s := by
   refine isCompact_iff_finite_subcover.2 fun U hUo hs => ?_
-  rcases ((noetherianSpace_iff_opens α).mp ‹_› ⟨⋃ i, U i, isOpen_unionᵢ hUo⟩).elim_finite_subcover U
+  rcases ((noetherianSpace_iff_opens α).mp ‹_› ⟨⋃ i, U i, isOpen_iUnion hUo⟩).elim_finite_subcover U
     hUo Set.Subset.rfl with ⟨t, ht⟩
   exact ⟨t, hs.trans ht⟩
 #align topological_space.noetherian_space.is_compact TopologicalSpace.NoetherianSpace.isCompact
@@ -150,13 +150,13 @@ theorem noetherian_univ_iff : NoetherianSpace (Set.univ : Set α) ↔ Noetherian
   noetherianSpace_iff_of_homeomorph (Homeomorph.Set.univ α)
 #align topological_space.noetherian_univ_iff TopologicalSpace.noetherian_univ_iff
 
-theorem NoetherianSpace.unionᵢ {ι : Type _} (f : ι → Set α) [Finite ι]
+theorem NoetherianSpace.iUnion {ι : Type _} (f : ι → Set α) [Finite ι]
     [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) := by
   simp_rw [noetherianSpace_set_iff] at hf ⊢
   intro t ht
-  rw [← Set.inter_eq_left_iff_subset.mpr ht, Set.inter_unionᵢ]
-  exact isCompact_unionᵢ fun i => hf i _ (Set.inter_subset_right _ _)
-#align topological_space.noetherian_space.Union TopologicalSpace.NoetherianSpace.unionᵢ
+  rw [← Set.inter_eq_left_iff_subset.mpr ht, Set.inter_iUnion]
+  exact isCompact_iUnion fun i => hf i _ (Set.inter_subset_right _ _)
+#align topological_space.noetherian_space.Union TopologicalSpace.NoetherianSpace.iUnion
 
 -- This is not an instance since it makes a loop with `t2_space_discrete`.
 theorem NoetherianSpace.discrete [NoetherianSpace α] [T2Space α] : DiscreteTopology α :=
@@ -176,7 +176,7 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 
 /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
 theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace α] (s : Closeds α) :
-    ∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = supₛ S := by
+    ∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = sSup S := by
   apply wellFounded_closeds.induction s; clear s
   intro s H
   rcases eq_or_ne s ⊥ with rfl | h₀
@@ -191,7 +191,7 @@ theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace 
       rcases H (s ⊓ z₁) (inf_lt_left.2 hz₁') with ⟨S₁, hSf₁, hS₁, h₁⟩
       rcases H (s ⊓ z₂) (inf_lt_left.2 hz₂') with ⟨S₂, hSf₂, hS₂, h₂⟩
       refine ⟨S₁ ∪ S₂, hSf₁.union hSf₂, Set.union_subset_iff.2 ⟨hS₁, hS₂⟩, ?_⟩
-      rwa [supₛ_union, ← h₁, ← h₂, ← inf_sup_left, left_eq_inf]
+      rwa [sSup_union, ← h₁, ← h₂, ← inf_sup_left, left_eq_inf]
 
 /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
 theorem NoetherianSpace.exists_finite_set_isClosed_irreducible [NoetherianSpace α]
@@ -201,12 +201,12 @@ theorem NoetherianSpace.exists_finite_set_isClosed_irreducible [NoetherianSpace
   rcases NoetherianSpace.exists_finite_set_closeds_irreducible s with ⟨S, hSf, hS, rfl⟩
   refine ⟨(↑) '' S, hSf.image _, Set.ball_image_iff.2 fun S _ => S.2, Set.ball_image_iff.2 hS, ?_⟩
   lift S to Finset (Closeds α) using hSf
-  simp [← Finset.sup_id_eq_supₛ, Closeds.coe_finset_sup]
+  simp [← Finset.sup_id_eq_sSup, Closeds.coe_finset_sup]
 
 /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
 theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Closeds α) :
     ∃ S : Finset (Closeds α), (∀ k : S, IsIrreducible (k : Set α)) ∧ s = S.sup id := by
-  simpa [Set.exists_finite_iff_finset, Finset.sup_id_eq_supₛ]
+  simpa [Set.exists_finite_iff_finset, Finset.sup_id_eq_sSup]
     using NoetherianSpace.exists_finite_set_closeds_irreducible s
 #align topological_space.noetherian_space.exists_finset_irreducible TopologicalSpace.NoetherianSpace.exists_finset_irreducible
 
@@ -216,7 +216,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     NoetherianSpace.exists_finite_set_isClosed_irreducible isClosed_univ
   refine hSf.subset fun s hs => ?_
   lift S to Finset (Set α) using hSf
-  rcases isIrreducible_iff_unionₛ_closed.1 hs.1 S hSc (hSU ▸ Set.subset_univ _) with ⟨t, htS, ht⟩
+  rcases isIrreducible_iff_sUnion_closed.1 hs.1 S hSc (hSU ▸ Set.subset_univ _) with ⟨t, htS, ht⟩
   rwa [ht.antisymm (hs.2 (hSi _ htS) ht)]
 #align topological_space.noetherian_space.finite_irreducible_components TopologicalSpace.NoetherianSpace.finite_irreducibleComponents
 

@@ -86,23 +86,22 @@ instance NoetherianSpace.set [NoetherianSpace α] (s : Set α) : NoetherianSpace
 
 variable (α)
 
-example (α : Type _) : Set α ≃o (Set α)ᵒᵈ := by refine' OrderIso.compl (Set α)
-
 open List in
 theorem noetherianSpace_TFAE :
     TFAE [NoetherianSpace α,
       WellFounded fun s t : Closeds α => s < t,
       ∀ s : Set α, IsCompact s,
       ∀ s : Opens α, IsCompact (s : Set α)] := by
-  have h12 : NoetherianSpace α ↔ WellFounded fun s t : Closeds α => s < t
+  tfae_have 1 ↔ 2
   · refine' (noetherianSpace_iff α).trans (Surjective.wellFounded_iff Opens.compl_bijective.2 _)
     exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
-  rw [← h12]
-  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil]
-  · exact id
+  tfae_have 1 ↔ 4
+  · exact noetherianSpace_iff_opens α
+  tfae_have 1 → 3
   · exact @NoetherianSpace.isCompact α _
+  tfae_have 3 → 4
   · exact fun h s => h s
-  · exact (noetherianSpace_iff_opens α).2
+  tfae_finish
 #align topological_space.noetherian_space_tfae TopologicalSpace.noetherianSpace_TFAE
 
 variable {α}