field_theory.krull_topology ⟷ Mathlib.FieldTheory.KrullTopology

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

@@ -120,7 +120,7 @@ theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L]
   ext f
   refine' ⟨fun _ => Subgroup.mem_top _, fun _ => _⟩
   rintro ⟨x, hx : x ∈ (⊥ : IntermediateField K L)⟩
-  rw [IntermediateField.mem_bot] at hx 
+  rw [IntermediateField.mem_bot] at hx
   rcases hx with ⟨y, rfl⟩
   exact f.commutes y
 #align intermediate_field.fixing_subgroup.bot IntermediateField.fixingSubgroup.bot
@@ -218,7 +218,7 @@ def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : GroupFilter
     have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp; rw [← AlgEquiv.invFun_eq_symm]; rfl⟩
     have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x :=
       by
-      rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg 
+      rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
       exact hg (σ⁻¹ x) h_in_F
     rw [h_g_fix]
     change σ (σ⁻¹ x) = x
@@ -285,7 +285,7 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
       let H := E.fixing_subgroup
       have h_basis : (H : Set (L ≃ₐ[K] L)) ∈ galGroupBasis K L := ⟨H, ⟨E, ⟨h_findim, rfl⟩⟩, rfl⟩
       have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
-      rw [mem_nhds_iff] at h_nhd 
+      rw [mem_nhds_iff] at h_nhd
       rcases h_nhd with ⟨W, hWH, hW_open, hW_1⟩
       refine'
         ⟨HSMul.hSMul f W, HSMul.hSMul g W,
@@ -293,11 +293,11 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
             _⟩⟩
       rw [Set.disjoint_left]
       rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩
-      rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h 
+      rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h
       have h_in_H : w1 * w2⁻¹ ∈ H := H.mul_mem (hWH hw1) (H.inv_mem (hWH hw2))
-      rw [h] at h_in_H 
-      change φ ∈ E.fixing_subgroup at h_in_H 
-      rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H 
+      rw [h] at h_in_H
+      change φ ∈ E.fixing_subgroup at h_in_H
+      rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H
       specialize h_in_H x
       have hxE : x ∈ E := by
         apply IntermediateField.subset_adjoin

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

@@ -108,7 +108,7 @@ def fixedByFinite (K L : Type _) [Field K] [Field L] [Algebra K L] : Set (Subgro
 /-- For an field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
 theorem IntermediateField.finiteDimensional_bot (K L : Type _) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
-  finiteDimensional_of_rank_eq_one IntermediateField.rank_bot
+  FiniteDimensional.of_rank_eq_one IntermediateField.rank_bot
 #align intermediate_field.finite_dimensional_bot IntermediateField.finiteDimensional_bot
 -/
 

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

@@ -274,7 +274,7 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
   {
     t2 := fun f g hfg => by
       let φ := f⁻¹ * g
-      cases' FunLike.exists_ne hfg with x hx
+      cases' DFunLike.exists_ne hfg with x hx
       have hφx : φ x ≠ x := by
         apply ne_of_apply_ne f
         change f (f.symm (g x)) ≠ f x
@@ -319,7 +319,7 @@ theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Al
   apply isTotallyDisconnected_of_isClopen_set
   intro σ τ h_diff
   have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne.def, inv_mul_eq_one]
-  rcases FunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
+  rcases DFunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)
   refine'

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

@@ -288,7 +288,7 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
       rw [mem_nhds_iff] at h_nhd 
       rcases h_nhd with ⟨W, hWH, hW_open, hW_1⟩
       refine'
-        ⟨leftCoset f W, leftCoset g W,
+        ⟨HSMul.hSMul f W, HSMul.hSMul g W,
           ⟨hW_open.left_coset f, hW_open.left_coset g, ⟨1, hW_1, mul_one _⟩, ⟨1, hW_1, mul_one _⟩,
             _⟩⟩
       rw [Set.disjoint_left]
@@ -323,7 +323,7 @@ theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Al
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)
   refine'
-    ⟨leftCoset σ E.fixing_subgroup,
+    ⟨HSMul.hSMul σ E.fixing_subgroup,
       ⟨E.fixing_subgroup_is_open.left_coset σ, E.fixing_subgroup_is_closed.left_coset σ⟩,
       ⟨1, E.fixing_subgroup.one_mem', mul_one σ⟩, _⟩
   simp only [mem_leftCoset_iff, SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff,

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

@@ -316,7 +316,7 @@ section TotallyDisconnected
 theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : IsTotallyDisconnected (Set.univ : Set (L ≃ₐ[K] L)) :=
   by
-  apply isTotallyDisconnected_of_clopen_set
+  apply isTotallyDisconnected_of_isClopen_set
   intro σ τ h_diff
   have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne.def, inv_mul_eq_one]
   rcases FunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

@@ -327,7 +327,7 @@ theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Al
       ⟨E.fixing_subgroup_is_open.left_coset σ, E.fixing_subgroup_is_closed.left_coset σ⟩,
       ⟨1, E.fixing_subgroup.one_mem', mul_one σ⟩, _⟩
   simp only [mem_leftCoset_iff, SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff,
-    not_forall]
+    Classical.not_forall]
   exact ⟨x, IntermediateField.mem_adjoin_simple_self K x, hx⟩
 #align krull_topology_totally_disconnected krullTopology_totallyDisconnected
 -/

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

@@ -3,10 +3,10 @@ Copyright (c) 2022 Sebastian Monnet. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
 -/
-import Mathbin.FieldTheory.Galois
-import Mathbin.Topology.Algebra.FilterBasis
-import Mathbin.Topology.Algebra.OpenSubgroup
-import Mathbin.Tactic.ByContra
+import FieldTheory.Galois
+import Topology.Algebra.FilterBasis
+import Topology.Algebra.OpenSubgroup
+import Tactic.ByContra
 
 #align_import field_theory.krull_topology from "leanprover-community/mathlib"@"e160cefedc932ce41c7049bf0c4b0f061d06216e"
 

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

@@ -171,7 +171,7 @@ def galBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : FilterBasis (L 
   Nonempty := ⟨⊤, ⊤, top_fixedByFinite, rfl⟩
   inter_sets := by
     rintro X Y ⟨H1, ⟨E1, h_E1, rfl⟩, rfl⟩ ⟨H2, ⟨E2, h_E2, rfl⟩, rfl⟩
-    use (IntermediateField.fixingSubgroup (E1 ⊔ E2)).carrier
+    use(IntermediateField.fixingSubgroup (E1 ⊔ E2)).carrier
     refine' ⟨⟨_, ⟨_, finiteDimensional_sup E1 E2 h_E1 h_E2, rfl⟩, rfl⟩, _⟩
     rw [Set.subset_inter_iff]
     exact

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Sebastian Monnet. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
-
-! This file was ported from Lean 3 source module field_theory.krull_topology
-! leanprover-community/mathlib commit e160cefedc932ce41c7049bf0c4b0f061d06216e
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.FieldTheory.Galois
 import Mathbin.Topology.Algebra.FilterBasis
 import Mathbin.Topology.Algebra.OpenSubgroup
 import Mathbin.Tactic.ByContra
 
+#align_import field_theory.krull_topology from "leanprover-community/mathlib"@"e160cefedc932ce41c7049bf0c4b0f061d06216e"
+
 /-!
 # Krull topology
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/0723536a0522d24fc2f159a096fb3304bef77472

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
 
 ! This file was ported from Lean 3 source module field_theory.krull_topology
-! leanprover-community/mathlib commit 039a089d2a4b93c761b234f3e5f5aeb752bac60f
+! leanprover-community/mathlib commit e160cefedc932ce41c7049bf0c4b0f061d06216e
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Tactic.ByContra
 /-!
 # Krull topology
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define the Krull topology on `L ≃ₐ[K] L` for an arbitrary field extension `L/K`. In order to do
 this, we first define a `group_filter_basis` on `L ≃ₐ[K] L`, whose sets are `E.fixing_subgroup` for
 all intermediate fields `E` with `E/K` finite dimensional.

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -60,19 +60,24 @@ all intermediate fields `E` with `E/K` finite dimensional.
 
 open scoped Classical
 
+#print IntermediateField.map_mono /-
 /-- Mapping intermediate fields along algebra equivalences preserves the partial order -/
 theorem IntermediateField.map_mono {K L M : Type _} [Field K] [Field L] [Field M] [Algebra K L]
     [Algebra K M] {E1 E2 : IntermediateField K L} (e : L ≃ₐ[K] M) (h12 : E1 ≤ E2) :
     E1.map e.toAlgHom ≤ E2.map e.toAlgHom :=
   Set.image_subset e h12
 #align intermediate_field.map_mono IntermediateField.map_mono
+-/
 
+#print IntermediateField.map_id /-
 /-- Mapping intermediate fields along the identity does not change them -/
 theorem IntermediateField.map_id {K L : Type _} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) : E.map (AlgHom.id K L) = E :=
   SetLike.coe_injective <| Set.image_id _
 #align intermediate_field.map_id IntermediateField.map_id
+-/
 
+#print im_finiteDimensional /-
 /-- Mapping a finite dimensional intermediate field along an algebra equivalence gives
 a finite-dimensional intermediate field. -/
 instance im_finiteDimensional {K L : Type _} [Field K] [Field L] [Algebra K L]
@@ -80,26 +85,34 @@ instance im_finiteDimensional {K L : Type _} [Field K] [Field L] [Algebra K L]
     FiniteDimensional K (E.map σ.toAlgHom) :=
   LinearEquiv.finiteDimensional (IntermediateField.intermediateFieldMap σ E).toLinearEquiv
 #align im_finite_dimensional im_finiteDimensional
+-/
 
+#print finiteExts /-
 /-- Given a field extension `L/K`, `finite_exts K L` is the set of
 intermediate field extensions `L/E/K` such that `E/K` is finite -/
 def finiteExts (K : Type _) [Field K] (L : Type _) [Field L] [Algebra K L] :
     Set (IntermediateField K L) :=
   {E | FiniteDimensional K E}
 #align finite_exts finiteExts
+-/
 
+#print fixedByFinite /-
 /-- Given a field extension `L/K`, `fixed_by_finite K L` is the set of
 subsets `Gal(L/E)` of `L ≃ₐ[K] L`, where `E/K` is finite -/
 def fixedByFinite (K L : Type _) [Field K] [Field L] [Algebra K L] : Set (Subgroup (L ≃ₐ[K] L)) :=
   IntermediateField.fixingSubgroup '' finiteExts K L
 #align fixed_by_finite fixedByFinite
+-/
 
+#print IntermediateField.finiteDimensional_bot /-
 /-- For an field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
 theorem IntermediateField.finiteDimensional_bot (K L : Type _) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
   finiteDimensional_of_rank_eq_one IntermediateField.rank_bot
 #align intermediate_field.finite_dimensional_bot IntermediateField.finiteDimensional_bot
+-/
 
+#print IntermediateField.fixingSubgroup.bot /-
 /-- This lemma says that `Gal(L/K) = L ≃ₐ[K] L` -/
 theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L] [Algebra K L] :
     IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ :=
@@ -111,13 +124,17 @@ theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L]
   rcases hx with ⟨y, rfl⟩
   exact f.commutes y
 #align intermediate_field.fixing_subgroup.bot IntermediateField.fixingSubgroup.bot
+-/
 
+#print top_fixedByFinite /-
 /-- If `L/K` is a field extension, then we have `Gal(L/K) ∈ fixed_by_finite K L` -/
 theorem top_fixedByFinite {K L : Type _} [Field K] [Field L] [Algebra K L] :
     ⊤ ∈ fixedByFinite K L :=
   ⟨⊥, IntermediateField.finiteDimensional_bot K L, IntermediateField.fixingSubgroup.bot⟩
 #align top_fixed_by_finite top_fixedByFinite
+-/
 
+#print finiteDimensional_sup /-
 /-- If `E1` and `E2` are finite-dimensional intermediate fields, then so is their compositum.
 This rephrases a result already in mathlib so that it is compatible with our type classes -/
 theorem finiteDimensional_sup {K L : Type _} [Field K] [Field L] [Algebra K L]
@@ -125,13 +142,17 @@ theorem finiteDimensional_sup {K L : Type _} [Field K] [Field L] [Algebra K L]
     FiniteDimensional K ↥(E1 ⊔ E2) :=
   IntermediateField.finiteDimensional_sup E1 E2
 #align finite_dimensional_sup finiteDimensional_sup
+-/
 
+#print IntermediateField.mem_fixingSubgroup_iff /-
 /-- An element of `L ≃ₐ[K] L` is in `Gal(L/E)` if and only if it fixes every element of `E`-/
 theorem IntermediateField.mem_fixingSubgroup_iff {K L : Type _} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) (σ : L ≃ₐ[K] L) : σ ∈ E.fixingSubgroup ↔ ∀ x : L, x ∈ E → σ x = x :=
   ⟨fun hσ x hx => hσ ⟨x, hx⟩, fun h ⟨x, hx⟩ => h x hx⟩
 #align intermediate_field.mem_fixing_subgroup_iff IntermediateField.mem_fixingSubgroup_iff
+-/
 
+#print IntermediateField.fixingSubgroup.antimono /-
 /-- The map `E ↦ Gal(L/E)` is inclusion-reversing -/
 theorem IntermediateField.fixingSubgroup.antimono {K L : Type _} [Field K] [Field L] [Algebra K L]
     {E1 E2 : IntermediateField K L} (h12 : E1 ≤ E2) : E2.fixingSubgroup ≤ E1.fixingSubgroup :=
@@ -139,7 +160,9 @@ theorem IntermediateField.fixingSubgroup.antimono {K L : Type _} [Field K] [Fiel
   rintro σ hσ ⟨x, hx⟩
   exact hσ ⟨x, h12 hx⟩
 #align intermediate_field.fixing_subgroup.antimono IntermediateField.fixingSubgroup.antimono
+-/
 
+#print galBasis /-
 /-- Given a field extension `L/K`, `gal_basis K L` is the filter basis on `L ≃ₐ[K] L` whose sets
 are `Gal(L/E)` for intermediate fields `E` with `E/K` finite dimensional -/
 def galBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L)
@@ -155,14 +178,18 @@ def galBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : FilterBasis (L 
       ⟨IntermediateField.fixingSubgroup.antimono le_sup_left,
         IntermediateField.fixingSubgroup.antimono le_sup_right⟩
 #align gal_basis galBasis
+-/
 
+#print mem_galBasis_iff /-
 /-- A subset of `L ≃ₐ[K] L` is a member of `gal_basis K L` if and only if it is the underlying set
 of `Gal(L/E)` for some finite subextension `E/K`-/
 theorem mem_galBasis_iff (K L : Type _) [Field K] [Field L] [Algebra K L] (U : Set (L ≃ₐ[K] L)) :
     U ∈ galBasis K L ↔ U ∈ Subgroup.carrier '' fixedByFinite K L :=
   Iff.rfl
 #align mem_gal_basis_iff mem_galBasis_iff
+-/
 
+#print galGroupBasis /-
 /-- For a field extension `L/K`, `gal_group_basis K L` is the group filter basis on `L ≃ₐ[K] L`
 whose sets are `Gal(L/E)` for finite subextensions `E/K` -/
 def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : GroupFilterBasis (L ≃ₐ[K] L)
@@ -197,13 +224,16 @@ def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : GroupFilter
     change σ (σ⁻¹ x) = x
     exact AlgEquiv.apply_symm_apply σ x
 #align gal_group_basis galGroupBasis
+-/
 
+#print krullTopology /-
 /-- For a field extension `L/K`, `krull_topology K L` is the topological space structure on
 `L ≃ₐ[K] L` induced by the group filter basis `gal_group_basis K L` -/
 instance krullTopology (K L : Type _) [Field K] [Field L] [Algebra K L] :
     TopologicalSpace (L ≃ₐ[K] L) :=
   GroupFilterBasis.topology (galGroupBasis K L)
 #align krull_topology krullTopology
+-/
 
 /-- For a field extension `L/K`, the Krull topology on `L ≃ₐ[K] L` makes it a topological group. -/
 instance (K L : Type _) [Field K] [Field L] [Algebra K L] : TopologicalGroup (L ≃ₐ[K] L) :=
@@ -213,6 +243,7 @@ section KrullT2
 
 open scoped Topology Filter
 
+#print IntermediateField.fixingSubgroup_isOpen /-
 /-- Let `L/E/K` be a tower of fields with `E/K` finite. Then `Gal(L/E)` is an open subgroup of
   `L ≃ₐ[K] L`. -/
 theorem IntermediateField.fixingSubgroup_isOpen {K L : Type _} [Field K] [Field L] [Algebra K L]
@@ -224,7 +255,9 @@ theorem IntermediateField.fixingSubgroup_isOpen {K L : Type _} [Field K] [Field
   have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
   exact Subgroup.isOpen_of_mem_nhds _ h_nhd
 #align intermediate_field.fixing_subgroup_is_open IntermediateField.fixingSubgroup_isOpen
+-/
 
+#print IntermediateField.fixingSubgroup_isClosed /-
 /-- Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is
   closed. -/
 theorem IntermediateField.fixingSubgroup_isClosed {K L : Type _} [Field K] [Field L] [Algebra K L]
@@ -232,7 +265,9 @@ theorem IntermediateField.fixingSubgroup_isClosed {K L : Type _} [Field K] [Fiel
     IsClosed (E.fixingSubgroup : Set (L ≃ₐ[K] L)) :=
   OpenSubgroup.isClosed ⟨E.fixingSubgroup, E.fixingSubgroup_isOpen⟩
 #align intermediate_field.fixing_subgroup_is_closed IntermediateField.fixingSubgroup_isClosed
+-/
 
+#print krullTopology_t2 /-
 /-- If `L/K` is an algebraic extension, then the Krull topology on `L ≃ₐ[K] L` is Hausdorff. -/
 theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : T2Space (L ≃ₐ[K] L) :=
@@ -269,14 +304,16 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
         apply Set.mem_singleton
       exact hφx (h_in_H hxE) }
 #align krull_topology_t2 krullTopology_t2
+-/
 
 end KrullT2
 
 section TotallyDisconnected
 
+#print krullTopology_totallyDisconnected /-
 /-- If `L/K` is an algebraic field extension, then the Krull topology on `L ≃ₐ[K] L` is
   totally disconnected. -/
-theorem krullTopology_totally_disconnected {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : IsTotallyDisconnected (Set.univ : Set (L ≃ₐ[K] L)) :=
   by
   apply isTotallyDisconnected_of_clopen_set
@@ -292,7 +329,8 @@ theorem krullTopology_totally_disconnected {K L : Type _} [Field K] [Field L] [A
   simp only [mem_leftCoset_iff, SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff,
     not_forall]
   exact ⟨x, IntermediateField.mem_adjoin_simple_self K x, hx⟩
-#align krull_topology_totally_disconnected krullTopology_totally_disconnected
+#align krull_topology_totally_disconnected krullTopology_totallyDisconnected
+-/
 
 end TotallyDisconnected
 

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

@@ -85,7 +85,7 @@ instance im_finiteDimensional {K L : Type _} [Field K] [Field L] [Algebra K L]
 intermediate field extensions `L/E/K` such that `E/K` is finite -/
 def finiteExts (K : Type _) [Field K] (L : Type _) [Field L] [Algebra K L] :
     Set (IntermediateField K L) :=
-  { E | FiniteDimensional K E }
+  {E | FiniteDimensional K E}
 #align finite_exts finiteExts
 
 /-- Given a field extension `L/K`, `fixed_by_finite K L` is the set of

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

@@ -107,7 +107,7 @@ theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L]
   ext f
   refine' ⟨fun _ => Subgroup.mem_top _, fun _ => _⟩
   rintro ⟨x, hx : x ∈ (⊥ : IntermediateField K L)⟩
-  rw [IntermediateField.mem_bot] at hx
+  rw [IntermediateField.mem_bot] at hx 
   rcases hx with ⟨y, rfl⟩
   exact f.commutes y
 #align intermediate_field.fixing_subgroup.bot IntermediateField.fixingSubgroup.bot
@@ -191,7 +191,7 @@ def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : GroupFilter
     have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp; rw [← AlgEquiv.invFun_eq_symm]; rfl⟩
     have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x :=
       by
-      rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
+      rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg 
       exact hg (σ⁻¹ x) h_in_F
     rw [h_g_fix]
     change σ (σ⁻¹ x) = x
@@ -250,7 +250,7 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
       let H := E.fixing_subgroup
       have h_basis : (H : Set (L ≃ₐ[K] L)) ∈ galGroupBasis K L := ⟨H, ⟨E, ⟨h_findim, rfl⟩⟩, rfl⟩
       have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
-      rw [mem_nhds_iff] at h_nhd
+      rw [mem_nhds_iff] at h_nhd 
       rcases h_nhd with ⟨W, hWH, hW_open, hW_1⟩
       refine'
         ⟨leftCoset f W, leftCoset g W,
@@ -258,11 +258,11 @@ theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
             _⟩⟩
       rw [Set.disjoint_left]
       rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩
-      rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h
+      rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h 
       have h_in_H : w1 * w2⁻¹ ∈ H := H.mul_mem (hWH hw1) (H.inv_mem (hWH hw2))
-      rw [h] at h_in_H
-      change φ ∈ E.fixing_subgroup at h_in_H
-      rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H
+      rw [h] at h_in_H 
+      change φ ∈ E.fixing_subgroup at h_in_H 
+      rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H 
       specialize h_in_H x
       have hxE : x ∈ E := by
         apply IntermediateField.subset_adjoin

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

@@ -58,7 +58,7 @@ all intermediate fields `E` with `E/K` finite dimensional.
 -/
 
 
-open Classical
+open scoped Classical
 
 /-- Mapping intermediate fields along algebra equivalences preserves the partial order -/
 theorem IntermediateField.map_mono {K L M : Type _} [Field K] [Field L] [Field M] [Algebra K L]
@@ -211,7 +211,7 @@ instance (K L : Type _) [Field K] [Field L] [Algebra K L] : TopologicalGroup (L
 
 section KrullT2
 
-open Topology Filter
+open scoped Topology Filter
 
 /-- Let `L/E/K` be a tower of fields with `E/K` finite. Then `Gal(L/E)` is an open subgroup of
   `L ≃ₐ[K] L`. -/

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

@@ -188,11 +188,7 @@ def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : GroupFilter
     rw [IntermediateField.mem_fixingSubgroup_iff]
     intro x hx
     change σ (g (σ⁻¹ x)) = x
-    have h_in_F : σ⁻¹ x ∈ F :=
-      ⟨x, hx, by
-        dsimp
-        rw [← AlgEquiv.invFun_eq_symm]
-        rfl⟩
+    have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp; rw [← AlgEquiv.invFun_eq_symm]; rfl⟩
     have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x :=
       by
       rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg

mathlib commit https://github.com/leanprover-community/mathlib/commit/06a655b5fcfbda03502f9158bbf6c0f1400886f9

@@ -211,7 +211,7 @@ instance krullTopology (K L : Type _) [Field K] [Field L] [Algebra K L] :
 
 /-- For a field extension `L/K`, the Krull topology on `L ≃ₐ[K] L` makes it a topological group. -/
 instance (K L : Type _) [Field K] [Field L] [Algebra K L] : TopologicalGroup (L ≃ₐ[K] L) :=
-  GroupFilterBasis.is_topologicalGroup (galGroupBasis K L)
+  GroupFilterBasis.isTopologicalGroup (galGroupBasis K L)
 
 section KrullT2
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/06a655b5fcfbda03502f9158bbf6c0f1400886f9

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
 
 ! This file was ported from Lean 3 source module field_theory.krull_topology
-! leanprover-community/mathlib commit 83f81aea33931a1edb94ce0f32b9a5d484de6978
+! leanprover-community/mathlib commit 039a089d2a4b93c761b234f3e5f5aeb752bac60f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -97,7 +97,7 @@ def fixedByFinite (K L : Type _) [Field K] [Field L] [Algebra K L] : Set (Subgro
 /-- For an field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
 theorem IntermediateField.finiteDimensional_bot (K L : Type _) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
-  finiteDimensional_of_dim_eq_one IntermediateField.dim_bot
+  finiteDimensional_of_rank_eq_one IntermediateField.rank_bot
 #align intermediate_field.finite_dimensional_bot IntermediateField.finiteDimensional_bot
 
 /-- This lemma says that `Gal(L/K) = L ≃ₐ[K] L` -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
 
 ! This file was ported from Lean 3 source module field_theory.krull_topology
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 83f81aea33931a1edb94ce0f32b9a5d484de6978
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -226,9 +226,7 @@ theorem IntermediateField.fixingSubgroup_isOpen {K L : Type _} [Field K] [Field
   have h_basis : E.fixing_subgroup.carrier ∈ galGroupBasis K L :=
     ⟨E.fixing_subgroup, ⟨E, ‹_›, rfl⟩, rfl⟩
   have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
-  rw [mem_nhds_iff] at h_nhd
-  rcases h_nhd with ⟨U, hU_le, hU_open, h1U⟩
-  exact Subgroup.isOpen_of_one_mem_interior ⟨U, ⟨hU_open, hU_le⟩, h1U⟩
+  exact Subgroup.isOpen_of_mem_nhds _ h_nhd
 #align intermediate_field.fixing_subgroup_is_open IntermediateField.fixingSubgroup_isOpen
 
 /-- Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is

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

@@ -262,7 +262,7 @@ theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Alg
     (h_int : Algebra.IsIntegral K L) : IsTotallyDisconnected (Set.univ : Set (L ≃ₐ[K] L)) := by
   apply isTotallyDisconnected_of_isClopen_set
   intro σ τ h_diff
-  have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne.def, inv_mul_eq_one]
+  have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne, inv_mul_eq_one]
   rcases DFunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)

From the FLT project.

@@ -275,3 +275,20 @@ theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Alg
 #align krull_topology_totally_disconnected krullTopology_totallyDisconnected
 
 end TotallyDisconnected
+
+@[simp] lemma IntermediateField.fixingSubgroup_top (K L : Type*) [Field K] [Field L] [Algebra K L] :
+    IntermediateField.fixingSubgroup (⊤ : IntermediateField K L) = ⊥ := by
+  ext
+  simp [mem_fixingSubgroup_iff, DFunLike.ext_iff]
+
+@[simp] lemma IntermediateField.fixingSubgroup_bot (K L : Type*) [Field K] [Field L] [Algebra K L] :
+    IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ := by
+  ext
+  simp [mem_fixingSubgroup_iff, mem_bot]
+
+instance krullTopology_discreteTopology_of_finiteDimensional (K L : Type) [Field K] [Field L]
+    [Algebra K L] [FiniteDimensional K L] : DiscreteTopology (L ≃ₐ[K] L) := by
+  rw [discreteTopology_iff_isOpen_singleton_one]
+  change IsOpen (⊥ : Subgroup (L ≃ₐ[K] L))
+  rw [← IntermediateField.fixingSubgroup_top]
+  exact IntermediateField.fixingSubgroup_isOpen ⊤

Rename lemmas to enable new-style dot notation or drop repeating FiniteDimensional.finiteDimensional_*. Restore old names as deprecated aliases.

@@ -86,7 +86,7 @@ def fixedByFinite (K L : Type*) [Field K] [Field L] [Algebra K L] : Set (Subgrou
 /-- For a field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
 theorem IntermediateField.finiteDimensional_bot (K L : Type*) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
-  finiteDimensional_of_rank_eq_one IntermediateField.rank_bot
+  .of_rank_eq_one IntermediateField.rank_bot
 #align intermediate_field.finite_dimensional_bot IntermediateField.finiteDimensional_bot
 
 /-- This lemma says that `Gal(L/K) = L ≃ₐ[K] L` -/

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

@@ -267,7 +267,7 @@ theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Alg
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)
   refine' ⟨σ • E.fixingSubgroup,
-    ⟨E.fixingSubgroup_isOpen.leftCoset σ, E.fixingSubgroup_isClosed.leftCoset σ⟩,
+    ⟨E.fixingSubgroup_isClosed.leftCoset σ, E.fixingSubgroup_isOpen.leftCoset σ⟩,
     ⟨1, E.fixingSubgroup.one_mem', mul_one σ⟩, _⟩
   simp only [mem_leftCoset_iff, SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff,
     not_forall]

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -222,7 +222,7 @@ theorem krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : T2Space (L ≃ₐ[K] L) :=
   { t2 := fun f g hfg => by
       let φ := f⁻¹ * g
-      cases' FunLike.exists_ne hfg with x hx
+      cases' DFunLike.exists_ne hfg with x hx
       have hφx : φ x ≠ x := by
         apply ne_of_apply_ne f
         change f (f.symm (g x)) ≠ f x
@@ -263,7 +263,7 @@ theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Alg
   apply isTotallyDisconnected_of_isClopen_set
   intro σ τ h_diff
   have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne.def, inv_mul_eq_one]
-  rcases FunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
+  rcases DFunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)
   refine' ⟨σ • E.fixingSubgroup,

• Redefine Set.image2 to use ∃ a ∈ s, ∃ b ∈ t, f a b = c instead of ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c.
• Redefine Set.seq as Set.image2. The new definition is equal to the old one but rw [Set.seq] gives a different result.
• Redefine Filter.map₂ to use ∃ u ∈ f, ∃ v ∈ g, image2 m u v ⊆ s instead of ∃ u v, u ∈ f ∧ v ∈ g ∧ ...
• Update lemmas like Set.mem_image2, Finset.mem_image₂, Set.mem_mul, Finset.mem_div etc

The two reasons to make the change are:

• ∃ a ∈ s, ∃ b ∈ t, _ is a simp-normal form, and
• it looks a bit nicer.

@@ -158,7 +158,7 @@ def galGroupBasis (K L : Type*) [Field K] [Field L] [Algebra K L] :
   mul' {U} hU :=
     ⟨U, hU, by
       rcases hU with ⟨H, _, rfl⟩
-      rintro x ⟨a, b, haH, hbH, rfl⟩
+      rintro x ⟨a, haH, b, hbH, rfl⟩
       exact H.mul_mem haH hbH⟩
   inv' {U} hU :=
     ⟨U, hU, by

Those two definitions are completely obsolete now that we have the pointwise API. This PR removes them but not the corresponding API. A much more tedious subsequent PR will be needed to merge the two API.

Note that I need to tweak simp lemmas to keep confluence since I'm merging two pairs of head keys together.

@@ -54,8 +54,7 @@ all intermediate fields `E` with `E/K` finite dimensional.
 - `krullTopology K L` is defined as an instance for type class inference.
 -/
 
-
-open scoped Classical
+open scoped Classical Pointwise
 
 /-- Mapping intermediate fields along the identity does not change them -/
 theorem IntermediateField.map_id {K L : Type*} [Field K] [Field L] [Algebra K L]
@@ -236,10 +235,11 @@ theorem krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
       have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
       rw [mem_nhds_iff] at h_nhd
       rcases h_nhd with ⟨W, hWH, hW_open, hW_1⟩
-      refine' ⟨leftCoset f W, leftCoset g W,
+      refine' ⟨f • W, g • W,
         ⟨hW_open.leftCoset f, hW_open.leftCoset g, ⟨1, hW_1, mul_one _⟩, ⟨1, hW_1, mul_one _⟩, _⟩⟩
       rw [Set.disjoint_left]
       rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩
+      dsimp at h
       rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h
       have h_in_H : w1 * w2⁻¹ ∈ H := H.mul_mem (hWH hw1) (H.inv_mem (hWH hw2))
       rw [h] at h_in_H
@@ -266,7 +266,7 @@ theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Alg
   rcases FunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
   let E := IntermediateField.adjoin K ({x} : Set L)
   haveI := IntermediateField.adjoin.finiteDimensional (h_int x)
-  refine' ⟨leftCoset σ E.fixingSubgroup,
+  refine' ⟨σ • E.fixingSubgroup,
     ⟨E.fixingSubgroup_isOpen.leftCoset σ, E.fixingSubgroup_isClosed.leftCoset σ⟩,
     ⟨1, E.fixingSubgroup.one_mem', mul_one σ⟩, _⟩
   simp only [mem_leftCoset_iff, SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff,

This PR renames the field Clopens.clopen' -> Clopens.isClopen', and the lemmas

• preimage_closed_of_closed -> ContinuousOn.preimage_isClosed_of_isClosed

as well as: ClopenUpperSet.clopen -> ClopenUpperSet.isClopen connectedComponent_eq_iInter_clopen -> connectedComponent_eq_iInter_isClopen connectedComponent_subset_iInter_clopen -> connectedComponent_subset_iInter_isClopen continuous_boolIndicator_iff_clopen -> continuous_boolIndicator_iff_isClopen continuousOn_boolIndicator_iff_clopen -> continuousOn_boolIndicator_iff_isClopen DiscreteQuotient.ofClopen -> DiscreteQuotient.ofIsClopen disjoint_or_subset_of_clopen -> disjoint_or_subset_of_isClopen exists_clopen_{lower,upper}of_not_le -> exists_isClopen{lower,upper}_of_not_le exists_clopen_of_cofiltered -> exists_isClopen_of_cofiltered exists_clopen_of_totally_separated -> exists_isClopen_of_totally_separated exists_clopen_upper_or_lower_of_ne -> exists_isClopen_upper_or_lower_of_ne IsPreconnected.subset_clopen -> IsPreconnected.subset_isClopen isTotallyDisconnected_of_clopen_set -> isTotallyDisconnected_of_isClopen_set LocallyConstant.ofClopen_fiber_one -> LocallyConstant.ofIsClopen_fiber_one LocallyConstant.ofClopen_fiber_zero -> LocallyConstant.ofIsClopen_fiber_zero LocallyConstant.ofClopen -> LocallyConstant.ofIsClopen preimage_clopen_of_clopen -> preimage_isClopen_of_isClopen TopologicalSpace.Clopens.clopen -> TopologicalSpace.Clopens.isClopen

@@ -260,7 +260,7 @@ section TotallyDisconnected
   totally disconnected. -/
 theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : IsTotallyDisconnected (Set.univ : Set (L ≃ₐ[K] L)) := by
-  apply isTotallyDisconnected_of_clopen_set
+  apply isTotallyDisconnected_of_isClopen_set
   intro σ τ h_diff
   have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne.def, inv_mul_eq_one]
   rcases FunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩

This PR adds API and changes the definition of the normal closure of $F\leq K\leq L$ to be an intermediate field of L/F, rather than an intermediate field of L/K. For example, I think it would be more common to say that the normal closure of $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2},\zeta_3)/\mathbb{Q}$ rather than $\mathbb{Q}(\sqrt[3]{2},\zeta_3)/\mathbb{Q}(\sqrt[3]{2})$. This change also means that the normal closure goes from being a dependent function (K : Type) → IntermediateField K L to being a non-dependent function Type → IntermediateField F L, making it easier to compare across the Galois corespondence.

Supersedes https://github.com/leanprover-community/mathlib/pull/18971

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Thomas Browning <tb65536@users.noreply.github.com>

@@ -57,13 +57,6 @@ all intermediate fields `E` with `E/K` finite dimensional.
 
 open scoped Classical
 
-/-- Mapping intermediate fields along algebra equivalences preserves the partial order -/
-theorem IntermediateField.map_mono {K L M : Type*} [Field K] [Field L] [Field M] [Algebra K L]
-    [Algebra K M] {E1 E2 : IntermediateField K L} (e : L ≃ₐ[K] M) (h12 : E1 ≤ E2) :
-    E1.map e.toAlgHom ≤ E2.map e.toAlgHom :=
-  Set.image_subset e h12
-#align intermediate_field.map_mono IntermediateField.map_mono
-
 /-- Mapping intermediate fields along the identity does not change them -/
 theorem IntermediateField.map_id {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) : E.map (AlgHom.id K L) = E :=

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

This has nice performance benefits.

@@ -58,21 +58,21 @@ all intermediate fields `E` with `E/K` finite dimensional.
 open scoped Classical
 
 /-- Mapping intermediate fields along algebra equivalences preserves the partial order -/
-theorem IntermediateField.map_mono {K L M : Type _} [Field K] [Field L] [Field M] [Algebra K L]
+theorem IntermediateField.map_mono {K L M : Type*} [Field K] [Field L] [Field M] [Algebra K L]
     [Algebra K M] {E1 E2 : IntermediateField K L} (e : L ≃ₐ[K] M) (h12 : E1 ≤ E2) :
     E1.map e.toAlgHom ≤ E2.map e.toAlgHom :=
   Set.image_subset e h12
 #align intermediate_field.map_mono IntermediateField.map_mono
 
 /-- Mapping intermediate fields along the identity does not change them -/
-theorem IntermediateField.map_id {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem IntermediateField.map_id {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) : E.map (AlgHom.id K L) = E :=
   SetLike.coe_injective <| Set.image_id _
 #align intermediate_field.map_id IntermediateField.map_id
 
 /-- Mapping a finite dimensional intermediate field along an algebra equivalence gives
 a finite-dimensional intermediate field. -/
-instance im_finiteDimensional {K L : Type _} [Field K] [Field L] [Algebra K L]
+instance im_finiteDimensional {K L : Type*} [Field K] [Field L] [Algebra K L]
     {E : IntermediateField K L} (σ : L ≃ₐ[K] L) [FiniteDimensional K E] :
     FiniteDimensional K (E.map σ.toAlgHom) :=
   LinearEquiv.finiteDimensional (IntermediateField.intermediateFieldMap σ E).toLinearEquiv
@@ -80,25 +80,25 @@ instance im_finiteDimensional {K L : Type _} [Field K] [Field L] [Algebra K L]
 
 /-- Given a field extension `L/K`, `finiteExts K L` is the set of
 intermediate field extensions `L/E/K` such that `E/K` is finite -/
-def finiteExts (K : Type _) [Field K] (L : Type _) [Field L] [Algebra K L] :
+def finiteExts (K : Type*) [Field K] (L : Type*) [Field L] [Algebra K L] :
     Set (IntermediateField K L) :=
   {E | FiniteDimensional K E}
 #align finite_exts finiteExts
 
 /-- Given a field extension `L/K`, `fixedByFinite K L` is the set of
 subsets `Gal(L/E)` of `L ≃ₐ[K] L`, where `E/K` is finite -/
-def fixedByFinite (K L : Type _) [Field K] [Field L] [Algebra K L] : Set (Subgroup (L ≃ₐ[K] L)) :=
+def fixedByFinite (K L : Type*) [Field K] [Field L] [Algebra K L] : Set (Subgroup (L ≃ₐ[K] L)) :=
   IntermediateField.fixingSubgroup '' finiteExts K L
 #align fixed_by_finite fixedByFinite
 
 /-- For a field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
-theorem IntermediateField.finiteDimensional_bot (K L : Type _) [Field K] [Field L] [Algebra K L] :
+theorem IntermediateField.finiteDimensional_bot (K L : Type*) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
   finiteDimensional_of_rank_eq_one IntermediateField.rank_bot
 #align intermediate_field.finite_dimensional_bot IntermediateField.finiteDimensional_bot
 
 /-- This lemma says that `Gal(L/K) = L ≃ₐ[K] L` -/
-theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L] [Algebra K L] :
+theorem IntermediateField.fixingSubgroup.bot {K L : Type*} [Field K] [Field L] [Algebra K L] :
     IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ := by
   ext f
   refine' ⟨fun _ => Subgroup.mem_top _, fun _ => _⟩
@@ -109,27 +109,27 @@ theorem IntermediateField.fixingSubgroup.bot {K L : Type _} [Field K] [Field L]
 #align intermediate_field.fixing_subgroup.bot IntermediateField.fixingSubgroup.bot
 
 /-- If `L/K` is a field extension, then we have `Gal(L/K) ∈ fixedByFinite K L` -/
-theorem top_fixedByFinite {K L : Type _} [Field K] [Field L] [Algebra K L] :
+theorem top_fixedByFinite {K L : Type*} [Field K] [Field L] [Algebra K L] :
     ⊤ ∈ fixedByFinite K L :=
   ⟨⊥, IntermediateField.finiteDimensional_bot K L, IntermediateField.fixingSubgroup.bot⟩
 #align top_fixed_by_finite top_fixedByFinite
 
 /-- If `E1` and `E2` are finite-dimensional intermediate fields, then so is their compositum.
 This rephrases a result already in mathlib so that it is compatible with our type classes -/
-theorem finiteDimensional_sup {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem finiteDimensional_sup {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E1 E2 : IntermediateField K L) (_ : FiniteDimensional K E1) (_ : FiniteDimensional K E2) :
     FiniteDimensional K (↥(E1 ⊔ E2)) :=
   IntermediateField.finiteDimensional_sup E1 E2
 #align finite_dimensional_sup finiteDimensional_sup
 
 /-- An element of `L ≃ₐ[K] L` is in `Gal(L/E)` if and only if it fixes every element of `E`-/
-theorem IntermediateField.mem_fixingSubgroup_iff {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem IntermediateField.mem_fixingSubgroup_iff {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) (σ : L ≃ₐ[K] L) : σ ∈ E.fixingSubgroup ↔ ∀ x : L, x ∈ E → σ x = x :=
   ⟨fun hσ x hx => hσ ⟨x, hx⟩, fun h ⟨x, hx⟩ => h x hx⟩
 #align intermediate_field.mem_fixing_subgroup_iff IntermediateField.mem_fixingSubgroup_iff
 
 /-- The map `E ↦ Gal(L/E)` is inclusion-reversing -/
-theorem IntermediateField.fixingSubgroup.antimono {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem IntermediateField.fixingSubgroup.antimono {K L : Type*} [Field K] [Field L] [Algebra K L]
     {E1 E2 : IntermediateField K L} (h12 : E1 ≤ E2) : E2.fixingSubgroup ≤ E1.fixingSubgroup := by
   rintro σ hσ ⟨x, hx⟩
   exact hσ ⟨x, h12 hx⟩
@@ -137,7 +137,7 @@ theorem IntermediateField.fixingSubgroup.antimono {K L : Type _} [Field K] [Fiel
 
 /-- Given a field extension `L/K`, `galBasis K L` is the filter basis on `L ≃ₐ[K] L` whose sets
 are `Gal(L/E)` for intermediate fields `E` with `E/K` finite dimensional -/
-def galBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L) where
+def galBasis (K L : Type*) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L) where
   sets := (fun g => g.carrier) '' fixedByFinite K L
   nonempty := ⟨⊤, ⊤, top_fixedByFinite, rfl⟩
   inter_sets := by
@@ -152,14 +152,14 @@ def galBasis (K L : Type _) [Field K] [Field L] [Algebra K L] : FilterBasis (L 
 
 /-- A subset of `L ≃ₐ[K] L` is a member of `galBasis K L` if and only if it is the underlying set
 of `Gal(L/E)` for some finite subextension `E/K`-/
-theorem mem_galBasis_iff (K L : Type _) [Field K] [Field L] [Algebra K L] (U : Set (L ≃ₐ[K] L)) :
+theorem mem_galBasis_iff (K L : Type*) [Field K] [Field L] [Algebra K L] (U : Set (L ≃ₐ[K] L)) :
     U ∈ galBasis K L ↔ U ∈ (fun g => g.carrier) '' fixedByFinite K L :=
   Iff.rfl
 #align mem_gal_basis_iff mem_galBasis_iff
 
 /-- For a field extension `L/K`, `galGroupBasis K L` is the group filter basis on `L ≃ₐ[K] L`
 whose sets are `Gal(L/E)` for finite subextensions `E/K` -/
-def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] :
+def galGroupBasis (K L : Type*) [Field K] [Field L] [Algebra K L] :
     GroupFilterBasis (L ≃ₐ[K] L) where
   toFilterBasis := galBasis K L
   one' := fun ⟨H, _, h2⟩ => h2 ▸ H.one_mem
@@ -193,13 +193,13 @@ def galGroupBasis (K L : Type _) [Field K] [Field L] [Algebra K L] :
 
 /-- For a field extension `L/K`, `krullTopology K L` is the topological space structure on
 `L ≃ₐ[K] L` induced by the group filter basis `galGroupBasis K L` -/
-instance krullTopology (K L : Type _) [Field K] [Field L] [Algebra K L] :
+instance krullTopology (K L : Type*) [Field K] [Field L] [Algebra K L] :
     TopologicalSpace (L ≃ₐ[K] L) :=
   GroupFilterBasis.topology (galGroupBasis K L)
 #align krull_topology krullTopology
 
 /-- For a field extension `L/K`, the Krull topology on `L ≃ₐ[K] L` makes it a topological group. -/
-instance (K L : Type _) [Field K] [Field L] [Algebra K L] : TopologicalGroup (L ≃ₐ[K] L) :=
+instance (K L : Type*) [Field K] [Field L] [Algebra K L] : TopologicalGroup (L ≃ₐ[K] L) :=
   GroupFilterBasis.isTopologicalGroup (galGroupBasis K L)
 
 section KrullT2
@@ -208,7 +208,7 @@ open scoped Topology Filter
 
 /-- Let `L/E/K` be a tower of fields with `E/K` finite. Then `Gal(L/E)` is an open subgroup of
   `L ≃ₐ[K] L`. -/
-theorem IntermediateField.fixingSubgroup_isOpen {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem IntermediateField.fixingSubgroup_isOpen {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) [FiniteDimensional K E] :
     IsOpen (E.fixingSubgroup : Set (L ≃ₐ[K] L)) := by
   have h_basis : E.fixingSubgroup.carrier ∈ galGroupBasis K L :=
@@ -219,14 +219,14 @@ theorem IntermediateField.fixingSubgroup_isOpen {K L : Type _} [Field K] [Field
 
 /-- Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is
   closed. -/
-theorem IntermediateField.fixingSubgroup_isClosed {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem IntermediateField.fixingSubgroup_isClosed {K L : Type*} [Field K] [Field L] [Algebra K L]
     (E : IntermediateField K L) [FiniteDimensional K E] :
     IsClosed (E.fixingSubgroup : Set (L ≃ₐ[K] L)) :=
   OpenSubgroup.isClosed ⟨E.fixingSubgroup, E.fixingSubgroup_isOpen⟩
 #align intermediate_field.fixing_subgroup_is_closed IntermediateField.fixingSubgroup_isClosed
 
 /-- If `L/K` is an algebraic extension, then the Krull topology on `L ≃ₐ[K] L` is Hausdorff. -/
-theorem krullTopology_t2 {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : T2Space (L ≃ₐ[K] L) :=
   { t2 := fun f g hfg => by
       let φ := f⁻¹ * g
@@ -265,7 +265,7 @@ section TotallyDisconnected
 
 /-- If `L/K` is an algebraic field extension, then the Krull topology on `L ≃ₐ[K] L` is
   totally disconnected. -/
-theorem krullTopology_totallyDisconnected {K L : Type _} [Field K] [Field L] [Algebra K L]
+theorem krullTopology_totallyDisconnected {K L : Type*} [Field K] [Field L] [Algebra K L]
     (h_int : Algebra.IsIntegral K L) : IsTotallyDisconnected (Set.univ : Set (L ≃ₐ[K] L)) := by
   apply isTotallyDisconnected_of_clopen_set
   intro σ τ h_diff

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Sebastian Monnet. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
-
-! This file was ported from Lean 3 source module field_theory.krull_topology
-! leanprover-community/mathlib commit 039a089d2a4b93c761b234f3e5f5aeb752bac60f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.FieldTheory.Galois
 import Mathlib.Topology.Algebra.FilterBasis
 import Mathlib.Topology.Algebra.OpenSubgroup
 import Mathlib.Tactic.ByContra
 
+#align_import field_theory.krull_topology from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
+
 /-!
 # Krull topology
 

@@ -94,7 +94,7 @@ def fixedByFinite (K L : Type _) [Field K] [Field L] [Algebra K L] : Set (Subgro
   IntermediateField.fixingSubgroup '' finiteExts K L
 #align fixed_by_finite fixedByFinite
 
-/-- For an field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
+/-- For a field extension `L/K`, the intermediate field `K` is finite-dimensional over `K` -/
 theorem IntermediateField.finiteDimensional_bot (K L : Type _) [Field K] [Field L] [Algebra K L] :
     FiniteDimensional K (⊥ : IntermediateField K L) :=
   finiteDimensional_of_rank_eq_one IntermediateField.rank_bot