topology.continuous_function.stone_weierstrass ⟷ Mathlib.Topology.ContinuousFunction.StoneWeierstrass

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

@@ -286,7 +286,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   -- We rewrite into this particular form,
   -- so that simp lemmas about inequalities involving `finset.inf'` can fire.
   rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros;
-      simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm']]
+      simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]]
   fconstructor
   · dsimp [k]
     simp only [Finset.inf'_lt_iff, ContinuousMap.inf'_apply]

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 -/
 import Topology.ContinuousFunction.Weierstrass
-import Data.IsROrC.Basic
+import Analysis.RCLike.Basic
 
 #align_import topology.continuous_function.stone_weierstrass from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
 
@@ -194,8 +194,8 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 open scoped Topology
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (f g «expr ∈ » L) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 #print ContinuousMap.sublattice_closure_eq_top /-
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
@@ -375,12 +375,12 @@ theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra
 
 end ContinuousMap
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
 -- Redefine `X`, since for the next few lemmas it need not be compact
-variable {𝕜 : Type _} {X : Type _} [IsROrC 𝕜] [TopologicalSpace X]
+variable {𝕜 : Type _} {X : Type _} [RCLike 𝕜] [TopologicalSpace X]
 
 namespace ContinuousMap
 
@@ -418,10 +418,10 @@ end ContinuousMap
 
 open ContinuousMap
 
-#print Subalgebra.SeparatesPoints.isROrC_to_real /-
+#print Subalgebra.SeparatesPoints.rclike_to_real /-
 /-- If a conjugation-invariant subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
 of its purely real-valued elements also separates points. -/
-theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜)}
+theorem Subalgebra.SeparatesPoints.rclike_to_real {A : Subalgebra 𝕜 C(X, 𝕜)}
     (hA : A.SeparatesPoints) (hA' : ConjInvariantSubalgebra (A.restrictScalars ℝ)) :
     ((A.restrictScalars ℝ).comap
         (ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints :=
@@ -438,18 +438,18 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
     simp only [coe_smul, coe_one, Pi.smul_apply, Pi.one_apply, Algebra.id.smul_eq_mul, mul_one,
       const_apply]
   -- Consider now the function `λ x, |f x - f x₂| ^ 2`
-  refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_norm_sq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
+  refine' ⟨_, ⟨(⟨RCLike.normSq, continuous_norm_sq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
     convert (A.restrict_scalars ℝ).hMul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
     ext1
     rw [mul_comm]
-    exact (IsROrC.mul_conj _).symm
+    exact (RCLike.mul_conj _).symm
   · -- And it also separates the points `x₁`, `x₂`
     have : f x₁ - f x₂ ≠ 0 := sub_ne_zero.mpr hf
     simpa only [comp_apply, coe_sub, coe_const, Pi.sub_apply, coe_mk, sub_self, map_zero, Ne.def,
       norm_sq_eq_zero] using this
-#align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.isROrC_to_real
+#align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.rclike_to_real
 -/
 
 variable [CompactSpace X]
@@ -487,17 +487,17 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
   -- In particular, for a function `f` in `C(X, 𝕜)`, the real and imaginary parts of `f` are in the
   -- closure of `A`
   intro f
-  let f_re : C(X, ℝ) := (⟨IsROrC.re, is_R_or_C.re_clm.continuous⟩ : C(𝕜, ℝ)).comp f
-  let f_im : C(X, ℝ) := (⟨IsROrC.im, is_R_or_C.im_clm.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_re : C(X, ℝ) := (⟨RCLike.re, is_R_or_C.re_clm.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_im : C(X, ℝ) := (⟨RCLike.im, is_R_or_C.im_clm.continuous⟩ : C(𝕜, ℝ)).comp f
   have h_f_re : I f_re ∈ A.topological_closure := key ⟨f_re, rfl⟩
   have h_f_im : I f_im ∈ A.topological_closure := key ⟨f_im, rfl⟩
   -- So `f_re + I • f_im` is in the closure of `A`
-  convert A.topological_closure.add_mem h_f_re (A.topological_closure.smul_mem h_f_im IsROrC.i)
+  convert A.topological_closure.add_mem h_f_re (A.topological_closure.smul_mem h_f_im RCLike.i)
   -- And this, of course, is just `f`
   ext
   apply Eq.symm
-  simp [I, mul_comm IsROrC.i _]
+  simp [I, mul_comm RCLike.i _]
 #align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPointsₓ
 
-end IsROrC
+end RCLike
 

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

@@ -223,7 +223,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     and finally using compactness to produce the desired function `h`
     as a maximum over finitely many `x` of a minimum over finitely many `y` of the `g x y`.
     -/
-  dsimp only [Set.SeparatesPointsStrongly] at sep 
+  dsimp only [Set.SeparatesPointsStrongly] at sep
   choose g hg w₁ w₂ using sep f
   -- For each `x y`, we define `U x y` to be `{z | f z - ε < g x y z}`,
   -- and observe this is a neighbourhood of `y`.
@@ -349,9 +349,9 @@ theorem exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalge
   by
   have w :=
     mem_closure_iff_frequently.mp (continuous_map_mem_subalgebra_closure_of_separates_points A w f)
-  rw [metric.nhds_basis_ball.frequently_iff] at w 
+  rw [metric.nhds_basis_ball.frequently_iff] at w
   obtain ⟨g, H, m⟩ := w ε Pos
-  rw [Metric.mem_ball, dist_eq_norm] at H 
+  rw [Metric.mem_ball, dist_eq_norm] at H
   exact ⟨⟨g, m⟩, H⟩
 #align continuous_map.exists_mem_subalgebra_near_continuous_map_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints
 -/
@@ -369,7 +369,7 @@ theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra
   by
   obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuous_map_of_separates_points A w ⟨f, c⟩ ε Pos
   use g
-  rwa [norm_lt_iff _ Pos] at b 
+  rwa [norm_lt_iff _ Pos] at b
 #align continuous_map.exists_mem_subalgebra_near_continuous_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
 -/
 
@@ -401,7 +401,7 @@ theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
   by
   rintro _ ⟨f, hf, rfl⟩
   change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ
-  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ at hf 
+  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ at hf
   rw [Subalgebra.mem_restrictScalars] at hf ⊢
   apply Algebra.adjoin_induction hf
   · exact fun g hg => Algebra.subset_adjoin (hS g hg)

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

@@ -386,17 +386,17 @@ namespace ContinuousMap
 
 /-- A real subalgebra of `C(X, 𝕜)` is `conj_invariant`, if it contains all its conjugates. -/
 def ConjInvariantSubalgebra (A : Subalgebra ℝ C(X, 𝕜)) : Prop :=
-  A.map (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) ≤ A
+  A.map (conjAe.toAlgHom.compLeftContinuous ℝ conjCLE.Continuous) ≤ A
 #align continuous_map.conj_invariant_subalgebra ContinuousMap.ConjInvariantSubalgebra
 
 theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjInvariantSubalgebra A)
-    {f : C(X, 𝕜)} (hf : f ∈ A) : (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ A :=
+    {f : C(X, 𝕜)} (hf : f ∈ A) : (conjAe.toAlgHom.compLeftContinuous ℝ conjCLE.Continuous) f ∈ A :=
   hA ⟨f, hf, rfl⟩
 #align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
 
 /-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
 theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
-    (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ S) :
+    (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCLE.Continuous) f ∈ S) :
     ConjInvariantSubalgebra ((Algebra.adjoin 𝕜 S).restrictScalars ℝ) :=
   by
   rintro _ ⟨f, hf, rfl⟩

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

@@ -3,8 +3,8 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 -/
-import Mathbin.Topology.ContinuousFunction.Weierstrass
-import Mathbin.Data.IsROrC.Basic
+import Topology.ContinuousFunction.Weierstrass
+import Data.IsROrC.Basic
 
 #align_import topology.continuous_function.stone_weierstrass from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
 
@@ -194,8 +194,8 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 open scoped Topology
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 #print ContinuousMap.sublattice_closure_eq_top /-
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -441,7 +441,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
   refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_norm_sq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
-    convert (A.restrict_scalars ℝ).mul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
+    convert (A.restrict_scalars ℝ).hMul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
     ext1
     rw [mul_comm]
     exact (IsROrC.mul_conj _).symm

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

@@ -384,21 +384,16 @@ variable {𝕜 : Type _} {X : Type _} [IsROrC 𝕜] [TopologicalSpace X]
 
 namespace ContinuousMap
 
-#print ContinuousMap.ConjInvariantSubalgebra /-
 /-- A real subalgebra of `C(X, 𝕜)` is `conj_invariant`, if it contains all its conjugates. -/
 def ConjInvariantSubalgebra (A : Subalgebra ℝ C(X, 𝕜)) : Prop :=
   A.map (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) ≤ A
 #align continuous_map.conj_invariant_subalgebra ContinuousMap.ConjInvariantSubalgebra
--/
 
-#print ContinuousMap.mem_conjInvariantSubalgebra /-
 theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjInvariantSubalgebra A)
     {f : C(X, 𝕜)} (hf : f ∈ A) : (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ A :=
   hA ⟨f, hf, rfl⟩
 #align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
--/
 
-#print ContinuousMap.subalgebraConjInvariant /-
 /-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
 theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
     (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ S) :
@@ -418,7 +413,6 @@ theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
     convert Subalgebra.mul_mem _ hf hg
     exact AlgHom.map_mul _ f g
 #align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebraConjInvariant
--/
 
 end ContinuousMap
 
@@ -460,12 +454,11 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
 
 variable [CompactSpace X]
 
-#print ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints /-
 /-- The Stone-Weierstrass approximation theorem, `is_R_or_C` version,
 that a subalgebra `A` of `C(X, 𝕜)`, where `X` is a compact topological space and `is_R_or_C 𝕜`,
 is dense if it is conjugation-invariant and separates points.
 -/
-theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints
+theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
     (A : Subalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints)
     (hA' : ConjInvariantSubalgebra (A.restrictScalars ℝ)) : A.topologicalClosure = ⊤ :=
   by
@@ -504,8 +497,7 @@ theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPo
   ext
   apply Eq.symm
   simp [I, mul_comm IsROrC.i _]
-#align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints
--/
+#align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPointsₓ
 
 end IsROrC
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
-
-! This file was ported from Lean 3 source module topology.continuous_function.stone_weierstrass
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.ContinuousFunction.Weierstrass
 import Mathbin.Data.IsROrC.Basic
 
+#align_import topology.continuous_function.stone_weierstrass from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
 /-!
 # The Stone-Weierstrass theorem
 
@@ -197,8 +194,8 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 open scoped Topology
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 #print ContinuousMap.sublattice_closure_eq_top /-
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)

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

@@ -57,18 +57,23 @@ variable {X : Type _} [TopologicalSpace X] [CompactSpace X]
 
 open scoped Polynomial
 
+#print ContinuousMap.attachBound /-
 /-- Turn a function `f : C(X, ℝ)` into a continuous map into `set.Icc (-‖f‖) (‖f‖)`,
 thereby explicitly attaching bounds.
 -/
 def attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖)
     where toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩
 #align continuous_map.attach_bound ContinuousMap.attachBound
+-/
 
+#print ContinuousMap.attachBound_apply_coe /-
 @[simp]
 theorem attachBound_apply_coe (f : C(X, ℝ)) (x : X) : ((attachBound f) x : ℝ) = f x :=
   rfl
 #align continuous_map.attach_bound_apply_coe ContinuousMap.attachBound_apply_coe
+-/
 
+#print ContinuousMap.polynomial_comp_attachBound /-
 theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
     (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =
       Polynomial.aeval f g :=
@@ -78,7 +83,9 @@ theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g :
     Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
     Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
 #align continuous_map.polynomial_comp_attach_bound ContinuousMap.polynomial_comp_attachBound
+-/
 
+#print ContinuousMap.polynomial_comp_attachBound_mem /-
 /-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial
 gives another function in `A`.
 
@@ -93,7 +100,9 @@ theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (
   rw [polynomial_comp_attach_bound]
   apply SetLike.coe_mem
 #align continuous_map.polynomial_comp_attach_bound_mem ContinuousMap.polynomial_comp_attachBound_mem
+-/
 
+#print ContinuousMap.comp_attachBound_mem_closure /-
 theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
     (p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound f) ∈ A.topologicalClosure :=
   by
@@ -116,7 +125,9 @@ theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
     Polynomial.toContinuousMapOnAlgHom_apply]
   apply polynomial_comp_attach_bound_mem
 #align continuous_map.comp_attach_bound_mem_closure ContinuousMap.comp_attachBound_mem_closure
+-/
 
+#print ContinuousMap.abs_mem_subalgebra_closure /-
 theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
     (f : C(X, ℝ)).abs ∈ A.topologicalClosure :=
   by
@@ -126,7 +137,9 @@ theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
   change abs.comp f' ∈ A.topological_closure
   apply comp_attach_bound_mem_closure
 #align continuous_map.abs_mem_subalgebra_closure ContinuousMap.abs_mem_subalgebra_closure
+-/
 
+#print ContinuousMap.inf_mem_subalgebra_closure /-
 theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
     (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure :=
   by
@@ -140,7 +153,9 @@ theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
       _
   exact_mod_cast abs_mem_subalgebra_closure A _
 #align continuous_map.inf_mem_subalgebra_closure ContinuousMap.inf_mem_subalgebra_closure
+-/
 
+#print ContinuousMap.inf_mem_closed_subalgebra /-
 theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
     (f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A :=
   by
@@ -150,7 +165,9 @@ theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
   erw [closure_eq_iff_isClosed]
   exact h
 #align continuous_map.inf_mem_closed_subalgebra ContinuousMap.inf_mem_closed_subalgebra
+-/
 
+#print ContinuousMap.sup_mem_subalgebra_closure /-
 theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
     (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure :=
   by
@@ -164,7 +181,9 @@ theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
       _
   exact_mod_cast abs_mem_subalgebra_closure A _
 #align continuous_map.sup_mem_subalgebra_closure ContinuousMap.sup_mem_subalgebra_closure
+-/
 
+#print ContinuousMap.sup_mem_closed_subalgebra /-
 theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
     (f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A :=
   by
@@ -174,11 +193,13 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
   erw [closure_eq_iff_isClosed]
   exact h
 #align continuous_map.sup_mem_closed_subalgebra ContinuousMap.sup_mem_closed_subalgebra
+-/
 
 open scoped Topology
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+#print ContinuousMap.sublattice_closure_eq_top /-
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     (inf_mem : ∀ (f) (_ : f ∈ L) (g) (_ : g ∈ L), f ⊓ g ∈ L)
@@ -278,7 +299,9 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     intro x xm
     apply lt_h
 #align continuous_map.sublattice_closure_eq_top ContinuousMap.sublattice_closure_eq_top
+-/
 
+#print ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints /-
 /-- The **Stone-Weierstrass Approximation Theorem**,
 that a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,
 is dense if it separates points.
@@ -300,7 +323,9 @@ theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra
         (Subalgebra.separatesPoints_monotone A.le_topological_closure w))
   · simp
 #align continuous_map.subalgebra_topological_closure_eq_top_of_separates_points ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints
+-/
 
+#print ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints /-
 /-- An alternative statement of the Stone-Weierstrass theorem.
 
 If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
@@ -312,7 +337,9 @@ theorem continuousMap_mem_subalgebra_closure_of_separatesPoints (A : Subalgebra
   rw [subalgebra_topological_closure_eq_top_of_separates_points A w]
   simp
 #align continuous_map.continuous_map_mem_subalgebra_closure_of_separates_points ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints
+-/
 
+#print ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints /-
 /-- An alternative statement of the Stone-Weierstrass theorem,
 for those who like their epsilons.
 
@@ -330,7 +357,9 @@ theorem exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalge
   rw [Metric.mem_ball, dist_eq_norm] at H 
   exact ⟨⟨g, m⟩, H⟩
 #align continuous_map.exists_mem_subalgebra_near_continuous_map_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints
+-/
 
+#print ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints /-
 /-- An alternative statement of the Stone-Weierstrass theorem,
 for those who like their epsilons and don't like bundled continuous functions.
 
@@ -345,6 +374,7 @@ theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra
   use g
   rwa [norm_lt_iff _ Pos] at b 
 #align continuous_map.exists_mem_subalgebra_near_continuous_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
+-/
 
 end ContinuousMap
 
@@ -357,16 +387,21 @@ variable {𝕜 : Type _} {X : Type _} [IsROrC 𝕜] [TopologicalSpace X]
 
 namespace ContinuousMap
 
+#print ContinuousMap.ConjInvariantSubalgebra /-
 /-- A real subalgebra of `C(X, 𝕜)` is `conj_invariant`, if it contains all its conjugates. -/
 def ConjInvariantSubalgebra (A : Subalgebra ℝ C(X, 𝕜)) : Prop :=
   A.map (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) ≤ A
 #align continuous_map.conj_invariant_subalgebra ContinuousMap.ConjInvariantSubalgebra
+-/
 
+#print ContinuousMap.mem_conjInvariantSubalgebra /-
 theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjInvariantSubalgebra A)
     {f : C(X, 𝕜)} (hf : f ∈ A) : (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ A :=
   hA ⟨f, hf, rfl⟩
 #align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
+-/
 
+#print ContinuousMap.subalgebraConjInvariant /-
 /-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
 theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
     (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ S) :
@@ -386,11 +421,13 @@ theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
     convert Subalgebra.mul_mem _ hf hg
     exact AlgHom.map_mul _ f g
 #align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebraConjInvariant
+-/
 
 end ContinuousMap
 
 open ContinuousMap
 
+#print Subalgebra.SeparatesPoints.isROrC_to_real /-
 /-- If a conjugation-invariant subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
 of its purely real-valued elements also separates points. -/
 theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜)}
@@ -422,9 +459,11 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
     simpa only [comp_apply, coe_sub, coe_const, Pi.sub_apply, coe_mk, sub_self, map_zero, Ne.def,
       norm_sq_eq_zero] using this
 #align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.isROrC_to_real
+-/
 
 variable [CompactSpace X]
 
+#print ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints /-
 /-- The Stone-Weierstrass approximation theorem, `is_R_or_C` version,
 that a subalgebra `A` of `C(X, 𝕜)`, where `X` is a compact topological space and `is_R_or_C 𝕜`,
 is dense if it is conjugation-invariant and separates points.
@@ -469,6 +508,7 @@ theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPo
   apply Eq.symm
   simp [I, mul_comm IsROrC.i _]
 #align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints
+-/
 
 end IsROrC
 

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

@@ -177,8 +177,8 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 open scoped Topology
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     (inf_mem : ∀ (f) (_ : f ∈ L) (g) (_ : g ∈ L), f ⊓ g ∈ L)

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 
 ! This file was ported from Lean 3 source module topology.continuous_function.stone_weierstrass
-! leanprover-community/mathlib commit 16e59248c0ebafabd5d071b1cd41743eb8698ffb
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Data.IsROrC.Basic
 /-!
 # The Stone-Weierstrass theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 If a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,
 separates points, then it is dense.
 

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

@@ -206,7 +206,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   choose g hg w₁ w₂ using sep f
   -- For each `x y`, we define `U x y` to be `{z | f z - ε < g x y z}`,
   -- and observe this is a neighbourhood of `y`.
-  let U : X → X → Set X := fun x y => { z | f z - ε < g x y z }
+  let U : X → X → Set X := fun x y => {z | f z - ε < g x y z}
   have U_nhd_y : ∀ x y, U x y ∈ 𝓝 y := by
     intro x y
     refine' IsOpen.mem_nhds _ _
@@ -239,7 +239,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     exact ⟨y, ym, zm⟩
   have h_eq : ∀ x, h x x = f x := by intro x; simp [coeFn_coe_base', w₁]
   -- For each `x`, we define `W x` to be `{z | h x z < f z + ε}`,
-  let W : ∀ x, Set X := fun x => { z | h x z < f z + ε }
+  let W : ∀ x, Set X := fun x => {z | h x z < f z + ε}
   -- This is still a neighbourhood of `x`.
   have W_nhd : ∀ x, W x ∈ 𝓝 x := by
     intro x
@@ -289,7 +289,8 @@ theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra
   apply SetLike.ext'
   let L := A.topological_closure
   have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topological_closure A.one_mem⟩
-  convert sublattice_closure_eq_top (L : Set C(X, ℝ)) n
+  convert
+    sublattice_closure_eq_top (L : Set C(X, ℝ)) n
       (fun f fm g gm => inf_mem_closed_subalgebra L A.is_closed_topological_closure ⟨f, fm⟩ ⟨g, gm⟩)
       (fun f fm g gm => sup_mem_closed_subalgebra L A.is_closed_topological_closure ⟨f, fm⟩ ⟨g, gm⟩)
       (Subalgebra.SeparatesPoints.strongly
@@ -409,7 +410,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
   refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_norm_sq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
-    convert(A.restrict_scalars ℝ).mul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
+    convert (A.restrict_scalars ℝ).mul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
     ext1
     rw [mul_comm]
     exact (IsROrC.mul_conj _).symm

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

@@ -202,7 +202,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     and finally using compactness to produce the desired function `h`
     as a maximum over finitely many `x` of a minimum over finitely many `y` of the `g x y`.
     -/
-  dsimp only [Set.SeparatesPointsStrongly] at sep
+  dsimp only [Set.SeparatesPointsStrongly] at sep 
   choose g hg w₁ w₂ using sep f
   -- For each `x y`, we define `U x y` to be `{z | f z - ε < g x y z}`,
   -- and observe this is a neighbourhood of `y`.
@@ -264,7 +264,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   intro z
   -- We rewrite into this particular form,
   -- so that simp lemmas about inequalities involving `finset.inf'` can fire.
-  rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros ;
+  rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros;
       simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm']]
   fconstructor
   · dsimp [k]
@@ -321,9 +321,9 @@ theorem exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalge
   by
   have w :=
     mem_closure_iff_frequently.mp (continuous_map_mem_subalgebra_closure_of_separates_points A w f)
-  rw [metric.nhds_basis_ball.frequently_iff] at w
+  rw [metric.nhds_basis_ball.frequently_iff] at w 
   obtain ⟨g, H, m⟩ := w ε Pos
-  rw [Metric.mem_ball, dist_eq_norm] at H
+  rw [Metric.mem_ball, dist_eq_norm] at H 
   exact ⟨⟨g, m⟩, H⟩
 #align continuous_map.exists_mem_subalgebra_near_continuous_map_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints
 
@@ -339,7 +339,7 @@ theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra
   by
   obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuous_map_of_separates_points A w ⟨f, c⟩ ε Pos
   use g
-  rwa [norm_lt_iff _ Pos] at b
+  rwa [norm_lt_iff _ Pos] at b 
 #align continuous_map.exists_mem_subalgebra_near_continuous_of_separates_points ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints
 
 end ContinuousMap
@@ -370,8 +370,8 @@ theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
   by
   rintro _ ⟨f, hf, rfl⟩
   change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ
-  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ at hf
-  rw [Subalgebra.mem_restrictScalars] at hf⊢
+  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ at hf 
+  rw [Subalgebra.mem_restrictScalars] at hf ⊢
   apply Algebra.adjoin_induction hf
   · exact fun g hg => Algebra.subset_adjoin (hS g hg)
   · exact fun c => Subalgebra.algebraMap_mem _ (starRingEnd 𝕜 c)

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

@@ -52,7 +52,7 @@ namespace ContinuousMap
 
 variable {X : Type _} [TopologicalSpace X] [CompactSpace X]
 
-open Polynomial
+open scoped Polynomial
 
 /-- Turn a function `f : C(X, ℝ)` into a continuous map into `set.Icc (-‖f‖) (‖f‖)`,
 thereby explicitly attaching bounds.
@@ -172,7 +172,7 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
   exact h
 #align continuous_map.sup_mem_closed_subalgebra ContinuousMap.sup_mem_closed_subalgebra
 
-open Topology
+open scoped Topology
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/

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

@@ -237,9 +237,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     dsimp [h]
     simp only [coeFn_coe_base', Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff]
     exact ⟨y, ym, zm⟩
-  have h_eq : ∀ x, h x x = f x := by
-    intro x
-    simp [coeFn_coe_base', w₁]
+  have h_eq : ∀ x, h x x = f x := by intro x; simp [coeFn_coe_base', w₁]
   -- For each `x`, we define `W x` to be `{z | h x z < f z + ε}`,
   let W : ∀ x, Set X := fun x => { z | h x z < f z + ε }
   -- This is still a neighbourhood of `x`.
@@ -266,9 +264,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   intro z
   -- We rewrite into this particular form,
   -- so that simp lemmas about inequalities involving `finset.inf'` can fire.
-  rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a
-      by
-      intros
+  rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by intros ;
       simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm']]
   fconstructor
   · dsimp [k]
@@ -406,7 +402,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
   have hFA : F ∈ A :=
     by
     refine' A.sub_mem hfA (@Eq.subst _ (· ∈ A) _ _ _ <| A.smul_mem A.one_mem <| f x₂)
-    ext1
+    ext1;
     simp only [coe_smul, coe_one, Pi.smul_apply, Pi.one_apply, Algebra.id.smul_eq_mul, mul_one,
       const_apply]
   -- Consider now the function `λ x, |f x - f x₂| ^ 2`

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -396,7 +396,7 @@ of its purely real-valued elements also separates points. -/
 theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜)}
     (hA : A.SeparatesPoints) (hA' : ConjInvariantSubalgebra (A.restrictScalars ℝ)) :
     ((A.restrictScalars ℝ).comap
-        (ofRealAm.compLeftContinuous ℝ continuous_of_real)).SeparatesPoints :=
+        (ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints :=
   by
   intro x₁ x₂ hx
   -- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

@@ -368,7 +368,7 @@ theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjIn
 #align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
 
 /-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
-theorem subalgebra_conj_invariant {S : Set C(X, 𝕜)}
+theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
     (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ S) :
     ConjInvariantSubalgebra ((Algebra.adjoin 𝕜 S).restrictScalars ℝ) :=
   by
@@ -385,7 +385,7 @@ theorem subalgebra_conj_invariant {S : Set C(X, 𝕜)}
   · intro f g hf hg
     convert Subalgebra.mul_mem _ hf hg
     exact AlgHom.map_mul _ f g
-#align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebra_conj_invariant
+#align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebraConjInvariant
 
 end ContinuousMap
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a4f99eae998680d3a2c240da4a2b16354c85ee49

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 
 ! This file was ported from Lean 3 source module topology.continuous_function.stone_weierstrass
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 16e59248c0ebafabd5d071b1cd41743eb8698ffb
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -202,10 +202,8 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     and finally using compactness to produce the desired function `h`
     as a maximum over finitely many `x` of a minimum over finitely many `y` of the `g x y`.
     -/
-  dsimp [Set.SeparatesPointsStrongly] at sep
-  let g : X → X → L := fun x y => (sep f x y).some
-  have w₁ : ∀ x y, g x y x = f x := fun x y => (sep f x y).choose_spec.1
-  have w₂ : ∀ x y, g x y y = f y := fun x y => (sep f x y).choose_spec.2
+  dsimp only [Set.SeparatesPointsStrongly] at sep
+  choose g hg w₁ w₂ using sep f
   -- For each `x y`, we define `U x y` to be `{z | f z - ε < g x y z}`,
   -- and observe this is a neighbourhood of `y`.
   let U : X → X → Set X := fun x y => { z | f z - ε < g x y z }
@@ -232,7 +230,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   -- and `h x x = f x`.
   let h : ∀ x, L := fun x =>
     ⟨(ys x).sup' (ys_nonempty x) fun y => (g x y : C(X, ℝ)),
-      Finset.sup'_mem _ sup_mem _ _ _ fun y _ => (g x y).2⟩
+      Finset.sup'_mem _ sup_mem _ _ _ fun y _ => hg x y⟩
   have lt_h : ∀ x z, f z - ε < h x z := by
     intro x z
     obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z
@@ -241,7 +239,6 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     exact ⟨y, ym, zm⟩
   have h_eq : ∀ x, h x x = f x := by
     intro x
-    simp only [coeFn_coe_base'] at w₁
     simp [coeFn_coe_base', w₁]
   -- For each `x`, we define `W x` to be `{z | h x z < f z + ε}`,
   let W : ∀ x, Set X := fun x => { z | h x z < f z + ε }

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

@@ -296,8 +296,7 @@ theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra
   apply SetLike.ext'
   let L := A.topological_closure
   have n : Set.Nonempty (L : Set C(X, ℝ)) := ⟨(1 : C(X, ℝ)), A.le_topological_closure A.one_mem⟩
-  convert
-    sublattice_closure_eq_top (L : Set C(X, ℝ)) n
+  convert sublattice_closure_eq_top (L : Set C(X, ℝ)) n
       (fun f fm g gm => inf_mem_closed_subalgebra L A.is_closed_topological_closure ⟨f, fm⟩ ⟨g, gm⟩)
       (fun f fm g gm => sup_mem_closed_subalgebra L A.is_closed_topological_closure ⟨f, fm⟩ ⟨g, gm⟩)
       (Subalgebra.SeparatesPoints.strongly
@@ -417,7 +416,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
   refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_norm_sq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
-    convert (A.restrict_scalars ℝ).mul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
+    convert(A.restrict_scalars ℝ).mul_mem (mem_conj_invariant_subalgebra hA' hFA) hFA
     ext1
     rw [mul_comm]
     exact (IsROrC.mul_conj _).symm

mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931

@@ -372,7 +372,7 @@ theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjIn
 #align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
 
 /-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
-theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
+theorem subalgebra_conj_invariant {S : Set C(X, 𝕜)}
     (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.Continuous) f ∈ S) :
     ConjInvariantSubalgebra ((Algebra.adjoin 𝕜 S).restrictScalars ℝ) :=
   by
@@ -389,7 +389,7 @@ theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
   · intro f g hf hg
     convert Subalgebra.mul_mem _ hf hg
     exact AlgHom.map_mul _ f g
-#align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebraConjInvariant
+#align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebra_conj_invariant
 
 end ContinuousMap
 

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

@@ -174,8 +174,8 @@ theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 open Topology
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (f g «expr ∈ » L) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (f g «expr ∈ » L) -/
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     (inf_mem : ∀ (f) (_ : f ∈ L) (g) (_ : g ∈ L), f ⊓ g ∈ L)

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

RCLike is an analytic typeclass, hence should be under Analysis

@@ -3,7 +3,7 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 -/
-import Mathlib.Data.RCLike.Basic
+import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Topology.Algebra.StarSubalgebra
 import Mathlib.Topology.ContinuousFunction.Weierstrass
 

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -3,7 +3,7 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 -/
-import Mathlib.Data.IsROrC.Basic
+import Mathlib.Data.RCLike.Basic
 import Mathlib.Topology.Algebra.StarSubalgebra
 import Mathlib.Topology.ContinuousFunction.Weierstrass
 
@@ -331,12 +331,12 @@ theorem exists_mem_subalgebra_near_continuous_of_separatesPoints (A : Subalgebra
 
 end ContinuousMap
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
 -- Redefine `X`, since for the next lemma it need not be compact
-variable {𝕜 : Type*} {X : Type*} [IsROrC 𝕜] [TopologicalSpace X]
+variable {𝕜 : Type*} {X : Type*} [RCLike 𝕜] [TopologicalSpace X]
 
 open ContinuousMap
 
@@ -350,7 +350,7 @@ which didn't exist at the time Stone-Weierstrass was written. -/
 
 /-- If a star subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
 of its purely real-valued elements also separates points. -/
-theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
+theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
     (hA : A.SeparatesPoints) :
       ((A.restrictScalars ℝ).comap
         (ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by
@@ -370,15 +370,15 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X,
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
     convert (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A)
     ext1
-    simp [← IsROrC.mul_conj]
+    simp [← RCLike.mul_conj]
   · -- And it also separates the points `x₁`, `x₂`
     simpa [F] using sub_ne_zero.mpr hf
-#align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.isROrC_to_real
+#align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.rclike_to_real
 
 variable [CompactSpace X]
 
-/-- The Stone-Weierstrass approximation theorem, `IsROrC` version, that a star subalgebra `A` of
-`C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if it separates
+/-- The Stone-Weierstrass approximation theorem, `RCLike` version, that a star subalgebra `A` of
+`C(X, 𝕜)`, where `X` is a compact topological space and `RCLike 𝕜`, is dense if it separates
 points. -/
 theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
     (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by
@@ -391,10 +391,10 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
     -- Let `A₀` be the subalgebra of `C(X, ℝ)` consisting of `A`'s purely real elements; it is the
     -- preimage of `A` under `I`.  In this argument we only need its submodule structure.
     let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I
-    -- By `Subalgebra.SeparatesPoints.isROrC_to_real`, this subalgebra also separates points, so
+    -- By `Subalgebra.SeparatesPoints.rclike_to_real`, this subalgebra also separates points, so
     -- we may apply the real Stone-Weierstrass result to it.
     have SW : A₀.topologicalClosure = ⊤ :=
-      haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real
+      haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.rclike_to_real
       congr_arg Subalgebra.toSubmodule this
     rw [← Submodule.map_top, ← SW]
     -- So it suffices to prove that the image under `I` of the closure of `A₀` is contained in the
@@ -405,21 +405,21 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
   -- In particular, for a function `f` in `C(X, 𝕜)`, the real and imaginary parts of `f` are in the
   -- closure of `A`
   intro f
-  let f_re : C(X, ℝ) := (⟨IsROrC.re, IsROrC.reCLM.continuous⟩ : C(𝕜, ℝ)).comp f
-  let f_im : C(X, ℝ) := (⟨IsROrC.im, IsROrC.imCLM.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_re : C(X, ℝ) := (⟨RCLike.re, RCLike.reCLM.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_im : C(X, ℝ) := (⟨RCLike.im, RCLike.imCLM.continuous⟩ : C(𝕜, ℝ)).comp f
   have h_f_re : I f_re ∈ A.topologicalClosure := key ⟨f_re, rfl⟩
   have h_f_im : I f_im ∈ A.topologicalClosure := key ⟨f_im, rfl⟩
   -- So `f_re + I • f_im` is in the closure of `A`
-  have := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I)
+  have := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im RCLike.I)
   rw [StarSubalgebra.mem_toSubalgebra] at this
   convert this
   -- And this, of course, is just `f`
   ext
   apply Eq.symm
-  simp [I, f_re, f_im, mul_comm IsROrC.I _]
+  simp [I, f_re, f_im, mul_comm RCLike.I _]
 #align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPointsₓ
 
-end IsROrC
+end RCLike
 
 section PolynomialFunctions
 
@@ -438,7 +438,7 @@ theorem polynomialFunctions.topologicalClosure (s : Set ℝ)
 
 /-- The star subalgebra generated by polynomials functions is dense in `C(s, 𝕜)` when `s` is
 compact and `𝕜` is either `ℝ` or `ℂ`. -/
-theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type*} [IsROrC 𝕜] (s : Set 𝕜)
+theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type*} [RCLike 𝕜] (s : Set 𝕜)
     [CompactSpace s] : (polynomialFunctions s).starClosure.topologicalClosure = ⊤ :=
   ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints _
     (Subalgebra.separatesPoints_monotone le_sup_left (polynomialFunctions_separatesPoints s))
@@ -459,7 +459,7 @@ theorem ContinuousMap.algHom_ext_map_X {A : Type*} [Ring A]
 /-- Continuous star algebra homomorphisms from `C(s, 𝕜)` into a star `𝕜`-algebra `A` which agree
 at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. -/
 @[ext]
-theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type*} [IsROrC 𝕜] [Ring A] [StarRing A]
+theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type*} [RCLike 𝕜] [Ring A] [StarRing A]
     [Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace s]
     {φ ψ : C(s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous φ) (hψ : Continuous ψ)
     (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by

Sometimes, that line can be golfed into the next line. Inspired by a comment of @loefflerd; any decisions are my own.

@@ -171,7 +171,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     (sup_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
     closure L = ⊤ := by
   -- We start by boiling down to a statement about close approximation.
-  apply eq_top_iff.mpr
+  rw [eq_top_iff]
   rintro f -
   refine'
     Filter.Frequently.mem_closure

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

@@ -235,7 +235,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     · -- Porting note: mathlib3 `continuity` found `continuous_set_coe`
       apply isOpen_lt (continuous_set_coe _ _)
       continuity
-    · dsimp only [Set.mem_setOf_eq]
+    · dsimp only [W, Set.mem_setOf_eq]
       rw [h_eq]
       exact lt_add_of_pos_right _ pos
   -- Since `X` is compact, there is some finset `ys t`
@@ -284,7 +284,7 @@ theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra
       (fun f fm g gm => sup_mem_closed_subalgebra L A.isClosed_topologicalClosure ⟨f, fm⟩ ⟨g, gm⟩)
       (Subalgebra.SeparatesPoints.strongly
         (Subalgebra.separatesPoints_monotone A.le_topologicalClosure w))
-  · simp
+  · simp [L]
 #align continuous_map.subalgebra_topological_closure_eq_top_of_separates_points ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints
 
 /-- An alternative statement of the Stone-Weierstrass theorem.
@@ -372,7 +372,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X,
     ext1
     simp [← IsROrC.mul_conj]
   · -- And it also separates the points `x₁`, `x₂`
-    simpa using sub_ne_zero.mpr hf
+    simpa [F] using sub_ne_zero.mpr hf
 #align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.isROrC_to_real
 
 variable [CompactSpace X]
@@ -416,7 +416,7 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
   -- And this, of course, is just `f`
   ext
   apply Eq.symm
-  simp [mul_comm IsROrC.I _]
+  simp [I, f_re, f_im, mul_comm IsROrC.I _]
 #align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPointsₓ
 
 end IsROrC

With multiple changes, it is a good time to check if existing set_option maxHeartbeats and set_option synthInstance.maxHeartbeats remain necessary. This brings the number of files with such down from 23 to 9. Most are straight deletions though I did change one proof.

@@ -377,7 +377,6 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X,
 
 variable [CompactSpace X]
 
-set_option synthInstance.maxHeartbeats 30000 in
 /-- The Stone-Weierstrass approximation theorem, `IsROrC` version, that a star subalgebra `A` of
 `C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if it separates
 points. -/

Rename

• Complex.equivRealProdClm → Complex.equivRealProdCLM;
  • TODO: should this one use CLE?
• Complex.reClm → Complex.reCLM;
• Complex.imClm → Complex.imCLM;
• Complex.conjLie → Complex.conjLIE;
• Complex.conjCle → Complex.conjCLE;
• Complex.ofRealLi → Complex.ofRealLI;
• Complex.ofRealClm → Complex.ofRealCLM;
• fderivInnerClm → fderivInnerCLM;
• LinearPMap.adjointDomainMkClm → LinearPMap.adjointDomainMkCLM;
• LinearPMap.adjointDomainMkClmExtend → LinearPMap.adjointDomainMkCLMExtend;
• IsROrC.reClm → IsROrC.reCLM;
• IsROrC.imClm → IsROrC.imCLM;
• IsROrC.conjLie → IsROrC.conjLIE;
• IsROrC.conjCle → IsROrC.conjCLE;
• IsROrC.ofRealLi → IsROrC.ofRealLI;
• IsROrC.ofRealClm → IsROrC.ofRealCLM;
• MeasureTheory.condexpL1Clm → MeasureTheory.condexpL1CLM;
• algebraMapClm → algebraMapCLM;
• WeakDual.CharacterSpace.toClm → WeakDual.CharacterSpace.toCLM;
• BoundedContinuousFunction.evalClm → BoundedContinuousFunction.evalCLM;
• ContinuousMap.evalClm → ContinuousMap.evalCLM;
• TrivSqZeroExt.fstClm → TrivSqZeroExt.fstClm;
• TrivSqZeroExt.sndClm → TrivSqZeroExt.sndCLM;
• TrivSqZeroExt.inlClm → TrivSqZeroExt.inlCLM;
• TrivSqZeroExt.inrClm → TrivSqZeroExt.inrCLM

and related theorems.

@@ -385,7 +385,7 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
     (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by
   rw [StarSubalgebra.eq_top_iff]
   -- Let `I` be the natural inclusion of `C(X, ℝ)` into `C(X, 𝕜)`
-  let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ofRealClm.compLeftContinuous ℝ X
+  let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ofRealCLM.compLeftContinuous ℝ X
   -- The main point of the proof is that its range (i.e., every real-valued function) is contained
   -- in the closure of `A`
   have key : LinearMap.range I ≤ (A.toSubmodule.restrictScalars ℝ).topologicalClosure := by
@@ -400,14 +400,14 @@ theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoint
     rw [← Submodule.map_top, ← SW]
     -- So it suffices to prove that the image under `I` of the closure of `A₀` is contained in the
     -- closure of `A`, which follows by abstract nonsense
-    have h₁ := A₀.topologicalClosure_map ((@ofRealClm 𝕜 _).compLeftContinuousCompact X)
+    have h₁ := A₀.topologicalClosure_map ((@ofRealCLM 𝕜 _).compLeftContinuousCompact X)
     have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I
     exact h₁.trans (Submodule.topologicalClosure_mono h₂)
   -- In particular, for a function `f` in `C(X, 𝕜)`, the real and imaginary parts of `f` are in the
   -- closure of `A`
   intro f
-  let f_re : C(X, ℝ) := (⟨IsROrC.re, IsROrC.reClm.continuous⟩ : C(𝕜, ℝ)).comp f
-  let f_im : C(X, ℝ) := (⟨IsROrC.im, IsROrC.imClm.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_re : C(X, ℝ) := (⟨IsROrC.re, IsROrC.reCLM.continuous⟩ : C(𝕜, ℝ)).comp f
+  let f_im : C(X, ℝ) := (⟨IsROrC.im, IsROrC.imCLM.continuous⟩ : C(𝕜, ℝ)).comp f
   have h_f_re : I f_re ∈ A.topologicalClosure := key ⟨f_re, rfl⟩
   have h_f_im : I f_im ∈ A.topologicalClosure := key ⟨f_im, rfl⟩
   -- So `f_re + I • f_im` is in the closure of `A`

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

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

@@ -3,8 +3,9 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
 -/
-import Mathlib.Topology.ContinuousFunction.Weierstrass
 import Mathlib.Data.IsROrC.Basic
+import Mathlib.Topology.Algebra.StarSubalgebra
+import Mathlib.Topology.ContinuousFunction.Weierstrass
 
 #align_import topology.continuous_function.stone_weierstrass from "leanprover-community/mathlib"@"16e59248c0ebafabd5d071b1cd41743eb8698ffb"
 

and other simple lemmas

From LeanAPAP

@@ -364,16 +364,14 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X,
     simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one,
       const_apply]
   -- Consider now the function `fun x ↦ |f x - f x₂| ^ 2`
-  refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_normSq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
+  refine' ⟨_, ⟨⟨(‖F ·‖ ^ 2), by continuity⟩, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
     convert (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A)
     ext1
-    exact (IsROrC.mul_conj (K := 𝕜) _).symm
+    simp [← IsROrC.mul_conj]
   · -- And it also separates the points `x₁`, `x₂`
-    have : f x₁ - f x₂ ≠ 0 := sub_ne_zero.mpr hf
-    simpa only [comp_apply, coe_sub, coe_const, sub_apply, coe_mk, sub_self, map_zero, Ne.def,
-      normSq_eq_zero, const_apply] using this
+    simpa using sub_ne_zero.mpr hf
 #align subalgebra.separates_points.is_R_or_C_to_real Subalgebra.SeparatesPoints.isROrC_to_real
 
 variable [CompactSpace X]

Delete ContinuousMap.abs in favor of the general construction in lattice ordered groups.

Part of #9411

@@ -111,7 +111,7 @@ theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
 #align continuous_map.comp_attach_bound_mem_closure ContinuousMap.comp_attachBound_mem_closure
 
 theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
-    (f : C(X, ℝ)).abs ∈ A.topologicalClosure := by
+    |(f : C(X, ℝ))| ∈ A.topologicalClosure := by
   let f' := attachBound (f : C(X, ℝ))
   let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }
   change abs.comp f' ∈ A.topologicalClosure
@@ -222,7 +222,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     intro x z
     obtain ⟨y, ym, zm⟩ := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z
     dsimp
-    simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff]
+    simp only [Subtype.coe_mk, coe_sup', Finset.sup'_apply, Finset.lt_sup'_iff]
     exact ⟨y, ym, zm⟩
   have h_eq : ∀ x, (h x : X → ℝ) x = f x := by intro x; simp [w₁]
   -- For each `x`, we define `W x` to be `{z | h x z < f z + ε}`,

@@ -208,14 +208,14 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   -- and still equal to `f x` at `x`.
   -- Since `X` is compact, for every `x` there is some finset `ys t`
   -- so the union of the `U x y` for `y ∈ ys x` still covers everything.
-  let ys : ∀ _, Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose
+  let ys : X → Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose
   let ys_w : ∀ x, ⋃ y ∈ ys x, U x y = ⊤ := fun x =>
     (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose_spec
   have ys_nonempty : ∀ x, (ys x).Nonempty := fun x =>
     Set.nonempty_of_union_eq_top_of_nonempty _ _ nX (ys_w x)
   -- Thus for each `x` we have the desired `h x : A` so `f z - ε < h x z` everywhere
   -- and `h x x = f x`.
-  let h : ∀ _, L := fun x =>
+  let h : X → L := fun x =>
     ⟨(ys x).sup' (ys_nonempty x) fun y => (g x y : C(X, ℝ)),
       Finset.sup'_mem _ sup_mem _ _ _ fun y _ => hg x y⟩
   have lt_h : ∀ x z, f z - ε < (h x : X → ℝ) z := by
@@ -226,7 +226,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     exact ⟨y, ym, zm⟩
   have h_eq : ∀ x, (h x : X → ℝ) x = f x := by intro x; simp [w₁]
   -- For each `x`, we define `W x` to be `{z | h x z < f z + ε}`,
-  let W : ∀ _, Set X := fun x => {z | (h x : X → ℝ) z < f z + ε}
+  let W : X → Set X := fun x => {z | (h x : X → ℝ) z < f z + ε}
   -- This is still a neighbourhood of `x`.
   have W_nhd : ∀ x, W x ∈ 𝓝 x := by
     intro x

Search for [∀∃].*(_ and manually replace some occurrences with more readable versions. In case of ∀, the new expressions are defeq to the old ones. In case of ∃, they differ by exists_prop.

In some rare cases, golf proofs that needed fixing.

@@ -166,8 +166,8 @@ open scoped Topology
 
 -- Here's the fun part of Stone-Weierstrass!
 theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
-    (inf_mem : ∀ (f) (_ : f ∈ L) (g) (_ : g ∈ L), f ⊓ g ∈ L)
-    (sup_mem : ∀ (f) (_ : f ∈ L) (g) (_ : g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
+    (inf_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊓ g ∈ L)
+    (sup_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
     closure L = ⊤ := by
   -- We start by boiling down to a statement about close approximation.
   apply eq_top_iff.mpr

@@ -380,7 +380,7 @@ variable [CompactSpace X]
 
 set_option synthInstance.maxHeartbeats 30000 in
 /-- The Stone-Weierstrass approximation theorem, `IsROrC` version, that a star subalgebra `A` of
-`C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if itseparates
+`C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if it separates
 points. -/
 theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
     (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -128,7 +128,7 @@ theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
           (A.le_topologicalClosure g.property))
         _)
       _
-  exact_mod_cast abs_mem_subalgebra_closure A _
+  exact mod_cast abs_mem_subalgebra_closure A _
 #align continuous_map.inf_mem_subalgebra_closure ContinuousMap.inf_mem_subalgebra_closure
 
 theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
@@ -150,7 +150,7 @@ theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
           (A.le_topologicalClosure g.property))
         _)
       _
-  exact_mod_cast abs_mem_subalgebra_closure A _
+  exact mod_cast abs_mem_subalgebra_closure A _
 #align continuous_map.sup_mem_subalgebra_closure ContinuousMap.sup_mem_subalgebra_closure
 
 theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))

We have turned to Type* instead of Type _, but many of them remained in mathlib because the straight replacement did not work. In general, having Type _ before the colon is a code smell, though, as it hides which types should be in the same universe and which shouldn't, and is not very robust.

This PR replaces most of the remaining Type _ before the colon (except those in category theory) by Type* or Type u. This has uncovered a few bugs (where declarations were not as polymorphic as they should be).

I had to increase heartbeats at two places when replacing Type _ by Type*, but I think it's worth it as it's really more robust.

@@ -335,7 +335,7 @@ section IsROrC
 open IsROrC
 
 -- Redefine `X`, since for the next lemma it need not be compact
-variable {𝕜 : Type _} {X : Type*} [IsROrC 𝕜] [TopologicalSpace X]
+variable {𝕜 : Type*} {X : Type*} [IsROrC 𝕜] [TopologicalSpace X]
 
 open ContinuousMap
 
@@ -378,6 +378,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X,
 
 variable [CompactSpace X]
 
+set_option synthInstance.maxHeartbeats 30000 in
 /-- The Stone-Weierstrass approximation theorem, `IsROrC` version, that a star subalgebra `A` of
 `C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if itseparates
 points. -/

This reverts commit f3695eb2.

@@ -70,6 +70,8 @@ theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g :
   simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,
     Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
     Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
+  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+  erw [ContinuousMap.attachBound_apply_coe]
 #align continuous_map.polynomial_comp_attach_bound ContinuousMap.polynomial_comp_attachBound
 
 /-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial

This reverts commit 26eb2b0a.

@@ -70,8 +70,6 @@ theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g :
   simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,
     Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
     Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
-  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-  erw [ContinuousMap.attachBound_apply_coe]
 #align continuous_map.polynomial_comp_attach_bound ContinuousMap.polynomial_comp_attachBound
 
 /-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial

This includes all the changes from #7606.

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

@@ -70,6 +70,8 @@ theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g :
   simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,
     Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
     Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
+  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+  erw [ContinuousMap.attachBound_apply_coe]
 #align continuous_map.polynomial_comp_attach_bound ContinuousMap.polynomial_comp_attachBound
 
 /-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial

@@ -54,8 +54,8 @@ open scoped Polynomial
 /-- Turn a function `f : C(X, ℝ)` into a continuous map into `Set.Icc (-‖f‖) (‖f‖)`,
 thereby explicitly attaching bounds.
 -/
-def attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖)
-    where toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩
+def attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖) where
+  toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩
 #align continuous_map.attach_bound ContinuousMap.attachBound
 
 @[simp]

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

This has nice performance benefits.

@@ -47,7 +47,7 @@ noncomputable section
 
 namespace ContinuousMap
 
-variable {X : Type _} [TopologicalSpace X] [CompactSpace X]
+variable {X : Type*} [TopologicalSpace X] [CompactSpace X]
 
 open scoped Polynomial
 
@@ -333,7 +333,7 @@ section IsROrC
 open IsROrC
 
 -- Redefine `X`, since for the next lemma it need not be compact
-variable {𝕜 : Type _} {X : Type _} [IsROrC 𝕜] [TopologicalSpace X]
+variable {𝕜 : Type _} {X : Type*} [IsROrC 𝕜] [TopologicalSpace X]
 
 open ContinuousMap
 
@@ -437,7 +437,7 @@ theorem polynomialFunctions.topologicalClosure (s : Set ℝ)
 
 /-- The star subalgebra generated by polynomials functions is dense in `C(s, 𝕜)` when `s` is
 compact and `𝕜` is either `ℝ` or `ℂ`. -/
-theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type _} [IsROrC 𝕜] (s : Set 𝕜)
+theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type*} [IsROrC 𝕜] (s : Set 𝕜)
     [CompactSpace s] : (polynomialFunctions s).starClosure.topologicalClosure = ⊤ :=
   ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints _
     (Subalgebra.separatesPoints_monotone le_sup_left (polynomialFunctions_separatesPoints s))
@@ -445,7 +445,7 @@ theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type _} [IsRO
 /-- Continuous algebra homomorphisms from `C(s, ℝ)` into an `ℝ`-algebra `A` which agree
 at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. -/
 @[ext]
-theorem ContinuousMap.algHom_ext_map_X {A : Type _} [Ring A]
+theorem ContinuousMap.algHom_ext_map_X {A : Type*} [Ring A]
     [Algebra ℝ A] [TopologicalSpace A] [T2Space A] {s : Set ℝ} [CompactSpace s]
     {φ ψ : C(s, ℝ) →ₐ[ℝ] A} (hφ : Continuous φ) (hψ : Continuous ψ)
     (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by
@@ -458,7 +458,7 @@ theorem ContinuousMap.algHom_ext_map_X {A : Type _} [Ring A]
 /-- Continuous star algebra homomorphisms from `C(s, 𝕜)` into a star `𝕜`-algebra `A` which agree
 at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. -/
 @[ext]
-theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type _} [IsROrC 𝕜] [Ring A] [StarRing A]
+theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type*} [IsROrC 𝕜] [Ring A] [StarRing A]
     [Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace s]
     {φ ψ : C(s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous φ) (hψ : Continuous ψ)
     (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by

Previously the following comment occured in Topology.ContinuousFunction.Algebra:

-- TODO: -- This lemma (and the next) could go all the way back in Algebra.Order.Field, -- except that it is tedious to prove without tactics. -- Rather than stranding it at some intermediate location, -- it's here, immediately prior to the point of use.

Subsequently, the theory of lattice ordered groups has been developed in Mathlib (Algebra.Order.LatticeGroup). This now provides the natural "intermediate location" for these lemmas, they are an immediate consequence of LatticeOrderedCommGroup.two_inf_eq_add_sub_abs_sub and LatticeOrderedCommGroup.two_sup_eq_add_add_abs_sub. In fact we can show that C(α, β) is itself a lattice ordered group and hence expressions for the inf and sup (inf_eq and sup_eq) can be deduced directly from LatticeOrderedCommGroup.two_inf_eq_add_sub_abs_sub and LatticeOrderedCommGroup.two_sup_eq_add_add_abs_sub.

This was previously submitted to Mathlib https://github.com/leanprover-community/mathlib/pull/18780

Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com> Co-authored-by: Christopher Hoskin <mans0954@users.noreply.github.com>

@@ -118,7 +118,7 @@ theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
 
 theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
     (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by
-  rw [inf_eq]
+  rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]
   refine'
     A.topologicalClosure.smul_mem
       (A.topologicalClosure.sub_mem
@@ -140,7 +140,7 @@ theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (
 
 theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
     (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by
-  rw [sup_eq]
+  rw [sup_eq_half_smul_add_add_abs_sub' ℝ]
   refine'
     A.topologicalClosure.smul_mem
       (A.topologicalClosure.add_mem

The Stone-Weierstrass theorem, including the version for IsROrC 𝕜, was developed prior to the introduction of StarSubalgebra. As such, in order to prove it, a predicate ConjInvariantSubalgebra was introduced for ℝ-subalgebras of C(X, 𝕜). This refactors the Stone-Weierstrass theorem to instead use the StarSubalgebra API and removes ContinuousMap.ConjInvariantSubalgebra entirely. In addition, we provide a few corollaries concerning polynomial functions which are missing from the library.

@@ -16,10 +16,10 @@ separates points, then it is dense.
 
 We argue as follows.
 
-* In any subalgebra `A` of `C(X, ℝ)`, if `f ∈ A`, then `abs f ∈ A.topological_closure`.
+* In any subalgebra `A` of `C(X, ℝ)`, if `f ∈ A`, then `abs f ∈ A.topologicalClosure`.
   This follows from the Weierstrass approximation theorem on `[-‖f‖, ‖f‖]` by
   approximating `abs` uniformly thereon by polynomials.
-* This ensures that `A.topological_closure` is actually a sublattice:
+* This ensures that `A.topologicalClosure` is actually a sublattice:
   if it contains `f` and `g`, then it contains the pointwise supremum `f ⊔ g`
   and the pointwise infimum `f ⊓ g`.
 * Any nonempty sublattice `L` of `C(X, ℝ)` which separates points is dense,
@@ -28,10 +28,10 @@ We argue as follows.
   By continuity these functions remain close to `f` on small patches around `x` and `y`.
   We use compactness to identify a certain finitely indexed infimum of finitely indexed supremums
   which is then close to `f` everywhere, obtaining the desired approximation.
-* Finally we put these pieces together. `L = A.topological_closure` is a nonempty sublattice
+* Finally we put these pieces together. `L = A.topologicalClosure` is a nonempty sublattice
   which separates points since `A` does, and so is dense (in fact equal to `⊤`).
 
-We then prove the complex version for self-adjoint subalgebras `A`, by separately approximating
+We then prove the complex version for star subalgebras `A`, by separately approximating
 the real and imaginary parts using the real subalgebra of real-valued functions in `A`
 (which still separates points, by taking the norm-square of a separating function).
 
@@ -51,7 +51,7 @@ variable {X : Type _} [TopologicalSpace X] [CompactSpace X]
 
 open scoped Polynomial
 
-/-- Turn a function `f : C(X, ℝ)` into a continuous map into `set.Icc (-‖f‖) (‖f‖)`,
+/-- Turn a function `f : C(X, ℝ)` into a continuous map into `Set.Icc (-‖f‖) (‖f‖)`,
 thereby explicitly attaching bounds.
 -/
 def attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖)
@@ -76,7 +76,7 @@ theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g :
 gives another function in `A`.
 
 This lemma proves something slightly more subtle than this:
-we take `f`, and think of it as a function into the restricted target `set.Icc (-‖f‖) ‖f‖)`,
+we take `f`, and think of it as a function into the restricted target `Set.Icc (-‖f‖) ‖f‖)`,
 and then postcompose with a polynomial function on that interval.
 This is in fact the same situation as above, and so also gives a function in `A`.
 -/
@@ -93,7 +93,7 @@ theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
     continuousMap_mem_polynomialFunctions_closure _ _ p
   -- and so there are polynomials arbitrarily close.
   have frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure
-  -- To prove `p.comp (attached_bound f)` is in the closure of `A`,
+  -- To prove `p.comp (attachBound f)` is in the closure of `A`,
   -- we show there are elements of `A` arbitrarily close.
   apply mem_closure_iff_frequently.mpr
   -- To show that, we pull back the polynomials close to `p`,
@@ -182,7 +182,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   /-
     The strategy now is to pick a family of continuous functions `g x y` in `A`
     with the property that `g x y x = f x` and `g x y y = f y`
-    (this is immediate from `h : separates_points_strongly`)
+    (this is immediate from `h : SeparatesPointsStrongly`)
     then use continuity to see that `g x y` is close to `f` near both `x` and `y`,
     and finally using compactness to produce the desired function `h`
     as a maximum over finitely many `x` of a minimum over finitely many `y` of the `g x y`.
@@ -250,7 +250,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   rw [dist_lt_iff pos]
   intro z
   -- We rewrite into this particular form,
-  -- so that simp lemmas about inequalities involving `finset.inf'` can fire.
+  -- so that simp lemmas about inequalities involving `Finset.inf'` can fire.
   rw [show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by
         intros; simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]]
   fconstructor
@@ -332,49 +332,24 @@ section IsROrC
 
 open IsROrC
 
--- Redefine `X`, since for the next few lemmas it need not be compact
+-- Redefine `X`, since for the next lemma it need not be compact
 variable {𝕜 : Type _} {X : Type _} [IsROrC 𝕜] [TopologicalSpace X]
 
-namespace ContinuousMap
+open ContinuousMap
 
-/-- A real subalgebra of `C(X, 𝕜)` is `conj_invariant`, if it contains all its conjugates. -/
-def ConjInvariantSubalgebra (A : Subalgebra ℝ C(X, 𝕜)) : Prop :=
-  A.map (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.continuous) ≤ A
-#align continuous_map.conj_invariant_subalgebra ContinuousMap.ConjInvariantSubalgebra
-
-theorem mem_conjInvariantSubalgebra {A : Subalgebra ℝ C(X, 𝕜)} (hA : ConjInvariantSubalgebra A)
-    {f : C(X, 𝕜)} (hf : f ∈ A) : (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.continuous) f ∈ A :=
-  hA ⟨f, hf, rfl⟩
-#align continuous_map.mem_conj_invariant_subalgebra ContinuousMap.mem_conjInvariantSubalgebra
-
-/-- If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. -/
-theorem subalgebraConjInvariant {S : Set C(X, 𝕜)}
-    (hS : ∀ f, f ∈ S → (conjAe.toAlgHom.compLeftContinuous ℝ conjCle.continuous) f ∈ S) :
-    ConjInvariantSubalgebra ((Algebra.adjoin 𝕜 S).restrictScalars ℝ) := by
-  rintro _ ⟨f, hf, rfl⟩
-  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ
-  change _ ∈ (Algebra.adjoin 𝕜 S).restrictScalars ℝ at hf
-  rw [Subalgebra.mem_restrictScalars] at hf ⊢
-  refine Algebra.adjoin_induction hf ?_ ?_ ?_ ?_
-  · exact fun g hg => Algebra.subset_adjoin (hS g hg)
-  · exact fun c => Subalgebra.algebraMap_mem _ (starRingEnd 𝕜 c)
-  · intro f g hf hg
-    convert Subalgebra.add_mem _ hf hg using 1
-    simp only [AlgEquiv.toAlgHom_eq_coe, map_add, RingHom.coe_coe]
-  · intro f g hf hg
-    convert Subalgebra.mul_mem _ hf hg using 1
-    simp only [AlgEquiv.toAlgHom_eq_coe, map_mul, RingHom.coe_coe]
-#align continuous_map.subalgebra_conj_invariant ContinuousMap.subalgebraConjInvariant
+/- a post-port refactor eliminated `conjInvariantSubalgebra`, which was only used to
+state and prove the Stone-Weierstrass theorem, in favor of using `StarSubalgebra`s,
+which didn't exist at the time Stone-Weierstrass was written. -/
+#noalign continuous_map.conj_invariant_subalgebra
+#noalign continuous_map.mem_conj_invariant_subalgebra
+#noalign continuous_map.subalgebra_conj_invariant
 
-end ContinuousMap
-
-open ContinuousMap
 
-/-- If a conjugation-invariant subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
+/-- If a star subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
 of its purely real-valued elements also separates points. -/
-theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜)}
-    (hA : A.SeparatesPoints) (hA' : ConjInvariantSubalgebra (A.restrictScalars ℝ)) :
-    ((A.restrictScalars ℝ).comap
+theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
+    (hA : A.SeparatesPoints) :
+      ((A.restrictScalars ℝ).comap
         (ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by
   intro x₁ x₂ hx
   -- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
@@ -390,10 +365,9 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
   refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_normSq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]
-    convert (A.restrictScalars ℝ).mul_mem (mem_conjInvariantSubalgebra hA' hFA) hFA
+    convert (A.restrictScalars ℝ).mul_mem hFA (star_mem hFA : star F ∈ A)
     ext1
-    rw [mul_comm]
-    exact (IsROrC.mul_conj _).symm
+    exact (IsROrC.mul_conj (K := 𝕜) _).symm
   · -- And it also separates the points `x₁`, `x₂`
     have : f x₁ - f x₂ ≠ 0 := sub_ne_zero.mpr hf
     simpa only [comp_apply, coe_sub, coe_const, sub_apply, coe_mk, sub_self, map_zero, Ne.def,
@@ -402,14 +376,12 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
 
 variable [CompactSpace X]
 
-/-- The Stone-Weierstrass approximation theorem, `is_R_or_C` version,
-that a subalgebra `A` of `C(X, 𝕜)`, where `X` is a compact topological space and `is_R_or_C 𝕜`,
-is dense if it is conjugation-invariant and separates points.
--/
-theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints
-    (A : Subalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints)
-    (hA' : ConjInvariantSubalgebra (A.restrictScalars ℝ)) : A.topologicalClosure = ⊤ := by
-  rw [Algebra.eq_top_iff]
+/-- The Stone-Weierstrass approximation theorem, `IsROrC` version, that a star subalgebra `A` of
+`C(X, 𝕜)`, where `X` is a compact topological space and `IsROrC 𝕜`, is dense if itseparates
+points. -/
+theorem ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
+    (A : StarSubalgebra 𝕜 C(X, 𝕜)) (hA : A.SeparatesPoints) : A.topologicalClosure = ⊤ := by
+  rw [StarSubalgebra.eq_top_iff]
   -- Let `I` be the natural inclusion of `C(X, ℝ)` into `C(X, 𝕜)`
   let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ofRealClm.compLeftContinuous ℝ X
   -- The main point of the proof is that its range (i.e., every real-valued function) is contained
@@ -418,11 +390,10 @@ theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPo
     -- Let `A₀` be the subalgebra of `C(X, ℝ)` consisting of `A`'s purely real elements; it is the
     -- preimage of `A` under `I`.  In this argument we only need its submodule structure.
     let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I
-    -- By `subalgebra.separates_points.complex_to_real`, this subalgebra also separates points, so
+    -- By `Subalgebra.SeparatesPoints.isROrC_to_real`, this subalgebra also separates points, so
     -- we may apply the real Stone-Weierstrass result to it.
     have SW : A₀.topologicalClosure = ⊤ :=
-      haveI :=
-        subalgebra_topologicalClosure_eq_top_of_separatesPoints _ (hA.isROrC_to_real hA')
+      haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real
       congr_arg Subalgebra.toSubmodule this
     rw [← Submodule.map_top, ← SW]
     -- So it suffices to prove that the image under `I` of the closure of `A₀` is contained in the
@@ -438,11 +409,63 @@ theorem ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPo
   have h_f_re : I f_re ∈ A.topologicalClosure := key ⟨f_re, rfl⟩
   have h_f_im : I f_im ∈ A.topologicalClosure := key ⟨f_im, rfl⟩
   -- So `f_re + I • f_im` is in the closure of `A`
-  convert A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I)
+  have := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I)
+  rw [StarSubalgebra.mem_toSubalgebra] at this
+  convert this
   -- And this, of course, is just `f`
   ext
   apply Eq.symm
   simp [mul_comm IsROrC.I _]
-#align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.subalgebra_isROrC_topologicalClosure_eq_top_of_separatesPoints
+#align continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPointsₓ
 
 end IsROrC
+
+section PolynomialFunctions
+
+open StarSubalgebra Polynomial
+open scoped Polynomial
+
+/-- Polynomial functions in are dense in `C(s, ℝ)` when `s` is compact.
+
+See `polynomialFunctions_closure_eq_top` for the special case `s = Set.Icc a b` which does not use
+the full Stone-Weierstrass theorem. Of course, that version could be used to prove this one as
+well. -/
+theorem polynomialFunctions.topologicalClosure (s : Set ℝ)
+    [CompactSpace s] : (polynomialFunctions s).topologicalClosure = ⊤ :=
+  ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints _
+    (polynomialFunctions_separatesPoints s)
+
+/-- The star subalgebra generated by polynomials functions is dense in `C(s, 𝕜)` when `s` is
+compact and `𝕜` is either `ℝ` or `ℂ`. -/
+theorem polynomialFunctions.starClosure_topologicalClosure {𝕜 : Type _} [IsROrC 𝕜] (s : Set 𝕜)
+    [CompactSpace s] : (polynomialFunctions s).starClosure.topologicalClosure = ⊤ :=
+  ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints _
+    (Subalgebra.separatesPoints_monotone le_sup_left (polynomialFunctions_separatesPoints s))
+
+/-- Continuous algebra homomorphisms from `C(s, ℝ)` into an `ℝ`-algebra `A` which agree
+at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. -/
+@[ext]
+theorem ContinuousMap.algHom_ext_map_X {A : Type _} [Ring A]
+    [Algebra ℝ A] [TopologicalSpace A] [T2Space A] {s : Set ℝ} [CompactSpace s]
+    {φ ψ : C(s, ℝ) →ₐ[ℝ] A} (hφ : Continuous φ) (hψ : Continuous ψ)
+    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by
+  suffices (⊤ : Subalgebra ℝ C(s, ℝ)) ≤ AlgHom.equalizer φ ψ from
+    AlgHom.ext fun x => this (by trivial)
+  rw [← polynomialFunctions.topologicalClosure s]
+  exact Subalgebra.topologicalClosure_minimal (polynomialFunctions s)
+    (polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ)
+
+/-- Continuous star algebra homomorphisms from `C(s, 𝕜)` into a star `𝕜`-algebra `A` which agree
+at `X : 𝕜[X]` (interpreted as a continuous map) are, in fact, equal. -/
+@[ext]
+theorem ContinuousMap.starAlgHom_ext_map_X {𝕜 A : Type _} [IsROrC 𝕜] [Ring A] [StarRing A]
+    [Algebra 𝕜 A] [TopologicalSpace A] [T2Space A] {s : Set 𝕜} [CompactSpace s]
+    {φ ψ : C(s, 𝕜) →⋆ₐ[𝕜] A} (hφ : Continuous φ) (hψ : Continuous ψ)
+    (h : φ (toContinuousMapOnAlgHom s X) = ψ (toContinuousMapOnAlgHom s X)) : φ = ψ := by
+  suffices (⊤ : StarSubalgebra 𝕜 C(s, 𝕜)) ≤ StarAlgHom.equalizer φ ψ from
+    StarAlgHom.ext fun x => this mem_top
+  rw [← polynomialFunctions.starClosure_topologicalClosure s]
+  exact StarSubalgebra.topologicalClosure_minimal
+    (polynomialFunctions.starClosure_le_equalizer s φ ψ h) (isClosed_eq hφ hψ)
+
+end PolynomialFunctions

• 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) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison, Heather Macbeth
-
-! This file was ported from Lean 3 source module topology.continuous_function.stone_weierstrass
-! leanprover-community/mathlib commit 16e59248c0ebafabd5d071b1cd41743eb8698ffb
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.ContinuousFunction.Weierstrass
 import Mathlib.Data.IsROrC.Basic
 
+#align_import topology.continuous_function.stone_weierstrass from "leanprover-community/mathlib"@"16e59248c0ebafabd5d071b1cd41743eb8698ffb"
+
 /-!
 # The Stone-Weierstrass theorem
 

@@ -210,7 +210,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   -- Since `X` is compact, for every `x` there is some finset `ys t`
   -- so the union of the `U x y` for `y ∈ ys x` still covers everything.
   let ys : ∀ _, Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose
-  let ys_w : ∀ x, (⋃ y ∈ ys x, U x y) = ⊤ := fun x =>
+  let ys_w : ∀ x, ⋃ y ∈ ys x, U x y = ⊤ := fun x =>
     (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose_spec
   have ys_nonempty : ∀ x, (ys x).Nonempty := fun x =>
     Set.nonempty_of_union_eq_top_of_nonempty _ _ nX (ys_w x)
@@ -241,7 +241,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   -- Since `X` is compact, there is some finset `ys t`
   -- so the union of the `W x` for `x ∈ xs` still covers everything.
   let xs : Finset X := (CompactSpace.elim_nhds_subcover W W_nhd).choose
-  let xs_w : (⋃ x ∈ xs, W x) = ⊤ := (CompactSpace.elim_nhds_subcover W W_nhd).choose_spec
+  let xs_w : ⋃ x ∈ xs, W x = ⊤ := (CompactSpace.elim_nhds_subcover W W_nhd).choose_spec
   have xs_nonempty : xs.Nonempty := Set.nonempty_of_union_eq_top_of_nonempty _ _ nX xs_w
   -- Finally our candidate function is the infimum over `x ∈ xs` of the `h x`.
   -- This function is then globally less than `f z + ε`.

Found with git grep -n "λ [a-zA-Z_ ]*,"

@@ -201,7 +201,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
     · apply isOpen_lt <;> continuity
     · rw [Set.mem_setOf_eq, w₂]
       exact sub_lt_self _ pos
-  -- Fixing `x` for a moment, we have a family of functions `λ y, g x y`
+  -- Fixing `x` for a moment, we have a family of functions `fun y ↦ g x y`
   -- which on different patches (the `U x y`) are greater than `f z - ε`.
   -- Taking the supremum of these functions
   -- indexed by a finite collection of patches which cover `X`
@@ -389,7 +389,7 @@ theorem Subalgebra.SeparatesPoints.isROrC_to_real {A : Subalgebra 𝕜 C(X, 𝕜
     ext1
     simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one,
       const_apply]
-  -- Consider now the function `λ x, |f x - f x₂| ^ 2`
+  -- Consider now the function `fun x ↦ |f x - f x₂| ^ 2`
   refine' ⟨_, ⟨(⟨IsROrC.normSq, continuous_normSq⟩ : C(𝕜, ℝ)).comp F, _, rfl⟩, _⟩
   · -- This is also an element of the subalgebra, and takes only real values
     rw [SetLike.mem_coe, Subalgebra.mem_comap]