topology.continuous_function.ideals ⟷ Mathlib.Topology.ContinuousFunction.Ideals

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

@@ -6,7 +6,7 @@ Authors: Jireh Loreaux
 import Topology.Algebra.Algebra
 import Topology.ContinuousFunction.Compact
 import Topology.UrysohnsLemma
-import Data.IsROrC.Basic
+import Analysis.RCLike.Basic
 import Analysis.NormedSpace.Units
 import Topology.Algebra.Module.CharacterSpace
 
@@ -206,11 +206,11 @@ theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idea
 
 end TopologicalRing
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
-variable {X 𝕜 : Type _} [IsROrC 𝕜] [TopologicalSpace X]
+variable {X 𝕜 : Type _} [RCLike 𝕜] [TopologicalSpace X]
 
 #print ContinuousMap.exists_mul_le_one_eqOn_ge /-
 /-- An auxiliary lemma used in the proof of `ideal_of_set_of_ideal_eq_closure` which may be useful
@@ -333,7 +333,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
       ext
       simp only [comp_apply, coe_mk, algebraMapCLM_coe, map_pow, coe_mul, coe_star, Pi.mul_apply,
         Pi.star_apply, star_def, ContinuousMap.coe_coe]
-      simpa only [norm_sq_eq_def', IsROrC.conj_mul, of_real_pow]
+      simpa only [norm_sq_eq_def', RCLike.conj_mul, of_real_pow]
   /- Get the function `g'` which is guaranteed to exist above. By the extreme value theorem and
     compactness of `t`, there is some `0 < c` such that `c ≤ g' x` for all `x ∈ t`. Then by
     `main_lemma_aux` there is some `g` for which `g * g'` is the desired function. -/
@@ -382,7 +382,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   exact
     ⟨⟨fun x => g x, continuous_of_real.comp (map_continuous g)⟩, by
       simpa only [coe_mk, of_real_eq_zero] using fun x hx => hgs (subset_closure hx), by
-      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
+      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, RCLike.ofReal_one] using
         one_ne_zero⟩
 #align continuous_map.set_of_ideal_of_set_eq_interior ContinuousMap.setOfIdeal_ofSet_eq_interior
 -/
@@ -463,7 +463,7 @@ theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X,
 #align continuous_map.ideal_is_maximal_iff ContinuousMap.ideal_isMaximal_iff
 -/
 
-end IsROrC
+end RCLike
 
 end ContinuousMap
 
@@ -505,7 +505,7 @@ theorem continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEv
 
 end ContinuousMapEval
 
-variable [CompactSpace X] [T2Space X] [IsROrC 𝕜]
+variable [CompactSpace X] [T2Space X] [RCLike 𝕜]
 
 #print WeakDual.CharacterSpace.continuousMapEval_bijective /-
 theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
@@ -517,9 +517,9 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [← Ne.def, DFunLike.ne_iff]
-    use(⟨coe, IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
+    use(⟨coe, RCLike.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuous_map_eval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
-      IsROrC.ofReal_inj] using
+      RCLike.ofReal_inj] using
       ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)
   · obtain ⟨x, hx⟩ := (ideal_is_maximal_iff (RingHom.ker φ)).mp inferInstance
     refine' ⟨x, ext_ker <| Ideal.ext fun f => _⟩

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

@@ -321,7 +321,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
         simpa only [zero_add] using add_lt_add_of_le_of_lt zero_le' (hgt₂ x hx)
     · intro x hx
       replace hx := htI.subset_compl_right hx
-      rw [compl_compl, mem_set_of_ideal] at hx 
+      rw [compl_compl, mem_set_of_ideal] at hx
       obtain ⟨g, hI, hgx⟩ := hx
       have := (map_continuous g).ContinuousAt.eventually_ne hgx
       refine'
@@ -340,7 +340,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   obtain ⟨g', hI', hgt'⟩ := this
   obtain ⟨c, hc, hgc'⟩ : ∃ (c : _) (hc : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
     t.eq_empty_or_nonempty.elim
-      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy )⟩) fun ht' =>
+      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
       let ⟨x, hx, hx'⟩ := ht.is_compact.exists_forall_le ht' (map_continuous g').ContinuousOn
       ⟨g' x, hgt' x hx, hx'⟩
   obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eq_on_ge g' hc
@@ -371,7 +371,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
         set.not_mem_compl_iff.mp (mt (@hf x) hfx))
       fun x hx => _
   -- If `x ∉ closure sᶜ`, we must produce `f : C(X, 𝕜)` which is zero on `sᶜ` and `f x ≠ 0`.
-  rw [← compl_compl (interior s), ← closure_compl] at hx 
+  rw [← compl_compl (interior s), ← closure_compl] at hx
   simp_rw [mem_set_of_ideal, mem_ideal_of_set]
   haveI : NormalSpace X := T4Space.of_compactSpace_t2Space
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
@@ -421,7 +421,7 @@ theorem idealOfSet_isMaximal_iff (s : Opens X) :
   by
   rw [Ideal.isMaximal_def]
   refine' (ideal_opens_gi X 𝕜).isCoatom_iff_ge_of_le (fun I hI => _) s
-  rw [← Ideal.isMaximal_def] at hI 
+  rw [← Ideal.isMaximal_def] at hI
   skip
   exact ideal_of_set_of_ideal_is_closed inferInstance
 #align continuous_map.ideal_of_set_is_maximal_iff ContinuousMap.idealOfSet_isMaximal_iff

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

@@ -282,7 +282,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
           _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
             simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
               sub_smul, Nonneg.coe_one, one_smul]
-          _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
+          _ ≤ 1 := (nnnorm_algebraMap_nnreal 𝕜 (1 - g x)).trans_le tsub_le_self
       calc
         ‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
             ‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=

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

@@ -260,34 +260,34 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     Indeed, then `‖f - f * ↑g‖ ≤ ‖f * (1 - ↑g)‖ ≤ ⨆ ‖f * (1 - ↑g) x‖`. When `x ∉ t`, `‖f x‖ < ε / 2`
     and `‖(1 - ↑g) x‖ ≤ 1`, and when `x ∈ t`, `(1 - ↑g) x = 0`, and clearly `f * ↑g ∈ I`. -/
   suffices
-    ∃ g : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.eq_on g 1
+    ∃ g : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.eq_on g 1
     by
     obtain ⟨g, hgI, hg, hgt⟩ := this
-    refine' ⟨f * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩
+    refine' ⟨f * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩
     rw [nndist_eq_nnnorm]
     refine' (nnnorm_lt_iff _ hε).2 fun x => _
     simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply]
     by_cases hx : x ∈ t
     ·
-      simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapClm_apply,
+      simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapCLM_apply,
         map_one, mul_one, sub_self, nnnorm_zero] using hε
     · refine' lt_of_le_of_lt _ (half_lt_self hε)
       have :=
         calc
-          ‖((1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
+          ‖((1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
               ‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ :=
             by
             simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
-              Pi.one_apply, Function.comp_apply, algebraMapClm_apply]
+              Pi.one_apply, Function.comp_apply, algebraMapCLM_apply]
           _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
             simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
               sub_smul, Nonneg.coe_one, one_smul]
           _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
       calc
-        ‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
-            ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
+        ‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
+            ‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
           by simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
-        _ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
+        _ ≤ ε / 2 * ‖(1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
           ((nnnorm_mul_le _ _).trans
             (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
         _ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _
@@ -297,12 +297,12 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     `fₓ x ≠ 0` for some `fₓ ∈ I` and so `λ y, ‖(star fₓ * fₓ) y‖₊` is strictly posiive in a
     neighborhood of `y`. Moreover, `(‖(star fₓ * fₓ) y‖₊ : 𝕜) = (star fₓ * fₓ) y`, so composition of
     this map with the natural embedding is just `star fₓ * fₓ ∈ I`. -/
-  have : ∃ g' : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x :=
+  have : ∃ g' : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x :=
     by
     refine'
       @IsCompact.induction_on _ _ _ ht.is_compact
         (fun s =>
-          ∃ g' : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x)
+          ∃ g' : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ s, 0 < g' x)
         _ _ _ _
     · refine' ⟨0, _, fun x hx => False.elim hx⟩
       convert I.zero_mem
@@ -331,7 +331,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
             pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩
       convert I.mul_mem_left (star g) hI
       ext
-      simp only [comp_apply, coe_mk, algebraMapClm_coe, map_pow, coe_mul, coe_star, Pi.mul_apply,
+      simp only [comp_apply, coe_mk, algebraMapCLM_coe, map_pow, coe_mul, coe_star, Pi.mul_apply,
         Pi.star_apply, star_def, ContinuousMap.coe_coe]
       simpa only [norm_sq_eq_def', IsROrC.conj_mul, of_real_pow]
   /- Get the function `g'` which is guaranteed to exist above. By the extreme value theorem and
@@ -345,9 +345,9 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
       ⟨g' x, hgt' x hx, hx'⟩
   obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eq_on_ge g' hc
   refine' ⟨g * g', _, hg, hgc.mono hgc'⟩
-  convert I.mul_mem_left ((algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
+  convert I.mul_mem_left ((algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
   ext
-  simp only [algebraMapClm_coe, ContinuousMap.coe_coe, comp_apply, coe_mul, Pi.mul_apply, map_mul]
+  simp only [algebraMapCLM_coe, ContinuousMap.coe_coe, comp_apply, coe_mul, Pi.mul_apply, map_mul]
 #align continuous_map.ideal_of_set_of_ideal_eq_closure ContinuousMap.idealOfSet_ofIdeal_eq_closure
 -/
 

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

@@ -177,7 +177,7 @@ theorem setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set
 theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
   Ideal.ext fun f => by
     simpa only [mem_ideal_of_set, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
-      FunLike.ext_iff] using Iff.rfl
+      DFunLike.ext_iff] using Iff.rfl
 #align continuous_map.ideal_of_empty_eq_bot ContinuousMap.idealOfEmpty_eq_bot
 -/
 
@@ -516,7 +516,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
     rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.is_closed {x})
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
-    rw [← Ne.def, FunLike.ne_iff]
+    rw [← Ne.def, DFunLike.ne_iff]
     use(⟨coe, IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuous_map_eval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
       IsROrC.ofReal_inj] using

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

@@ -420,7 +420,7 @@ theorem idealOfSet_isMaximal_iff (s : Opens X) :
     (idealOfSet 𝕜 (s : Set X)).IsMaximal ↔ IsCoatom s :=
   by
   rw [Ideal.isMaximal_def]
-  refine' (ideal_opens_gi X 𝕜).isCoatom_iff (fun I hI => _) s
+  refine' (ideal_opens_gi X 𝕜).isCoatom_iff_ge_of_le (fun I hI => _) s
   rw [← Ideal.isMaximal_def] at hI 
   skip
   exact ideal_of_set_of_ideal_is_closed inferInstance

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

@@ -377,7 +377,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
     with the natural embedding `ℝ ↪ 𝕜` to produce the desired `f`. -/
   obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
-    exists_continuous_zero_one_of_closed isClosed_closure isClosed_singleton
+    exists_continuous_zero_one_of_isClosed isClosed_closure isClosed_singleton
       (set.disjoint_singleton_right.mpr hx)
   exact
     ⟨⟨fun x => g x, continuous_of_real.comp (map_continuous g)⟩, by
@@ -513,7 +513,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
   refine' ⟨fun x y hxy => _, fun φ => _⟩
   · contrapose! hxy
     haveI := @T4Space.of_compactSpace_t2Space X _ _ _
-    rcases exists_continuous_zero_one_of_closed (isClosed_singleton : _root_.is_closed {x})
+    rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.is_closed {x})
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [← Ne.def, FunLike.ne_iff]

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

@@ -3,12 +3,12 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathbin.Topology.Algebra.Algebra
-import Mathbin.Topology.ContinuousFunction.Compact
-import Mathbin.Topology.UrysohnsLemma
-import Mathbin.Data.IsROrC.Basic
-import Mathbin.Analysis.NormedSpace.Units
-import Mathbin.Topology.Algebra.Module.CharacterSpace
+import Topology.Algebra.Algebra
+import Topology.ContinuousFunction.Compact
+import Topology.UrysohnsLemma
+import Data.IsROrC.Basic
+import Analysis.NormedSpace.Units
+import Topology.Algebra.Module.CharacterSpace
 
 #align_import topology.continuous_function.ideals from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
 

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

@@ -373,7 +373,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   -- If `x ∉ closure sᶜ`, we must produce `f : C(X, 𝕜)` which is zero on `sᶜ` and `f x ≠ 0`.
   rw [← compl_compl (interior s), ← closure_compl] at hx 
   simp_rw [mem_set_of_ideal, mem_ideal_of_set]
-  haveI : NormalSpace X := normalOfCompactT2
+  haveI : NormalSpace X := T4Space.of_compactSpace_t2Space
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
     with the natural embedding `ℝ ↪ 𝕜` to produce the desired `f`. -/
   obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
@@ -512,7 +512,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
   by
   refine' ⟨fun x y hxy => _, fun φ => _⟩
   · contrapose! hxy
-    haveI := @normalOfCompactT2 X _ _ _
+    haveI := @T4Space.of_compactSpace_t2Space X _ _ _
     rcases exists_continuous_zero_one_of_closed (isClosed_singleton : _root_.is_closed {x})
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩

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

@@ -517,7 +517,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [← Ne.def, FunLike.ne_iff]
-    use (⟨coe, IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
+    use(⟨coe, IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuous_map_eval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
       IsROrC.ofReal_inj] using
       ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module topology.continuous_function.ideals
-! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Algebra.Algebra
 import Mathbin.Topology.ContinuousFunction.Compact
@@ -15,6 +10,8 @@ import Mathbin.Data.IsROrC.Basic
 import Mathbin.Analysis.NormedSpace.Units
 import Mathbin.Topology.Algebra.Module.CharacterSpace
 
+#align_import topology.continuous_function.ideals from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
+
 /-!
 # Ideals of continuous functions
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/728ef9dbb281241906f25cbeb30f90d83e0bb451

@@ -493,7 +493,7 @@ def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜))
     ⟨{  toFun := fun f => f x
         map_add' := fun f g => rfl
         map_smul' := fun z f => rfl
-        cont := continuous_eval_const' x }, by rw [character_space.eq_set_map_one_map_mul];
+        cont := continuous_eval_const x }, by rw [character_space.eq_set_map_one_map_mul];
       exact ⟨rfl, fun f g => rfl⟩⟩
   continuous_toFun := Continuous.subtype_mk (continuous_of_continuous_eval map_continuous) _
 #align weak_dual.character_space.continuous_map_eval WeakDual.CharacterSpace.continuousMapEval

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.ideals
-! leanprover-community/mathlib commit c2258f7bf086b17eac0929d635403780c39e239f
+! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Topology.Algebra.Module.CharacterSpace
 /-!
 # Ideals of continuous functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 For a topological semiring `R` and a topological space `X` there is a Galois connection between
 `ideal C(X, R)` and `set X` given by sending each `I : ideal C(X, R)` to
 `{x : X | ∀ f ∈ I, f x = 0}ᶜ` and mapping `s : set X` to the ideal with carrier

mathlib commit https://github.com/leanprover-community/mathlib/commit/8efcf8022aac8e01df8d302dcebdbc25d6a886c8

@@ -87,6 +87,7 @@ variable [TopologicalSpace R] [TopologicalSemiring R]
 
 variable (R)
 
+#print ContinuousMap.idealOfSet /-
 /-- Given a topological ring `R` and `s : set X`, construct the ideal in `C(X, R)` of functions
 which vanish on the complement of `s`. -/
 def idealOfSet (s : Set X) : Ideal C(X, R)
@@ -96,7 +97,9 @@ def idealOfSet (s : Set X) : Ideal C(X, R)
   zero_mem' _ _ := rfl
   smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
 #align continuous_map.ideal_of_set ContinuousMap.idealOfSet
+-/
 
+#print ContinuousMap.idealOfSet_closed /-
 theorem idealOfSet_closed [LocallyCompactSpace X] [T2Space R] (s : Set X) :
     IsClosed (idealOfSet R s : Set C(X, R)) :=
   by
@@ -105,34 +108,46 @@ theorem idealOfSet_closed [LocallyCompactSpace X] [T2Space R] (s : Set X) :
     isClosed_iInter fun x =>
       isClosed_iInter fun hx => isClosed_eq (continuous_eval_const' x) continuous_const
 #align continuous_map.ideal_of_set_closed ContinuousMap.idealOfSet_closed
+-/
 
 variable {R}
 
+#print ContinuousMap.mem_idealOfSet /-
 theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
     f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 :=
   Iff.rfl
 #align continuous_map.mem_ideal_of_set ContinuousMap.mem_idealOfSet
+-/
 
+#print ContinuousMap.not_mem_idealOfSet /-
 theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by
   simp_rw [mem_ideal_of_set, exists_prop]; push_neg
 #align continuous_map.not_mem_ideal_of_set ContinuousMap.not_mem_idealOfSet
+-/
 
+#print ContinuousMap.setOfIdeal /-
 /-- Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
 ideal vanishes on the complement. -/
 def setOfIdeal (I : Ideal C(X, R)) : Set X :=
   {x : X | ∀ f ∈ I, (f : C(X, R)) x = 0}ᶜ
 #align continuous_map.set_of_ideal ContinuousMap.setOfIdeal
+-/
 
+#print ContinuousMap.not_mem_setOfIdeal /-
 theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
     x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 := by
   rw [← Set.mem_compl_iff, set_of_ideal, compl_compl, Set.mem_setOf]
 #align continuous_map.not_mem_set_of_ideal ContinuousMap.not_mem_setOfIdeal
+-/
 
+#print ContinuousMap.mem_setOfIdeal /-
 theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
     x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 := by
   simp_rw [set_of_ideal, Set.mem_compl_iff, Set.mem_setOf, exists_prop]; push_neg
 #align continuous_map.mem_set_of_ideal ContinuousMap.mem_setOfIdeal
+-/
 
+#print ContinuousMap.setOfIdeal_open /-
 theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) :=
   by
   simp only [set_of_ideal, Set.setOf_forall, isOpen_compl_iff]
@@ -140,33 +155,43 @@ theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I)
     isClosed_iInter fun f =>
       isClosed_iInter fun hf => isClosed_eq (map_continuous f) continuous_const
 #align continuous_map.set_of_ideal_open ContinuousMap.setOfIdeal_open
+-/
 
+#print ContinuousMap.opensOfIdeal /-
 /-- The open set `set_of_ideal I` realized as a term of `opens X`. -/
 @[simps]
 def opensOfIdeal [T2Space R] (I : Ideal C(X, R)) : Opens X :=
   ⟨setOfIdeal I, setOfIdeal_open I⟩
 #align continuous_map.opens_of_ideal ContinuousMap.opensOfIdeal
+-/
 
+#print ContinuousMap.setOfTop_eq_univ /-
 @[simp]
-theorem set_of_top_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set.univ :=
+theorem setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set.univ :=
   Set.univ_subset_iff.mp fun x hx => mem_setOfIdeal.mpr ⟨1, Submodule.mem_top, one_ne_zero⟩
-#align continuous_map.set_of_top_eq_univ ContinuousMap.set_of_top_eq_univ
+#align continuous_map.set_of_top_eq_univ ContinuousMap.setOfTop_eq_univ
+-/
 
+#print ContinuousMap.idealOfEmpty_eq_bot /-
 @[simp]
-theorem ideal_of_empty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
+theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
   Ideal.ext fun f => by
     simpa only [mem_ideal_of_set, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
       FunLike.ext_iff] using Iff.rfl
-#align continuous_map.ideal_of_empty_eq_bot ContinuousMap.ideal_of_empty_eq_bot
+#align continuous_map.ideal_of_empty_eq_bot ContinuousMap.idealOfEmpty_eq_bot
+-/
 
+#print ContinuousMap.mem_idealOfSet_compl_singleton /-
 @[simp]
 theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
     f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 := by
   simp only [mem_ideal_of_set, compl_compl, Set.mem_singleton_iff, forall_eq]
 #align continuous_map.mem_ideal_of_set_compl_singleton ContinuousMap.mem_idealOfSet_compl_singleton
+-/
 
 variable (X R)
 
+#print ContinuousMap.ideal_gc /-
 theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) :=
   by
   refine' fun I s => ⟨fun h f hf => _, fun h x hx => _⟩
@@ -177,6 +202,7 @@ theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idea
     by_contra hx'
     exact not_mem_ideal_of_set.mpr ⟨x, hx', hfx⟩ (h hf)
 #align continuous_map.ideal_gc ContinuousMap.ideal_gc
+-/
 
 end TopologicalRing
 
@@ -186,6 +212,7 @@ open IsROrC
 
 variable {X 𝕜 : Type _} [IsROrC 𝕜] [TopologicalSpace X]
 
+#print ContinuousMap.exists_mul_le_one_eqOn_ge /-
 /-- An auxiliary lemma used in the proof of `ideal_of_set_of_ideal_eq_closure` which may be useful
 on its own. -/
 theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
@@ -200,11 +227,13 @@ theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c)
       Pi.one_apply, sup_eq_left.mpr (set.mem_set_of.mp hx)] using
       inv_mul_cancel (hc.trans_le hx).ne'⟩
 #align continuous_map.exists_mul_le_one_eq_on_ge ContinuousMap.exists_mul_le_one_eqOn_ge
+-/
 
 variable [CompactSpace X] [T2Space X]
 
+#print ContinuousMap.idealOfSet_ofIdeal_eq_closure /-
 @[simp]
-theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
+theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     idealOfSet 𝕜 (setOfIdeal I) = I.closure :=
   by
   /- Since `ideal_of_set 𝕜 (set_of_ideal I)` is closed and contains `I`, it contains `I.closure`.
@@ -319,17 +348,21 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
   convert I.mul_mem_left ((algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
   ext
   simp only [algebraMapClm_coe, ContinuousMap.coe_coe, comp_apply, coe_mul, Pi.mul_apply, map_mul]
-#align continuous_map.ideal_of_set_of_ideal_eq_closure ContinuousMap.idealOfSet_of_ideal_eq_closure
+#align continuous_map.ideal_of_set_of_ideal_eq_closure ContinuousMap.idealOfSet_ofIdeal_eq_closure
+-/
 
-theorem idealOfSet_of_ideal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
+#print ContinuousMap.idealOfSet_ofIdeal_isClosed /-
+theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
     idealOfSet 𝕜 (setOfIdeal I) = I :=
-  (idealOfSet_of_ideal_eq_closure I).trans (Ideal.ext <| Set.ext_iff.mp hI.closure_eq)
-#align continuous_map.ideal_of_set_of_ideal_is_closed ContinuousMap.idealOfSet_of_ideal_isClosed
+  (idealOfSet_ofIdeal_eq_closure I).trans (Ideal.ext <| Set.ext_iff.mp hI.closure_eq)
+#align continuous_map.ideal_of_set_of_ideal_is_closed ContinuousMap.idealOfSet_ofIdeal_isClosed
+-/
 
 variable (𝕜)
 
+#print ContinuousMap.setOfIdeal_ofSet_eq_interior /-
 @[simp]
-theorem setOfIdeal_of_set_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s) = interior s :=
+theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s) = interior s :=
   by
   refine'
     Set.Subset.antisymm
@@ -351,32 +384,38 @@ theorem setOfIdeal_of_set_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜
       simpa only [coe_mk, of_real_eq_zero] using fun x hx => hgs (subset_closure hx), by
       simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
         one_ne_zero⟩
-#align continuous_map.set_of_ideal_of_set_eq_interior ContinuousMap.setOfIdeal_of_set_eq_interior
+#align continuous_map.set_of_ideal_of_set_eq_interior ContinuousMap.setOfIdeal_ofSet_eq_interior
+-/
 
-theorem setOfIdeal_of_set_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (idealOfSet 𝕜 s) = s :=
-  (setOfIdeal_of_set_eq_interior 𝕜 s).trans hs.interior_eq
-#align continuous_map.set_of_ideal_of_set_of_is_open ContinuousMap.setOfIdeal_of_set_of_isOpen
+#print ContinuousMap.setOfIdeal_ofSet_of_isOpen /-
+theorem setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (idealOfSet 𝕜 s) = s :=
+  (setOfIdeal_ofSet_eq_interior 𝕜 s).trans hs.interior_eq
+#align continuous_map.set_of_ideal_of_set_of_is_open ContinuousMap.setOfIdeal_ofSet_of_isOpen
+-/
 
 variable (X)
 
+#print ContinuousMap.idealOpensGI /-
 /-- The Galois insertion `continuous_map.opens_of_ideal : ideal C(X, 𝕜) → opens X` and
 `λ s, continuous_map.ideal_of_set ↑s`. -/
 @[simps]
-def idealOpensGi : GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s
+def idealOpensGI : GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s
     where
   choice I hI := opensOfIdeal I.closure
   gc I s := ideal_gc X 𝕜 I s
-  le_l_u s := (setOfIdeal_of_set_of_isOpen 𝕜 s.IsOpen).ge
+  le_l_u s := (setOfIdeal_ofSet_of_isOpen 𝕜 s.IsOpen).ge
   choice_eq I hI :=
     congr_arg _ <|
       Ideal.ext
         (Set.ext_iff.mp
           (isClosed_of_closure_subset <|
-              (idealOfSet_of_ideal_eq_closure I ▸ hI : I.closure ≤ I)).closure_eq)
-#align continuous_map.ideal_opens_gi ContinuousMap.idealOpensGi
+              (idealOfSet_ofIdeal_eq_closure I ▸ hI : I.closure ≤ I)).closure_eq)
+#align continuous_map.ideal_opens_gi ContinuousMap.idealOpensGI
+-/
 
 variable {X}
 
+#print ContinuousMap.idealOfSet_isMaximal_iff /-
 theorem idealOfSet_isMaximal_iff (s : Opens X) :
     (idealOfSet 𝕜 (s : Set X)).IsMaximal ↔ IsCoatom s :=
   by
@@ -386,13 +425,17 @@ theorem idealOfSet_isMaximal_iff (s : Opens X) :
   skip
   exact ideal_of_set_of_ideal_is_closed inferInstance
 #align continuous_map.ideal_of_set_is_maximal_iff ContinuousMap.idealOfSet_isMaximal_iff
+-/
 
-theorem ideal_of_compl_singleton_isMaximal (x : X) : (idealOfSet 𝕜 ({x}ᶜ : Set X)).IsMaximal :=
+#print ContinuousMap.idealOf_compl_singleton_isMaximal /-
+theorem idealOf_compl_singleton_isMaximal (x : X) : (idealOfSet 𝕜 ({x}ᶜ : Set X)).IsMaximal :=
   (idealOfSet_isMaximal_iff 𝕜 (Closeds.singleton x).compl).mpr <| Opens.isCoatom_iff.mpr ⟨x, rfl⟩
-#align continuous_map.ideal_of_compl_singleton_is_maximal ContinuousMap.ideal_of_compl_singleton_isMaximal
+#align continuous_map.ideal_of_compl_singleton_is_maximal ContinuousMap.idealOf_compl_singleton_isMaximal
+-/
 
 variable {𝕜}
 
+#print ContinuousMap.setOfIdeal_eq_compl_singleton /-
 theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal] :
     ∃ x : X, setOfIdeal I = {x}ᶜ :=
   by
@@ -401,7 +444,9 @@ theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal]
   obtain ⟨x, hx⟩ := opens.is_coatom_iff.1 ((ideal_of_set_is_maximal_iff 𝕜 (opens_of_ideal I)).1 h)
   exact ⟨x, congr_arg coe hx⟩
 #align continuous_map.set_of_ideal_eq_compl_singleton ContinuousMap.setOfIdeal_eq_compl_singleton
+-/
 
+#print ContinuousMap.ideal_isMaximal_iff /-
 theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
     I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 ({x}ᶜ) = I :=
   by
@@ -416,6 +461,7 @@ theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X,
       simpa only [ideal_of_set_of_ideal_eq_closure, Ideal.closure_eq_of_isClosed] using
         congr_arg (ideal_of_set 𝕜) hx.symm⟩
 #align continuous_map.ideal_is_maximal_iff ContinuousMap.ideal_isMaximal_iff
+-/
 
 end IsROrC
 
@@ -435,6 +481,7 @@ variable [LocallyCompactSpace X] [CommRing 𝕜] [TopologicalSpace 𝕜] [Topolo
 
 variable [Nontrivial 𝕜] [NoZeroDivisors 𝕜]
 
+#print WeakDual.CharacterSpace.continuousMapEval /-
 /-- The natural continuous map from a locally compact topological space `X` to the
 `character_space 𝕜 C(X, 𝕜)` which sends `x : X` to point evaluation at `x`. -/
 def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜))
@@ -446,17 +493,21 @@ def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜))
         cont := continuous_eval_const' x }, by rw [character_space.eq_set_map_one_map_mul];
       exact ⟨rfl, fun f g => rfl⟩⟩
   continuous_toFun := Continuous.subtype_mk (continuous_of_continuous_eval map_continuous) _
-#align weak_dual.character_space.continuous_map_eval WeakDual.characterSpace.continuousMapEval
+#align weak_dual.character_space.continuous_map_eval WeakDual.CharacterSpace.continuousMapEval
+-/
 
+#print WeakDual.CharacterSpace.continuousMapEval_apply_apply /-
 @[simp]
 theorem continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEval X 𝕜 x f = f x :=
   rfl
-#align weak_dual.character_space.continuous_map_eval_apply_apply WeakDual.characterSpace.continuousMapEval_apply_apply
+#align weak_dual.character_space.continuous_map_eval_apply_apply WeakDual.CharacterSpace.continuousMapEval_apply_apply
+-/
 
 end ContinuousMapEval
 
 variable [CompactSpace X] [T2Space X] [IsROrC 𝕜]
 
+#print WeakDual.CharacterSpace.continuousMapEval_bijective /-
 theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
   by
   refine' ⟨fun x y hxy => _, fun φ => _⟩
@@ -474,15 +525,18 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
     refine' ⟨x, ext_ker <| Ideal.ext fun f => _⟩
     simpa only [RingHom.mem_ker, continuous_map_eval_apply_apply, mem_ideal_of_set_compl_singleton,
       RingHom.mem_ker] using set_like.ext_iff.mp hx f
-#align weak_dual.character_space.continuous_map_eval_bijective WeakDual.characterSpace.continuousMapEval_bijective
+#align weak_dual.character_space.continuous_map_eval_bijective WeakDual.CharacterSpace.continuousMapEval_bijective
+-/
 
+#print WeakDual.CharacterSpace.homeoEval /-
 /-- This is the natural homeomorphism between a compact Hausdorff space `X` and the
 `character_space 𝕜 C(X, 𝕜)`. -/
 noncomputable def homeoEval : X ≃ₜ characterSpace 𝕜 C(X, 𝕜) :=
   @Continuous.homeoOfEquivCompactToT2 _ _ _ _ _ _
     { Equiv.ofBijective _ (continuousMapEval_bijective X 𝕜) with toFun := continuousMapEval X 𝕜 }
     (map_continuous (continuousMapEval X 𝕜))
-#align weak_dual.character_space.homeo_eval WeakDual.characterSpace.homeoEval
+#align weak_dual.character_space.homeo_eval WeakDual.CharacterSpace.homeoEval
+-/
 
 end CharacterSpace
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -254,7 +254,6 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
             simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
               sub_smul, Nonneg.coe_one, one_smul]
           _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
-          
       calc
         ‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
             ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
@@ -263,7 +262,6 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
           ((nnnorm_mul_le _ _).trans
             (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
         _ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _
-        
   /- There is some `g' : C(X, ℝ≥0)` which is strictly positive on `t` such that the composition
     `↑g` with the natural embedding of `ℝ≥0` into `𝕜` lies in `I`. This follows from compactness of
     `t` and that we can do it in any neighborhood of a point `x ∈ t`. Indeed, since `x ∈ t`, then

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

@@ -91,7 +91,7 @@ variable (R)
 which vanish on the complement of `s`. -/
 def idealOfSet (s : Set X) : Ideal C(X, R)
     where
-  carrier := { f : C(X, R) | ∀ x ∈ sᶜ, f x = 0 }
+  carrier := {f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}
   add_mem' f g hf hg x hx := by simp only [hf x hx, hg x hx, coe_add, Pi.add_apply, add_zero]
   zero_mem' _ _ := rfl
   smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
@@ -120,7 +120,7 @@ theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔
 /-- Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
 ideal vanishes on the complement. -/
 def setOfIdeal (I : Ideal C(X, R)) : Set X :=
-  { x : X | ∀ f ∈ I, (f : C(X, R)) x = 0 }ᶜ
+  {x : X | ∀ f ∈ I, (f : C(X, R)) x = 0}ᶜ
 #align continuous_map.set_of_ideal ContinuousMap.setOfIdeal
 
 theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
@@ -189,7 +189,7 @@ variable {X 𝕜 : Type _} [IsROrC 𝕜] [TopologicalSpace X]
 /-- An auxiliary lemma used in the proof of `ideal_of_set_of_ideal_eq_closure` which may be useful
 on its own. -/
 theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
-    ∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ { x : X | c ≤ f x }.EqOn (g * f) 1 :=
+    ∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ {x : X | c ≤ f x}.EqOn (g * f) 1 :=
   ⟨{  toFun := (f ⊔ const X c)⁻¹
       continuous_toFun :=
         ((map_continuous f).sup <| map_continuous _).inv₀ fun _ => (hc.trans_le le_sup_right).ne' },
@@ -219,7 +219,7 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
   simp_rw [dist_nndist]
   norm_cast
   -- Let `t := {x : X | ε / 2 ≤ ‖f x‖₊}}` which is closed and disjoint from `set_of_ideal I`.
-  set t := { x : X | ε / 2 ≤ ‖f x‖₊ }
+  set t := {x : X | ε / 2 ≤ ‖f x‖₊}
   have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm
   have htI : Disjoint t (set_of_ideal Iᶜ) :=
     by
@@ -298,7 +298,7 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
       obtain ⟨g, hI, hgx⟩ := hx
       have := (map_continuous g).ContinuousAt.eventually_ne hgx
       refine'
-        ⟨{ y : X | g y ≠ 0 } ∩ t,
+        ⟨{y : X | g y ≠ 0} ∩ t,
           mem_nhds_within_iff_exists_mem_nhds_inter.mpr ⟨_, this, Set.Subset.rfl⟩,
           ⟨⟨fun x => ‖g x‖₊ ^ 2, (map_continuous g).nnnorm.pow 2⟩, _, fun x hx =>
             pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩

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

@@ -294,7 +294,7 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
         simpa only [zero_add] using add_lt_add_of_le_of_lt zero_le' (hgt₂ x hx)
     · intro x hx
       replace hx := htI.subset_compl_right hx
-      rw [compl_compl, mem_set_of_ideal] at hx
+      rw [compl_compl, mem_set_of_ideal] at hx 
       obtain ⟨g, hI, hgx⟩ := hx
       have := (map_continuous g).ContinuousAt.eventually_ne hgx
       refine'
@@ -311,9 +311,9 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
     compactness of `t`, there is some `0 < c` such that `c ≤ g' x` for all `x ∈ t`. Then by
     `main_lemma_aux` there is some `g` for which `g * g'` is the desired function. -/
   obtain ⟨g', hI', hgt'⟩ := this
-  obtain ⟨c, hc, hgc'⟩ : ∃ (c : _)(hc : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
+  obtain ⟨c, hc, hgc'⟩ : ∃ (c : _) (hc : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
     t.eq_empty_or_nonempty.elim
-      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
+      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy )⟩) fun ht' =>
       let ⟨x, hx, hx'⟩ := ht.is_compact.exists_forall_le ht' (map_continuous g').ContinuousOn
       ⟨g' x, hgt' x hx, hx'⟩
   obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eq_on_ge g' hc
@@ -340,7 +340,7 @@ theorem setOfIdeal_of_set_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜
         set.not_mem_compl_iff.mp (mt (@hf x) hfx))
       fun x hx => _
   -- If `x ∉ closure sᶜ`, we must produce `f : C(X, 𝕜)` which is zero on `sᶜ` and `f x ≠ 0`.
-  rw [← compl_compl (interior s), ← closure_compl] at hx
+  rw [← compl_compl (interior s), ← closure_compl] at hx 
   simp_rw [mem_set_of_ideal, mem_ideal_of_set]
   haveI : NormalSpace X := normalOfCompactT2
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
@@ -384,7 +384,7 @@ theorem idealOfSet_isMaximal_iff (s : Opens X) :
   by
   rw [Ideal.isMaximal_def]
   refine' (ideal_opens_gi X 𝕜).isCoatom_iff (fun I hI => _) s
-  rw [← Ideal.isMaximal_def] at hI
+  rw [← Ideal.isMaximal_def] at hI 
   skip
   exact ideal_of_set_of_ideal_is_closed inferInstance
 #align continuous_map.ideal_of_set_is_maximal_iff ContinuousMap.idealOfSet_isMaximal_iff

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

@@ -73,7 +73,7 @@ ideal, continuous function, compact, Hausdorff
 -/
 
 
-open NNReal
+open scoped NNReal
 
 namespace ContinuousMap
 

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

@@ -113,10 +113,8 @@ theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
   Iff.rfl
 #align continuous_map.mem_ideal_of_set ContinuousMap.mem_idealOfSet
 
-theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 :=
-  by
-  simp_rw [mem_ideal_of_set, exists_prop]
-  push_neg
+theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by
+  simp_rw [mem_ideal_of_set, exists_prop]; push_neg
 #align continuous_map.not_mem_ideal_of_set ContinuousMap.not_mem_idealOfSet
 
 /-- Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
@@ -131,10 +129,8 @@ theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
 #align continuous_map.not_mem_set_of_ideal ContinuousMap.not_mem_setOfIdeal
 
 theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
-    x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 :=
-  by
-  simp_rw [set_of_ideal, Set.mem_compl_iff, Set.mem_setOf, exists_prop]
-  push_neg
+    x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 := by
+  simp_rw [set_of_ideal, Set.mem_compl_iff, Set.mem_setOf, exists_prop]; push_neg
 #align continuous_map.mem_set_of_ideal ContinuousMap.mem_setOfIdeal
 
 theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) :=
@@ -286,8 +282,7 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
       ext
       simp only [coe_zero, Pi.zero_apply, ContinuousMap.coe_coe, ContinuousMap.coe_comp, map_zero,
         Pi.comp_zero]
-    · rintro s₁ s₂ hs ⟨g, hI, hgt⟩
-      exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩
+    · rintro s₁ s₂ hs ⟨g, hI, hgt⟩; exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩
     · rintro s₁ s₂ ⟨g₁, hI₁, hgt₁⟩ ⟨g₂, hI₂, hgt₂⟩
       refine' ⟨g₁ + g₂, _, fun x hx => _⟩
       · convert I.add_mem hI₁ hI₂
@@ -450,9 +445,7 @@ def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜))
     ⟨{  toFun := fun f => f x
         map_add' := fun f g => rfl
         map_smul' := fun z f => rfl
-        cont := continuous_eval_const' x },
-      by
-      rw [character_space.eq_set_map_one_map_mul]
+        cont := continuous_eval_const' x }, by rw [character_space.eq_set_map_one_map_mul];
       exact ⟨rfl, fun f g => rfl⟩⟩
   continuous_toFun := Continuous.subtype_mk (continuous_of_continuous_eval map_continuous) _
 #align weak_dual.character_space.continuous_map_eval WeakDual.characterSpace.continuousMapEval

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.ideals
-! leanprover-community/mathlib commit 25580801f04aed44a0daf912d3760d0eaaf6d1bb
+! leanprover-community/mathlib commit c2258f7bf086b17eac0929d635403780c39e239f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,7 +18,7 @@ import Mathbin.Topology.Algebra.Module.CharacterSpace
 /-!
 # Ideals of continuous functions
 
-For a topological ring `R` and a topological space `X` there is a Galois connection between
+For a topological semiring `R` and a topological space `X` there is a Galois connection between
 `ideal C(X, R)` and `set X` given by sending each `I : ideal C(X, R)` to
 `{x : X | ∀ f ∈ I, f x = 0}ᶜ` and mapping `s : set X` to the ideal with carrier
 `{f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}`, and we call these maps `continuous_map.set_of_ideal` and
@@ -81,7 +81,9 @@ open TopologicalSpace
 
 section TopologicalRing
 
-variable {X R : Type _} [TopologicalSpace X] [Ring R] [TopologicalSpace R] [TopologicalRing R]
+variable {X R : Type _} [TopologicalSpace X] [Semiring R]
+
+variable [TopologicalSpace R] [TopologicalSemiring R]
 
 variable (R)
 

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

@@ -100,8 +100,8 @@ theorem idealOfSet_closed [LocallyCompactSpace X] [T2Space R] (s : Set X) :
   by
   simp only [ideal_of_set, Submodule.coe_set_mk, Set.setOf_forall]
   exact
-    isClosed_interᵢ fun x =>
-      isClosed_interᵢ fun hx => isClosed_eq (continuous_eval_const' x) continuous_const
+    isClosed_iInter fun x =>
+      isClosed_iInter fun hx => isClosed_eq (continuous_eval_const' x) continuous_const
 #align continuous_map.ideal_of_set_closed ContinuousMap.idealOfSet_closed
 
 variable {R}
@@ -139,8 +139,8 @@ theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I)
   by
   simp only [set_of_ideal, Set.setOf_forall, isOpen_compl_iff]
   exact
-    isClosed_interᵢ fun f =>
-      isClosed_interᵢ fun hf => isClosed_eq (map_continuous f) continuous_const
+    isClosed_iInter fun f =>
+      isClosed_iInter fun hf => isClosed_eq (map_continuous f) continuous_const
 #align continuous_map.set_of_ideal_open ContinuousMap.setOfIdeal_open
 
 /-- The open set `set_of_ideal I` realized as a term of `opens X`. -/

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

@@ -354,7 +354,7 @@ theorem setOfIdeal_of_set_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜
   exact
     ⟨⟨fun x => g x, continuous_of_real.comp (map_continuous g)⟩, by
       simpa only [coe_mk, of_real_eq_zero] using fun x hx => hgs (subset_closure hx), by
-      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.of_real_one] using
+      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
         one_ne_zero⟩
 #align continuous_map.set_of_ideal_of_set_eq_interior ContinuousMap.setOfIdeal_of_set_eq_interior
 
@@ -473,9 +473,9 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) :=
         (isClosed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [← Ne.def, FunLike.ne_iff]
-    use (⟨coe, IsROrC.continuous_of_real⟩ : C(ℝ, 𝕜)).comp f
+    use (⟨coe, IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuous_map_eval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
-      IsROrC.of_real_inj] using
+      IsROrC.ofReal_inj] using
       ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)
   · obtain ⟨x, hx⟩ := (ideal_is_maximal_iff (RingHom.ker φ)).mp inferInstance
     refine' ⟨x, ext_ker <| Ideal.ext fun f => _⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/3905fa80e62c0898131285baab35559fbc4e5cda

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module topology.continuous_function.ideals
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit 25580801f04aed44a0daf912d3760d0eaaf6d1bb
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -309,7 +309,7 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
       ext
       simp only [comp_apply, coe_mk, algebraMapClm_coe, map_pow, coe_mul, coe_star, Pi.mul_apply,
         Pi.star_apply, star_def, ContinuousMap.coe_coe]
-      simpa only [norm_sq_eq_def', conj_mul_eq_norm_sq_left, of_real_pow]
+      simpa only [norm_sq_eq_def', IsROrC.conj_mul, of_real_pow]
   /- Get the function `g'` which is guaranteed to exist above. By the extreme value theorem and
     compactness of `t`, there is some `0 < c` such that `c ≤ g' x` for all `x ∈ t`. Then by
     `main_lemma_aux` there is some `g` for which `g * g'` is the desired function. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -92,7 +92,7 @@ def idealOfSet (s : Set X) : Ideal C(X, R)
   carrier := { f : C(X, R) | ∀ x ∈ sᶜ, f x = 0 }
   add_mem' f g hf hg x hx := by simp only [hf x hx, hg x hx, coe_add, Pi.add_apply, add_zero]
   zero_mem' _ _ := rfl
-  smul_mem' c f hf x hx := mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
+  smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
 #align continuous_map.ideal_of_set ContinuousMap.idealOfSet
 
 theorem idealOfSet_closed [LocallyCompactSpace X] [T2Space R] (s : Set X) :

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

@@ -262,8 +262,8 @@ theorem idealOfSet_of_ideal_eq_closure (I : Ideal C(X, 𝕜)) :
             ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
           by simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
         _ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
-          (nnnorm_mul_le _ _).trans
-            (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _)
+          ((nnnorm_mul_le _ _).trans
+            (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
         _ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _
         
   /- There is some `g' : C(X, ℝ≥0)` which is strictly positive on `t` such that the composition

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

These are changes from #11997, the latest adaptation PR for nightly-2024-04-07, which can be made directly on master.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -437,7 +437,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
       ⟨f, fx, fy, -⟩
     rw [DFunLike.ne_iff]
     use (⟨fun (x : ℝ) => (x : 𝕜), RCLike.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
-    simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
+    simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne,
       RCLike.ofReal_inj] using
       ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)
   · obtain ⟨x, hx⟩ := (ideal_isMaximal_iff (RingHom.ker φ)).mp inferInstance

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -198,7 +198,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     idealOfSet 𝕜 (setOfIdeal I) = I.closure := by
   /- Since `idealOfSet 𝕜 (setOfIdeal I)` is closed and contains `I`, it contains `I.closure`.
     For the reverse inclusion, given `f ∈ idealOfSet 𝕜 (setOfIdeal I)` and `(ε : ℝ≥0) > 0` it
-    suffices to show that `f` is within `ε` of `I`.-/
+    suffices to show that `f` is within `ε` of `I`. -/
   refine' le_antisymm _
       ((idealOfSet_closed 𝕜 <| setOfIdeal I).closure_subset_iff.mpr fun f hf x hx =>
         not_mem_setOfIdeal.mp hx hf)

RCLike is an analytic typeclass, hence should be under Analysis

@@ -6,7 +6,7 @@ Authors: Jireh Loreaux
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Topology.ContinuousFunction.Compact
 import Mathlib.Topology.UrysohnsLemma
-import Mathlib.Data.RCLike.Basic
+import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Analysis.NormedSpace.Units
 import Mathlib.Topology.Algebra.Module.CharacterSpace
 

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.

@@ -6,7 +6,7 @@ Authors: Jireh Loreaux
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Topology.ContinuousFunction.Compact
 import Mathlib.Topology.UrysohnsLemma
-import Mathlib.Data.IsROrC.Basic
+import Mathlib.Data.RCLike.Basic
 import Mathlib.Analysis.NormedSpace.Units
 import Mathlib.Topology.Algebra.Module.CharacterSpace
 
@@ -22,7 +22,7 @@ For a topological semiring `R` and a topological space `X` there is a Galois con
 `ContinuousMap.idealOfSet`. As long as `R` is Hausdorff, `ContinuousMap.setOfIdeal I` is open,
 and if, in addition, `X` is locally compact, then `ContinuousMap.setOfIdeal s` is closed.
 
-When `R = 𝕜` with `IsROrC 𝕜` and `X` is compact Hausdorff, then this Galois connection can be
+When `R = 𝕜` with `RCLike 𝕜` and `X` is compact Hausdorff, then this Galois connection can be
 improved to a true Galois correspondence (i.e., order isomorphism) between the type `opens X` and
 the subtype of closed ideals of `C(X, 𝕜)`. Because we do not have a bundled type of closed ideals,
 we simply register this as a Galois insertion between `Ideal C(X, 𝕜)` and `opens X`, which is
@@ -32,7 +32,7 @@ ideals corresponding to (complements of) singletons in `X`.
 In addition, when `X` is locally compact and `𝕜` is a nontrivial topological integral domain, then
 there is a natural continuous map from `X` to `WeakDual.characterSpace 𝕜 C(X, 𝕜)` given by point
 evaluation, which is herein called `WeakDual.CharacterSpace.continuousMapEval`. Again, when `X` is
-compact Hausdorff and `IsROrC 𝕜`, more can be obtained. In particular, in that context this map is
+compact Hausdorff and `RCLike 𝕜`, more can be obtained. In particular, in that context this map is
 bijective, and since the domain is compact and the codomain is Hausdorff, it is a homeomorphism,
 herein called `WeakDual.CharacterSpace.homeoEval`.
 
@@ -47,15 +47,15 @@ herein called `WeakDual.CharacterSpace.homeoEval`.
   topological space `X` to the `WeakDual.characterSpace 𝕜 C(X, 𝕜)` which sends `x : X` to point
   evaluation at `x`, with modest hypothesis on `𝕜`.
 * `WeakDual.CharacterSpace.homeoEval`: this is `WeakDual.CharacterSpace.continuousMapEval`
-  upgraded to a homeomorphism when `X` is compact Hausdorff and `IsROrC 𝕜`.
+  upgraded to a homeomorphism when `X` is compact Hausdorff and `RCLike 𝕜`.
 
 ## Main statements
 
 * `ContinuousMap.idealOfSet_ofIdeal_eq_closure`: when `X` is compact Hausdorff and
-  `IsROrC 𝕜`, `idealOfSet 𝕜 (setOfIdeal I) = I.closure` for any ideal `I : Ideal C(X, 𝕜)`.
-* `ContinuousMap.setOfIdeal_ofSet_eq_interior`: when `X` is compact Hausdorff and `IsROrC 𝕜`,
+  `RCLike 𝕜`, `idealOfSet 𝕜 (setOfIdeal I) = I.closure` for any ideal `I : Ideal C(X, 𝕜)`.
+* `ContinuousMap.setOfIdeal_ofSet_eq_interior`: when `X` is compact Hausdorff and `RCLike 𝕜`,
   `setOfIdeal (idealOfSet 𝕜 s) = interior s` for any `s : Set X`.
-* `ContinuousMap.ideal_isMaximal_iff`: when `X` is compact Hausdorff and `IsROrC 𝕜`, a closed
+* `ContinuousMap.ideal_isMaximal_iff`: when `X` is compact Hausdorff and `RCLike 𝕜`, a closed
   ideal of `C(X, 𝕜)` is maximal if and only if it is `idealOfSet 𝕜 {x}ᶜ` for some `x : X`.
 
 ## Implementation details
@@ -170,11 +170,11 @@ theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idea
 
 end TopologicalRing
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
-variable {X 𝕜 : Type*} [IsROrC 𝕜] [TopologicalSpace X]
+variable {X 𝕜 : Type*} [RCLike 𝕜] [TopologicalSpace X]
 
 /-- An auxiliary lemma used in the proof of `ContinuousMap.idealOfSet_ofIdeal_eq_closure` which may
 be useful on its own. -/
@@ -283,7 +283,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
       ext
       simp only [comp_apply, ContinuousMap.coe_coe, coe_mk, algebraMapCLM_toFun, map_pow,
         mul_apply, star_apply, star_def]
-      simp only [normSq_eq_def', IsROrC.conj_mul, ofReal_pow]
+      simp only [normSq_eq_def', RCLike.conj_mul, ofReal_pow]
       rfl
   /- Get the function `g'` which is guaranteed to exist above. By the extreme value theorem and
     compactness of `t`, there is some `0 < c` such that `c ≤ g' x` for all `x ∈ t`. Then by
@@ -327,7 +327,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   exact
     ⟨⟨fun x => g x, continuous_ofReal.comp (map_continuous g)⟩, by
       simpa only [coe_mk, ofReal_eq_zero] using fun x hx => hgs (subset_closure hx), by
-      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, IsROrC.ofReal_one] using
+      simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, RCLike.ofReal_one] using
         one_ne_zero⟩
 #align continuous_map.set_of_ideal_of_set_eq_interior ContinuousMap.setOfIdeal_ofSet_eq_interior
 
@@ -391,7 +391,7 @@ theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X,
         congr_arg (idealOfSet 𝕜) hx.symm⟩
 #align continuous_map.ideal_is_maximal_iff ContinuousMap.ideal_isMaximal_iff
 
-end IsROrC
+end RCLike
 
 end ContinuousMap
 
@@ -427,7 +427,7 @@ theorem continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEv
 
 end ContinuousMapEval
 
-variable [CompactSpace X] [T2Space X] [IsROrC 𝕜]
+variable [CompactSpace X] [T2Space X] [RCLike 𝕜]
 
 theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
   refine' ⟨fun x y hxy => _, fun φ => _⟩
@@ -436,9 +436,9 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
         (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [DFunLike.ne_iff]
-    use (⟨fun (x : ℝ) => (x : 𝕜), IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
+    use (⟨fun (x : ℝ) => (x : 𝕜), RCLike.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
-      IsROrC.ofReal_inj] using
+      RCLike.ofReal_inj] using
       ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)
   · obtain ⟨x, hx⟩ := (ideal_isMaximal_iff (RingHom.ker φ)).mp inferInstance
     refine' ⟨x, CharacterSpace.ext_ker <| Ideal.ext fun f => _⟩

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -79,9 +79,7 @@ open TopologicalSpace
 section TopologicalRing
 
 variable {X R : Type*} [TopologicalSpace X] [Semiring R]
-
 variable [TopologicalSpace R] [TopologicalSemiring R]
-
 variable (R)
 
 /-- Given a topological ring `R` and `s : Set X`, construct the ideal in `C(X, R)` of functions
@@ -408,7 +406,6 @@ variable (X 𝕜 : Type*) [TopologicalSpace X]
 section ContinuousMapEval
 
 variable [LocallyCompactSpace X] [CommRing 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜]
-
 variable [Nontrivial 𝕜] [NoZeroDivisors 𝕜]
 
 /-- The natural continuous map from a locally compact topological space `X` to the

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

@@ -214,7 +214,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm
   have htI : Disjoint t (setOfIdeal I)ᶜ := by
     refine' Set.subset_compl_iff_disjoint_left.mp fun x hx => _
-    simpa only [Set.mem_setOf, Set.mem_compl_iff, not_le] using
+    simpa only [t, Set.mem_setOf, Set.mem_compl_iff, not_le] using
       (nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε)
   /- It suffices to produce `g : C(X, ℝ≥0)` which takes values in `[0,1]` and is constantly `1` on
     `t` such that when composed with the natural embedding of `ℝ≥0` into `𝕜` lies in the ideal `I`.

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

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

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

@@ -376,7 +376,7 @@ theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal]
   have h : (idealOfSet 𝕜 (setOfIdeal I)).IsMaximal :=
     (idealOfSet_ofIdeal_isClosed (inferInstance : IsClosed (I : Set C(X, 𝕜)))).symm ▸ hI
   obtain ⟨x, hx⟩ := Opens.isCoatom_iff.1 ((idealOfSet_isMaximal_iff 𝕜 (opensOfIdeal I)).1 h)
-  refine ⟨x, congr_arg (fun (s : Opens X) => (s : Set X)) hx⟩
+  exact ⟨x, congr_arg (fun (s : Opens X) => (s : Set X)) hx⟩
 #align continuous_map.set_of_ideal_eq_compl_singleton ContinuousMap.setOfIdeal_eq_compl_singleton
 
 theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :

@@ -240,7 +240,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
           _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
             simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, ge_iff_le,
               NNReal.coe_sub (hg x), NNReal.coe_one, sub_smul, one_smul]
-          _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
+          _ ≤ 1 := (nnnorm_algebraMap_nnreal 𝕜 (1 - g x)).trans_le tsub_le_self
       calc
         ‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
             ‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by

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.

@@ -221,31 +221,31 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     Indeed, then `‖f - f * ↑g‖ ≤ ‖f * (1 - ↑g)‖ ≤ ⨆ ‖f * (1 - ↑g) x‖`. When `x ∉ t`, `‖f x‖ < ε / 2`
     and `‖(1 - ↑g) x‖ ≤ 1`, and when `x ∈ t`, `(1 - ↑g) x = 0`, and clearly `f * ↑g ∈ I`. -/
   suffices
-    ∃ g : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.EqOn g 1 by
+    ∃ g : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.EqOn g 1 by
     obtain ⟨g, hgI, hg, hgt⟩ := this
-    refine' ⟨f * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩
+    refine' ⟨f * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩
     rw [nndist_eq_nnnorm]
     refine' (nnnorm_lt_iff _ hε).2 fun x => _
     simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply]
     by_cases hx : x ∈ t
-    · simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapClm_apply,
+    · simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapCLM_apply,
         map_one, mul_one, sub_self, nnnorm_zero] using hε
     · refine' lt_of_le_of_lt _ (half_lt_self hε)
       have :=
         calc
-          ‖((1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
+          ‖((1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
               ‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
             simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
-              Pi.one_apply, Function.comp_apply, algebraMapClm_apply]
+              Pi.one_apply, Function.comp_apply, algebraMapCLM_apply]
           _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
             simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, ge_iff_le,
               NNReal.coe_sub (hg x), NNReal.coe_one, sub_smul, one_smul]
           _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
       calc
-        ‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
-            ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
+        ‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
+            ‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
           simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
-        _ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
+        _ ≤ ε / 2 * ‖(1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
           ((nnnorm_mul_le _ _).trans
             (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
         _ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _
@@ -255,7 +255,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     `fₓ x ≠ 0` for some `fₓ ∈ I` and so `fun y ↦ ‖(star fₓ * fₓ) y‖₊` is strictly posiive in a
     neighborhood of `y`. Moreover, `(‖(star fₓ * fₓ) y‖₊ : 𝕜) = (star fₓ * fₓ) y`, so composition of
     this map with the natural embedding is just `star fₓ * fₓ ∈ I`. -/
-  have : ∃ g' : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x := by
+  have : ∃ g' : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x := by
     refine' ht.isCompact.induction_on _ _ _ _
     · refine' ⟨0, _, fun x hx => False.elim hx⟩
       convert I.zero_mem
@@ -283,7 +283,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
             pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩
       convert I.mul_mem_left (star g) hI
       ext
-      simp only [comp_apply, ContinuousMap.coe_coe, coe_mk, algebraMapClm_toFun, map_pow,
+      simp only [comp_apply, ContinuousMap.coe_coe, coe_mk, algebraMapCLM_toFun, map_pow,
         mul_apply, star_apply, star_def]
       simp only [normSq_eq_def', IsROrC.conj_mul, ofReal_pow]
       rfl
@@ -298,9 +298,9 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
       ⟨g' x, hgt' x hx, hx'⟩
   obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eqOn_ge g' hc
   refine' ⟨g * g', _, hg, hgc.mono hgc'⟩
-  convert I.mul_mem_left ((algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
+  convert I.mul_mem_left ((algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
   ext
-  simp only [algebraMapClm_coe, comp_apply, mul_apply, ContinuousMap.coe_coe, map_mul]
+  simp only [algebraMapCLM_coe, comp_apply, mul_apply, ContinuousMap.coe_coe, map_mul]
 #align continuous_map.ideal_of_set_of_ideal_eq_closure ContinuousMap.idealOfSet_ofIdeal_eq_closure
 
 theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :

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>

@@ -149,7 +149,7 @@ theorem setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set
 theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ :=
   Ideal.ext fun f => by
     simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
-      FunLike.ext_iff, zero_apply]
+      DFunLike.ext_iff, zero_apply]
 #align continuous_map.ideal_of_empty_eq_bot ContinuousMap.idealOfEmpty_eq_bot
 
 @[simp]
@@ -438,7 +438,7 @@ theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
     rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
         (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
-    rw [FunLike.ne_iff]
+    rw [DFunLike.ne_iff]
     use (⟨fun (x : ℝ) => (x : 𝕜), IsROrC.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
     simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne.def,
       IsROrC.ofReal_inj] using

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.

@@ -291,7 +291,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     compactness of `t`, there is some `0 < c` such that `c ≤ g' x` for all `x ∈ t`. Then by
     `exists_mul_le_one_eqOn_ge` there is some `g` for which `g * g'` is the desired function. -/
   obtain ⟨g', hI', hgt'⟩ := this
-  obtain ⟨c, hc, hgc'⟩ : ∃ (c : _) (_ : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
+  obtain ⟨c, hc, hgc'⟩ : ∃ c > 0, ∀ y : X, y ∈ t → c ≤ g' y :=
     t.eq_empty_or_nonempty.elim
       (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
       let ⟨x, hx, hx'⟩ := ht.isCompact.exists_forall_le ht' (map_continuous g').continuousOn

Three particular examples which caught my eye; not exhaustive.

@@ -324,7 +324,7 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
     with the natural embedding `ℝ ↪ 𝕜` to produce the desired `f`. -/
   obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
-    exists_continuous_zero_one_of_closed isClosed_closure isClosed_singleton
+    exists_continuous_zero_one_of_isClosed isClosed_closure isClosed_singleton
       (Set.disjoint_singleton_right.mpr hx)
   exact
     ⟨⟨fun x => g x, continuous_ofReal.comp (map_continuous g)⟩, by
@@ -435,7 +435,7 @@ variable [CompactSpace X] [T2Space X] [IsROrC 𝕜]
 theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
   refine' ⟨fun x y hxy => _, fun φ => _⟩
   · contrapose! hxy
-    rcases exists_continuous_zero_one_of_closed (isClosed_singleton : _root_.IsClosed {x})
+    rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
         (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩
     rw [FunLike.ne_iff]

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -243,8 +243,8 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
           _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
       calc
         ‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
-            ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
-          by simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
+            ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
+          simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
         _ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
           ((nnnorm_mul_le _ _).trans
             (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))

Purely cosmetic PR.

@@ -293,7 +293,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   obtain ⟨g', hI', hgt'⟩ := this
   obtain ⟨c, hc, hgc'⟩ : ∃ (c : _) (_ : 0 < c), ∀ y : X, y ∈ t → c ≤ g' y :=
     t.eq_empty_or_nonempty.elim
-      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy )⟩) fun ht' =>
+      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
       let ⟨x, hx, hx'⟩ := ht.isCompact.exists_forall_le ht' (map_continuous g').continuousOn
       ⟨g' x, hgt' x hx, hx'⟩
   obtain ⟨g, hg, hgc⟩ := exists_mul_le_one_eqOn_ge g' hc

• Rename NormalSpace to T4Space.
• Add NormalSpace, a version without the T1Space assumption.
• Adjust some theorems.
• Supersedes thus closes #6892.
• Add some instance cycles, see #2030

@@ -321,7 +321,6 @@ theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s
   -- If `x ∉ closure sᶜ`, we must produce `f : C(X, 𝕜)` which is zero on `sᶜ` and `f x ≠ 0`.
   rw [← compl_compl (interior s), ← closure_compl] at hx
   simp_rw [mem_setOfIdeal, mem_idealOfSet]
-  haveI : NormalSpace X := normalOfCompactT2
   /- Apply Urysohn's lemma to get `g : C(X, ℝ)` which is zero on `sᶜ` and `g x ≠ 0`, then compose
     with the natural embedding `ℝ ↪ 𝕜` to produce the desired `f`. -/
   obtain ⟨g, hgs, hgx : Set.EqOn g 1 {x}, -⟩ :=
@@ -436,7 +435,6 @@ variable [CompactSpace X] [T2Space X] [IsROrC 𝕜]
 theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by
   refine' ⟨fun x y hxy => _, fun φ => _⟩
   · contrapose! hxy
-    haveI := @normalOfCompactT2 X _ _ _
     rcases exists_continuous_zero_one_of_closed (isClosed_singleton : _root_.IsClosed {x})
         (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
       ⟨f, fx, fy, -⟩

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -90,7 +90,7 @@ def idealOfSet (s : Set X) : Ideal C(X, R) where
   carrier := {f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}
   add_mem' {f g} hf hg x hx := by simp [hf x hx, hg x hx, coe_add, Pi.add_apply, add_zero]
   zero_mem' _ _ := rfl
-  smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
+  smul_mem' c f hf x hx := mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
 #align continuous_map.ideal_of_set ContinuousMap.idealOfSet
 
 theorem idealOfSet_closed [T2Space R] (s : Set X) :

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -108,7 +108,7 @@ theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
 #align continuous_map.mem_ideal_of_set ContinuousMap.mem_idealOfSet
 
 theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by
-  simp_rw [mem_idealOfSet, exists_prop]; push_neg; rfl
+  simp_rw [mem_idealOfSet]; push_neg; rfl
 #align continuous_map.not_mem_ideal_of_set ContinuousMap.not_mem_idealOfSet
 
 /-- Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
@@ -124,7 +124,7 @@ theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
 
 theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
     x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 := by
-  simp_rw [setOfIdeal, Set.mem_compl_iff, Set.mem_setOf, exists_prop]; push_neg; rfl
+  simp_rw [setOfIdeal, Set.mem_compl_iff, Set.mem_setOf]; push_neg; rfl
 #align continuous_map.mem_set_of_ideal ContinuousMap.mem_setOfIdeal
 
 theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) := by

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

This has nice performance benefits.

@@ -78,7 +78,7 @@ open TopologicalSpace
 
 section TopologicalRing
 
-variable {X R : Type _} [TopologicalSpace X] [Semiring R]
+variable {X R : Type*} [TopologicalSpace X] [Semiring R]
 
 variable [TopologicalSpace R] [TopologicalSemiring R]
 
@@ -176,7 +176,7 @@ section IsROrC
 
 open IsROrC
 
-variable {X 𝕜 : Type _} [IsROrC 𝕜] [TopologicalSpace X]
+variable {X 𝕜 : Type*} [IsROrC 𝕜] [TopologicalSpace X]
 
 /-- An auxiliary lemma used in the proof of `ContinuousMap.idealOfSet_ofIdeal_eq_closure` which may
 be useful on its own. -/
@@ -404,7 +404,7 @@ namespace CharacterSpace
 
 open Function ContinuousMap
 
-variable (X 𝕜 : Type _) [TopologicalSpace X]
+variable (X 𝕜 : Type*) [TopologicalSpace X]
 
 section ContinuousMapEval
 

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module topology.continuous_function.ideals
-! leanprover-community/mathlib commit c2258f7bf086b17eac0929d635403780c39e239f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Topology.ContinuousFunction.Compact
@@ -15,6 +10,8 @@ import Mathlib.Data.IsROrC.Basic
 import Mathlib.Analysis.NormedSpace.Units
 import Mathlib.Topology.Algebra.Module.CharacterSpace
 
+#align_import topology.continuous_function.ideals from "leanprover-community/mathlib"@"c2258f7bf086b17eac0929d635403780c39e239f"
+
 /-!
 # Ideals of continuous functions
 

We had continuity of fun f : C(X, Y) ↦ f a in two cases:

• X is a locally compact space;
• X is a compact space and Y is a metric space.

In fact, it is true in general topological spaces.

@@ -96,11 +96,11 @@ def idealOfSet (s : Set X) : Ideal C(X, R) where
   smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
 #align continuous_map.ideal_of_set ContinuousMap.idealOfSet
 
-theorem idealOfSet_closed [LocallyCompactSpace X] [T2Space R] (s : Set X) :
+theorem idealOfSet_closed [T2Space R] (s : Set X) :
     IsClosed (idealOfSet R s : Set C(X, R)) := by
   simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]
   exact isClosed_iInter fun x => isClosed_iInter fun _ =>
-    isClosed_eq (continuous_eval_const' x) continuous_const
+    isClosed_eq (continuous_eval_const x) continuous_const
 #align continuous_map.ideal_of_set_closed ContinuousMap.idealOfSet_closed
 
 variable {R}
@@ -422,7 +422,7 @@ def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜)) where
     ⟨{  toFun := fun f => f x
         map_add' := fun f g => rfl
         map_smul' := fun z f => rfl
-        cont := continuous_eval_const' x }, by
+        cont := continuous_eval_const x }, by
         rw [CharacterSpace.eq_set_map_one_map_mul]; exact ⟨rfl, fun f g => rfl⟩⟩
   continuous_toFun := Continuous.subtype_mk (continuous_of_continuous_eval map_continuous) _
 #align weak_dual.character_space.continuous_map_eval WeakDual.CharacterSpace.continuousMapEval

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -215,7 +215,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   -- Let `t := {x : X | ε / 2 ≤ ‖f x‖₊}}` which is closed and disjoint from `set_of_ideal I`.
   set t := {x : X | ε / 2 ≤ ‖f x‖₊}
   have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm
-  have htI : Disjoint t (setOfIdeal Iᶜ) := by
+  have htI : Disjoint t (setOfIdeal I)ᶜ := by
     refine' Set.subset_compl_iff_disjoint_left.mp fun x hx => _
     simpa only [Set.mem_setOf, Set.mem_compl_iff, not_le] using
       (nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε)
@@ -384,7 +384,7 @@ theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal]
 #align continuous_map.set_of_ideal_eq_compl_singleton ContinuousMap.setOfIdeal_eq_compl_singleton
 
 theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
-    I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 ({x}ᶜ) = I := by
+    I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 {x}ᶜ = I := by
   refine'
     ⟨_, fun h =>
       let ⟨x, hx⟩ := h

@@ -263,7 +263,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
     · refine' ⟨0, _, fun x hx => False.elim hx⟩
       convert I.zero_mem
       ext
-      simp only [comp_apply, zero_apply, coe_mk, map_zero]
+      simp only [comp_apply, zero_apply, ContinuousMap.coe_coe, map_zero]
     · rintro s₁ s₂ hs ⟨g, hI, hgt⟩; exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩
     · rintro s₁ s₂ ⟨g₁, hI₁, hgt₁⟩ ⟨g₂, hI₂, hgt₂⟩
       refine' ⟨g₁ + g₂, _, fun x hx => _⟩
@@ -286,7 +286,8 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
             pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩
       convert I.mul_mem_left (star g) hI
       ext
-      simp only [comp_apply, coe_mk, algebraMapClm_toFun, map_pow, mul_apply, star_apply, star_def]
+      simp only [comp_apply, ContinuousMap.coe_coe, coe_mk, algebraMapClm_toFun, map_pow,
+        mul_apply, star_apply, star_def]
       simp only [normSq_eq_def', IsROrC.conj_mul, ofReal_pow]
       rfl
   /- Get the function `g'` which is guaranteed to exist above. By the extreme value theorem and
@@ -302,7 +303,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   refine' ⟨g * g', _, hg, hgc.mono hgc'⟩
   convert I.mul_mem_left ((algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) hI'
   ext
-  simp only [algebraMapClm_coe, comp_apply, mul_apply, coe_mk, map_mul]
+  simp only [algebraMapClm_coe, comp_apply, mul_apply, ContinuousMap.coe_coe, map_mul]
 #align continuous_map.ideal_of_set_of_ideal_eq_closure ContinuousMap.idealOfSet_ofIdeal_eq_closure
 
 theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :

@@ -91,7 +91,7 @@ variable (R)
 which vanish on the complement of `s`. -/
 def idealOfSet (s : Set X) : Ideal C(X, R) where
   carrier := {f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}
-  add_mem' := @fun f g hf hg x hx => by simp [hf x hx, hg x hx, coe_add, Pi.add_apply, add_zero]
+  add_mem' {f g} hf hg x hx := by simp [hf x hx, hg x hx, coe_add, Pi.add_apply, add_zero]
   zero_mem' _ _ := rfl
   smul_mem' c f hf x hx := MulZeroClass.mul_zero (c x) ▸ congr_arg (fun y => c x * y) (hf x hx)
 #align continuous_map.ideal_of_set ContinuousMap.idealOfSet
@@ -342,7 +342,7 @@ theorem setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (ide
 
 variable (X)
 
-/-- The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → opens X` and
+/-- The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → Opens X` and
 `fun s ↦ ContinuousMap.idealOfSet ↑s`. -/
 @[simps]
 def idealOpensGI : GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s
@@ -365,7 +365,6 @@ theorem idealOfSet_isMaximal_iff (s : Opens X) :
   rw [Ideal.isMaximal_def]
   refine' (idealOpensGI X 𝕜).isCoatom_iff (fun I hI => _) s
   rw [← Ideal.isMaximal_def] at hI
-  skip
   exact idealOfSet_ofIdeal_isClosed inferInstance
 #align continuous_map.ideal_of_set_is_maximal_iff ContinuousMap.idealOfSet_isMaximal_iff
 

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

@@ -45,7 +45,7 @@ herein called `WeakDual.CharacterSpace.homeoEval`.
 * `ContinuousMap.setOfIdeal`: complement of the set on which all functions in the ideal vanish.
 * `ContinuousMap.opensOfIdeal`: `ContinuousMap.setOfIdeal` as a term of `opens X`.
 * `ContinuousMap.idealOpensGI`: The Galois insertion `ContinuousMap.opensOfIdeal` and
-  `λ s, ContinuousMap.idealOfSet ↑s`.
+  `fun s ↦ ContinuousMap.idealOfSet ↑s`.
 * `WeakDual.CharacterSpace.continuousMapEval`: the natural continuous map from a locally compact
   topological space `X` to the `WeakDual.characterSpace 𝕜 C(X, 𝕜)` which sends `x : X` to point
   evaluation at `x`, with modest hypothesis on `𝕜`.
@@ -255,7 +255,7 @@ theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
   /- There is some `g' : C(X, ℝ≥0)` which is strictly positive on `t` such that the composition
     `↑g` with the natural embedding of `ℝ≥0` into `𝕜` lies in `I`. This follows from compactness of
     `t` and that we can do it in any neighborhood of a point `x ∈ t`. Indeed, since `x ∈ t`, then
-    `fₓ x ≠ 0` for some `fₓ ∈ I` and so `λ y, ‖(star fₓ * fₓ) y‖₊` is strictly posiive in a
+    `fₓ x ≠ 0` for some `fₓ ∈ I` and so `fun y ↦ ‖(star fₓ * fₓ) y‖₊` is strictly posiive in a
     neighborhood of `y`. Moreover, `(‖(star fₓ * fₓ) y‖₊ : 𝕜) = (star fₓ * fₓ) y`, so composition of
     this map with the natural embedding is just `star fₓ * fₓ ∈ I`. -/
   have : ∃ g' : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x := by
@@ -343,7 +343,7 @@ theorem setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (ide
 variable (X)
 
 /-- The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → opens X` and
-`λ s, ContinuousMap.idealOfSet ↑s`. -/
+`fun s ↦ ContinuousMap.idealOfSet ↑s`. -/
 @[simps]
 def idealOpensGI : GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s
     where

Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>