topology.tietze_extension ⟷ Mathlib.Topology.TietzeExtension

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: Yury G. Kudryashov
 -/
 import Analysis.SpecificLimits.Basic
-import Data.Set.Intervals.IsoIoo
+import Order.Interval.Set.IsoIoo
 import Topology.Order.MonotoneContinuity
 import Topology.UrysohnsBounded
 

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

@@ -5,7 +5,7 @@ Authors: Yury G. Kudryashov
 -/
 import Analysis.SpecificLimits.Basic
 import Data.Set.Intervals.IsoIoo
-import Topology.Algebra.Order.MonotoneContinuity
+import Topology.Order.MonotoneContinuity
 import Topology.UrysohnsBounded
 
 #align_import topology.tietze_extension from "leanprover-community/mathlib"@"61b5e2755ccb464b68d05a9acf891ae04992d09d"
@@ -121,7 +121,8 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     by
     intro n; induction' n with n ihn
     · simp [g0]
-    · rw [g_succ n, add_comp_continuous, ← dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]
+    · rw [g_succ n, add_comp_continuous, ← dist_sub_right, add_sub_cancel_left, pow_succ',
+        mul_assoc]
       refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))
       rwa [← dist_eq_norm']
   have hg_dist : ∀ n, dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n :=
@@ -129,7 +130,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     intro n
     calc
       dist (g n) (g (n + 1)) = ‖F (f - (g n).comp_continuous e)‖ := by
-        rw [g_succ, dist_eq_norm', add_sub_cancel']
+        rw [g_succ, dist_eq_norm', add_sub_cancel_left]
       _ ≤ ‖f - (g n).comp_continuous e‖ / 3 := (hF_norm _)
       _ = 1 / 3 * dist ((g n).comp_continuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
       _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
@@ -276,7 +277,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       ·
         calc
           g y + dg y ≤ c + (c - a) := add_le_add hc (dgmem _).2
-          _ = b := by rw [hsub, add_sub_cancel'_right]
+          _ = b := by rw [hsub, add_sub_cancel]
   /- Now we deal with the case `∀ x, f x ≠ b`. The proof is the same as in the first case, with
     minor modifications that make it hard to deduplicate code. -/
   choose xl hxl hgb using hg_mem

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

@@ -390,7 +390,7 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
   have : ord_connected t' := by
     constructor; rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ z hz
     lift z to Ioo (-1 : ℝ) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz
-    change z ∈ Icc (h x) (h y) at hz ; rw [← h.image_Icc] at hz 
+    change z ∈ Icc (h x) (h y) at hz; rw [← h.image_Icc] at hz
     rcases hz with ⟨z, hz, rfl⟩
     exact ⟨z, hs.out hx hy hz, rfl⟩
   have hFt : ∀ x, F x ∈ t' := fun x => mem_image_of_mem _ (hf x)

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

@@ -408,16 +408,16 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
 #align continuous_map.exists_extension_forall_mem_of_closed_embedding ContinuousMap.exists_extension_forall_mem_of_closedEmbedding
 -/
 
-#print ContinuousMap.exists_extension_of_closedEmbedding /-
+#print ContinuousMap.exists_extension' /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a continuous real-valued function on `X`. Then there exists a
 continuous real-valued function `g : C(Y, ℝ)` such that `g ∘ e = f`. -/
-theorem exists_extension_of_closedEmbedding (f : C(X, ℝ)) (e : X → Y) (he : ClosedEmbedding e) :
+theorem exists_extension' (f : C(X, ℝ)) (e : X → Y) (he : ClosedEmbedding e) :
     ∃ g : C(Y, ℝ), g ∘ e = f :=
   (exists_extension_forall_mem_of_closedEmbedding f (fun x => mem_univ _) univ_nonempty he).imp
     fun g => And.right
-#align continuous_map.exists_extension_of_closed_embedding ContinuousMap.exists_extension_of_closedEmbedding
+#align continuous_map.exists_extension_of_closed_embedding ContinuousMap.exists_extension'
 -/
 
 #print ContinuousMap.exists_restrict_eq_forall_mem_of_closed /-
@@ -436,17 +436,17 @@ theorem exists_restrict_eq_forall_mem_of_closed {s : Set Y} (f : C(s, ℝ)) {t :
 #align continuous_map.exists_restrict_eq_forall_mem_of_closed ContinuousMap.exists_restrict_eq_forall_mem_of_closed
 -/
 
-#print ContinuousMap.exists_restrict_eq_of_closed /-
+#print ContinuousMap.exists_restrict_eq /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
 on `s`. Then there exists a continuous real-valued function `g : C(Y, ℝ)` such that
 `g.restrict s = f`. -/
-theorem exists_restrict_eq_of_closed {s : Set Y} (f : C(s, ℝ)) (hs : IsClosed s) :
+theorem exists_restrict_eq {s : Set Y} (f : C(s, ℝ)) (hs : IsClosed s) :
     ∃ g : C(Y, ℝ), g.restrict s = f :=
   let ⟨g, hg, hgf⟩ :=
     exists_restrict_eq_forall_mem_of_closed f (fun _ => mem_univ _) univ_nonempty hs
   ⟨g, hgf⟩
-#align continuous_map.exists_restrict_eq_of_closed ContinuousMap.exists_restrict_eq_of_closed
+#align continuous_map.exists_restrict_eq_of_closed ContinuousMap.exists_restrict_eq
 -/
 
 end ContinuousMap

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

@@ -144,7 +144,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)
     refine' squeeze_zero (fun _ => dist_nonneg) hgf _
     rw [← MulZeroClass.zero_mul ‖f‖]
-    refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1
+    refine' (tendsto_pow_atTop_nhds_zero_of_lt_one _ _).mul tendsto_const_nhds <;> norm_num1
   refine' ⟨limUnder at_top g, le_antisymm _ _, hge⟩
   · rw [← dist_zero_left, ← g0]
     refine'

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

@@ -359,7 +359,7 @@ theorem exists_forall_mem_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (
   rcases exists_extension_forall_mem_of_closed_embedding f hf hne
       (closedEmbedding_subtype_val hs) with
     ⟨g, hg, hgf⟩
-  exact ⟨g, hg, FunLike.coe_injective hgf⟩
+  exact ⟨g, hg, DFunLike.coe_injective hgf⟩
 #align bounded_continuous_function.exists_forall_mem_restrict_eq_of_closed BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed
 -/
 

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 -/
-import Mathbin.Analysis.SpecificLimits.Basic
-import Mathbin.Data.Set.Intervals.IsoIoo
-import Mathbin.Topology.Algebra.Order.MonotoneContinuity
-import Mathbin.Topology.UrysohnsBounded
+import Analysis.SpecificLimits.Basic
+import Data.Set.Intervals.IsoIoo
+import Topology.Algebra.Order.MonotoneContinuity
+import Topology.UrysohnsBounded
 
 #align_import topology.tietze_extension from "leanprover-community/mathlib"@"61b5e2755ccb464b68d05a9acf891ae04992d09d"
 

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

@@ -219,9 +219,9 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   inhabit X
   -- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
   obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a
-  exact ⟨_, isGLB_ciInf (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1⟩
+  exact ⟨_, isGLB_ciInf (Real.isBounded_iff_bddBelow_bddAbove.1 f.bounded_range).1⟩
   obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b
-  exact ⟨_, isLUB_ciSup (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2⟩
+  exact ⟨_, isLUB_ciSup (Real.isBounded_iff_bddBelow_bddAbove.1 f.bounded_range).2⟩
   -- Then `f x ∈ [a, b]` for all `x`
   have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.1 ⟨x, rfl⟩, hb.1 ⟨x, rfl⟩⟩
   -- Rule out the trivial case `a = b`

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module topology.tietze_extension
-! leanprover-community/mathlib commit 61b5e2755ccb464b68d05a9acf891ae04992d09d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecificLimits.Basic
 import Mathbin.Data.Set.Intervals.IsoIoo
 import Mathbin.Topology.Algebra.Order.MonotoneContinuity
 import Mathbin.Topology.UrysohnsBounded
 
+#align_import topology.tietze_extension from "leanprover-community/mathlib"@"61b5e2755ccb464b68d05a9acf891ae04992d09d"
+
 /-!
 # Tietze extension theorem
 

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

@@ -51,6 +51,7 @@ noncomputable section
 
 namespace BoundedContinuousFunction
 
+#print BoundedContinuousFunction.tietze_extension_step /-
 /-- One step in the proof of the Tietze extension theorem. If `e : C(X, Y)` is a closed embedding
 of a topological space into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous
 function, then there exists a bounded continuous function `g : Y →ᵇ ℝ` of the norm `‖g‖ ≤ ‖f‖ / 3`
@@ -102,7 +103,9 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
             rw [hg₂ (mem_image_of_mem _ hle₂), abs_sub_comm, abs_of_nonneg (sub_nonneg.2 hle₂)]
           _ ≤ 2 / 3 * ‖f‖ := by linarith
 #align bounded_continuous_function.tietze_extension_step BoundedContinuousFunction.tietze_extension_step
+-/
 
+#print BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding' /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
 embedding and bundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
 into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
@@ -152,7 +155,9 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     field_simp [show (3 - 2 : ℝ) = 1 by norm_num1]
   · rw [← hge]; exact norm_comp_continuous_le _ _
 #align bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding' BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding'
+-/
 
+#print BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
 embedding and unbundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
 into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
@@ -163,7 +168,9 @@ theorem exists_extension_norm_eq_of_closedEmbedding (f : X →ᵇ ℝ) {e : X 
   rcases exists_extension_norm_eq_of_closed_embedding' f ⟨e, he.continuous⟩ he with ⟨g, hg, rfl⟩
   exact ⟨g, hg, rfl⟩
 #align bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding
+-/
 
+#print BoundedContinuousFunction.exists_norm_eq_restrict_eq_of_closed /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
 set. If `f` is a bounded continuous real-valued function defined on a closed set in a normal
 topological space, then it can be extended to a bounded continuous function of the same norm defined
@@ -173,7 +180,9 @@ theorem exists_norm_eq_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (hs
   exists_extension_norm_eq_of_closedEmbedding' f ((ContinuousMap.id _).restrict s)
     (closedEmbedding_subtype_val hs)
 #align bounded_continuous_function.exists_norm_eq_restrict_eq_of_closed BoundedContinuousFunction.exists_norm_eq_restrict_eq_of_closed
+-/
 
+#print BoundedContinuousFunction.exists_extension_forall_mem_Icc_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
 embedding and a bounded continuous function that takes values in a non-trivial closed interval.
 See also `exists_extension_forall_mem_of_closed_embedding` for a more general statement that works
@@ -198,7 +207,9 @@ theorem exists_extension_forall_mem_Icc_of_closedEmbedding (f : X →ᵇ ℝ) {a
     have : g (e x) = f x - (a + b) / 2 := congr_fun hge x
     simp [this]
 #align bounded_continuous_function.exists_extension_forall_mem_Icc_of_closed_embedding BoundedContinuousFunction.exists_extension_forall_mem_Icc_of_closedEmbedding
+-/
 
+#print BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Then there
@@ -313,7 +324,9 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
     · refine' ⟨xl y, xu, _, hyxu.le⟩
       simp [dg0 (Or.inr hc), hxl]
 #align bounded_continuous_function.exists_extension_forall_exists_le_ge_of_closed_embedding BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_closedEmbedding
+-/
 
+#print BoundedContinuousFunction.exists_extension_forall_mem_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Let `t` be
@@ -333,7 +346,9 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : X →ᵇ ℝ) {t : S
   rcases hg y with ⟨xl, xu, h⟩
   exact hs.out (hf _) (hf _) h
 #align bounded_continuous_function.exists_extension_forall_mem_of_closed_embedding BoundedContinuousFunction.exists_extension_forall_mem_of_closedEmbedding
+-/
 
+#print BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed /-
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
 set. Let `s` be a closed set in a normal topological space `Y`. Let `f` be a bounded continuous
 real-valued function on `s`. Let `t` be a nonempty convex set of real numbers (we use
@@ -349,11 +364,13 @@ theorem exists_forall_mem_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (
     ⟨g, hg, hgf⟩
   exact ⟨g, hg, FunLike.coe_injective hgf⟩
 #align bounded_continuous_function.exists_forall_mem_restrict_eq_of_closed BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed
+-/
 
 end BoundedContinuousFunction
 
 namespace ContinuousMap
 
+#print ContinuousMap.exists_extension_forall_mem_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a continuous real-valued function on `X`. Let `t` be a nonempty
@@ -392,7 +409,9 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
     exact hgG.2 hay.symm
   · ext x; exact hgG.2 (congr_fun hGF _)
 #align continuous_map.exists_extension_forall_mem_of_closed_embedding ContinuousMap.exists_extension_forall_mem_of_closedEmbedding
+-/
 
+#print ContinuousMap.exists_extension_of_closedEmbedding /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a continuous real-valued function on `X`. Then there exists a
@@ -402,7 +421,9 @@ theorem exists_extension_of_closedEmbedding (f : C(X, ℝ)) (e : X → Y) (he :
   (exists_extension_forall_mem_of_closedEmbedding f (fun x => mem_univ _) univ_nonempty he).imp
     fun g => And.right
 #align continuous_map.exists_extension_of_closed_embedding ContinuousMap.exists_extension_of_closedEmbedding
+-/
 
+#print ContinuousMap.exists_restrict_eq_forall_mem_of_closed /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
 on `s`. Let `t` be a nonempty convex set of real numbers (we use `ord_connected` instead of `convex`
@@ -416,6 +437,7 @@ theorem exists_restrict_eq_forall_mem_of_closed {s : Set Y} (f : C(s, ℝ)) {t :
     exists_extension_forall_mem_of_closedEmbedding f ht hne (closedEmbedding_subtype_val hs)
   ⟨g, hgt, coe_injective hgf⟩
 #align continuous_map.exists_restrict_eq_forall_mem_of_closed ContinuousMap.exists_restrict_eq_forall_mem_of_closed
+-/
 
 #print ContinuousMap.exists_restrict_eq_of_closed /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let

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

@@ -90,20 +90,17 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
         |g (e x) - f x| = -‖f‖ / 3 - f x := by
           rw [hg₁ (mem_image_of_mem _ hle₁), abs_of_nonneg (sub_nonneg.2 hle₁)]
         _ ≤ 2 / 3 * ‖f‖ := by linarith
-        
     · cases' le_total (f x) (‖f‖ / 3) with hle₂ hle₂
       · simp only [neg_div] at *
         calc
           dist (g (e x)) (f x) ≤ |g (e x)| + |f x| := dist_le_norm_add_norm _ _
           _ ≤ ‖f‖ / 3 + ‖f‖ / 3 := (add_le_add (abs_le.2 <| hgf _) (abs_le.2 ⟨hle₁, hle₂⟩))
           _ = 2 / 3 * ‖f‖ := by linarith
-          
       ·
         calc
           |g (e x) - f x| = f x - ‖f‖ / 3 := by
             rw [hg₂ (mem_image_of_mem _ hle₂), abs_sub_comm, abs_of_nonneg (sub_nonneg.2 hle₂)]
           _ ≤ 2 / 3 * ‖f‖ := by linarith
-          
 #align bounded_continuous_function.tietze_extension_step BoundedContinuousFunction.tietze_extension_step
 
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
@@ -137,7 +134,6 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
       _ = 1 / 3 * dist ((g n).comp_continuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
       _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
       _ = 1 / 3 * ‖f‖ * (2 / 3) ^ n := by ac_rfl
-      
   have hg_cau : CauchySeq g := cauchySeq_of_le_geometric _ _ (by norm_num1) hg_dist
   have :
     tendsto (fun n => (g n).comp_continuous e) at_top
@@ -264,7 +260,6 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
           calc
             0 < c - g y := sub_pos.2 hac
             _ = dg y := (dga rfl).symm
-            
         · exact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)
       rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, hax, hxy⟩
       refine' ⟨x, hxy.le, _⟩
@@ -274,7 +269,6 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         calc
           g y + dg y ≤ c + (c - a) := add_le_add hc (dgmem _).2
           _ = b := by rw [hsub, add_sub_cancel'_right]
-          
   /- Now we deal with the case `∀ x, f x ≠ b`. The proof is the same as in the first case, with
     minor modifications that make it hard to deduplicate code. -/
   choose xl hxl hgb using hg_mem
@@ -301,7 +295,6 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         calc
           0 < g y - c := sub_pos.2 hcb
           _ = dg y := (dgb rfl).symm
-          
       · exact ((sub_le_self_iff _).2 (dgmem _).1).trans_lt hlt
     rcases hb.exists_between hyb with ⟨_, ⟨xu, rfl⟩, hyxu, hxub⟩
     cases' lt_or_le c (g y) with hc hc
@@ -310,13 +303,11 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         calc
           f x = c - (b - c) := by rw [← hsub, sub_sub_cancel]
           _ ≤ g y - dg y := sub_le_sub hc.le (dgmem _).2
-          
       · have hay : a < (g - dg) y := by
           calc
             a = c - (b - c) := by rw [← hsub, sub_sub_cancel]
             _ < g y - (b - c) := (sub_lt_sub_right hc _)
             _ ≤ g y - dg y := sub_le_sub_left (dgmem _).2 _
-            
         rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, ha, hxy⟩
         exact ⟨x, xu, hxy.le, hyxu.le⟩
     · refine' ⟨xl y, xu, _, hyxu.le⟩

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

@@ -385,7 +385,7 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
   have : ord_connected t' := by
     constructor; rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ z hz
     lift z to Ioo (-1 : ℝ) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz
-    change z ∈ Icc (h x) (h y) at hz; rw [← h.image_Icc] at hz
+    change z ∈ Icc (h x) (h y) at hz ; rw [← h.image_Icc] at hz 
     rcases hz with ⟨z, hz, rfl⟩
     exact ⟨z, hs.out hx hy hz, rfl⟩
   have hFt : ∀ x, F x ∈ t' := fun x => mem_image_of_mem _ (hf x)

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 
 ! This file was ported from Lean 3 source module topology.tietze_extension
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 61b5e2755ccb464b68d05a9acf891ae04992d09d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Topology.UrysohnsBounded
 /-!
 # Tietze extension theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove a few version of the Tietze extension theorem. The theorem says that a
 continuous function `s → ℝ` defined on a closed set in a normal topological space `Y` can be
 extended to a continuous function on the whole space. Moreover, if all values of the original
@@ -107,7 +110,7 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
 embedding and bundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
 into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
 a bounded continuous function `g : Y →ᵇ ℝ` of the same norm such that `g ∘ e = f`. -/
-theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(X, Y))
+theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y))
     (he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.comp_continuous e = f :=
   by
   /- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
@@ -152,7 +155,7 @@ theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(
       (dist_le_of_le_geometric_of_tendsto₀ _ _ (by norm_num1) hg_dist hg_cau.tendsto_lim).trans_eq _
     field_simp [show (3 - 2 : ℝ) = 1 by norm_num1]
   · rw [← hge]; exact norm_comp_continuous_le _ _
-#align bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding' BoundedContinuousFunction.exists_extension_norm_eq_of_closed_embedding'
+#align bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding' BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding'
 
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
 embedding and unbundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
@@ -171,7 +174,7 @@ topological space, then it can be extended to a bounded continuous function of t
 on the whole space. -/
 theorem exists_norm_eq_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (hs : IsClosed s) :
     ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.restrict s = f :=
-  exists_extension_norm_eq_of_closed_embedding' f ((ContinuousMap.id _).restrict s)
+  exists_extension_norm_eq_of_closedEmbedding' f ((ContinuousMap.id _).restrict s)
     (closedEmbedding_subtype_val hs)
 #align bounded_continuous_function.exists_norm_eq_restrict_eq_of_closed BoundedContinuousFunction.exists_norm_eq_restrict_eq_of_closed
 
@@ -423,6 +426,7 @@ theorem exists_restrict_eq_forall_mem_of_closed {s : Set Y} (f : C(s, ℝ)) {t :
   ⟨g, hgt, coe_injective hgf⟩
 #align continuous_map.exists_restrict_eq_forall_mem_of_closed ContinuousMap.exists_restrict_eq_forall_mem_of_closed
 
+#print ContinuousMap.exists_restrict_eq_of_closed /-
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
 on `s`. Then there exists a continuous real-valued function `g : C(Y, ℝ)` such that
@@ -433,6 +437,7 @@ theorem exists_restrict_eq_of_closed {s : Set Y} (f : C(s, ℝ)) (hs : IsClosed
     exists_restrict_eq_forall_mem_of_closed f (fun _ => mem_univ _) univ_nonempty hs
   ⟨g, hgf⟩
 #align continuous_map.exists_restrict_eq_of_closed ContinuousMap.exists_restrict_eq_of_closed
+-/
 
 end ContinuousMap
 

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

@@ -42,7 +42,7 @@ variable {X Y : Type _} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y
 
 open Metric Set Filter
 
-open BoundedContinuousFunction Topology
+open scoped BoundedContinuousFunction Topology
 
 noncomputable section
 

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

@@ -58,9 +58,8 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
   have h3 : (0 : ℝ) < 3 := by norm_num1
   have h23 : 0 < (2 / 3 : ℝ) := by norm_num1
   -- In the trivial case `f = 0`, we take `g = 0`
-  rcases eq_or_ne f 0 with (rfl | hf)
-  · use 0
-    simp
+  rcases eq_or_ne f 0 with (rfl | hf);
+  · use 0; simp
   replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf
   /- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻¹' (Ici (‖f‖ / 3)))`
     are disjoint, hence by Urysohn's lemma there exists a function `g` that is equal to `-‖f‖ / 3`
@@ -120,8 +119,7 @@ theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(
     Function.iterate_succ_apply' _ _ _
   have hgf : ∀ n, dist ((g n).comp_continuous e) f ≤ (2 / 3) ^ n * ‖f‖ :=
     by
-    intro n
-    induction' n with n ihn
+    intro n; induction' n with n ihn
     · simp [g0]
     · rw [g_succ n, add_comp_continuous, ← dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]
       refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))
@@ -153,8 +151,7 @@ theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(
     refine'
       (dist_le_of_le_geometric_of_tendsto₀ _ _ (by norm_num1) hg_dist hg_cau.tendsto_lim).trans_eq _
     field_simp [show (3 - 2 : ℝ) = 1 by norm_num1]
-  · rw [← hge]
-    exact norm_comp_continuous_le _ _
+  · rw [← hge]; exact norm_comp_continuous_le _ _
 #align bounded_continuous_function.exists_extension_norm_eq_of_closed_embedding' BoundedContinuousFunction.exists_extension_norm_eq_of_closed_embedding'
 
 /-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
@@ -224,16 +221,12 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   have hle : a ≤ b := (hmem default).1.trans (hmem default).2
   rcases hle.eq_or_lt with (rfl | hlt)
   · have : ∀ x, f x = a := by simpa using hmem
-    use const Y a
-    simp [this, Function.funext_iff]
+    use const Y a; simp [this, Function.funext_iff]
   -- Put `c = (a + b) / 2`. Then `a < c < b` and `c - a = b - c`.
   set c := (a + b) / 2
   have hac : a < c := left_lt_add_div_two.2 hlt
   have hcb : c < b := add_div_two_lt_right.2 hlt
-  have hsub : c - a = b - c := by
-    simp only [c]
-    field_simp
-    ring
+  have hsub : c - a = b - c := by simp only [c]; field_simp; ring
   /- Due to `exists_extension_forall_mem_Icc_of_closed_embedding`, there exists an extension `g`
     such that `g y ∈ [a, b]` for all `y`. However, if `a` and/or `b` do not belong to the range of
     `f`, then we need to ensure that these points do not belong to the range of `g`. This is done
@@ -259,8 +252,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         (is_closed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with
       ⟨dg, dg0, dga, dgmem⟩
     replace hgf : ∀ x, (g + dg) (e x) = f x
-    · intro x
-      simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
+    · intro x; simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
     refine' ⟨g + dg, fun y => _, funext hgf⟩
     · have hay : a < (g + dg) y :=
         by
@@ -297,8 +289,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       (is_closed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with
     ⟨dg, dg0, dgb, dgmem⟩
   replace hgf : ∀ x, (g - dg) (e x) = f x
-  · intro x
-    simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
+  · intro x; simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
   refine' ⟨g - dg, fun y => _, funext hgf⟩
   · have hyb : (g - dg) y < b :=
       by
@@ -389,11 +380,9 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
   set t' : Set ℝ := coe ∘ h '' t
   have ht_sub : t' ⊆ Ioo (-1 : ℝ) 1 := image_subset_iff.2 fun x hx => (h x).2
   have : ord_connected t' := by
-    constructor
-    rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ z hz
+    constructor; rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ z hz
     lift z to Ioo (-1 : ℝ) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz
-    change z ∈ Icc (h x) (h y) at hz
-    rw [← h.image_Icc] at hz
+    change z ∈ Icc (h x) (h y) at hz; rw [← h.image_Icc] at hz
     rcases hz with ⟨z, hz, rfl⟩
     exact ⟨z, hs.out hx hy hz, rfl⟩
   have hFt : ∀ x, F x ∈ t' := fun x => mem_image_of_mem _ (hf x)
@@ -407,8 +396,7 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
   · rcases hG y with ⟨a, ha, hay⟩
     convert ha
     exact hgG.2 hay.symm
-  · ext x
-    exact hgG.2 (congr_fun hGF _)
+  · ext x; exact hgG.2 (congr_fun hGF _)
 #align continuous_map.exists_extension_forall_mem_of_closed_embedding ContinuousMap.exists_extension_forall_mem_of_closedEmbedding
 
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed

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

@@ -215,9 +215,9 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   inhabit X
   -- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
   obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a
-  exact ⟨_, isGLB_cinfᵢ (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1⟩
+  exact ⟨_, isGLB_ciInf (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1⟩
   obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b
-  exact ⟨_, isLUB_csupᵢ (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2⟩
+  exact ⟨_, isLUB_ciSup (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2⟩
   -- Then `f x ∈ [a, b]` for all `x`
   have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.1 ⟨x, rfl⟩, hb.1 ⟨x, rfl⟩⟩
   -- Rule out the trivial case `a = b`

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

@@ -146,7 +146,7 @@ theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(
     by
     refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)
     refine' squeeze_zero (fun _ => dist_nonneg) hgf _
-    rw [← zero_mul ‖f‖]
+    rw [← MulZeroClass.zero_mul ‖f‖]
     refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1
   refine' ⟨limUnder at_top g, le_antisymm _ _, hge⟩
   · rw [← dist_zero_left, ← g0]

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

@@ -93,7 +93,7 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
       · simp only [neg_div] at *
         calc
           dist (g (e x)) (f x) ≤ |g (e x)| + |f x| := dist_le_norm_add_norm _ _
-          _ ≤ ‖f‖ / 3 + ‖f‖ / 3 := add_le_add (abs_le.2 <| hgf _) (abs_le.2 ⟨hle₁, hle₂⟩)
+          _ ≤ ‖f‖ / 3 + ‖f‖ / 3 := (add_le_add (abs_le.2 <| hgf _) (abs_le.2 ⟨hle₁, hle₂⟩))
           _ = 2 / 3 * ‖f‖ := by linarith
           
       ·
@@ -132,9 +132,9 @@ theorem exists_extension_norm_eq_of_closed_embedding' (f : X →ᵇ ℝ) (e : C(
     calc
       dist (g n) (g (n + 1)) = ‖F (f - (g n).comp_continuous e)‖ := by
         rw [g_succ, dist_eq_norm', add_sub_cancel']
-      _ ≤ ‖f - (g n).comp_continuous e‖ / 3 := hF_norm _
+      _ ≤ ‖f - (g n).comp_continuous e‖ / 3 := (hF_norm _)
       _ = 1 / 3 * dist ((g n).comp_continuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
-      _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := mul_le_mul_of_nonneg_left (hgf n) (by norm_num1)
+      _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
       _ = 1 / 3 * ‖f‖ * (2 / 3) ^ n := by ac_rfl
       
   have hg_cau : CauchySeq g := cauchySeq_of_le_geometric _ _ (by norm_num1) hg_dist
@@ -320,7 +320,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       · have hay : a < (g - dg) y := by
           calc
             a = c - (b - c) := by rw [← hsub, sub_sub_cancel]
-            _ < g y - (b - c) := sub_lt_sub_right hc _
+            _ < g y - (b - c) := (sub_lt_sub_right hc _)
             _ ≤ g y - dg y := sub_le_sub_left (dgmem _).2 _
             
         rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, ha, hxy⟩

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

A PR accompanying #12339.

Zulip discussion

@@ -317,9 +317,9 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   inhabit X
   -- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
   obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a
-  exact ⟨_, isGLB_ciInf (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).1⟩
+  · exact ⟨_, isGLB_ciInf (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).1⟩
   obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b
-  exact ⟨_, isLUB_ciSup (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).2⟩
+  · exact ⟨_, isLUB_ciSup (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).2⟩
   -- Then `f x ∈ [a, b]` for all `x`
   have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.1 ⟨x, rfl⟩, hb.1 ⟨x, rfl⟩⟩
   -- Rule out the trivial case `a = b`

Move Set.Ixx, Finset.Ixx, Multiset.Ixx together under two different folders:

• Order.Interval for their definition and basic properties
• Algebra.Order.Interval for their algebraic properties

Move the definitions of Multiset.Ixx to what is now Order.Interval.Multiset. I believe we could just delete this file in a later PR as nothing uses it (and I already had doubts when defining Multiset.Ixx three years ago).

Move the algebraic results out of what is now Order.Interval.Finset.Basic to a new file Algebra.Order.Interval.Finset.Basic.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
-import Mathlib.Data.Set.Intervals.IsoIoo
+import Mathlib.Order.Interval.Set.IsoIoo
 import Mathlib.Topology.Order.MonotoneContinuity
 import Mathlib.Topology.UrysohnsBounded
 

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

@@ -238,9 +238,9 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     calc
       dist (g n) (g (n + 1)) = ‖F (f - (g n).compContinuous e)‖ := by
         rw [g_succ, dist_eq_norm', add_sub_cancel_left]
-      _ ≤ ‖f - (g n).compContinuous e‖ / 3 := (hF_norm _)
+      _ ≤ ‖f - (g n).compContinuous e‖ / 3 := hF_norm _
       _ = 1 / 3 * dist ((g n).compContinuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
-      _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
+      _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := mul_le_mul_of_nonneg_left (hgf n) (by norm_num1)
       _ = 1 / 3 * ‖f‖ * (2 / 3) ^ n := by ac_rfl
   have hg_cau : CauchySeq g := cauchySeq_of_le_geometric _ _ (by norm_num1) hg_dist
   have :
@@ -411,7 +411,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       · have hay : a < (g - dg) y := by
           calc
             a = c - (b - c) := by rw [← hsub, sub_sub_cancel]
-            _ < g y - (b - c) := (sub_lt_sub_right hc _)
+            _ < g y - (b - c) := sub_lt_sub_right hc _
             _ ≤ g y - dg y := sub_le_sub_left (dgmem _).2 _
         rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, _, hxy⟩
         exact ⟨x, xu, hxy.le, hyxu.le⟩

A mix of various changes; generated with a script and manually tweaked.

@@ -502,7 +502,7 @@ attribute [deprecated] exists_extension_of_closedEmbedding -- deprecated since 2
 
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
-on `s`. Let `t` be a nonempty convex set of real numbers (we use `ord_connected` instead of `Convex`
+on `s`. Let `t` be a nonempty convex set of real numbers (we use `OrdConnected` instead of `Convex`
 to automatically deduce this argument by typeclass search) such that `f x ∈ t` for all `x : s`. Then
 there exists a continuous real-valued function `g : C(Y, ℝ)` such that `g y ∈ t` for all `y` and
 `g.restrict s = f`. -/

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

@@ -130,7 +130,7 @@ instance Unique.instTietzeExtension {Y : Type v} [TopologicalSpace Y] [Unique Y]
     TietzeExtension.{u, v} Y where
   exists_restrict_eq' _ _ f := ⟨.const _ default, by ext x; exact Subsingleton.elim _ _⟩
 
-/-- Any retract of a `TietzeExtension` space is one itself.-/
+/-- Any retract of a `TietzeExtension` space is one itself. -/
 theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y]
     [TopologicalSpace Z] [TietzeExtension.{u, w} Z] (ι : C(Y, Z)) (r : C(Z, Y))
     (h : r.comp ι = .id Y) : TietzeExtension.{u, v} Y where
@@ -142,7 +142,7 @@ theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y
     rw [← r.comp_assoc ι, h, f.id_comp] at this
     congrm($this x)
 
-/-- Any homeomorphism from a `TietzeExtension` space is one itself.-/
+/-- Any homeomorphism from a `TietzeExtension` space is one itself. -/
 theorem TietzeExtension.of_homeo {Y : Type v} {Z : Type w} [TopologicalSpace Y]
     [TopologicalSpace Z] [TietzeExtension.{u, w} Z] (e : Y ≃ₜ Z) :
     TietzeExtension.{u, v} Y :=

@@ -502,7 +502,7 @@ attribute [deprecated] exists_extension_of_closedEmbedding -- deprecated since 2
 
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
-on `s`. Let `t` be a nonempty convex set of real numbers (we use `ord_connected` instead of `convex`
+on `s`. Let `t` be a nonempty convex set of real numbers (we use `ord_connected` instead of `Convex`
 to automatically deduce this argument by typeclass search) such that `f x ∈ t` for all `x : s`. Then
 there exists a continuous real-valued function `g : C(Y, ℝ)` such that `g y ∈ t` for all `y` and
 `g.restrict s = f`. -/

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -230,7 +230,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     intro n
     induction' n with n ihn
     · simp [g0]
-    · rw [g_succ n, add_compContinuous, ← dist_sub_right, add_sub_cancel_left, pow_succ, mul_assoc]
+    · rw [g_succ n, add_compContinuous, ← dist_sub_right, add_sub_cancel_left, pow_succ', mul_assoc]
       refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))
       rwa [← dist_eq_norm']
   have hg_dist : ∀ n, dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n := by

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

According to git, the moves are:

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

@@ -5,7 +5,7 @@ Authors: Yury G. Kudryashov
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Data.Set.Intervals.IsoIoo
-import Mathlib.Topology.Algebra.Order.MonotoneContinuity
+import Mathlib.Topology.Order.MonotoneContinuity
 import Mathlib.Topology.UrysohnsBounded
 
 #align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

| Statement | New name | Old name | |

@@ -230,14 +230,14 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     intro n
     induction' n with n ihn
     · simp [g0]
-    · rw [g_succ n, add_compContinuous, ← dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]
+    · rw [g_succ n, add_compContinuous, ← dist_sub_right, add_sub_cancel_left, pow_succ, mul_assoc]
       refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))
       rwa [← dist_eq_norm']
   have hg_dist : ∀ n, dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n := by
     intro n
     calc
       dist (g n) (g (n + 1)) = ‖F (f - (g n).compContinuous e)‖ := by
-        rw [g_succ, dist_eq_norm', add_sub_cancel']
+        rw [g_succ, dist_eq_norm', add_sub_cancel_left]
       _ ≤ ‖f - (g n).compContinuous e‖ / 3 := (hF_norm _)
       _ = 1 / 3 * dist ((g n).compContinuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
       _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
@@ -374,7 +374,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       · simp [dg0 (Or.inr hc), (hg_mem y).2]
       · calc
           g y + dg y ≤ c + (c - a) := add_le_add hc (dgmem _).2
-          _ = b := by rw [hsub, add_sub_cancel'_right]
+          _ = b := by rw [hsub, add_sub_cancel]
   /- Now we deal with the case `∀ x, f x ≠ b`. The proof is the same as in the first case, with
     minor modifications that make it hard to deduplicate code. -/
   choose xl hxl hgb using hg_mem

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -73,7 +73,7 @@ continuous function `g : C(X₂, Y)` such that `g ∘ e = f`. -/
 theorem ContinuousMap.exists_extension (f : C(X₁, Y)) :
     ∃ (g : C(X, Y)), g.comp ⟨e, he.continuous⟩ = f := by
   let e' : X₁ ≃ₜ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding
-  obtain ⟨g, hg⟩ := (f.comp e'.symm).exists_restrict_eq he.closed_range
+  obtain ⟨g, hg⟩ := (f.comp e'.symm).exists_restrict_eq he.isClosed_range
   exact ⟨g, by ext x; simpa using congr($(hg) ⟨e' x, x, rfl⟩)⟩
 
 /-- **Tietze extension theorem** for `TietzeExtension` spaces. Let `e` be a closed embedding of a
@@ -354,7 +354,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         exact ha' ⟨x, (congr_fun hgf x).symm⟩
       · exact Set.disjoint_singleton_right.2 hac.not_le
     rcases exists_bounded_mem_Icc_of_closed_of_le
-        (he.closed_range.union <| isClosed_Ici.preimage g.continuous)
+        (he.isClosed_range.union <| isClosed_Ici.preimage g.continuous)
         (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with
       ⟨dg, dg0, dga, dgmem⟩
     replace hgf : ∀ x, (g + dg) (e x) = f x := by
@@ -387,7 +387,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       exact hb' ⟨x, (congr_fun hgf x).symm⟩
     · exact Set.disjoint_singleton_right.2 hcb.not_le
   rcases exists_bounded_mem_Icc_of_closed_of_le
-      (he.closed_range.union <| isClosed_Iic.preimage g.continuous)
+      (he.isClosed_range.union <| isClosed_Iic.preimage g.continuous)
       (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with
     ⟨dg, dg0, dgb, dgmem⟩
   replace hgf : ∀ x, (g - dg) (e x) = f x := by

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

@@ -333,7 +333,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   have hac : a < c := left_lt_add_div_two.2 hlt
   have hcb : c < b := add_div_two_lt_right.2 hlt
   have hsub : c - a = b - c := by
-    field_simp
+    field_simp [c]
     ring
   /- Due to `exists_extension_forall_mem_Icc_of_closedEmbedding`, there exists an extension `g`
     such that `g y ∈ [a, b]` for all `y`. However, if `a` and/or `b` do not belong to the range of

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.

@@ -431,7 +431,7 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : X →ᵇ ℝ) {t : S
     ∃ g : Y →ᵇ ℝ, (∀ y, g y ∈ t) ∧ g ∘ e = f := by
   cases isEmpty_or_nonempty X
   · rcases hne with ⟨c, hc⟩
-    refine' ⟨const Y c, fun _ => hc, funext fun x => isEmptyElim x⟩
+    exact ⟨const Y c, fun _ => hc, funext fun x => isEmptyElim x⟩
   rcases exists_extension_forall_exists_le_ge_of_closedEmbedding f he with ⟨g, hg, hgf⟩
   refine' ⟨g, fun y => _, hgf⟩
   rcases hg y with ⟨xl, xu, h⟩

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

This follows on from #6964.

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

@@ -357,8 +357,8 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         (he.closed_range.union <| isClosed_Ici.preimage g.continuous)
         (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with
       ⟨dg, dg0, dga, dgmem⟩
-    replace hgf : ∀ x, (g + dg) (e x) = f x
-    · intro x
+    replace hgf : ∀ x, (g + dg) (e x) = f x := by
+      intro x
       simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
     refine' ⟨g + dg, fun y => _, funext hgf⟩
     · have hay : a < (g + dg) y := by
@@ -390,8 +390,8 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       (he.closed_range.union <| isClosed_Iic.preimage g.continuous)
       (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with
     ⟨dg, dg0, dgb, dgmem⟩
-  replace hgf : ∀ x, (g - dg) (e x) = f x
-  · intro x
+  replace hgf : ∀ x, (g - dg) (e x) = f x := by
+    intro x
     simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
   refine' ⟨g - dg, fun y => _, funext hgf⟩
   · have hyb : (g - dg) y < b := by

This adds a class TietzeExtension to encode the Tietze extension theorem as a type class that can be satisfied by various types. For now, we provide instances for ℝ, ℝ≥0, ℂ, IsROrC 𝕜, pi types, product types, as well as constructors via homeomorphisms or retracts of a TietzeExtension space, and also a constructor for finite dimensional topological vector spaces.

@@ -24,6 +24,12 @@ The proof mostly follows <https://ncatlab.org/nlab/show/Tietze+extension+theorem
 gap in the proof for unbounded functions, see
 `exists_extension_forall_exists_le_ge_of_closedEmbedding`.
 
+In addition we provide a class `TietzeExtension` encoding the idea that a topological space
+satisfies the Tietze extension theorem. This allows us to get a version of the Tietze extension
+theorem that simultaneously applies to `ℝ`, `ℝ × ℝ`, `ℂ`, `ι → ℝ`, `ℝ≥0` et cetera. At some point
+in the future, it may be desirable to provide instead a more general approach via
+*absolute retracts*, but the current implementation covers the most common use cases easily.
+
 ## Implementation notes
 
 We first prove the theorems for a closed embedding `e : X → Y` of a topological space into a normal
@@ -34,6 +40,117 @@ topological space, then specialize them to the case `X = s : Set Y`, `e = (↑)`
 Tietze extension theorem, Urysohn's lemma, normal topological space
 -/
 
+/-!  ### The `TietzeExtension` class -/
+
+section TietzeExtensionClass
+
+universe u u₁ u₂ v w
+
+-- TODO: define *absolute retracts* and then prove they satisfy Tietze extension.
+-- Then make instances of that instead and remove this class.
+/-- A class encoding the concept that a space satisfies the Tietze extension property. -/
+class TietzeExtension (Y : Type v) [TopologicalSpace Y] : Prop where
+  exists_restrict_eq' {X : Type u} [TopologicalSpace X] [NormalSpace X] (s : Set X)
+    (hs : IsClosed s) (f : C(s, Y)) : ∃ (g : C(X, Y)), g.restrict s = f
+
+variable {X₁ : Type u₁} [TopologicalSpace X₁]
+variable {X : Type u} [TopologicalSpace X] [NormalSpace X] {s : Set X} (hs : IsClosed s)
+variable {e : X₁ → X} (he : ClosedEmbedding e)
+variable {Y : Type v} [TopologicalSpace Y] [TietzeExtension.{u, v} Y]
+
+/-- **Tietze extension theorem** for `TietzeExtension` spaces, a version for a closed set. Let
+`s` be a closed set in a normal topological space `X₂`. Let `f` be a continuous function
+on `s` with values in a `TietzeExtension` space `Y`. Then there exists a continuous function
+`g : C(X₂, Y)` such that `g.restrict s = f`. -/
+theorem ContinuousMap.exists_restrict_eq (f : C(s, Y)) : ∃ (g : C(X, Y)), g.restrict s = f :=
+  TietzeExtension.exists_restrict_eq' s hs f
+#align continuous_map.exists_restrict_eq_of_closed ContinuousMap.exists_restrict_eq
+
+/-- **Tietze extension theorem** for `TietzeExtension` spaces. Let `e` be a closed embedding of a
+nonempty topological space `X₁` into a normal topological space `X₂`. Let `f` be a continuous
+function on `X₁` with values in a `TietzeExtension` space `Y`. Then there exists a
+continuous function `g : C(X₂, Y)` such that `g ∘ e = f`. -/
+theorem ContinuousMap.exists_extension (f : C(X₁, Y)) :
+    ∃ (g : C(X, Y)), g.comp ⟨e, he.continuous⟩ = f := by
+  let e' : X₁ ≃ₜ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding
+  obtain ⟨g, hg⟩ := (f.comp e'.symm).exists_restrict_eq he.closed_range
+  exact ⟨g, by ext x; simpa using congr($(hg) ⟨e' x, x, rfl⟩)⟩
+
+/-- **Tietze extension theorem** for `TietzeExtension` spaces. Let `e` be a closed embedding of a
+nonempty topological space `X₁` into a normal topological space `X₂`. Let `f` be a continuous
+function on `X₁` with values in a `TietzeExtension` space `Y`. Then there exists a
+continuous function `g : C(X₂, Y)` such that `g ∘ e = f`.
+
+This version is provided for convenience and backwards compatibility. Here the composition is
+phrased in terms of bare functions. -/
+theorem ContinuousMap.exists_extension' (f : C(X₁, Y)) : ∃ (g : C(X, Y)), g ∘ e = f :=
+  f.exists_extension he |>.imp fun g hg ↦ by ext x; congrm($(hg) x)
+#align continuous_map.exists_extension_of_closed_embedding ContinuousMap.exists_extension'
+
+/-- This theorem is not intended to be used directly because it is rare for a set alone to
+satisfy `[TietzeExtension t]`. For example, `Metric.ball` in `ℝ` only satisfies it when
+the radius is strictly positive, so finding this as an instance will fail.
+
+Instead, it is intended to be used as a constructor for theorems about sets which *do* satisfy
+`[TietzeExtension t]` under some hypotheses. -/
+theorem ContinuousMap.exists_forall_mem_restrict_eq {Y : Type v} [TopologicalSpace Y] (f : C(s, Y))
+    {t : Set Y} (hf : ∀ x, f x ∈ t) [ht : TietzeExtension.{u, v} t] :
+    ∃ (g : C(X, Y)), (∀ x, g x ∈ t) ∧ g.restrict s = f := by
+  obtain ⟨g, hg⟩ := mk _ (map_continuous f |>.codRestrict hf) |>.exists_restrict_eq hs
+  exact ⟨comp ⟨Subtype.val, by continuity⟩ g, by simp, by ext x; congrm(($(hg) x : Y))⟩
+
+/-- This theorem is not intended to be used directly because it is rare for a set alone to
+satisfy `[TietzeExtension t]`. For example, `Metric.ball` in `ℝ` only satisfies it when
+the radius is strictly positive, so finding this as an instance will fail.
+
+Instead, it is intended to be used as a constructor for theorems about sets which *do* satisfy
+`[TietzeExtension t]` under some hypotheses. -/
+theorem ContinuousMap.exists_extension_forall_mem {Y : Type v} [TopologicalSpace Y] (f : C(X₁, Y))
+    {t : Set Y} (hf : ∀ x, f x ∈ t) [ht : TietzeExtension.{u, v} t] :
+    ∃ (g : C(X, Y)), (∀ x, g x ∈ t) ∧ g.comp ⟨e, he.continuous⟩ = f := by
+  obtain ⟨g, hg⟩ := mk _ (map_continuous f |>.codRestrict hf) |>.exists_extension he
+  exact ⟨comp ⟨Subtype.val, by continuity⟩ g, by simp, by ext x; congrm(($(hg) x : Y))⟩
+
+instance Pi.instTietzeExtension {ι : Type*} {Y : ι → Type v} [∀ i, TopologicalSpace (Y i)]
+    [∀ i, TietzeExtension (Y i)] : TietzeExtension (∀ i, Y i) where
+  exists_restrict_eq' s hs f := by
+    obtain ⟨g', hg'⟩ := Classical.skolem.mp <| fun i ↦
+      ContinuousMap.exists_restrict_eq hs (ContinuousMap.piEquiv _ _ |>.symm f i)
+    exact ⟨ContinuousMap.piEquiv _ _ g', by ext x i; congrm($(hg' i) x)⟩
+
+instance Prod.instTietzeExtension {Y : Type v} {Z : Type w} [TopologicalSpace Y]
+    [TietzeExtension.{u, v} Y] [TopologicalSpace Z] [TietzeExtension.{u, w} Z] :
+    TietzeExtension (Y × Z) where
+  exists_restrict_eq' s hs f := by
+    obtain ⟨g₁, hg₁⟩ := (ContinuousMap.fst.comp f).exists_restrict_eq hs
+    obtain ⟨g₂, hg₂⟩ := (ContinuousMap.snd.comp f).exists_restrict_eq hs
+    exact ⟨g₁.prodMk g₂, by ext1 x; congrm(($(hg₁) x), $(hg₂) x)⟩
+
+instance Unique.instTietzeExtension {Y : Type v} [TopologicalSpace Y] [Unique Y] :
+    TietzeExtension.{u, v} Y where
+  exists_restrict_eq' _ _ f := ⟨.const _ default, by ext x; exact Subsingleton.elim _ _⟩
+
+/-- Any retract of a `TietzeExtension` space is one itself.-/
+theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y]
+    [TopologicalSpace Z] [TietzeExtension.{u, w} Z] (ι : C(Y, Z)) (r : C(Z, Y))
+    (h : r.comp ι = .id Y) : TietzeExtension.{u, v} Y where
+  exists_restrict_eq' s hs f := by
+    obtain ⟨g, hg⟩ := (ι.comp f).exists_restrict_eq hs
+    use r.comp g
+    ext1 x
+    have := congr(r.comp $(hg))
+    rw [← r.comp_assoc ι, h, f.id_comp] at this
+    congrm($this x)
+
+/-- Any homeomorphism from a `TietzeExtension` space is one itself.-/
+theorem TietzeExtension.of_homeo {Y : Type v} {Z : Type w} [TopologicalSpace Y]
+    [TopologicalSpace Z] [TietzeExtension.{u, w} Z] (e : Y ≃ₜ Z) :
+    TietzeExtension.{u, v} Y :=
+  .of_retract (e : C(Y, Z)) (e.symm : C(Z, Y)) <| by simp
+
+end TietzeExtensionClass
+
+/-! The Tietze extension theorem for `ℝ`. -/
 
 variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y]
 
@@ -380,15 +497,8 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
     exact hgG.2 (congr_fun hGF _)
 #align continuous_map.exists_extension_forall_mem_of_closed_embedding ContinuousMap.exists_extension_forall_mem_of_closedEmbedding
 
-/-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed
-embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
-topological space `Y`. Let `f` be a continuous real-valued function on `X`. Then there exists a
-continuous real-valued function `g : C(Y, ℝ)` such that `g ∘ e = f`. -/
-theorem exists_extension_of_closedEmbedding (f : C(X, ℝ)) (e : X → Y) (he : ClosedEmbedding e) :
-    ∃ g : C(Y, ℝ), g ∘ e = f :=
-  (exists_extension_forall_mem_of_closedEmbedding f (fun _ => mem_univ _) univ_nonempty he).imp
-    fun _ => And.right
-#align continuous_map.exists_extension_of_closed_embedding ContinuousMap.exists_extension_of_closedEmbedding
+alias exists_extension_of_closedEmbedding := exists_extension'
+attribute [deprecated] exists_extension_of_closedEmbedding -- deprecated since 2024-01-16
 
 /-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
 `s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
@@ -404,15 +514,21 @@ theorem exists_restrict_eq_forall_mem_of_closed {s : Set Y} (f : C(s, ℝ)) {t :
   ⟨g, hgt, coe_injective hgf⟩
 #align continuous_map.exists_restrict_eq_forall_mem_of_closed ContinuousMap.exists_restrict_eq_forall_mem_of_closed
 
-/-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed set. Let
-`s` be a closed set in a normal topological space `Y`. Let `f` be a continuous real-valued function
-on `s`. Then there exists a continuous real-valued function `g : C(Y, ℝ)` such that
-`g.restrict s = f`. -/
-theorem exists_restrict_eq_of_closed {s : Set Y} (f : C(s, ℝ)) (hs : IsClosed s) :
-    ∃ g : C(Y, ℝ), g.restrict s = f :=
-  let ⟨g, _, hgf⟩ :=
-    exists_restrict_eq_forall_mem_of_closed f (fun _ => mem_univ _) univ_nonempty hs
-  ⟨g, hgf⟩
-#align continuous_map.exists_restrict_eq_of_closed ContinuousMap.exists_restrict_eq_of_closed
+alias exists_restrict_eq_of_closed := exists_restrict_eq
+attribute [deprecated] exists_restrict_eq_of_closed -- deprecated since 2024-01-16
 
 end ContinuousMap
+
+/-- **Tietze extension theorem** for real-valued continuous maps.
+`ℝ` is a `TietzeExtension` space. -/
+instance Real.instTietzeExtension : TietzeExtension ℝ where
+  exists_restrict_eq' _s hs f :=
+    f.exists_restrict_eq_forall_mem_of_closed (fun _ => mem_univ _) univ_nonempty hs |>.imp
+      fun _ ↦ (And.right ·)
+
+open NNReal in
+/-- **Tietze extension theorem** for nonnegative real-valued continuous maps.
+`ℝ≥0` is a `TietzeExtension` space. -/
+instance NNReal.instTietzeExtension : TietzeExtension ℝ≥0 :=
+  .of_retract ⟨((↑) : ℝ≥0 → ℝ), by continuity⟩ ⟨Real.toNNReal, continuous_real_toNNReal⟩ <| by
+    ext; simp

See here on Zulip.

This PR changes a bunch of names containing nhds_0 or/and lt_1 to nhds_zero or/and lt_one.

@@ -134,7 +134,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
     refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)
     refine' squeeze_zero (fun _ => dist_nonneg) hgf _
     rw [← zero_mul ‖f‖]
-    refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1
+    refine' (tendsto_pow_atTop_nhds_zero_of_lt_one _ _).mul tendsto_const_nhds <;> norm_num1
   refine' ⟨limUnder atTop g, le_antisymm _ _, hge⟩
   · rw [← dist_zero_left, ← g0]
     refine'

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>

@@ -333,7 +333,7 @@ theorem exists_forall_mem_restrict_eq_of_closed {s : Set Y} (f : s →ᵇ ℝ) (
   rcases exists_extension_forall_mem_of_closedEmbedding f hf hne
       (closedEmbedding_subtype_val hs) with
     ⟨g, hg, hgf⟩
-  exact ⟨g, hg, FunLike.coe_injective hgf⟩
+  exact ⟨g, hg, DFunLike.coe_injective hgf⟩
 #align bounded_continuous_function.exists_forall_mem_restrict_eq_of_closed BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed
 
 end BoundedContinuousFunction

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

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

@@ -77,13 +77,13 @@ theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbed
   · refine' (dist_le <| mul_nonneg h23.le hf.le).mpr fun x => _
     have hfx : -‖f‖ ≤ f x ∧ f x ≤ ‖f‖ := by
       simpa only [Real.norm_eq_abs, abs_le] using f.norm_coe_le_norm x
-    cases' le_total (f x) (-‖f‖ / 3) with hle₁ hle₁
+    rcases le_total (f x) (-‖f‖ / 3) with hle₁ | hle₁
     · calc
         |g (e x) - f x| = -‖f‖ / 3 - f x := by
           rw [hg₁ (mem_image_of_mem _ hle₁), Function.const_apply,
             abs_of_nonneg (sub_nonneg.2 hle₁)]
         _ ≤ 2 / 3 * ‖f‖ := by linarith
-    · cases' le_total (f x) (‖f‖ / 3) with hle₂ hle₂
+    · rcases le_total (f x) (‖f‖ / 3) with hle₂ | hle₂
       · simp only [neg_div] at *
         calc
           dist (g (e x)) (f x) ≤ |g (e x)| + |f x| := dist_le_norm_add_norm _ _
@@ -253,7 +253,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
         · exact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)
       rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, _, hxy⟩
       refine' ⟨x, hxy.le, _⟩
-      cases' le_total c (g y) with hc hc
+      rcases le_total c (g y) with hc | hc
       · simp [dg0 (Or.inr hc), (hg_mem y).2]
       · calc
           g y + dg y ≤ c + (c - a) := add_le_add hc (dgmem _).2

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

@@ -245,7 +245,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
       simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
     refine' ⟨g + dg, fun y => _, funext hgf⟩
     · have hay : a < (g + dg) y := by
-        rcases(hg_mem y).1.eq_or_lt with (rfl | hlt)
+        rcases (hg_mem y).1.eq_or_lt with (rfl | hlt)
         · refine' (lt_add_iff_pos_right _).2 _
           calc
             0 < c - g y := sub_pos.2 hac
@@ -278,7 +278,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
     simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
   refine' ⟨g - dg, fun y => _, funext hgf⟩
   · have hyb : (g - dg) y < b := by
-      rcases(hgb y).eq_or_lt with (rfl | hlt)
+      rcases (hgb y).eq_or_lt with (rfl | hlt)
       · refine' (sub_lt_self_iff _).2 _
         calc
           0 < g y - c := sub_pos.2 hcb

Use Bornology.IsBounded instead.

@@ -200,9 +200,9 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
   inhabit X
   -- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
   obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a
-  exact ⟨_, isGLB_ciInf (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1⟩
+  exact ⟨_, isGLB_ciInf (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).1⟩
   obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b
-  exact ⟨_, isLUB_ciSup (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2⟩
+  exact ⟨_, isLUB_ciSup (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).2⟩
   -- Then `f x ∈ [a, b]` for all `x`
   have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.1 ⟨x, rfl⟩, hb.1 ⟨x, rfl⟩⟩
   -- Rule out the trivial case `a = b`
@@ -353,9 +353,8 @@ theorem exists_extension_forall_mem_of_closedEmbedding (f : C(X, ℝ)) {t : Set
   let F : X →ᵇ ℝ :=
     { toFun := (↑) ∘ h ∘ f
       continuous_toFun := continuous_subtype_val.comp (h.continuous.comp f.continuous)
-      map_bounded' :=
-        bounded_range_iff.1
-          ((bounded_Ioo (-1 : ℝ) 1).mono <| forall_range_iff.2 fun x => (h (f x)).2) }
+      map_bounded' := isBounded_range_iff.1
+        ((isBounded_Ioo (-1 : ℝ) 1).subset <| range_subset_iff.2 fun x => (h (f x)).2) }
   let t' : Set ℝ := (↑) ∘ h '' t
   have ht_sub : t' ⊆ Ioo (-1 : ℝ) 1 := image_subset_iff.2 fun x _ => (h x).2
   have : OrdConnected t' := by

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).

@@ -133,7 +133,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
   have hge : (limUnder atTop g).compContinuous e = f := by
     refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)
     refine' squeeze_zero (fun _ => dist_nonneg) hgf _
-    rw [← MulZeroClass.zero_mul ‖f‖]
+    rw [← zero_mul ‖f‖]
     refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1
   refine' ⟨limUnder atTop g, le_antisymm _ _, hge⟩
   · rw [← dist_zero_left, ← g0]

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

This has nice performance benefits.

@@ -35,7 +35,7 @@ Tietze extension theorem, Urysohn's lemma, normal topological space
 -/
 
 
-variable {X Y : Type _} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y]
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace Y]
 
 open Metric Set Filter
 

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module topology.tietze_extension
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Data.Set.Intervals.IsoIoo
 import Mathlib.Topology.Algebra.Order.MonotoneContinuity
 import Mathlib.Topology.UrysohnsBounded
 
+#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # Tietze extension theorem
 

@@ -108,7 +108,7 @@ theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X
   /- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
     between the previous approximation and `f`. -/
   choose F hF_norm hF_dist using fun f : X →ᵇ ℝ => tietze_extension_step f e he
-  set g : ℕ → Y →ᵇ ℝ := fun n => ((fun g => g + F (f - g.compContinuous e))^[n]) 0
+  set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
   have g0 : g 0 = 0 := rfl
   have g_succ : ∀ n, g (n + 1) = g n + F (f - (g n).compContinuous e) := fun n =>
     Function.iterate_succ_apply' _ _ _

Co-authored-by: Komyyy <pol_tta@outlook.jp>