analysis.calculus.inverse ⟷ Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv

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

@@ -3,7 +3,7 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 -/
-import Analysis.Calculus.ContDiff
+import Analysis.Calculus.ContDiff.Basic
 import Analysis.NormedSpace.Banach
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"575b4ea3738b017e30fb205cb9b4a8742e5e82b6"
@@ -182,8 +182,7 @@ theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) :
 #print ApproximatesLinearOn.lipschitz /-
 protected theorem lipschitz (hf : ApproximatesLinearOn f f' s c) :
     LipschitzWith (‖f'‖₊ + c) (s.restrict f) := by
-  simpa only [restrict_apply, add_sub_cancel'_right] using
-    (f'.lipschitz.restrict s).add hf.lipschitz_sub
+  simpa only [restrict_apply, add_sub_cancel] using (f'.lipschitz.restrict s).add hf.lipschitz_sub
 #align approximates_linear_on.lipschitz ApproximatesLinearOn.lipschitz
 -/
 
@@ -217,7 +216,7 @@ variable {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F}
 right inverse, then it is locally onto: a ball of an explicit radius is included in the image
 of the map. -/
 theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c)
-    (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
+    (f'symm : f'.NonlinearRightInverseₓ) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
     SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) :=
   by
   intro y hy
@@ -251,7 +250,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   have A : ∀ z, dist (g z) z ≤ f'symm.nnnorm * dist (f z) y :=
     by
     intro z
-    rw [dist_eq_norm, hg, add_sub_cancel', dist_eq_norm']
+    rw [dist_eq_norm, hg, add_sub_cancel_left, dist_eq_norm']
     exact f'symm.bound _
   -- Second bound: if `z` and `g z` are in the set with good control, then `f (g z)` becomes closer
   -- to `y` than `f z` was (this uses the linear approximation property, and is the reason for the
@@ -266,7 +265,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       dist (f (g z)) y = ‖f (z + v) - y‖ := by rw [dist_eq_norm]
       _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ := by congr 1; abel
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
-        simp only [ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
+        simp only [ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel_left,
           sub_add_cancel]
       _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) :=
@@ -375,7 +374,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
 -/
 
 #print ApproximatesLinearOn.open_image /-
-theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
+theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverseₓ)
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) :=
   by
   cases' hc with hE hc; · skip; apply isOpen_discrete
@@ -389,9 +388,9 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print ApproximatesLinearOn.image_mem_nhds /-
-theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
+theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverseₓ)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=
   by
   obtain ⟨t, hts, ht, xt⟩ : ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs
@@ -401,8 +400,8 @@ theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.Nonline
 -/
 
 #print ApproximatesLinearOn.map_nhds_eq /-
-theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E}
-    (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) :=
+theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverseₓ)
+    {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) :=
   by
   refine'
     le_antisymm ((hf.continuous_on x (mem_of_mem_nhds hs)).ContinuousAt hs) (le_map fun t ht => _)
@@ -572,14 +571,14 @@ theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedS
     ∃ u : E → F, LipschitzWith (lipschitzExtensionConstant F * c) u ∧ eq_on (f - f') u s :=
     hf.lipschitz_on_with.extend_finite_dimension
   let g : E → F := fun x => f' x + u x
-  have fg : eq_on f g s := fun x hx => by simp_rw [g, ← uf hx, Pi.sub_apply, add_sub_cancel'_right]
+  have fg : eq_on f g s := fun x hx => by simp_rw [g, ← uf hx, Pi.sub_apply, add_sub_cancel]
   have hg : ApproximatesLinearOn g (f' : E →L[ℝ] F) univ (lipschitzExtensionConstant F * c) :=
     by
     apply LipschitzOnWith.approximatesLinearOn
     rw [lipschitzOn_univ]
     convert hu
     ext x
-    simp only [add_sub_cancel', ContinuousLinearEquiv.coe_coe, Pi.sub_apply]
+    simp only [add_sub_cancel_left, ContinuousLinearEquiv.coe_coe, Pi.sub_apply]
   haveI : FiniteDimensional ℝ E := f'.symm.to_linear_equiv.finite_dimensional
   exact ⟨hg.to_homeomorph g hc, fg⟩
 #align approximates_linear_on.exists_homeomorph_extension ApproximatesLinearOn.exists_homeomorph_extension
@@ -819,12 +818,12 @@ theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a)
 
 end HasStrictFDerivAt
 
-#print open_map_of_strict_fderiv_equiv /-
+#print isOpenMap_of_hasStrictFDerivAt_equiv /-
 /-- If a function has an invertible strict derivative at all points, then it is an open map. -/
-theorem open_map_of_strict_fderiv_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
+theorem isOpenMap_of_hasStrictFDerivAt_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
     (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
-#align open_map_of_strict_fderiv_equiv open_map_of_strict_fderiv_equiv
+#align open_map_of_strict_fderiv_equiv isOpenMap_of_hasStrictFDerivAt_equiv
 -/
 
 /-!
@@ -873,12 +872,12 @@ theorem to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f
 
 end HasStrictDerivAt
 
-#print open_map_of_strict_deriv /-
+#print isOpenMap_of_hasStrictDerivAt /-
 /-- If a function has a non-zero strict derivative at all points, then it is an open map. -/
-theorem open_map_of_strict_deriv [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
+theorem isOpenMap_of_hasStrictDerivAt [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
     (hf : ∀ x, HasStrictDerivAt f (f' x) x) (h0 : ∀ x, f' x ≠ 0) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => ((hf x).map_nhds_eq (h0 x)).ge
-#align open_map_of_strict_deriv open_map_of_strict_deriv
+#align open_map_of_strict_deriv isOpenMap_of_hasStrictDerivAt
 -/
 
 /-!
@@ -889,7 +888,7 @@ theorem open_map_of_strict_deriv [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
 
 namespace ContDiffAt
 
-variable {𝕂 : Type _} [IsROrC 𝕂]
+variable {𝕂 : Type _} [RCLike 𝕂]
 
 variable {E' : Type _} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
 

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

@@ -225,10 +225,10 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   · refine' ⟨b, by simp [ε0], _⟩
     have : dist y (f b) ≤ 0 :=
       (mem_closed_ball.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
-    simp only [dist_le_zero] at this 
+    simp only [dist_le_zero] at this
     rw [this]
   have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
-  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc 
+  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
   have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
   /- We have to show that `y` can be written as `f x` for some `x ∈ closed_ball b ε`.
@@ -538,7 +538,7 @@ def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   open_source := hs
   open_target :=
     hf.open_image f'.toNonlinearRightInverse hs
-      (by rwa [f'.to_linear_equiv.to_equiv.subsingleton_congr] at hc )
+      (by rwa [f'.to_linear_equiv.to_equiv.subsingleton_congr] at hc)
   continuous_toFun := hf.ContinuousOn
   continuous_invFun := hf.inverse_continuousOn hc
 #align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toPartialHomeomorph
@@ -648,7 +648,7 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x hx y hy => _⟩
     simp [@Subsingleton.elim E hE x y]
   have := hf.def hc
-  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this 
+  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
 #align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds

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

@@ -367,8 +367,8 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     rw [tendsto_iff_dist_tendsto_zero]
     refine' squeeze_zero (fun n => dist_nonneg) (fun n => (D n).1) _
     simpa using
-      (tendsto_pow_atTop_nhds_0_of_lt_1 (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _))
-            Icf').mul
+      (tendsto_pow_atTop_nhds_zero_of_lt_one
+            (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _)) Icf').mul
         tendsto_const_nhds
   exact tendsto_nhds_unique T1 T2
 #align approximates_linear_on.surj_on_closed_ball_of_nonlinear_right_inverse ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse

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

@@ -472,21 +472,21 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
 #align approximates_linear_on.surjective ApproximatesLinearOn.surjective
 -/
 
-#print ApproximatesLinearOn.toLocalEquiv /-
+#print ApproximatesLinearOn.toPartialEquiv /-
 /-- A map approximating a linear equivalence on a set defines a local equivalence on this set.
 Should not be used outside of this file, because it is superseded by `to_local_homeomorph` below.
 
 This is a first step towards the inverse function. -/
-def toLocalEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) : LocalEquiv E F :=
-  (hf.InjOn hc).toLocalEquiv _ _
-#align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toLocalEquiv
+def toPartialEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) : PartialEquiv E F :=
+  (hf.InjOn hc).toPartialEquiv _ _
+#align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toPartialEquiv
 -/
 
 #print ApproximatesLinearOn.inverse_continuousOn /-
 /-- The inverse function is continuous on `f '' s`. Use properties of `local_homeomorph` instead. -/
 theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toLocalEquiv hc).symm (f '' s) :=
+    (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toPartialEquiv hc).symm (f '' s) :=
   by
   apply continuousOn_iff_continuous_restrict.2
   refine' ((hf.antilipschitz hc).to_rightInvOn' _ (hf.to_local_equiv hc).right_inv').Continuous
@@ -497,7 +497,7 @@ theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F)
 #print ApproximatesLinearOn.to_inv /-
 /-- The inverse function is approximated linearly on `f '' s` by `f'.symm`. -/
 theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) :
-    ApproximatesLinearOn (hf.toLocalEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
+    ApproximatesLinearOn (hf.toPartialEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
       (N * (N⁻¹ - c)⁻¹ * c) :=
   by
   intro x hx y hy
@@ -534,7 +534,7 @@ returns a local homeomorph with `to_fun = f` and `source = s`. -/
 def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : PartialHomeomorph E F
     where
-  toLocalEquiv := hf.toLocalEquiv hc
+  toPartialEquiv := hf.toPartialEquiv hc
   open_source := hs
   open_target :=
     hf.open_image f'.toNonlinearRightInverse hs

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

@@ -528,11 +528,11 @@ section
 
 variable (f s)
 
-#print ApproximatesLinearOn.toLocalHomeomorph /-
+#print ApproximatesLinearOn.toPartialHomeomorph /-
 /-- Given a function `f` that approximates a linear equivalence on an open set `s`,
 returns a local homeomorph with `to_fun = f` and `source = s`. -/
-def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : LocalHomeomorph E F
+def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : PartialHomeomorph E F
     where
   toLocalEquiv := hf.toLocalEquiv hc
   open_source := hs
@@ -541,7 +541,7 @@ def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
       (by rwa [f'.to_linear_equiv.to_equiv.subsingleton_congr] at hc )
   continuous_toFun := hf.ContinuousOn
   continuous_invFun := hf.inverse_continuousOn hc
-#align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
+#align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toPartialHomeomorph
 -/
 
 #print ApproximatesLinearOn.toHomeomorph /-
@@ -587,36 +587,37 @@ theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedS
 
 end
 
-#print ApproximatesLinearOn.toLocalHomeomorph_coe /-
+#print ApproximatesLinearOn.toPartialHomeomorph_coe /-
 @[simp]
-theorem toLocalHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+theorem toPartialHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs : E → F) = f :=
+    (hf.toPartialHomeomorph f s hc hs : E → F) = f :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toLocalHomeomorph_coe
+#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toPartialHomeomorph_coe
 -/
 
-#print ApproximatesLinearOn.toLocalHomeomorph_source /-
+#print ApproximatesLinearOn.toPartialHomeomorph_source /-
 @[simp]
-theorem toLocalHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : (hf.toLocalHomeomorph f s hc hs).source = s :=
+theorem toPartialHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
+    (hf.toPartialHomeomorph f s hc hs).source = s :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toLocalHomeomorph_source
+#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toPartialHomeomorph_source
 -/
 
-#print ApproximatesLinearOn.toLocalHomeomorph_target /-
+#print ApproximatesLinearOn.toPartialHomeomorph_target /-
 @[simp]
-theorem toLocalHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+theorem toPartialHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs).target = f '' s :=
+    (hf.toPartialHomeomorph f s hc hs).target = f '' s :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toLocalHomeomorph_target
+#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toPartialHomeomorph_target
 -/
 
 #print ApproximatesLinearOn.closedBall_subset_target /-
 theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
-    closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toLocalHomeomorph f s hc hs).target :=
+    closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toPartialHomeomorph f s hc hs).target :=
   (hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse ε0 hε).mono hε
     (Subset.refl _)
 #align approximates_linear_on.closed_ball_subset_target ApproximatesLinearOn.closedBall_subset_target
@@ -686,48 +687,48 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[
 
 variable (f)
 
-#print HasStrictFDerivAt.toLocalHomeomorph /-
+#print HasStrictFDerivAt.toPartialHomeomorph /-
 /-- Given a function with an invertible strict derivative at `a`, returns a `local_homeomorph`
 with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
 The other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function
 of this `local_homeomorph` has derivative `f'.symm`. -/
-def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : LocalHomeomorph E F :=
-  ApproximatesLinearOn.toLocalHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
+def toPartialHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : PartialHomeomorph E F :=
+  ApproximatesLinearOn.toPartialHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).snd
     (f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
       NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.2
-#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toLocalHomeomorph
+#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toPartialHomeomorph
 -/
 
 variable {f}
 
-#print HasStrictFDerivAt.toLocalHomeomorph_coe /-
+#print HasStrictFDerivAt.toPartialHomeomorph_coe /-
 @[simp]
-theorem toLocalHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    (hf.toLocalHomeomorph f : E → F) = f :=
+theorem toPartialHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    (hf.toPartialHomeomorph f : E → F) = f :=
   rfl
-#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toLocalHomeomorph_coe
+#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toPartialHomeomorph_coe
 -/
 
-#print HasStrictFDerivAt.mem_toLocalHomeomorph_source /-
-theorem mem_toLocalHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    a ∈ (hf.toLocalHomeomorph f).source :=
+#print HasStrictFDerivAt.mem_toPartialHomeomorph_source /-
+theorem mem_toPartialHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    a ∈ (hf.toPartialHomeomorph f).source :=
   (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.1
-#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toLocalHomeomorph_source
+#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toPartialHomeomorph_source
 -/
 
-#print HasStrictFDerivAt.image_mem_toLocalHomeomorph_target /-
-theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    f a ∈ (hf.toLocalHomeomorph f).target :=
-  (hf.toLocalHomeomorph f).map_source hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toLocalHomeomorph_target
+#print HasStrictFDerivAt.image_mem_toPartialHomeomorph_target /-
+theorem image_mem_toPartialHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    f a ∈ (hf.toPartialHomeomorph f).target :=
+  (hf.toPartialHomeomorph f).map_source hf.mem_toPartialHomeomorph_source
+#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toPartialHomeomorph_target
 -/
 
 #print HasStrictFDerivAt.map_nhds_eq_of_equiv /-
 theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     map f (𝓝 a) = 𝓝 (f a) :=
-  (hf.toLocalHomeomorph f).map_nhds_eq hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).map_nhds_eq hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFDerivAt.map_nhds_eq_of_equiv
 -/
 
@@ -737,7 +738,7 @@ variable (f f' a)
 /-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
 to `f`. -/
 def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
-  (hf.toLocalHomeomorph f).symm
+  (hf.toPartialHomeomorph f).symm
 #align has_strict_fderiv_at.local_inverse HasStrictFDerivAt.localInverse
 -/
 
@@ -745,7 +746,7 @@ variable {f f' a}
 
 #print HasStrictFDerivAt.localInverse_def /-
 theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    hf.localInverse f _ _ = (hf.toLocalHomeomorph f).symm :=
+    hf.localInverse f _ _ = (hf.toPartialHomeomorph f).symm :=
   rfl
 #align has_strict_fderiv_at.local_inverse_def HasStrictFDerivAt.localInverse_def
 -/
@@ -753,7 +754,7 @@ theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
 #print HasStrictFDerivAt.eventually_left_inverse /-
 theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x :=
-  (hf.toLocalHomeomorph f).eventually_left_inverse hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).eventually_left_inverse hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.eventually_left_inverse HasStrictFDerivAt.eventually_left_inverse
 -/
 
@@ -768,21 +769,21 @@ theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F)
 #print HasStrictFDerivAt.eventually_right_inverse /-
 theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y :=
-  (hf.toLocalHomeomorph f).eventually_right_inverse' hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).eventually_right_inverse' hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.eventually_right_inverse HasStrictFDerivAt.eventually_right_inverse
 -/
 
 #print HasStrictFDerivAt.localInverse_continuousAt /-
 theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ContinuousAt (hf.localInverse f f' a) (f a) :=
-  (hf.toLocalHomeomorph f).continuousAt_symm hf.image_mem_toLocalHomeomorph_target
+  (hf.toPartialHomeomorph f).continuousAt_symm hf.image_mem_toPartialHomeomorph_target
 #align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFDerivAt.localInverse_continuousAt
 -/
 
 #print HasStrictFDerivAt.localInverse_tendsto /-
 theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 a) :=
-  (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).tendsto_symm hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_tendsto HasStrictFDerivAt.localInverse_tendsto
 -/
 
@@ -790,7 +791,7 @@ theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a)
 theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y :=
   eventuallyEq_of_left_inv_of_right_inv hg hf.eventually_right_inverse <|
-    (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
+    (hf.toPartialHomeomorph f).tendsto_symm hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
 -/
 
@@ -799,7 +800,7 @@ theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {
 then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -/
 theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
-  (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
+  (hf.toPartialHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toPartialHomeomorph_target <| by
     simpa [← local_inverse_def] using hf
 #align has_strict_fderiv_at.to_local_inverse HasStrictFDerivAt.to_localInverse
 -/
@@ -896,40 +897,40 @@ variable {F' : Type _} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
 
 variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
 
-#print ContDiffAt.toLocalHomeomorph /-
+#print ContDiffAt.toPartialHomeomorph /-
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
 derivative at `a`, returns a `local_homeomorph` with `to_fun = f` and `a ∈ source`. -/
-def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
-    (hn : 1 ≤ n) : LocalHomeomorph E' F' :=
-  (hf.hasStrictFDerivAt' hf' hn).toLocalHomeomorph f
-#align cont_diff_at.to_local_homeomorph ContDiffAt.toLocalHomeomorph
+def toPartialHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : PartialHomeomorph E' F' :=
+  (hf.hasStrictFDerivAt' hf' hn).toPartialHomeomorph f
+#align cont_diff_at.to_local_homeomorph ContDiffAt.toPartialHomeomorph
 -/
 
 variable {f}
 
-#print ContDiffAt.toLocalHomeomorph_coe /-
+#print ContDiffAt.toPartialHomeomorph_coe /-
 @[simp]
-theorem toLocalHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
+theorem toPartialHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    (hf.toLocalHomeomorph f hf' hn : E' → F') = f :=
+    (hf.toPartialHomeomorph f hf' hn : E' → F') = f :=
   rfl
-#align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toLocalHomeomorph_coe
+#align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toPartialHomeomorph_coe
 -/
 
-#print ContDiffAt.mem_toLocalHomeomorph_source /-
-theorem mem_toLocalHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
+#print ContDiffAt.mem_toPartialHomeomorph_source /-
+theorem mem_toPartialHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    a ∈ (hf.toLocalHomeomorph f hf' hn).source :=
-  (hf.hasStrictFDerivAt' hf' hn).mem_toLocalHomeomorph_source
-#align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toLocalHomeomorph_source
+    a ∈ (hf.toPartialHomeomorph f hf' hn).source :=
+  (hf.hasStrictFDerivAt' hf' hn).mem_toPartialHomeomorph_source
+#align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toPartialHomeomorph_source
 -/
 
-#print ContDiffAt.image_mem_toLocalHomeomorph_target /-
-theorem image_mem_toLocalHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
+#print ContDiffAt.image_mem_toPartialHomeomorph_target /-
+theorem image_mem_toPartialHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    f a ∈ (hf.toLocalHomeomorph f hf' hn).target :=
-  (hf.hasStrictFDerivAt' hf' hn).image_mem_toLocalHomeomorph_target
-#align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toLocalHomeomorph_target
+    f a ∈ (hf.toPartialHomeomorph f hf' hn).target :=
+  (hf.hasStrictFDerivAt' hf' hn).image_mem_toPartialHomeomorph_target
+#align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toPartialHomeomorph_target
 -/
 
 #print ContDiffAt.localInverse /-

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 -/
-import Mathbin.Analysis.Calculus.ContDiff
-import Mathbin.Analysis.NormedSpace.Banach
+import Analysis.Calculus.ContDiff
+import Analysis.NormedSpace.Banach
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"575b4ea3738b017e30fb205cb9b4a8742e5e82b6"
 
@@ -389,7 +389,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print ApproximatesLinearOn.image_mem_nhds /-
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -576,7 +576,7 @@ theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedS
   have hg : ApproximatesLinearOn g (f' : E →L[ℝ] F) univ (lipschitzExtensionConstant F * c) :=
     by
     apply LipschitzOnWith.approximatesLinearOn
-    rw [lipschitz_on_univ]
+    rw [lipschitzOn_univ]
     convert hu
     ext x
     simp only [add_sub_cancel', ContinuousLinearEquiv.coe_coe, Pi.sub_apply]

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

@@ -163,8 +163,8 @@ theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜
 #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
 -/
 
-alias approximates_linear_on_iff_lipschitz_on_with ↔ LipschitzOnWith
-  _root_.lipschitz_on_with.approximates_linear_on
+alias ⟨LipschitzOnWith, _root_.lipschitz_on_with.approximates_linear_on⟩ :=
+  approximates_linear_on_iff_lipschitz_on_with
 #align approximates_linear_on.lipschitz_on_with ApproximatesLinearOn.lipschitzOnWith
 #align lipschitz_on_with.approximates_linear_on LipschitzOnWith.approximatesLinearOn
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.inverse
-! leanprover-community/mathlib commit 575b4ea3738b017e30fb205cb9b4a8742e5e82b6
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.ContDiff
 import Mathbin.Analysis.NormedSpace.Banach
 
+#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"575b4ea3738b017e30fb205cb9b4a8742e5e82b6"
+
 /-!
 # Inverse function theorem
 
@@ -392,7 +389,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print ApproximatesLinearOn.image_mem_nhds /-
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=

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

@@ -124,10 +124,12 @@ def ApproximatesLinearOn (f : E → F) (f' : E →L[𝕜] F) (s : Set E) (c : 
 #align approximates_linear_on ApproximatesLinearOn
 -/
 
+#print approximatesLinearOn_empty /-
 @[simp]
 theorem approximatesLinearOn_empty (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0) :
     ApproximatesLinearOn f f' ∅ c := by simp [ApproximatesLinearOn]
 #align approximates_linear_on_empty approximatesLinearOn_empty
+-/
 
 namespace ApproximatesLinearOn
 
@@ -141,15 +143,20 @@ section
 
 variable {f' : E →L[𝕜] F} {s t : Set E} {c c' : ℝ≥0}
 
+#print ApproximatesLinearOn.mono_num /-
 theorem mono_num (hc : c ≤ c') (hf : ApproximatesLinearOn f f' s c) :
     ApproximatesLinearOn f f' s c' := fun x hx y hy =>
   le_trans (hf x hx y hy) (mul_le_mul_of_nonneg_right hc <| norm_nonneg _)
 #align approximates_linear_on.mono_num ApproximatesLinearOn.mono_num
+-/
 
+#print ApproximatesLinearOn.mono_set /-
 theorem mono_set (hst : s ⊆ t) (hf : ApproximatesLinearOn f f' t c) :
     ApproximatesLinearOn f f' s c := fun x hx y hy => hf x (hst hx) y (hst hy)
 #align approximates_linear_on.mono_set ApproximatesLinearOn.mono_set
+-/
 
+#print ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith /-
 theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜] F} {s : Set E}
     {c : ℝ≥0} : ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - f') s :=
   by
@@ -157,12 +164,14 @@ theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜
     simp only [map_sub, Pi.sub_apply]; abel
   simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
 #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
+-/
 
 alias approximates_linear_on_iff_lipschitz_on_with ↔ LipschitzOnWith
   _root_.lipschitz_on_with.approximates_linear_on
 #align approximates_linear_on.lipschitz_on_with ApproximatesLinearOn.lipschitzOnWith
 #align lipschitz_on_with.approximates_linear_on LipschitzOnWith.approximatesLinearOn
 
+#print ApproximatesLinearOn.lipschitz_sub /-
 theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) :
     LipschitzWith c fun x : s => f x - f' x :=
   by
@@ -171,20 +180,27 @@ theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) :
   convert hf x x.2 y y.2 using 2
   rw [f'.map_sub]; abel
 #align approximates_linear_on.lipschitz_sub ApproximatesLinearOn.lipschitz_sub
+-/
 
+#print ApproximatesLinearOn.lipschitz /-
 protected theorem lipschitz (hf : ApproximatesLinearOn f f' s c) :
     LipschitzWith (‖f'‖₊ + c) (s.restrict f) := by
   simpa only [restrict_apply, add_sub_cancel'_right] using
     (f'.lipschitz.restrict s).add hf.lipschitz_sub
 #align approximates_linear_on.lipschitz ApproximatesLinearOn.lipschitz
+-/
 
+#print ApproximatesLinearOn.continuous /-
 protected theorem continuous (hf : ApproximatesLinearOn f f' s c) : Continuous (s.restrict f) :=
   hf.lipschitz.Continuous
 #align approximates_linear_on.continuous ApproximatesLinearOn.continuous
+-/
 
+#print ApproximatesLinearOn.continuousOn /-
 protected theorem continuousOn (hf : ApproximatesLinearOn f f' s c) : ContinuousOn f s :=
   continuousOn_iff_continuous_restrict.2 hf.Continuous
 #align approximates_linear_on.continuous_on ApproximatesLinearOn.continuousOn
+-/
 
 end
 
@@ -197,10 +213,9 @@ equivalence, for the local inverse theorem, but also whenever the approximating
 by Banach's open mapping theorem. -/
 
 
-include cs
-
 variable {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F}
 
+#print ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse /-
 /-- If a function is linearly approximated by a continuous linear map with a (possibly nonlinear)
 right inverse, then it is locally onto: a ball of an explicit radius is included in the image
 of the map. -/
@@ -360,7 +375,9 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         tendsto_const_nhds
   exact tendsto_nhds_unique T1 T2
 #align approximates_linear_on.surj_on_closed_ball_of_nonlinear_right_inverse ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse
+-/
 
+#print ApproximatesLinearOn.open_image /-
 theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) :=
   by
@@ -373,8 +390,10 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
     (hf.surj_on_closed_ball_of_nonlinear_right_inverse f'symm (le_of_lt ε0) hε).mono hε
       (subset.refl _)
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print ApproximatesLinearOn.image_mem_nhds /-
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=
   by
@@ -382,7 +401,9 @@ theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.Nonline
   have := IsOpen.mem_nhds ((hf.mono_set hts).open_image f'symm ht hc) (mem_image_of_mem _ xt)
   exact mem_of_superset this (image_subset _ hts)
 #align approximates_linear_on.image_mem_nhds ApproximatesLinearOn.image_mem_nhds
+-/
 
+#print ApproximatesLinearOn.map_nhds_eq /-
 theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E}
     (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) :=
   by
@@ -392,6 +413,7 @@ theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearR
     (hf.mono_set (inter_subset_left s t)).image_mem_nhds f'symm (inter_mem hs ht) hc
   exact mem_of_superset this (image_subset _ (inter_subset_right _ _))
 #align approximates_linear_on.map_nhds_eq ApproximatesLinearOn.map_nhds_eq
+-/
 
 end LocallyOnto
 
@@ -404,9 +426,9 @@ We also assume that either `E = {0}`, or `c < ‖f'⁻¹‖⁻¹`. We use `N` as
 
 variable {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0}
 
--- mathport name: exprN
 local notation "N" => ‖(f'.symm : F →L[𝕜] E)‖₊
 
+#print ApproximatesLinearOn.antilipschitz /-
 protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : AntilipschitzWith (N⁻¹ - c)⁻¹ (s.restrict f) :=
   by
@@ -416,17 +438,23 @@ protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜]
   convert (f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
   simp [restrict]
 #align approximates_linear_on.antilipschitz ApproximatesLinearOn.antilipschitz
+-/
 
+#print ApproximatesLinearOn.injective /-
 protected theorem injective (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : Injective (s.restrict f) :=
   (hf.antilipschitz hc).Injective
 #align approximates_linear_on.injective ApproximatesLinearOn.injective
+-/
 
+#print ApproximatesLinearOn.injOn /-
 protected theorem injOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : InjOn f s :=
   injOn_iff_injective.2 <| hf.Injective hc
 #align approximates_linear_on.inj_on ApproximatesLinearOn.injOn
+-/
 
+#print ApproximatesLinearOn.surjective /-
 protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
     (hc : Subsingleton E ∨ c < N⁻¹) : Surjective f :=
   by
@@ -445,7 +473,9 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
     refine' ((tendsto_id.const_mul_at_top hc').Frequently hp.frequently).mono _
     exact fun R h y hy => h hy
 #align approximates_linear_on.surjective ApproximatesLinearOn.surjective
+-/
 
+#print ApproximatesLinearOn.toLocalEquiv /-
 /-- A map approximating a linear equivalence on a set defines a local equivalence on this set.
 Should not be used outside of this file, because it is superseded by `to_local_homeomorph` below.
 
@@ -454,7 +484,9 @@ def toLocalEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : LocalEquiv E F :=
   (hf.InjOn hc).toLocalEquiv _ _
 #align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toLocalEquiv
+-/
 
+#print ApproximatesLinearOn.inverse_continuousOn /-
 /-- The inverse function is continuous on `f '' s`. Use properties of `local_homeomorph` instead. -/
 theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toLocalEquiv hc).symm (f '' s) :=
@@ -463,7 +495,9 @@ theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F)
   refine' ((hf.antilipschitz hc).to_rightInvOn' _ (hf.to_local_equiv hc).right_inv').Continuous
   exact fun x hx => (hf.to_local_equiv hc).map_target hx
 #align approximates_linear_on.inverse_continuous_on ApproximatesLinearOn.inverse_continuousOn
+-/
 
+#print ApproximatesLinearOn.to_inv /-
 /-- The inverse function is approximated linearly on `f '' s` by `f'.symm`. -/
 theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) :
     ApproximatesLinearOn (hf.toLocalEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
@@ -491,13 +525,13 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
     _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by simp only [norm_sub_rev, Nonneg.coe_mul];
       ring
 #align approximates_linear_on.to_inv ApproximatesLinearOn.to_inv
-
-include cs
+-/
 
 section
 
 variable (f s)
 
+#print ApproximatesLinearOn.toLocalHomeomorph /-
 /-- Given a function `f` that approximates a linear equivalence on an open set `s`,
 returns a local homeomorph with `to_fun = f` and `source = s`. -/
 def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
@@ -511,7 +545,9 @@ def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   continuous_toFun := hf.ContinuousOn
   continuous_invFun := hf.inverse_continuousOn hc
 #align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
+-/
 
+#print ApproximatesLinearOn.toHomeomorph /-
 /-- A function `f` that approximates a linear equivalence on the whole space is a homeomorphism. -/
 def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
     (hc : Subsingleton E ∨ c < N⁻¹) : E ≃ₜ F :=
@@ -521,9 +557,9 @@ def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
   rw [image_univ, range_iff_surjective]
   exact hf.surjective hc
 #align approximates_linear_on.to_homeomorph ApproximatesLinearOn.toHomeomorph
+-/
 
-omit cs
-
+#print ApproximatesLinearOn.exists_homeomorph_extension /-
 /-- In a real vector space, a function `f` that approximates a linear equivalence on a subset `s`
 can be extended to a homeomorphism of the whole space. -/
 theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
@@ -550,35 +586,44 @@ theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedS
   haveI : FiniteDimensional ℝ E := f'.symm.to_linear_equiv.finite_dimensional
   exact ⟨hg.to_homeomorph g hc, fg⟩
 #align approximates_linear_on.exists_homeomorph_extension ApproximatesLinearOn.exists_homeomorph_extension
+-/
 
 end
 
+#print ApproximatesLinearOn.toLocalHomeomorph_coe /-
 @[simp]
 theorem toLocalHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
     (hf.toLocalHomeomorph f s hc hs : E → F) = f :=
   rfl
 #align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toLocalHomeomorph_coe
+-/
 
+#print ApproximatesLinearOn.toLocalHomeomorph_source /-
 @[simp]
 theorem toLocalHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : (hf.toLocalHomeomorph f s hc hs).source = s :=
   rfl
 #align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toLocalHomeomorph_source
+-/
 
+#print ApproximatesLinearOn.toLocalHomeomorph_target /-
 @[simp]
 theorem toLocalHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
     (hf.toLocalHomeomorph f s hc hs).target = f '' s :=
   rfl
 #align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toLocalHomeomorph_target
+-/
 
+#print ApproximatesLinearOn.closedBall_subset_target /-
 theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
     closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toLocalHomeomorph f s hc hs).target :=
   (hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse ε0 hε).mono hε
     (Subset.refl _)
 #align approximates_linear_on.closed_ball_subset_target ApproximatesLinearOn.closedBall_subset_target
+-/
 
 end ApproximatesLinearOn
 
@@ -594,6 +639,7 @@ function. -/
 
 namespace HasStrictFDerivAt
 
+#print HasStrictFDerivAt.approximates_deriv_on_nhds /-
 /-- If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`
 with constant `c` on some neighborhood of `a`. -/
 theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
@@ -608,7 +654,9 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
 #align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds
+-/
 
+#print HasStrictFDerivAt.map_nhds_eq_of_surj /-
 theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
     (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
     map f (𝓝 a) = 𝓝 (f a) :=
@@ -622,9 +670,11 @@ theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f
   apply hs.map_nhds_eq f'symm s_nhds (Or.inr (NNReal.half_lt_self _))
   simp [ne_of_gt f'symm_pos]
 #align has_strict_fderiv_at.map_nhds_eq_of_surj HasStrictFDerivAt.map_nhds_eq_of_surj
+-/
 
 variable [cs : CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E}
 
+#print HasStrictFDerivAt.approximates_deriv_on_open_nhds /-
 theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∃ (s : Set E) (hs : a ∈ s ∧ IsOpen s),
       ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) :=
@@ -635,8 +685,7 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[
     hf.approximates_deriv_on_nhds <|
       f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
 #align has_strict_fderiv_at.approximates_deriv_on_open_nhds HasStrictFDerivAt.approximates_deriv_on_open_nhds
-
-include cs
+-/
 
 variable (f)
 
@@ -656,26 +705,34 @@ def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : Local
 
 variable {f}
 
+#print HasStrictFDerivAt.toLocalHomeomorph_coe /-
 @[simp]
 theorem toLocalHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     (hf.toLocalHomeomorph f : E → F) = f :=
   rfl
 #align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toLocalHomeomorph_coe
+-/
 
+#print HasStrictFDerivAt.mem_toLocalHomeomorph_source /-
 theorem mem_toLocalHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     a ∈ (hf.toLocalHomeomorph f).source :=
   (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.1
 #align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toLocalHomeomorph_source
+-/
 
+#print HasStrictFDerivAt.image_mem_toLocalHomeomorph_target /-
 theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     f a ∈ (hf.toLocalHomeomorph f).target :=
   (hf.toLocalHomeomorph f).map_source hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toLocalHomeomorph_target
+-/
 
+#print HasStrictFDerivAt.map_nhds_eq_of_equiv /-
 theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     map f (𝓝 a) = 𝓝 (f a) :=
   (hf.toLocalHomeomorph f).map_nhds_eq hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFDerivAt.map_nhds_eq_of_equiv
+-/
 
 variable (f f' a)
 
@@ -689,43 +746,58 @@ def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
 
 variable {f f' a}
 
+#print HasStrictFDerivAt.localInverse_def /-
 theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     hf.localInverse f _ _ = (hf.toLocalHomeomorph f).symm :=
   rfl
 #align has_strict_fderiv_at.local_inverse_def HasStrictFDerivAt.localInverse_def
+-/
 
+#print HasStrictFDerivAt.eventually_left_inverse /-
 theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x :=
   (hf.toLocalHomeomorph f).eventually_left_inverse hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.eventually_left_inverse HasStrictFDerivAt.eventually_left_inverse
+-/
 
+#print HasStrictFDerivAt.localInverse_apply_image /-
 @[simp]
 theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     hf.localInverse f f' a (f a) = a :=
   hf.eventually_left_inverse.self_of_nhds
 #align has_strict_fderiv_at.local_inverse_apply_image HasStrictFDerivAt.localInverse_apply_image
+-/
 
+#print HasStrictFDerivAt.eventually_right_inverse /-
 theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y :=
   (hf.toLocalHomeomorph f).eventually_right_inverse' hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.eventually_right_inverse HasStrictFDerivAt.eventually_right_inverse
+-/
 
+#print HasStrictFDerivAt.localInverse_continuousAt /-
 theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ContinuousAt (hf.localInverse f f' a) (f a) :=
   (hf.toLocalHomeomorph f).continuousAt_symm hf.image_mem_toLocalHomeomorph_target
 #align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFDerivAt.localInverse_continuousAt
+-/
 
+#print HasStrictFDerivAt.localInverse_tendsto /-
 theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 a) :=
   (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_tendsto HasStrictFDerivAt.localInverse_tendsto
+-/
 
+#print HasStrictFDerivAt.localInverse_unique /-
 theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y :=
   eventuallyEq_of_left_inv_of_right_inv hg hf.eventually_right_inverse <|
     (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
+-/
 
+#print HasStrictFDerivAt.to_localInverse /-
 /-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
 then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -/
 theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
@@ -733,7 +805,9 @@ theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
   (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
     simpa [← local_inverse_def] using hf
 #align has_strict_fderiv_at.to_local_inverse HasStrictFDerivAt.to_localInverse
+-/
 
+#print HasStrictFDerivAt.to_local_left_inverse /-
 /-- If `f : E → F` has an invertible derivative `f'` at `a` in the sense of strict differentiability
 and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
 
@@ -743,14 +817,17 @@ theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a)
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) (f a) :=
   hf.to_localInverse.congr_of_eventuallyEq <| (hf.localInverse_unique hg).mono fun _ => Eq.symm
 #align has_strict_fderiv_at.to_local_left_inverse HasStrictFDerivAt.to_local_left_inverse
+-/
 
 end HasStrictFDerivAt
 
+#print open_map_of_strict_fderiv_equiv /-
 /-- If a function has an invertible strict derivative at all points, then it is an open map. -/
 theorem open_map_of_strict_fderiv_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
     (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
 #align open_map_of_strict_fderiv_equiv open_map_of_strict_fderiv_equiv
+-/
 
 /-!
 ### Inverse function theorem, 1D case
@@ -765,38 +842,46 @@ namespace HasStrictDerivAt
 
 variable [cs : CompleteSpace 𝕜] {f : 𝕜 → 𝕜} {f' a : 𝕜} (hf : HasStrictDerivAt f f' a) (hf' : f' ≠ 0)
 
-include cs
-
 variable (f f' a)
 
+#print HasStrictDerivAt.localInverse /-
 /-- A function that is inverse to `f` near `a`. -/
 @[reducible]
 def localInverse : 𝕜 → 𝕜 :=
   (hf.hasStrictFDerivAt_equiv hf').localInverse _ _ _
 #align has_strict_deriv_at.local_inverse HasStrictDerivAt.localInverse
+-/
 
 variable {f f' a}
 
+#print HasStrictDerivAt.map_nhds_eq /-
 theorem map_nhds_eq : map f (𝓝 a) = 𝓝 (f a) :=
   (hf.hasStrictFDerivAt_equiv hf').map_nhds_eq_of_equiv
 #align has_strict_deriv_at.map_nhds_eq HasStrictDerivAt.map_nhds_eq
+-/
 
+#print HasStrictDerivAt.to_localInverse /-
 theorem to_localInverse : HasStrictDerivAt (hf.localInverse f f' a hf') f'⁻¹ (f a) :=
   (hf.hasStrictFDerivAt_equiv hf').to_localInverse
 #align has_strict_deriv_at.to_local_inverse HasStrictDerivAt.to_localInverse
+-/
 
+#print HasStrictDerivAt.to_local_left_inverse /-
 theorem to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
     HasStrictDerivAt g f'⁻¹ (f a) :=
   (hf.hasStrictFDerivAt_equiv hf').to_local_left_inverse hg
 #align has_strict_deriv_at.to_local_left_inverse HasStrictDerivAt.to_local_left_inverse
+-/
 
 end HasStrictDerivAt
 
+#print open_map_of_strict_deriv /-
 /-- If a function has a non-zero strict derivative at all points, then it is an open map. -/
 theorem open_map_of_strict_deriv [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
     (hf : ∀ x, HasStrictDerivAt f (f' x) x) (h0 : ∀ x, f' x ≠ 0) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => ((hf x).map_nhds_eq (h0 x)).ge
 #align open_map_of_strict_deriv open_map_of_strict_deriv
+-/
 
 /-!
 ### Inverse function theorem, smooth case
@@ -814,46 +899,59 @@ variable {F' : Type _} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
 
 variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
 
+#print ContDiffAt.toLocalHomeomorph /-
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
 derivative at `a`, returns a `local_homeomorph` with `to_fun = f` and `a ∈ source`. -/
 def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : LocalHomeomorph E' F' :=
   (hf.hasStrictFDerivAt' hf' hn).toLocalHomeomorph f
 #align cont_diff_at.to_local_homeomorph ContDiffAt.toLocalHomeomorph
+-/
 
 variable {f}
 
+#print ContDiffAt.toLocalHomeomorph_coe /-
 @[simp]
 theorem toLocalHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     (hf.toLocalHomeomorph f hf' hn : E' → F') = f :=
   rfl
 #align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toLocalHomeomorph_coe
+-/
 
+#print ContDiffAt.mem_toLocalHomeomorph_source /-
 theorem mem_toLocalHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     a ∈ (hf.toLocalHomeomorph f hf' hn).source :=
   (hf.hasStrictFDerivAt' hf' hn).mem_toLocalHomeomorph_source
 #align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toLocalHomeomorph_source
+-/
 
+#print ContDiffAt.image_mem_toLocalHomeomorph_target /-
 theorem image_mem_toLocalHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     f a ∈ (hf.toLocalHomeomorph f hf' hn).target :=
   (hf.hasStrictFDerivAt' hf' hn).image_mem_toLocalHomeomorph_target
 #align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toLocalHomeomorph_target
+-/
 
+#print ContDiffAt.localInverse /-
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
 at `a`, returns a function that is locally inverse to `f`. -/
 def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : F' → E' :=
   (hf.hasStrictFDerivAt' hf' hn).localInverse f f' a
 #align cont_diff_at.local_inverse ContDiffAt.localInverse
+-/
 
+#print ContDiffAt.localInverse_apply_image /-
 theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
   (hf.hasStrictFDerivAt' hf' hn).localInverse_apply_image
 #align cont_diff_at.local_inverse_apply_image ContDiffAt.localInverse_apply_image
+-/
 
+#print ContDiffAt.to_localInverse /-
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
 at `a`, the inverse function (produced by `cont_diff.to_local_homeomorph`) is
 also `cont_diff`. -/
@@ -868,6 +966,7 @@ theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
   · convert hf'
   · convert hf
 #align cont_diff_at.to_local_inverse ContDiffAt.to_localInverse
+-/
 
 end ContDiffAt
 

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

@@ -262,7 +262,6 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         apply mul_le_mul_of_nonneg_left _ (NNReal.coe_nonneg c)
         simpa [hv, dist_eq_norm'] using f'symm.bound (y - f z)
       _ = c * f'symm.nnnorm * dist (f z) y := by ring
-      
   -- Third bound: a complicated bound on `dist w b` (that will show up in the induction) is enough
   -- to check that `w` is in the ball on which one has controls. Will be used to check that `u n`
   -- belongs to this ball for all `n`.
@@ -289,7 +288,6 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         rw [mul_one]
         exact mul_le_mul_of_nonneg_left (mem_closed_ball'.1 hy) (NNReal.coe_nonneg _)
       _ = ε * (1 - c * f'symm.nnnorm) := by field_simp; ring
-      
   /- Main inductive control: `f (u n)` becomes exponentially close to `y`, and therefore
     `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive
     bound on `dist (u n) b`, from which one checks that `u n` remains in the ball on which we
@@ -319,7 +317,6 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
             f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n.succ) / (1 - c * f'symm.nnnorm) *
               dist (f b) y :=
           by field_simp [Jcf']; ring
-        
     refine' ⟨_, Ign⟩
     calc
       dist (f (g (u n))) y ≤ c * f'symm.nnnorm * dist (f (u n)) y :=
@@ -327,7 +324,6 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       _ ≤ c * f'symm.nnnorm * ((c * f'symm.nnnorm) ^ n * dist (f b) y) :=
         (mul_le_mul_of_nonneg_left IH.1 (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _)))
       _ = (c * f'symm.nnnorm) ^ n.succ * dist (f b) y := by ring
-      
   -- Deduce from the inductive bound that `uₙ` is a Cauchy sequence, therefore converging.
   have : CauchySeq u :=
     haveI :
@@ -340,7 +336,6 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         _ ≤ f'symm.nnnorm * ((c * f'symm.nnnorm) ^ n * dist (f b) y) :=
           (mul_le_mul_of_nonneg_left (D n).1 (NNReal.coe_nonneg _))
         _ = f'symm.nnnorm * dist (f b) y * (c * f'symm.nnnorm) ^ n := by ring
-        
     cauchySeq_of_le_geometric _ _ Icf' this
   obtain ⟨x, hx⟩ : ∃ x, tendsto u at_top (𝓝 x) := cauchySeq_tendsto_of_complete this
   -- As all the `uₙ` belong to the ball `closed_ball b ε`, so does their limit `x`.
@@ -495,7 +490,6 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
       exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩
     _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by simp only [norm_sub_rev, Nonneg.coe_mul];
       ring
-    
 #align approximates_linear_on.to_inv ApproximatesLinearOn.to_inv
 
 include cs

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

@@ -379,7 +379,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
       (subset.refl _)
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.inverse
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! leanprover-community/mathlib commit 575b4ea3738b017e30fb205cb9b4a8742e5e82b6
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.NormedSpace.Banach
 /-!
 # Inverse function theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we prove the inverse function theorem. It says that if a map `f : E → F`
 has an invertible strict derivative `f'` at `a`, then it is locally invertible,
 and the inverse function has derivative `f' ⁻¹`.

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

@@ -109,6 +109,7 @@ lemmas. This approach makes it possible
 -/
 
 
+#print ApproximatesLinearOn /-
 /-- We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`,
 if `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` whenever `x, y ∈ s`.
 
@@ -118,6 +119,7 @@ on a specific set. -/
 def ApproximatesLinearOn (f : E → F) (f' : E →L[𝕜] F) (s : Set E) (c : ℝ≥0) : Prop :=
   ∀ x ∈ s, ∀ y ∈ s, ‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖
 #align approximates_linear_on ApproximatesLinearOn
+-/
 
 @[simp]
 theorem approximatesLinearOn_empty (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0) :
@@ -641,6 +643,7 @@ include cs
 
 variable (f)
 
+#print HasStrictFDerivAt.toLocalHomeomorph /-
 /-- Given a function with an invertible strict derivative at `a`, returns a `local_homeomorph`
 with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
 The other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function
@@ -652,6 +655,7 @@ def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : Local
       NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.2
 #align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toLocalHomeomorph
+-/
 
 variable {f}
 
@@ -678,11 +682,13 @@ theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a)
 
 variable (f f' a)
 
+#print HasStrictFDerivAt.localInverse /-
 /-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
 to `f`. -/
 def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
   (hf.toLocalHomeomorph f).symm
 #align has_strict_fderiv_at.local_inverse HasStrictFDerivAt.localInverse
+-/
 
 variable {f f' a}
 

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

@@ -153,8 +153,8 @@ theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜
   simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
 #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
 
-alias approximates_linear_on_iff_lipschitz_on_with ↔
-  LipschitzOnWith _root_.lipschitz_on_with.approximates_linear_on
+alias approximates_linear_on_iff_lipschitz_on_with ↔ LipschitzOnWith
+  _root_.lipschitz_on_with.approximates_linear_on
 #align approximates_linear_on.lipschitz_on_with ApproximatesLinearOn.lipschitzOnWith
 #align lipschitz_on_with.approximates_linear_on LipschitzOnWith.approximatesLinearOn
 
@@ -413,7 +413,7 @@ protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜]
   cases' hc with hE hc
   · haveI : Subsingleton s := ⟨fun x y => Subtype.eq <| @Subsingleton.elim _ hE _ _⟩
     exact AntilipschitzWith.of_subsingleton
-  convert(f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
+  convert (f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
   simp [restrict]
 #align approximates_linear_on.antilipschitz ApproximatesLinearOn.antilipschitz
 
@@ -815,7 +815,7 @@ variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
 derivative at `a`, returns a `local_homeomorph` with `to_fun = f` and `a ∈ source`. -/
 def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : LocalHomeomorph E' F' :=
-  (hf.has_strict_fderiv_at' hf' hn).toLocalHomeomorph f
+  (hf.hasStrictFDerivAt' hf' hn).toLocalHomeomorph f
 #align cont_diff_at.to_local_homeomorph ContDiffAt.toLocalHomeomorph
 
 variable {f}
@@ -830,25 +830,25 @@ theorem toLocalHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
 theorem mem_toLocalHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     a ∈ (hf.toLocalHomeomorph f hf' hn).source :=
-  (hf.has_strict_fderiv_at' hf' hn).mem_toLocalHomeomorph_source
+  (hf.hasStrictFDerivAt' hf' hn).mem_toLocalHomeomorph_source
 #align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toLocalHomeomorph_source
 
 theorem image_mem_toLocalHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     f a ∈ (hf.toLocalHomeomorph f hf' hn).target :=
-  (hf.has_strict_fderiv_at' hf' hn).image_mem_toLocalHomeomorph_target
+  (hf.hasStrictFDerivAt' hf' hn).image_mem_toLocalHomeomorph_target
 #align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toLocalHomeomorph_target
 
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
 at `a`, returns a function that is locally inverse to `f`. -/
 def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : F' → E' :=
-  (hf.has_strict_fderiv_at' hf' hn).localInverse f f' a
+  (hf.hasStrictFDerivAt' hf' hn).localInverse f f' a
 #align cont_diff_at.local_inverse ContDiffAt.localInverse
 
 theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
     (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
-  (hf.has_strict_fderiv_at' hf' hn).localInverse_apply_image
+  (hf.hasStrictFDerivAt' hf' hn).localInverse_apply_image
 #align cont_diff_at.local_inverse_apply_image ContDiffAt.localInverse_apply_image
 
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative

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

@@ -208,10 +208,10 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   · refine' ⟨b, by simp [ε0], _⟩
     have : dist y (f b) ≤ 0 :=
       (mem_closed_ball.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
-    simp only [dist_le_zero] at this
+    simp only [dist_le_zero] at this 
     rw [this]
   have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
-  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
+  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc 
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
   have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
   /- We have to show that `y` can be written as `f x` for some `x ∈ closed_ball b ε`.
@@ -313,7 +313,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         _ =
             f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n.succ) / (1 - c * f'symm.nnnorm) *
               dist (f b) y :=
-          by field_simp [Jcf'] ; ring
+          by field_simp [Jcf']; ring
         
     refine' ⟨_, Ign⟩
     calc
@@ -365,7 +365,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) :=
   by
   cases' hc with hE hc; · skip; apply isOpen_discrete
-  simp only [isOpen_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs⊢
+  simp only [isOpen_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs ⊢
   intro x hx
   rcases hs x hx with ⟨ε, ε0, hε⟩
   refine' ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, _⟩
@@ -378,7 +378,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=
   by
-  obtain ⟨t, hts, ht, xt⟩ : ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs
+  obtain ⟨t, hts, ht, xt⟩ : ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs
   have := IsOpen.mem_nhds ((hf.mono_set hts).open_image f'symm ht hc) (mem_image_of_mem _ xt)
   exact mem_of_superset this (image_subset _ hts)
 #align approximates_linear_on.image_mem_nhds ApproximatesLinearOn.image_mem_nhds
@@ -508,7 +508,7 @@ def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   open_source := hs
   open_target :=
     hf.open_image f'.toNonlinearRightInverse hs
-      (by rwa [f'.to_linear_equiv.to_equiv.subsingleton_congr] at hc)
+      (by rwa [f'.to_linear_equiv.to_equiv.subsingleton_congr] at hc )
   continuous_toFun := hf.ContinuousOn
   continuous_invFun := hf.inverse_continuousOn hc
 #align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
@@ -605,7 +605,7 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x hx y hy => _⟩
     simp [@Subsingleton.elim E hE x y]
   have := hf.def hc
-  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
+  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this 
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
 #align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds
@@ -627,7 +627,7 @@ theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f
 variable [cs : CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E}
 
 theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    ∃ (s : Set E)(hs : a ∈ s ∧ IsOpen s),
+    ∃ (s : Set E) (hs : a ∈ s ∧ IsOpen s),
       ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) :=
   by
   refine' ((nhds_basis_opens a).exists_iff _).1 _

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

@@ -62,7 +62,7 @@ derivative, strictly differentiable, continuously differentiable, smooth, invers
 
 open Function Set Filter Metric
 
-open Topology Classical NNReal
+open scoped Topology Classical NNReal
 
 noncomputable section
 

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

@@ -148,11 +148,8 @@ theorem mono_set (hst : s ⊆ t) (hf : ApproximatesLinearOn f f' t c) :
 theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜] F} {s : Set E}
     {c : ℝ≥0} : ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - f') s :=
   by
-  have : ∀ x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y :=
-    by
-    intro x y
-    simp only [map_sub, Pi.sub_apply]
-    abel
+  have : ∀ x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y := by intro x y;
+    simp only [map_sub, Pi.sub_apply]; abel
   simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
 #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
 
@@ -213,15 +210,10 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       (mem_closed_ball.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
     simp only [dist_le_zero] at this
     rw [this]
-  have If' : (0 : ℝ) < f'symm.nnnorm := by
-    rw [← inv_pos]
-    exact (NNReal.coe_nonneg _).trans_lt hc
+  have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
   have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
-  have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 :=
-    by
-    apply ne_of_gt
-    linarith
+  have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
   /- We have to show that `y` can be written as `f x` for some `x ∈ closed_ball b ε`.
     The idea of the proof is to apply the Banach contraction principle to the map
     `g : x ↦ x + f'symm (y - f x)`, as a fixed point of this map satisfies `f x = y`.
@@ -255,10 +247,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     set v := f'symm (y - f z) with hv
     calc
       dist (f (g z)) y = ‖f (z + v) - y‖ := by rw [dist_eq_norm]
-      _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ :=
-        by
-        congr 1
-        abel
+      _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ := by congr 1; abel
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
         simp only [ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
           sub_add_cancel]
@@ -280,9 +269,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     by
     intro n w hw
     apply hw.trans
-    rw [div_mul_eq_mul_div, div_le_iff]
-    swap
-    · linarith
+    rw [div_mul_eq_mul_div, div_le_iff]; swap; · linarith
     calc
       (f'symm.nnnorm : ℝ) * (1 - (c * f'symm.nnnorm) ^ n) * dist (f b) y =
           f'symm.nnnorm * dist (f b) y * (1 - (c * f'symm.nnnorm) ^ n) :=
@@ -296,9 +283,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         by
         rw [mul_one]
         exact mul_le_mul_of_nonneg_left (mem_closed_ball'.1 hy) (NNReal.coe_nonneg _)
-      _ = ε * (1 - c * f'symm.nnnorm) := by
-        field_simp
-        ring
+      _ = ε * (1 - c * f'symm.nnnorm) := by field_simp; ring
       
   /- Main inductive control: `f (u n)` becomes exponentially close to `y`, and therefore
     `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive
@@ -311,8 +296,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
           f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) * dist (f b) y :=
     by
     intro n
-    induction' n with n IH
-    · simp [hu, le_refl]
+    induction' n with n IH; · simp [hu, le_refl]
     rw [usucc]
     have Ign :
       dist (g (u n)) b ≤
@@ -329,9 +313,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         _ =
             f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n.succ) / (1 - c * f'symm.nnnorm) *
               dist (f b) y :=
-          by
-          field_simp [Jcf']
-          ring
+          by field_simp [Jcf'] ; ring
         
     refine' ⟨_, Ign⟩
     calc
@@ -382,9 +364,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
 theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) :=
   by
-  cases' hc with hE hc;
-  · skip
-    apply isOpen_discrete
+  cases' hc with hE hc; · skip; apply isOpen_discrete
   simp only [isOpen_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs⊢
   intro x hx
   rcases hs x hx with ⟨ε, ε0, hε⟩
@@ -454,9 +434,7 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
   · haveI : Subsingleton F := (Equiv.subsingleton_congr f'.to_linear_equiv.to_equiv).1 hE
     exact surjective_to_subsingleton _
   · apply forall_of_forall_mem_closed_ball (fun y : F => ∃ a, f a = y) (f 0) _
-    have hc' : (0 : ℝ) < N⁻¹ - c := by
-      rw [sub_pos]
-      exact hc
+    have hc' : (0 : ℝ) < N⁻¹ - c := by rw [sub_pos]; exact hc
     let p : ℝ → Prop := fun R => closed_ball (f 0) R ⊆ Set.range f
     have hp : ∀ᶠ r : ℝ in at_top, p ((N⁻¹ - c) * r) :=
       by
@@ -510,9 +488,7 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
       apply_rules [mul_le_mul_of_nonneg_left, NNReal.coe_nonneg]
       rw [← dist_eq_norm, ← dist_eq_norm]
       exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩
-    _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ :=
-      by
-      simp only [norm_sub_rev, Nonneg.coe_mul]
+    _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by simp only [norm_sub_rev, Nonneg.coe_mul];
       ring
     
 #align approximates_linear_on.to_inv ApproximatesLinearOn.to_inv

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -617,12 +617,12 @@ of `a`, and we can apply `approximates_linear_on.to_local_homeomorph` to constru
 function. -/
 
 
-namespace HasStrictFderivAt
+namespace HasStrictFDerivAt
 
 /-- If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`
 with constant `c` on some neighborhood of `a`. -/
 theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
-    (hf : HasStrictFderivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
+    (hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
     ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c :=
   by
   cases' hc with hE hc
@@ -632,10 +632,10 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
-#align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFderivAt.approximates_deriv_on_nhds
+#align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds
 
 theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
-    (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
+    (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
     map f (𝓝 a) = 𝓝 (f a) :=
   by
   let f'symm := f'.nonlinear_right_inverse_of_surjective h
@@ -646,11 +646,11 @@ theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f
     hf.approximates_deriv_on_nhds (Or.inr cpos)
   apply hs.map_nhds_eq f'symm s_nhds (Or.inr (NNReal.half_lt_self _))
   simp [ne_of_gt f'symm_pos]
-#align has_strict_fderiv_at.map_nhds_eq_of_surj HasStrictFderivAt.map_nhds_eq_of_surj
+#align has_strict_fderiv_at.map_nhds_eq_of_surj HasStrictFDerivAt.map_nhds_eq_of_surj
 
 variable [cs : CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E}
 
-theorem approximates_deriv_on_open_nhds (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∃ (s : Set E)(hs : a ∈ s ∧ IsOpen s),
       ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) :=
   by
@@ -659,7 +659,7 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFderivAt f (f' : E →L[
   exact
     hf.approximates_deriv_on_nhds <|
       f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
-#align has_strict_fderiv_at.approximates_deriv_on_open_nhds HasStrictFderivAt.approximates_deriv_on_open_nhds
+#align has_strict_fderiv_at.approximates_deriv_on_open_nhds HasStrictFDerivAt.approximates_deriv_on_open_nhds
 
 include cs
 
@@ -669,107 +669,107 @@ variable (f)
 with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
 The other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function
 of this `local_homeomorph` has derivative `f'.symm`. -/
-def toLocalHomeomorph (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) : LocalHomeomorph E F :=
+def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : LocalHomeomorph E F :=
   ApproximatesLinearOn.toLocalHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).snd
     (f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
       NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.2
-#align has_strict_fderiv_at.to_local_homeomorph HasStrictFderivAt.toLocalHomeomorph
+#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toLocalHomeomorph
 
 variable {f}
 
 @[simp]
-theorem toLocalHomeomorph_coe (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem toLocalHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     (hf.toLocalHomeomorph f : E → F) = f :=
   rfl
-#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFderivAt.toLocalHomeomorph_coe
+#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toLocalHomeomorph_coe
 
-theorem mem_toLocalHomeomorph_source (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem mem_toLocalHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     a ∈ (hf.toLocalHomeomorph f).source :=
   (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.1
-#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFderivAt.mem_toLocalHomeomorph_source
+#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toLocalHomeomorph_source
 
-theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     f a ∈ (hf.toLocalHomeomorph f).target :=
   (hf.toLocalHomeomorph f).map_source hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFderivAt.image_mem_toLocalHomeomorph_target
+#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toLocalHomeomorph_target
 
-theorem map_nhds_eq_of_equiv (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     map f (𝓝 a) = 𝓝 (f a) :=
   (hf.toLocalHomeomorph f).map_nhds_eq hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFderivAt.map_nhds_eq_of_equiv
+#align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFDerivAt.map_nhds_eq_of_equiv
 
 variable (f f' a)
 
 /-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
 to `f`. -/
-def localInverse (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) : F → E :=
+def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
   (hf.toLocalHomeomorph f).symm
-#align has_strict_fderiv_at.local_inverse HasStrictFderivAt.localInverse
+#align has_strict_fderiv_at.local_inverse HasStrictFDerivAt.localInverse
 
 variable {f f' a}
 
-theorem localInverse_def (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     hf.localInverse f _ _ = (hf.toLocalHomeomorph f).symm :=
   rfl
-#align has_strict_fderiv_at.local_inverse_def HasStrictFderivAt.localInverse_def
+#align has_strict_fderiv_at.local_inverse_def HasStrictFDerivAt.localInverse_def
 
-theorem eventually_left_inverse (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x :=
   (hf.toLocalHomeomorph f).eventually_left_inverse hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.eventually_left_inverse HasStrictFderivAt.eventually_left_inverse
+#align has_strict_fderiv_at.eventually_left_inverse HasStrictFDerivAt.eventually_left_inverse
 
 @[simp]
-theorem localInverse_apply_image (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     hf.localInverse f f' a (f a) = a :=
   hf.eventually_left_inverse.self_of_nhds
-#align has_strict_fderiv_at.local_inverse_apply_image HasStrictFderivAt.localInverse_apply_image
+#align has_strict_fderiv_at.local_inverse_apply_image HasStrictFDerivAt.localInverse_apply_image
 
-theorem eventually_right_inverse (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y :=
   (hf.toLocalHomeomorph f).eventually_right_inverse' hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.eventually_right_inverse HasStrictFderivAt.eventually_right_inverse
+#align has_strict_fderiv_at.eventually_right_inverse HasStrictFDerivAt.eventually_right_inverse
 
-theorem localInverse_continuousAt (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ContinuousAt (hf.localInverse f f' a) (f a) :=
   (hf.toLocalHomeomorph f).continuousAt_symm hf.image_mem_toLocalHomeomorph_target
-#align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFderivAt.localInverse_continuousAt
+#align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFDerivAt.localInverse_continuousAt
 
-theorem localInverse_tendsto (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
+theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 a) :=
   (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.local_inverse_tendsto HasStrictFderivAt.localInverse_tendsto
+#align has_strict_fderiv_at.local_inverse_tendsto HasStrictFDerivAt.localInverse_tendsto
 
-theorem localInverse_unique (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) {g : F → E}
+theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y :=
   eventuallyEq_of_left_inv_of_right_inv hg hf.eventually_right_inverse <|
     (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.local_inverse_unique HasStrictFderivAt.localInverse_unique
+#align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
 
 /-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
 then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -/
-theorem to_localInverse (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) :
-    HasStrictFderivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
-  (hf.toLocalHomeomorph f).hasStrictFderivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
+theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
+  (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
     simpa [← local_inverse_def] using hf
-#align has_strict_fderiv_at.to_local_inverse HasStrictFderivAt.to_localInverse
+#align has_strict_fderiv_at.to_local_inverse HasStrictFDerivAt.to_localInverse
 
 /-- If `f : E → F` has an invertible derivative `f'` at `a` in the sense of strict differentiability
 and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
 
 For a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[complete_space E]`
 see `of_local_left_inverse`.  -/
-theorem to_local_left_inverse (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) {g : F → E}
-    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFderivAt g (f'.symm : F →L[𝕜] E) (f a) :=
+theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
+    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) (f a) :=
   hf.to_localInverse.congr_of_eventuallyEq <| (hf.localInverse_unique hg).mono fun _ => Eq.symm
-#align has_strict_fderiv_at.to_local_left_inverse HasStrictFderivAt.to_local_left_inverse
+#align has_strict_fderiv_at.to_local_left_inverse HasStrictFDerivAt.to_local_left_inverse
 
-end HasStrictFderivAt
+end HasStrictFDerivAt
 
 /-- If a function has an invertible strict derivative at all points, then it is an open map. -/
 theorem open_map_of_strict_fderiv_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
-    (hf : ∀ x, HasStrictFderivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
+    (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
 #align open_map_of_strict_fderiv_equiv open_map_of_strict_fderiv_equiv
 
@@ -793,22 +793,22 @@ variable (f f' a)
 /-- A function that is inverse to `f` near `a`. -/
 @[reducible]
 def localInverse : 𝕜 → 𝕜 :=
-  (hf.hasStrictFderivAt_equiv hf').localInverse _ _ _
+  (hf.hasStrictFDerivAt_equiv hf').localInverse _ _ _
 #align has_strict_deriv_at.local_inverse HasStrictDerivAt.localInverse
 
 variable {f f' a}
 
 theorem map_nhds_eq : map f (𝓝 a) = 𝓝 (f a) :=
-  (hf.hasStrictFderivAt_equiv hf').map_nhds_eq_of_equiv
+  (hf.hasStrictFDerivAt_equiv hf').map_nhds_eq_of_equiv
 #align has_strict_deriv_at.map_nhds_eq HasStrictDerivAt.map_nhds_eq
 
 theorem to_localInverse : HasStrictDerivAt (hf.localInverse f f' a hf') f'⁻¹ (f a) :=
-  (hf.hasStrictFderivAt_equiv hf').to_localInverse
+  (hf.hasStrictFDerivAt_equiv hf').to_localInverse
 #align has_strict_deriv_at.to_local_inverse HasStrictDerivAt.to_localInverse
 
 theorem to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
     HasStrictDerivAt g f'⁻¹ (f a) :=
-  (hf.hasStrictFderivAt_equiv hf').to_local_left_inverse hg
+  (hf.hasStrictFDerivAt_equiv hf').to_local_left_inverse hg
 #align has_strict_deriv_at.to_local_left_inverse HasStrictDerivAt.to_local_left_inverse
 
 end HasStrictDerivAt
@@ -837,7 +837,7 @@ variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
 
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
 derivative at `a`, returns a `local_homeomorph` with `to_fun = f` and `a ∈ source`. -/
-def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a)
+def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : LocalHomeomorph E' F' :=
   (hf.has_strict_fderiv_at' hf' hn).toLocalHomeomorph f
 #align cont_diff_at.to_local_homeomorph ContDiffAt.toLocalHomeomorph
@@ -846,32 +846,32 @@ variable {f}
 
 @[simp]
 theorem toLocalHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     (hf.toLocalHomeomorph f hf' hn : E' → F') = f :=
   rfl
 #align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toLocalHomeomorph_coe
 
 theorem mem_toLocalHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     a ∈ (hf.toLocalHomeomorph f hf' hn).source :=
   (hf.has_strict_fderiv_at' hf' hn).mem_toLocalHomeomorph_source
 #align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toLocalHomeomorph_source
 
 theorem image_mem_toLocalHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     f a ∈ (hf.toLocalHomeomorph f hf' hn).target :=
   (hf.has_strict_fderiv_at' hf' hn).image_mem_toLocalHomeomorph_target
 #align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toLocalHomeomorph_target
 
 /-- Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
 at `a`, returns a function that is locally inverse to `f`. -/
-def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a)
+def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
     (hn : 1 ≤ n) : F' → E' :=
   (hf.has_strict_fderiv_at' hf' hn).localInverse f f' a
 #align cont_diff_at.local_inverse ContDiffAt.localInverse
 
 theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
   (hf.has_strict_fderiv_at' hf' hn).localInverse_apply_image
 #align cont_diff_at.local_inverse_apply_image ContDiffAt.localInverse_apply_image
 
@@ -879,7 +879,7 @@ theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
 at `a`, the inverse function (produced by `cont_diff.to_local_homeomorph`) is
 also `cont_diff`. -/
 theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFderivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
+    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
     ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) :=
   by
   have := hf.local_inverse_apply_image hf' hn

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -4,14 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.inverse
-! leanprover-community/mathlib commit b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.ContDiff
-import Mathbin.Tactic.RingExp
 import Mathbin.Analysis.NormedSpace.Banach
-import Mathbin.Topology.LocalHomeomorph
 
 /-!
 # Inverse function theorem

mathlib commit https://github.com/leanprover-community/mathlib/commit/172bf2812857f5e56938cc148b7a539f52f84ca9

@@ -484,7 +484,7 @@ theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F)
     (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toLocalEquiv hc).symm (f '' s) :=
   by
   apply continuousOn_iff_continuous_restrict.2
-  refine' ((hf.antilipschitz hc).to_right_inv_on' _ (hf.to_local_equiv hc).right_inv').Continuous
+  refine' ((hf.antilipschitz hc).to_rightInvOn' _ (hf.to_local_equiv hc).right_inv').Continuous
   exact fun x hx => (hf.to_local_equiv hc).map_target hx
 #align approximates_linear_on.inverse_continuous_on ApproximatesLinearOn.inverse_continuousOn
 

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

@@ -435,7 +435,7 @@ protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜]
   cases' hc with hE hc
   · haveI : Subsingleton s := ⟨fun x y => Subtype.eq <| @Subsingleton.elim _ hE _ _⟩
     exact AntilipschitzWith.of_subsingleton
-  convert (f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
+  convert(f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
   simp [restrict]
 #align approximates_linear_on.antilipschitz ApproximatesLinearOn.antilipschitz
 

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

@@ -264,7 +264,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
         simp only [ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
           sub_add_cancel]
-      _ ≤ c * ‖z + v - z‖ := hf _ (hε hgz) _ (hε hz)
+      _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) :=
         by
         apply mul_le_mul_of_nonneg_left _ (NNReal.coe_nonneg c)
@@ -322,12 +322,12 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
           dist (f b) y :=
       calc
         dist (g (u n)) b ≤ dist (g (u n)) (u n) + dist (u n) b := dist_triangle _ _ _
-        _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := add_le_add (A _) le_rfl
+        _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := (add_le_add (A _) le_rfl)
         _ ≤
             f'symm.nnnorm * ((c * f'symm.nnnorm) ^ n * dist (f b) y) +
               f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) *
                 dist (f b) y :=
-          add_le_add (mul_le_mul_of_nonneg_left IH.1 (NNReal.coe_nonneg _)) IH.2
+          (add_le_add (mul_le_mul_of_nonneg_left IH.1 (NNReal.coe_nonneg _)) IH.2)
         _ =
             f'symm.nnnorm * (1 - (c * f'symm.nnnorm) ^ n.succ) / (1 - c * f'symm.nnnorm) *
               dist (f b) y :=
@@ -340,7 +340,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       dist (f (g (u n))) y ≤ c * f'symm.nnnorm * dist (f (u n)) y :=
         B _ (C n _ IH.2) (C n.succ _ Ign)
       _ ≤ c * f'symm.nnnorm * ((c * f'symm.nnnorm) ^ n * dist (f b) y) :=
-        mul_le_mul_of_nonneg_left IH.1 (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _))
+        (mul_le_mul_of_nonneg_left IH.1 (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _)))
       _ = (c * f'symm.nnnorm) ^ n.succ * dist (f b) y := by ring
       
   -- Deduce from the inductive bound that `uₙ` is a Cauchy sequence, therefore converging.
@@ -351,9 +351,9 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       intro n
       calc
         dist (u n) (u (n + 1)) = dist (g (u n)) (u n) := by rw [usucc, dist_comm]
-        _ ≤ f'symm.nnnorm * dist (f (u n)) y := A _
+        _ ≤ f'symm.nnnorm * dist (f (u n)) y := (A _)
         _ ≤ f'symm.nnnorm * ((c * f'symm.nnnorm) ^ n * dist (f b) y) :=
-          mul_le_mul_of_nonneg_left (D n).1 (NNReal.coe_nonneg _)
+          (mul_le_mul_of_nonneg_left (D n).1 (NNReal.coe_nonneg _))
         _ = f'symm.nnnorm * dist (f b) y * (c * f'symm.nnnorm) ^ n := by ring
         
     cauchySeq_of_le_geometric _ _ Icf' this
@@ -396,7 +396,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
       (subset.refl _)
 #align approximates_linear_on.open_image ApproximatesLinearOn.open_image
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) :=
   by
@@ -506,7 +506,7 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
       congr 2
       simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub]
       abel
-    _ ≤ N * (c * ‖y' - x'‖) := mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _)
+    _ ≤ N * (c * ‖y' - x'‖) := (mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _))
     _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) :=
       by
       apply_rules [mul_le_mul_of_nonneg_left, NNReal.coe_nonneg]

mathlib commit https://github.com/leanprover-community/mathlib/commit/271bf175e6c51b8d31d6c0107b7bb4a967c7277e

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.calculus.inverse
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -643,7 +643,7 @@ theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f
   let f'symm := f'.nonlinear_right_inverse_of_surjective h
   set c : ℝ≥0 := f'symm.nnnorm⁻¹ / 2 with hc
   have f'symm_pos : 0 < f'symm.nnnorm := f'.nonlinear_right_inverse_of_surjective_nnnorm_pos h
-  have cpos : 0 < c := by simp [hc, NNReal.half_pos, NNReal.inv_pos, f'symm_pos]
+  have cpos : 0 < c := by simp [hc, half_pos, inv_pos, f'symm_pos]
   obtain ⟨s, s_nhds, hs⟩ : ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c :=
     hf.approximates_deriv_on_nhds (Or.inr cpos)
   apply hs.map_nhds_eq f'symm s_nhds (Or.inr (NNReal.half_lt_self _))
@@ -660,8 +660,7 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFderivAt f (f' : E →L[
   exact fun s t => ApproximatesLinearOn.mono_set
   exact
     hf.approximates_deriv_on_nhds <|
-      f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
-        NNReal.half_pos <| NNReal.inv_pos.2 <| hf'
+      f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
 #align has_strict_fderiv_at.approximates_deriv_on_open_nhds HasStrictFderivAt.approximates_deriv_on_open_nhds
 
 include cs
@@ -676,7 +675,7 @@ def toLocalHomeomorph (hf : HasStrictFderivAt f (f' : E →L[𝕜] F) a) : Local
   ApproximatesLinearOn.toLocalHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).snd
     (f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
-      NNReal.half_lt_self <| ne_of_gt <| NNReal.inv_pos.2 <| hf')
+      NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).fst.2
 #align has_strict_fderiv_at.to_local_homeomorph HasStrictFderivAt.toLocalHomeomorph
 

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

• consistently use the YYYY-MM-DD format
• when easily possible, put the date on the same line as the deprecated attribute
• when easily possible, format the entire declaration on the same line

Why these changes?

• consistency makes it easier for tools to parse this information
• compactness: I don't see a good reason for these declarations taking up more space than needed; as I understand it, deprecated lemmas are not supposed to be used in mathlib anyway
• putting the date on the same line as the attribute makes it easier to discover un-dated deprecations; they also ease writing a tool to replace these by a machine-readable version using leanprover/lean4#3968

@@ -217,5 +217,5 @@ theorem isOpenMap_of_hasStrictFDerivAt_equiv [CompleteSpace E] {f : E → F} {f'
     (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
 #align open_map_of_strict_fderiv_equiv isOpenMap_of_hasStrictFDerivAt_equiv
-@[deprecated] alias open_map_of_strict_fderiv_equiv :=
-  isOpenMap_of_hasStrictFDerivAt_equiv -- 2024-03-23
+@[deprecated] -- 2024-03-23
+alias open_map_of_strict_fderiv_equiv := isOpenMap_of_hasStrictFDerivAt_equiv

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

@@ -194,7 +194,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
         simp only [v, ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel_left,
           sub_add_cancel]
-      _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
+      _ ≤ c * ‖z + v - z‖ := hf _ (hε hgz) _ (hε hz)
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) := by
         gcongr
         simpa [dist_eq_norm'] using f'symm.bound (y - f z)
@@ -233,7 +233,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         (1 - c * f'symm.nnnorm) * dist (f b) y :=
       calc
         dist (g (u n)) b ≤ dist (g (u n)) (u n) + dist (u n) b := dist_triangle _ _ _
-        _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := (add_le_add (A _) le_rfl)
+        _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := add_le_add (A _) le_rfl
         _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) +
               f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) *
                 dist (f b) y := by gcongr; exact IH.1; exact IH.2
@@ -253,7 +253,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     refine cauchySeq_of_le_geometric _ (↑f'symm.nnnorm * dist (f b) y) Icf' fun n ↦ ?_
     calc
       dist (u n) (u (n + 1)) = dist (g (u n)) (u n) := by rw [usucc, dist_comm]
-      _ ≤ f'symm.nnnorm * dist (f (u n)) y := (A _)
+      _ ≤ f'symm.nnnorm * dist (f (u n)) y := A _
       _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) := by
         gcongr
         exact (D n).1
@@ -386,7 +386,7 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
       congr 2
       simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub]
       abel
-    _ ≤ N * (c * ‖y' - x'‖) := (mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _))
+    _ ≤ N * (c * ‖y' - x'‖) := mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _)
     _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by
       gcongr
       rw [← dist_eq_norm, ← dist_eq_norm]

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

@@ -77,7 +77,7 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   cases' hc with hE hc
   · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => _⟩
     simp [@Subsingleton.elim E hE x y]
-  have := hf.definition hc
+  have := hf.def hc
   rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩

• open_map_of_strict_deriv -> isOpenMap_of_hasStrictDerivAt
• open_map_of_strict_fderiv_equiv -> isOpenMap_of_hasStrictFderivAt_equiv These lemmas have conclusion isOpenMap, hence should be named accordingly.

@@ -213,7 +213,9 @@ theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a)
 end HasStrictFDerivAt
 
 /-- If a function has an invertible strict derivative at all points, then it is an open map. -/
-theorem open_map_of_strict_fderiv_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
+theorem isOpenMap_of_hasStrictFDerivAt_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
     (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
   isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
-#align open_map_of_strict_fderiv_equiv open_map_of_strict_fderiv_equiv
+#align open_map_of_strict_fderiv_equiv isOpenMap_of_hasStrictFDerivAt_equiv
+@[deprecated] alias open_map_of_strict_fderiv_equiv :=
+  isOpenMap_of_hasStrictFDerivAt_equiv -- 2024-03-23

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 | |

@@ -117,7 +117,7 @@ theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) :
 
 protected theorem lipschitz (hf : ApproximatesLinearOn f f' s c) :
     LipschitzWith (‖f'‖₊ + c) (s.restrict f) := by
-  simpa only [restrict_apply, add_sub_cancel'_right] using
+  simpa only [restrict_apply, add_sub_cancel] using
     (f'.lipschitz.restrict s).add hf.lipschitz_sub
 #align approximates_linear_on.lipschitz ApproximatesLinearOn.lipschitz
 
@@ -178,7 +178,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   -- First bound: if `f z` is close to `y`, then `g z` is close to `z` (i.e., almost a fixed point).
   have A : ∀ z, dist (g z) z ≤ f'symm.nnnorm * dist (f z) y := by
     intro z
-    rw [dist_eq_norm, hg, add_sub_cancel', dist_eq_norm']
+    rw [dist_eq_norm, hg, add_sub_cancel_left, dist_eq_norm']
     exact f'symm.bound _
   -- Second bound: if `z` and `g z` are in the set with good control, then `f (g z)` becomes closer
   -- to `y` than `f z` was (this uses the linear approximation property, and is the reason for the
@@ -192,7 +192,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       dist (f (g z)) y = ‖f (z + v) - y‖ := by rw [dist_eq_norm]
       _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ := by congr 1; abel
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
-        simp only [v, ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
+        simp only [v, ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel_left,
           sub_add_cancel]
       _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) := by

This will become an error in 2024-03-16 nightly, possibly not permanently.

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -77,7 +77,7 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   cases' hc with hE hc
   · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => _⟩
     simp [@Subsingleton.elim E hE x y]
-  have := hf.def hc
+  have := hf.definition hc
   rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩

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)

@@ -47,15 +47,10 @@ open scoped Topology Classical NNReal
 noncomputable section
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-
 variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
-
 variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
-
 variable {ε : ℝ}
 
 open Asymptotics Filter Metric Set

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)

@@ -52,15 +52,10 @@ open scoped Topology Classical NNReal
 noncomputable section
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-
 variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
-
 variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
-
 variable {ε : ℝ}
 
 open Filter Metric Set

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -286,7 +286,7 @@ theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRi
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) := by
   cases' hc with hE hc
   · exact isOpen_discrete _
-  simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, ball_image_iff] at hs ⊢
+  simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, forall_mem_image] at hs ⊢
   intro x hx
   rcases hs x hx with ⟨ε, ε0, hε⟩
   refine' ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, _⟩

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

@@ -197,7 +197,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
       dist (f (g z)) y = ‖f (z + v) - y‖ := by rw [dist_eq_norm]
       _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ := by congr 1; abel
       _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by
-        simp only [ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
+        simp only [v, ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel',
           sub_add_cancel]
       _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) := by

Split the 2300-line behemoth OperatorNorm.lean into 8 smaller files, of which the largest is 600 lines.

@@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Sébastien Gouëzel
 -/
 import Mathlib.Analysis.NormedSpace.Banach
+import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
+import Mathlib.Topology.PartialHomeomorph
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
 /-!
@@ -61,7 +63,7 @@ variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
 
 variable {ε : ℝ}
 
-open Asymptotics Filter Metric Set
+open Filter Metric Set
 
 open ContinuousLinearMap (id)
 

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>

@@ -252,8 +252,8 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         apply IH.1
       _ = ((c : ℝ) * f'symm.nnnorm) ^ n.succ * dist (f b) y := by simp only [pow_succ']; ring
   -- Deduce from the inductive bound that `uₙ` is a Cauchy sequence, therefore converging.
-  have : CauchySeq u
-  · refine cauchySeq_of_le_geometric _ (↑f'symm.nnnorm * dist (f b) y) Icf' fun n ↦ ?_
+  have : CauchySeq u := by
+    refine cauchySeq_of_le_geometric _ (↑f'symm.nnnorm * dist (f b) y) Icf' fun n ↦ ?_
     calc
       dist (u n) (u (n + 1)) = dist (g (u n)) (u n) := by rw [usucc, dist_comm]
       _ ≤ f'symm.nnnorm * dist (f (u n)) y := (A _)

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.

@@ -276,7 +276,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   have T2 : Tendsto (f ∘ u) atTop (𝓝 y) := by
     rw [tendsto_iff_dist_tendsto_zero]
     refine' squeeze_zero (fun _ => dist_nonneg) (fun n => (D n).1) _
-    simpa using (tendsto_pow_atTop_nhds_0_of_lt_1 (by positivity) Icf').mul tendsto_const_nhds
+    simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one (by positivity) Icf').mul tendsto_const_nhds
   exact tendsto_nhds_unique T1 T2
 #align approximates_linear_on.surj_on_closed_ball_of_nonlinear_right_inverse ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse
 

Almost all of them should speak about partial homeomorphisms instead. In two cases, I decided removing the "local" was clearer than adding "partial".

Follow-up to #8982; complements #9238.

@@ -355,7 +355,7 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
     exact fun R h y hy => h hy
 #align approximates_linear_on.surjective ApproximatesLinearOn.surjective
 
-/-- A map approximating a linear equivalence on a set defines a local equivalence on this set.
+/-- A map approximating a linear equivalence on a set defines a partial equivalence on this set.
 Should not be used outside of this file, because it is superseded by `toPartialHomeomorph` below.
 
 This is a first step towards the inverse function. -/
@@ -403,7 +403,7 @@ section
 variable (f s)
 
 /-- Given a function `f` that approximates a linear equivalence on an open set `s`,
-returns a local homeomorph with `toFun = f` and `source = s`. -/
+returns a partial homeomorphism with `toFun = f` and `source = s`. -/
 def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : PartialHomeomorph E F where
   toPartialEquiv := hf.toPartialEquiv hc

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.

@@ -152,7 +152,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
     SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) := by
   intro y hy
-  cases' le_or_lt (f'symm.nnnorm : ℝ)⁻¹ c with hc hc
+  rcases le_or_lt (f'symm.nnnorm : ℝ)⁻¹ c with hc | hc
   · refine' ⟨b, by simp [ε0], _⟩
     have : dist y (f b) ≤ 0 :=
       (mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)

The current name is misleading: there's no open set involved; it's just an equivalence between subsets of domain and target. zulip discussion

PEquiv is similarly named: this is fine, as they're different designs for the same concept.

Co-authored-by: Michael Rothgang <rothgami@math.hu-berlin.de>

@@ -18,9 +18,12 @@ When `f'` is onto, we show that `f` is locally onto.
 When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism
 between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toPartialHomeomorph` to
 be a `PartialHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
+between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toPartialHomeomorph` to
+be a `PartialHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
 
 Maps of this type naturally appear in the proof of the inverse function theorem (see next section),
 and `ApproximatesLinearOn.toPartialHomeomorph` will imply that the locally inverse function
+and `ApproximatesLinearOn.toPartialHomeomorph` will imply that the locally inverse function
 exists.
 
 We define this auxiliary notion to split the proof of the inverse function theorem into small
@@ -356,25 +359,25 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
 Should not be used outside of this file, because it is superseded by `toPartialHomeomorph` below.
 
 This is a first step towards the inverse function. -/
-def toLocalEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) : LocalEquiv E F :=
-  (hf.injOn hc).toLocalEquiv _ _
-#align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toLocalEquiv
+def toPartialEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) : PartialEquiv E F :=
+  (hf.injOn hc).toPartialEquiv _ _
+#align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toPartialEquiv
 
 /-- The inverse function is continuous on `f '' s`.
 Use properties of `PartialHomeomorph` instead. -/
 theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toLocalEquiv hc).symm (f '' s) := by
+    (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toPartialEquiv hc).symm (f '' s) := by
   apply continuousOn_iff_continuous_restrict.2
-  refine' ((hf.antilipschitz hc).to_rightInvOn' _ (hf.toLocalEquiv hc).right_inv').continuous
-  exact fun x hx => (hf.toLocalEquiv hc).map_target hx
+  refine' ((hf.antilipschitz hc).to_rightInvOn' _ (hf.toPartialEquiv hc).right_inv').continuous
+  exact fun x hx => (hf.toPartialEquiv hc).map_target hx
 #align approximates_linear_on.inverse_continuous_on ApproximatesLinearOn.inverse_continuousOn
 
 /-- The inverse function is approximated linearly on `f '' s` by `f'.symm`. -/
 theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) :
-    ApproximatesLinearOn (hf.toLocalEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
+    ApproximatesLinearOn (hf.toPartialEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
       (N * (N⁻¹ - c)⁻¹ * c) := fun x hx y hy ↦ by
-  set A := hf.toLocalEquiv hc
+  set A := hf.toPartialEquiv hc
   have Af : ∀ z, A z = f z := fun z => rfl
   rcases (mem_image _ _ _).1 hx with ⟨x', x's, rfl⟩
   rcases (mem_image _ _ _).1 hy with ⟨y', y's, rfl⟩
@@ -403,7 +406,7 @@ variable (f s)
 returns a local homeomorph with `toFun = f` and `source = s`. -/
 def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : PartialHomeomorph E F where
-  toLocalEquiv := hf.toLocalEquiv hc
+  toPartialEquiv := hf.toPartialEquiv hc
   open_source := hs
   open_target := hf.open_image f'.toNonlinearRightInverse hs <| by
     rwa [f'.toEquiv.subsingleton_congr] at hc

LocalHomeomorph evokes a "local homeomorphism": this is not what this means. Instead, this is a homeomorphism on an open set of the domain (extended to the whole space, by the junk value pattern). Hence, partial homeomorphism is more appropriate, and avoids confusion with IsLocallyHomeomorph.

A future PR will rename LocalEquiv to PartialEquiv.

Zulip discussion

@@ -15,12 +15,12 @@ In this file we prove the inverse function theorem. It says that if a map `f : E
 has an invertible strict derivative `f'` at `a`, then it is locally invertible,
 and the inverse function has derivative `f' ⁻¹`.
 
-We define `HasStrictFDerivAt.toLocalHomeomorph` that repacks a function `f`
-with a `hf : HasStrictFDerivAt f f' a`, `f' : E ≃L[𝕜] F`, into a `LocalHomeomorph`.
-The `toFun` of this `LocalHomeomorph` is defeq to `f`, so one can apply theorems
-about `LocalHomeomorph` to `hf.toLocalHomeomorph f`, and get statements about `f`.
+We define `HasStrictFDerivAt.toPartialHomeomorph` that repacks a function `f`
+with a `hf : HasStrictFDerivAt f f' a`, `f' : E ≃L[𝕜] F`, into a `PartialHomeomorph`.
+The `toFun` of this `PartialHomeomorph` is defeq to `f`, so one can apply theorems
+about `PartialHomeomorph` to `hf.toPartialHomeomorph f`, and get statements about `f`.
 
-Then we define `HasStrictFDerivAt.localInverse` to be the `invFun` of this `LocalHomeomorph`,
+Then we define `HasStrictFDerivAt.localInverse` to be the `invFun` of this `PartialHomeomorph`,
 and prove two versions of the inverse function theorem:
 
 * `HasStrictFDerivAt.to_localInverse`: if `f` has an invertible derivative `f'` at `a` in the
@@ -69,7 +69,7 @@ open ContinuousLinearMap (id)
 Let `f : E → F` be a map defined on a complete vector
 space `E`. Assume that `f` has an invertible derivative `f' : E ≃L[𝕜] F` at `a : E` in the strict
 sense. Then `f` approximates `f'` in the sense of `ApproximatesLinearOn` on an open neighborhood
-of `a`, and we can apply `ApproximatesLinearOn.toLocalHomeomorph` to construct the inverse
+of `a`, and we can apply `ApproximatesLinearOn.toPartialHomeomorph` to construct the inverse
 function. -/
 
 namespace HasStrictFDerivAt
@@ -115,39 +115,39 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[
 
 variable (f)
 
-/-- Given a function with an invertible strict derivative at `a`, returns a `LocalHomeomorph`
+/-- Given a function with an invertible strict derivative at `a`, returns a `PartialHomeomorph`
 with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
 The other part `HasStrictFDerivAt.to_localInverse` states that the inverse function
-of this `LocalHomeomorph` has derivative `f'.symm`. -/
-def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : LocalHomeomorph E F :=
-  ApproximatesLinearOn.toLocalHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
+of this `PartialHomeomorph` has derivative `f'.symm`. -/
+def toPartialHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : PartialHomeomorph E F :=
+  ApproximatesLinearOn.toPartialHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).2.2
     (f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
       NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
     (Classical.choose_spec hf.approximates_deriv_on_open_nhds).2.1
-#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toLocalHomeomorph
+#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toPartialHomeomorph
 
 variable {f}
 
 @[simp]
-theorem toLocalHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    (hf.toLocalHomeomorph f : E → F) = f :=
+theorem toPartialHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    (hf.toPartialHomeomorph f : E → F) = f :=
   rfl
-#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toLocalHomeomorph_coe
+#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toPartialHomeomorph_coe
 
-theorem mem_toLocalHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    a ∈ (hf.toLocalHomeomorph f).source :=
+theorem mem_toPartialHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    a ∈ (hf.toPartialHomeomorph f).source :=
   (Classical.choose_spec hf.approximates_deriv_on_open_nhds).1
-#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toLocalHomeomorph_source
+#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toPartialHomeomorph_source
 
-theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    f a ∈ (hf.toLocalHomeomorph f).target :=
-  (hf.toLocalHomeomorph f).map_source hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toLocalHomeomorph_target
+theorem image_mem_toPartialHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
+    f a ∈ (hf.toPartialHomeomorph f).target :=
+  (hf.toPartialHomeomorph f).map_source hf.mem_toPartialHomeomorph_source
+#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toPartialHomeomorph_target
 
 theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     map f (𝓝 a) = 𝓝 (f a) :=
-  (hf.toLocalHomeomorph f).map_nhds_eq hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).map_nhds_eq hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFDerivAt.map_nhds_eq_of_equiv
 
 variable (f f' a)
@@ -155,19 +155,19 @@ variable (f f' a)
 /-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
 to `f`. -/
 def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
-  (hf.toLocalHomeomorph f).symm
+  (hf.toPartialHomeomorph f).symm
 #align has_strict_fderiv_at.local_inverse HasStrictFDerivAt.localInverse
 
 variable {f f' a}
 
 theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    hf.localInverse f _ _ = (hf.toLocalHomeomorph f).symm :=
+    hf.localInverse f _ _ = (hf.toPartialHomeomorph f).symm :=
   rfl
 #align has_strict_fderiv_at.local_inverse_def HasStrictFDerivAt.localInverse_def
 
 theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x :=
-  (hf.toLocalHomeomorph f).eventually_left_inverse hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).eventually_left_inverse hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.eventually_left_inverse HasStrictFDerivAt.eventually_left_inverse
 
 @[simp]
@@ -178,30 +178,30 @@ theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F)
 
 theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y :=
-  (hf.toLocalHomeomorph f).eventually_right_inverse' hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).eventually_right_inverse' hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.eventually_right_inverse HasStrictFDerivAt.eventually_right_inverse
 
 theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     ContinuousAt (hf.localInverse f f' a) (f a) :=
-  (hf.toLocalHomeomorph f).continuousAt_symm hf.image_mem_toLocalHomeomorph_target
+  (hf.toPartialHomeomorph f).continuousAt_symm hf.image_mem_toPartialHomeomorph_target
 #align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFDerivAt.localInverse_continuousAt
 
 theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 a) :=
-  (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
+  (hf.toPartialHomeomorph f).tendsto_symm hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_tendsto HasStrictFDerivAt.localInverse_tendsto
 
 theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y :=
   eventuallyEq_of_left_inv_of_right_inv hg hf.eventually_right_inverse <|
-    (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
+    (hf.toPartialHomeomorph f).tendsto_symm hf.mem_toPartialHomeomorph_source
 #align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
 
 /-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
 then the inverse function `hf.localInverse f` has derivative `f'.symm` at `f a`. -/
 theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
-  (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
+  (hf.toPartialHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toPartialHomeomorph_target <| by
     simpa [← localInverse_def] using hf
 #align has_strict_fderiv_at.to_local_inverse HasStrictFDerivAt.to_localInverse
 

LocalHomeomorph evokes a "local homeomorphism": this is not what this means. Instead, this is a homeomorphism on an open set of the domain (extended to the whole space, by the junk value pattern). Hence, partial homeomorphism is more appropriate, and avoids confusion with IsLocallyHomeomorph.

A future PR will rename LocalEquiv to PartialEquiv.

Zulip discussion

@@ -16,11 +16,11 @@ behave like `f a + f' (x - a)` near each `a ∈ s`.
 When `f'` is onto, we show that `f` is locally onto.
 
 When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism
-between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toLocalHomeomorph` to
-be a `LocalHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
+between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toPartialHomeomorph` to
+be a `PartialHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
 
 Maps of this type naturally appear in the proof of the inverse function theorem (see next section),
-and `ApproximatesLinearOn.toLocalHomeomorph` will imply that the locally inverse function
+and `ApproximatesLinearOn.toPartialHomeomorph` will imply that the locally inverse function
 exists.
 
 We define this auxiliary notion to split the proof of the inverse function theorem into small
@@ -353,7 +353,7 @@ protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f'
 #align approximates_linear_on.surjective ApproximatesLinearOn.surjective
 
 /-- A map approximating a linear equivalence on a set defines a local equivalence on this set.
-Should not be used outside of this file, because it is superseded by `toLocalHomeomorph` below.
+Should not be used outside of this file, because it is superseded by `toPartialHomeomorph` below.
 
 This is a first step towards the inverse function. -/
 def toLocalEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
@@ -361,7 +361,8 @@ def toLocalEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   (hf.injOn hc).toLocalEquiv _ _
 #align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toLocalEquiv
 
-/-- The inverse function is continuous on `f '' s`. Use properties of `LocalHomeomorph` instead. -/
+/-- The inverse function is continuous on `f '' s`.
+Use properties of `PartialHomeomorph` instead. -/
 theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toLocalEquiv hc).symm (f '' s) := by
   apply continuousOn_iff_continuous_restrict.2
@@ -400,41 +401,42 @@ variable (f s)
 
 /-- Given a function `f` that approximates a linear equivalence on an open set `s`,
 returns a local homeomorph with `toFun = f` and `source = s`. -/
-def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : LocalHomeomorph E F where
+def toPartialHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : PartialHomeomorph E F where
   toLocalEquiv := hf.toLocalEquiv hc
   open_source := hs
   open_target := hf.open_image f'.toNonlinearRightInverse hs <| by
     rwa [f'.toEquiv.subsingleton_congr] at hc
   continuousOn_toFun := hf.continuousOn
   continuousOn_invFun := hf.inverse_continuousOn hc
-#align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
+#align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toPartialHomeomorph
 
 @[simp]
-theorem toLocalHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+theorem toPartialHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs : E → F) = f :=
+    (hf.toPartialHomeomorph f s hc hs : E → F) = f :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toLocalHomeomorph_coe
+#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toPartialHomeomorph_coe
 
 @[simp]
-theorem toLocalHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : (hf.toLocalHomeomorph f s hc hs).source = s :=
+theorem toPartialHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
+    (hf.toPartialHomeomorph f s hc hs).source = s :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toLocalHomeomorph_source
+#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toPartialHomeomorph_source
 
 @[simp]
-theorem toLocalHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+theorem toPartialHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs).target = f '' s :=
+    (hf.toPartialHomeomorph f s hc hs).target = f '' s :=
   rfl
-#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toLocalHomeomorph_target
+#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toPartialHomeomorph_target
 
 /-- A function `f` that approximates a linear equivalence on the whole space is a homeomorphism. -/
 def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
     (hc : Subsingleton E ∨ c < N⁻¹) : E ≃ₜ F := by
-  refine' (hf.toLocalHomeomorph _ _ hc isOpen_univ).toHomeomorphOfSourceEqUnivTargetEqUniv rfl _
-  rw [toLocalHomeomorph_target, image_univ, range_iff_surjective]
+  refine' (hf.toPartialHomeomorph _ _ hc isOpen_univ).toHomeomorphOfSourceEqUnivTargetEqUniv rfl _
+  rw [toPartialHomeomorph_target, image_univ, range_iff_surjective]
   exact hf.surjective hc
 #align approximates_linear_on.to_homeomorph ApproximatesLinearOn.toHomeomorph
 
@@ -442,7 +444,7 @@ end
 
 theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
-    closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toLocalHomeomorph f s hc hs).target :=
+    closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toPartialHomeomorph f s hc hs).target :=
   (hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse ε0 hε).mono hε
     Subset.rfl
 #align approximates_linear_on.closed_ball_subset_target ApproximatesLinearOn.closedBall_subset_target

@@ -1,64 +1,45 @@
 /-
 Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
+Authors: Yury Kudryashov, Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
-import Mathlib.Analysis.Calculus.ContDiff.IsROrC
 import Mathlib.Analysis.NormedSpace.Banach
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
-
 /-!
-# Inverse function theorem
-
-In this file we prove the inverse function theorem. It says that if a map `f : E → F`
-has an invertible strict derivative `f'` at `a`, then it is locally invertible,
-and the inverse function has derivative `f' ⁻¹`.
+# Non-linear maps close to affine maps
 
-We define `HasStrictDerivAt.toLocalHomeomorph` that repacks a function `f`
-with a `hf : HasStrictFDerivAt f f' a`, `f' : E ≃L[𝕜] F`, into a `LocalHomeomorph`.
-The `toFun` of this `LocalHomeomorph` is defeq to `f`, so one can apply theorems
-about `LocalHomeomorph` to `hf.toLocalHomeomorph f`, and get statements about `f`.
+In this file we study a map `f` such that `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` on an open set
+`s`, where `f' : E →L[𝕜] F` is a continuous linear map and `c` is suitably small. Maps of this type
+behave like `f a + f' (x - a)` near each `a ∈ s`.
 
-Then we define `HasStrictFDerivAt.localInverse` to be the `invFun` of this `LocalHomeomorph`,
-and prove two versions of the inverse function theorem:
+When `f'` is onto, we show that `f` is locally onto.
 
-* `HasStrictFDerivAt.to_localInverse`: if `f` has an invertible derivative `f'` at `a` in the
-  strict sense (`hf`), then `hf.localInverse f f' a` has derivative `f'.symm` at `f a` in the
-  strict sense;
+When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism
+between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toLocalHomeomorph` to
+be a `LocalHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
 
-* `HasStrictFDerivAt.to_local_left_inverse`: if `f` has an invertible derivative `f'` at `a` in
-  the strict sense and `g` is locally left inverse to `f` near `a`, then `g` has derivative
-  `f'.symm` at `f a` in the strict sense.
+Maps of this type naturally appear in the proof of the inverse function theorem (see next section),
+and `ApproximatesLinearOn.toLocalHomeomorph` will imply that the locally inverse function
+exists.
 
-In the one-dimensional case we reformulate these theorems in terms of `HasStrictDerivAt` and
-`f'⁻¹`.
+We define this auxiliary notion to split the proof of the inverse function theorem into small
+lemmas. This approach makes it possible
 
-We also reformulate the theorems in terms of `ContDiff`, to give that `C^k` (respectively,
-smooth) inputs give `C^k` (smooth) inverses.  These versions require that continuous
-differentiability implies strict differentiability; this is false over a general field, true over
-`ℝ` or `ℂ` and implemented here assuming `IsROrC 𝕂`.
+- to prove a lower estimate on the size of the domain of the inverse function;
 
-Some related theorems, providing the derivative and higher regularity assuming that we already know
-the inverse function, are formulated in the `Analysis/Calculus/FDeriv` and `Analysis/Calculus/Deriv`
-folders, and in `ContDiff.lean`.
+- to reuse parts of the proofs in the case if a function is not strictly differentiable. E.g., for a
+  function `f : E × F → G` with estimates on `f x y₁ - f x y₂` but not on `f x₁ y - f x₂ y`.
 
 ## Notations
 
-In the section about `ApproximatesLinearOn` we introduce some `local notation` to make formulas
-shorter:
+We introduce some `local notation` to make formulas shorter:
 
 * by `N` we denote `‖f'⁻¹‖`;
 * by `g` we denote the auxiliary contracting map `x ↦ x + f'.symm (y - f x)` used to prove that
   `{x | f x = y}` is nonempty.
-
-## Tags
-
-derivative, strictly differentiable, continuously differentiable, smooth, inverse function
 -/
 
-
 open Function Set Filter Metric
 
 open scoped Topology Classical NNReal
@@ -81,33 +62,6 @@ open Asymptotics Filter Metric Set
 
 open ContinuousLinearMap (id)
 
-/-!
-### Non-linear maps close to affine maps
-
-In this section we study a map `f` such that `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` on an open set
-`s`, where `f' : E →L[𝕜] F` is a continuous linear map and `c` is suitably small. Maps of this type
-behave like `f a + f' (x - a)` near each `a ∈ s`.
-
-When `f'` is onto, we show that `f` is locally onto.
-
-When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism
-between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toLocalHomeomorph` to
-be a `LocalHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`.
-
-Maps of this type naturally appear in the proof of the inverse function theorem (see next section),
-and `ApproximatesLinearOn.toLocalHomeomorph` will imply that the locally inverse function
-exists.
-
-We define this auxiliary notion to split the proof of the inverse function theorem into small
-lemmas. This approach makes it possible
-
-- to prove a lower estimate on the size of the domain of the inverse function;
-
-- to reuse parts of the proofs in the case if a function is not strictly differentiable. E.g., for a
-  function `f : E × F → G` with estimates on `f x y₁ - f x y₂` but not on `f x₁ y - f x₂ y`.
--/
-
-
 /-- We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`,
 if `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` whenever `x, y ∈ s`.
 
@@ -484,31 +438,6 @@ def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
   exact hf.surjective hc
 #align approximates_linear_on.to_homeomorph ApproximatesLinearOn.toHomeomorph
 
-/-- In a real vector space, a function `f` that approximates a linear equivalence on a subset `s`
-can be extended to a homeomorphism of the whole space. -/
-theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {s : Set E}
-    {f : E → F} {f' : E ≃L[ℝ] F} {c : ℝ≥0} (hf : ApproximatesLinearOn f (f' : E →L[ℝ] F) s c)
-    (hc : Subsingleton E ∨ lipschitzExtensionConstant F * c < ‖(f'.symm : F →L[ℝ] E)‖₊⁻¹) :
-    ∃ g : E ≃ₜ F, EqOn f g s := by
-  -- the difference `f - f'` is Lipschitz on `s`. It can be extended to a Lipschitz function `u`
-  -- on the whole space, with a slightly worse Lipschitz constant. Then `f' + u` will be the
-  -- desired homeomorphism.
-  obtain ⟨u, hu, uf⟩ :
-    ∃ u : E → F, LipschitzWith (lipschitzExtensionConstant F * c) u ∧ EqOn (f - ⇑f') u s :=
-    hf.lipschitzOnWith.extend_finite_dimension
-  let g : E → F := fun x => f' x + u x
-  have fg : EqOn f g s := fun x hx => by simp_rw [← uf hx, Pi.sub_apply, add_sub_cancel'_right]
-  have hg : ApproximatesLinearOn g (f' : E →L[ℝ] F) univ (lipschitzExtensionConstant F * c) := by
-    apply LipschitzOnWith.approximatesLinearOn
-    rw [lipschitzOn_univ]
-    convert hu
-    ext x
-    simp only [add_sub_cancel', ContinuousLinearEquiv.coe_coe, Pi.sub_apply]
-  haveI : FiniteDimensional ℝ E := f'.symm.finiteDimensional
-  exact ⟨hg.toHomeomorph g hc, fg⟩
-#align approximates_linear_on.exists_homeomorph_extension ApproximatesLinearOn.exists_homeomorph_extension
-
 end
 
 theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
@@ -519,278 +448,3 @@ theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜]
 #align approximates_linear_on.closed_ball_subset_target ApproximatesLinearOn.closedBall_subset_target
 
 end ApproximatesLinearOn
-
-/-!
-### Inverse function theorem
-
-Now we prove the inverse function theorem. Let `f : E → F` be a map defined on a complete vector
-space `E`. Assume that `f` has an invertible derivative `f' : E ≃L[𝕜] F` at `a : E` in the strict
-sense. Then `f` approximates `f'` in the sense of `ApproximatesLinearOn` on an open neighborhood
-of `a`, and we can apply `ApproximatesLinearOn.toLocalHomeomorph` to construct the inverse
-function. -/
-
-namespace HasStrictFDerivAt
-
-/-- If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`
-with constant `c` on some neighborhood of `a`. -/
-theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
-    (hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
-    ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c := by
-  cases' hc with hE hc
-  · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => _⟩
-    simp [@Subsingleton.elim E hE x y]
-  have := hf.def hc
-  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
-  rcases this with ⟨s, has, hs⟩
-  exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
-#align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds
-
-theorem map_nhds_eq_of_surj [CompleteSpace E] [CompleteSpace F] {f : E → F} {f' : E →L[𝕜] F} {a : E}
-    (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) (h : LinearMap.range f' = ⊤) :
-    map f (𝓝 a) = 𝓝 (f a) := by
-  let f'symm := f'.nonlinearRightInverseOfSurjective h
-  set c : ℝ≥0 := f'symm.nnnorm⁻¹ / 2 with hc
-  have f'symm_pos : 0 < f'symm.nnnorm := f'.nonlinearRightInverseOfSurjective_nnnorm_pos h
-  have cpos : 0 < c := by simp [hc, half_pos, inv_pos, f'symm_pos]
-  obtain ⟨s, s_nhds, hs⟩ : ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c :=
-    hf.approximates_deriv_on_nhds (Or.inr cpos)
-  apply hs.map_nhds_eq f'symm s_nhds (Or.inr (NNReal.half_lt_self _))
-  simp [ne_of_gt f'symm_pos]
-#align has_strict_fderiv_at.map_nhds_eq_of_surj HasStrictFDerivAt.map_nhds_eq_of_surj
-
-variable [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E}
-
-theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    ∃ s : Set E, a ∈ s ∧ IsOpen s ∧
-      ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) := by
-  simp only [← and_assoc]
-  refine ((nhds_basis_opens a).exists_iff fun s t => ApproximatesLinearOn.mono_set).1 ?_
-  exact
-    hf.approximates_deriv_on_nhds <|
-      f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
-#align has_strict_fderiv_at.approximates_deriv_on_open_nhds HasStrictFDerivAt.approximates_deriv_on_open_nhds
-
-variable (f)
-
-/-- Given a function with an invertible strict derivative at `a`, returns a `LocalHomeomorph`
-with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
-The other part `HasStrictFDerivAt.to_localInverse` states that the inverse function
-of this `LocalHomeomorph` has derivative `f'.symm`. -/
-def toLocalHomeomorph (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : LocalHomeomorph E F :=
-  ApproximatesLinearOn.toLocalHomeomorph f (Classical.choose hf.approximates_deriv_on_open_nhds)
-    (Classical.choose_spec hf.approximates_deriv_on_open_nhds).2.2
-    (f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' =>
-      NNReal.half_lt_self <| ne_of_gt <| inv_pos.2 hf')
-    (Classical.choose_spec hf.approximates_deriv_on_open_nhds).2.1
-#align has_strict_fderiv_at.to_local_homeomorph HasStrictFDerivAt.toLocalHomeomorph
-
-variable {f}
-
-@[simp]
-theorem toLocalHomeomorph_coe (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    (hf.toLocalHomeomorph f : E → F) = f :=
-  rfl
-#align has_strict_fderiv_at.to_local_homeomorph_coe HasStrictFDerivAt.toLocalHomeomorph_coe
-
-theorem mem_toLocalHomeomorph_source (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    a ∈ (hf.toLocalHomeomorph f).source :=
-  (Classical.choose_spec hf.approximates_deriv_on_open_nhds).1
-#align has_strict_fderiv_at.mem_to_local_homeomorph_source HasStrictFDerivAt.mem_toLocalHomeomorph_source
-
-theorem image_mem_toLocalHomeomorph_target (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    f a ∈ (hf.toLocalHomeomorph f).target :=
-  (hf.toLocalHomeomorph f).map_source hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.image_mem_to_local_homeomorph_target HasStrictFDerivAt.image_mem_toLocalHomeomorph_target
-
-theorem map_nhds_eq_of_equiv (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    map f (𝓝 a) = 𝓝 (f a) :=
-  (hf.toLocalHomeomorph f).map_nhds_eq hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.map_nhds_eq_of_equiv HasStrictFDerivAt.map_nhds_eq_of_equiv
-
-variable (f f' a)
-
-/-- Given a function `f` with an invertible derivative, returns a function that is locally inverse
-to `f`. -/
-def localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : F → E :=
-  (hf.toLocalHomeomorph f).symm
-#align has_strict_fderiv_at.local_inverse HasStrictFDerivAt.localInverse
-
-variable {f f' a}
-
-theorem localInverse_def (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    hf.localInverse f _ _ = (hf.toLocalHomeomorph f).symm :=
-  rfl
-#align has_strict_fderiv_at.local_inverse_def HasStrictFDerivAt.localInverse_def
-
-theorem eventually_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    ∀ᶠ x in 𝓝 a, hf.localInverse f f' a (f x) = x :=
-  (hf.toLocalHomeomorph f).eventually_left_inverse hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.eventually_left_inverse HasStrictFDerivAt.eventually_left_inverse
-
-@[simp]
-theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    hf.localInverse f f' a (f a) = a :=
-  hf.eventually_left_inverse.self_of_nhds
-#align has_strict_fderiv_at.local_inverse_apply_image HasStrictFDerivAt.localInverse_apply_image
-
-theorem eventually_right_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    ∀ᶠ y in 𝓝 (f a), f (hf.localInverse f f' a y) = y :=
-  (hf.toLocalHomeomorph f).eventually_right_inverse' hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.eventually_right_inverse HasStrictFDerivAt.eventually_right_inverse
-
-theorem localInverse_continuousAt (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    ContinuousAt (hf.localInverse f f' a) (f a) :=
-  (hf.toLocalHomeomorph f).continuousAt_symm hf.image_mem_toLocalHomeomorph_target
-#align has_strict_fderiv_at.local_inverse_continuous_at HasStrictFDerivAt.localInverse_continuousAt
-
-theorem localInverse_tendsto (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    Tendsto (hf.localInverse f f' a) (𝓝 <| f a) (𝓝 a) :=
-  (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.local_inverse_tendsto HasStrictFDerivAt.localInverse_tendsto
-
-theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
-    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : ∀ᶠ y in 𝓝 (f a), g y = localInverse f f' a hf y :=
-  eventuallyEq_of_left_inv_of_right_inv hg hf.eventually_right_inverse <|
-    (hf.toLocalHomeomorph f).tendsto_symm hf.mem_toLocalHomeomorph_source
-#align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
-
-/-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
-then the inverse function `hf.localInverse f` has derivative `f'.symm` at `f a`. -/
-theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
-    HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
-  (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by
-    simpa [← localInverse_def] using hf
-#align has_strict_fderiv_at.to_local_inverse HasStrictFDerivAt.to_localInverse
-
-/-- If `f : E → F` has an invertible derivative `f'` at `a` in the sense of strict differentiability
-and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
-
-For a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[CompleteSpace E]`
-see `of_local_left_inverse`.  -/
-theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
-    (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) (f a) :=
-  hf.to_localInverse.congr_of_eventuallyEq <| (hf.localInverse_unique hg).mono fun _ => Eq.symm
-#align has_strict_fderiv_at.to_local_left_inverse HasStrictFDerivAt.to_local_left_inverse
-
-end HasStrictFDerivAt
-
-/-- If a function has an invertible strict derivative at all points, then it is an open map. -/
-theorem open_map_of_strict_fderiv_equiv [CompleteSpace E] {f : E → F} {f' : E → E ≃L[𝕜] F}
-    (hf : ∀ x, HasStrictFDerivAt f (f' x : E →L[𝕜] F) x) : IsOpenMap f :=
-  isOpenMap_iff_nhds_le.2 fun x => (hf x).map_nhds_eq_of_equiv.ge
-#align open_map_of_strict_fderiv_equiv open_map_of_strict_fderiv_equiv
-
-/-!
-### Inverse function theorem, 1D case
-
-In this case we prove a version of the inverse function theorem for maps `f : 𝕜 → 𝕜`.
-We use `ContinuousLinearEquiv.unitsEquivAut` to translate `HasStrictDerivAt f f' a` and
-`f' ≠ 0` into `HasStrictFDerivAt f (_ : 𝕜 ≃L[𝕜] 𝕜) a`.
--/
-
-
-namespace HasStrictDerivAt
-
-variable [CompleteSpace 𝕜] {f : 𝕜 → 𝕜} {f' a : 𝕜} (hf : HasStrictDerivAt f f' a) (hf' : f' ≠ 0)
-
-variable (f f' a)
-
-/-- A function that is inverse to `f` near `a`. -/
-@[reducible]
-def localInverse : 𝕜 → 𝕜 :=
-  (hf.hasStrictFDerivAt_equiv hf').localInverse _ _ _
-#align has_strict_deriv_at.local_inverse HasStrictDerivAt.localInverse
-
-variable {f f' a}
-
-theorem map_nhds_eq : map f (𝓝 a) = 𝓝 (f a) :=
-  (hf.hasStrictFDerivAt_equiv hf').map_nhds_eq_of_equiv
-#align has_strict_deriv_at.map_nhds_eq HasStrictDerivAt.map_nhds_eq
-
-theorem to_localInverse : HasStrictDerivAt (hf.localInverse f f' a hf') f'⁻¹ (f a) :=
-  (hf.hasStrictFDerivAt_equiv hf').to_localInverse
-#align has_strict_deriv_at.to_local_inverse HasStrictDerivAt.to_localInverse
-
-theorem to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
-    HasStrictDerivAt g f'⁻¹ (f a) :=
-  (hf.hasStrictFDerivAt_equiv hf').to_local_left_inverse hg
-#align has_strict_deriv_at.to_local_left_inverse HasStrictDerivAt.to_local_left_inverse
-
-end HasStrictDerivAt
-
-/-- If a function has a non-zero strict derivative at all points, then it is an open map. -/
-theorem open_map_of_strict_deriv [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
-    (hf : ∀ x, HasStrictDerivAt f (f' x) x) (h0 : ∀ x, f' x ≠ 0) : IsOpenMap f :=
-  isOpenMap_iff_nhds_le.2 fun x => ((hf x).map_nhds_eq (h0 x)).ge
-#align open_map_of_strict_deriv open_map_of_strict_deriv
-
-/-!
-### Inverse function theorem, smooth case
-
--/
-
-
-namespace ContDiffAt
-
-variable {𝕂 : Type*} [IsROrC 𝕂]
-
-variable {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
-
-variable {F' : Type*} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
-
-variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
-
-/-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
-derivative at `a`, returns a `LocalHomeomorph` with `to_fun = f` and `a ∈ source`. -/
-def toLocalHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
-    (hn : 1 ≤ n) : LocalHomeomorph E' F' :=
-  (hf.hasStrictFDerivAt' hf' hn).toLocalHomeomorph f
-#align cont_diff_at.to_local_homeomorph ContDiffAt.toLocalHomeomorph
-
-variable {f}
-
-@[simp]
-theorem toLocalHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    (hf.toLocalHomeomorph f hf' hn : E' → F') = f :=
-  rfl
-#align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toLocalHomeomorph_coe
-
-theorem mem_toLocalHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    a ∈ (hf.toLocalHomeomorph f hf' hn).source :=
-  (hf.hasStrictFDerivAt' hf' hn).mem_toLocalHomeomorph_source
-#align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toLocalHomeomorph_source
-
-theorem image_mem_toLocalHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    f a ∈ (hf.toLocalHomeomorph f hf' hn).target :=
-  (hf.hasStrictFDerivAt' hf' hn).image_mem_toLocalHomeomorph_target
-#align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toLocalHomeomorph_target
-
-/-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
-at `a`, returns a function that is locally inverse to `f`. -/
-def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a)
-    (hn : 1 ≤ n) : F' → E' :=
-  (hf.hasStrictFDerivAt' hf' hn).localInverse f f' a
-#align cont_diff_at.local_inverse ContDiffAt.localInverse
-
-theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
-  (hf.hasStrictFDerivAt' hf' hn).localInverse_apply_image
-#align cont_diff_at.local_inverse_apply_image ContDiffAt.localInverse_apply_image
-
-/-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
-at `a`, the inverse function (produced by `ContDiff.toLocalHomeomorph`) is
-also `ContDiff`. -/
-theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
-    (hf' : HasFDerivAt f (f' : E' →L[𝕂] F') a) (hn : 1 ≤ n) :
-    ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) := by
-  have := hf.localInverse_apply_image hf' hn
-  apply (hf.toLocalHomeomorph f hf' hn).contDiffAt_symm
-    (image_mem_toLocalHomeomorph_target hf hf' hn)
-  · convert hf'
-  · convert hf
-#align cont_diff_at.to_local_inverse ContDiffAt.to_localInverse
-
-end ContDiffAt

They have type ContinuousOn ..., hence should be named accordingly. Suggested by @fpvandoorn in #8736.

@@ -452,8 +452,8 @@ def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   open_source := hs
   open_target := hf.open_image f'.toNonlinearRightInverse hs <| by
     rwa [f'.toEquiv.subsingleton_congr] at hc
-  continuous_toFun := hf.continuousOn
-  continuous_invFun := hf.inverse_continuousOn hc
+  continuousOn_toFun := hf.continuousOn
+  continuousOn_invFun := hf.inverse_continuousOn hc
 #align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
 
 @[simp]

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

@@ -3,7 +3,8 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.ContDiff.Basic
+import Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
+import Mathlib.Analysis.Calculus.ContDiff.IsROrC
 import Mathlib.Analysis.NormedSpace.Banach
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"

No changes to content, or splitting, just a rename so there is room for more files.

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

@@ -3,7 +3,7 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.ContDiff
+import Mathlib.Analysis.Calculus.ContDiff.Basic
 import Mathlib.Analysis.NormedSpace.Banach
 
 #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"

@@ -204,7 +204,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
   have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
-  /- We have to show that `y` can be written as `f x` for some `x ∈ closed_ball b ε`.
+  /- We have to show that `y` can be written as `f x` for some `x ∈ closedBall b ε`.
     The idea of the proof is to apply the Banach contraction principle to the map
     `g : x ↦ x + f'symm (y - f x)`, as a fixed point of this map satisfies `f x = y`.
     When `f'symm` is a genuine linear inverse, `g` is a contracting map. In our case, since `f'symm`
@@ -304,11 +304,11 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         exact (D n).1
       _ = f'symm.nnnorm * dist (f b) y * ((c : ℝ) * f'symm.nnnorm) ^ n := by ring
   obtain ⟨x, hx⟩ : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete this
-  -- As all the `uₙ` belong to the ball `closed_ball b ε`, so does their limit `x`.
+  -- As all the `uₙ` belong to the ball `closedBall b ε`, so does their limit `x`.
   have xmem : x ∈ closedBall b ε :=
     isClosed_ball.mem_of_tendsto hx (eventually_of_forall fun n => C n _ (D n).2)
   refine' ⟨x, xmem, _⟩
-  -- It remains to check that `f x = y`. This follows from continuity of `f` on `closed_ball b ε`
+  -- It remains to check that `f x = y`. This follows from continuity of `f` on `closedBall b ε`
   -- and from the fact that `f uₙ` is converging to `y` by construction.
   have hx' : Tendsto u atTop (𝓝[closedBall b ε] x) := by
     simp only [nhdsWithin, tendsto_inf, hx, true_and_iff, ge_iff_le, tendsto_principal]
@@ -324,7 +324,8 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
 
 theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
     (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) := by
-  cases' hc with hE hc; · skip; apply isOpen_discrete
+  cases' hc with hE hc
+  · exact isOpen_discrete _
   simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, ball_image_iff] at hs ⊢
   intro x hx
   rcases hs x hx with ⟨ε, ε0, hε⟩
@@ -359,15 +360,13 @@ We also assume that either `E = {0}`, or `c < ‖f'⁻¹‖⁻¹`. We use `N` as
 
 variable {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0}
 
--- mathport name: exprN
 local notation "N" => ‖(f'.symm : F →L[𝕜] E)‖₊
 
 protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) : AntilipschitzWith (N⁻¹ - c)⁻¹ (s.restrict f) := by
   cases' hc with hE hc
-  · haveI : Subsingleton s := ⟨fun x y => Subtype.eq <| @Subsingleton.elim _ hE _ _⟩
-    exact AntilipschitzWith.of_subsingleton
-  convert(f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
+  · exact AntilipschitzWith.of_subsingleton
+  convert (f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc
   simp [restrict]
 #align approximates_linear_on.antilipschitz ApproximatesLinearOn.antilipschitz
 
@@ -445,7 +444,7 @@ section
 variable (f s)
 
 /-- Given a function `f` that approximates a linear equivalence on an open set `s`,
-returns a local homeomorph with `to_fun = f` and `source = s`. -/
+returns a local homeomorph with `toFun = f` and `source = s`. -/
 def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : LocalHomeomorph E F where
   toLocalEquiv := hf.toLocalEquiv hc
@@ -456,12 +455,31 @@ def toLocalHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
   continuous_invFun := hf.inverse_continuousOn hc
 #align approximates_linear_on.to_local_homeomorph ApproximatesLinearOn.toLocalHomeomorph
 
+@[simp]
+theorem toLocalHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
+    (hf.toLocalHomeomorph f s hc hs : E → F) = f :=
+  rfl
+#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toLocalHomeomorph_coe
+
+@[simp]
+theorem toLocalHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : (hf.toLocalHomeomorph f s hc hs).source = s :=
+  rfl
+#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toLocalHomeomorph_source
+
+@[simp]
+theorem toLocalHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
+    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
+    (hf.toLocalHomeomorph f s hc hs).target = f '' s :=
+  rfl
+#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toLocalHomeomorph_target
+
 /-- A function `f` that approximates a linear equivalence on the whole space is a homeomorphism. -/
 def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
     (hc : Subsingleton E ∨ c < N⁻¹) : E ≃ₜ F := by
   refine' (hf.toLocalHomeomorph _ _ hc isOpen_univ).toHomeomorphOfSourceEqUnivTargetEqUniv rfl _
-  change f '' univ = univ
-  rw [image_univ, range_iff_surjective]
+  rw [toLocalHomeomorph_target, image_univ, range_iff_surjective]
   exact hf.surjective hc
 #align approximates_linear_on.to_homeomorph ApproximatesLinearOn.toHomeomorph
 
@@ -492,31 +510,11 @@ theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSp
 
 end
 
-@[simp]
-theorem toLocalHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs : E → F) = f :=
-  rfl
-#align approximates_linear_on.to_local_homeomorph_coe ApproximatesLinearOn.toLocalHomeomorph_coe
-
-@[simp]
-theorem toLocalHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) : (hf.toLocalHomeomorph f s hc hs).source = s :=
-  rfl
-#align approximates_linear_on.to_local_homeomorph_source ApproximatesLinearOn.toLocalHomeomorph_source
-
-@[simp]
-theorem toLocalHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
-    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
-    (hf.toLocalHomeomorph f s hc hs).target = f '' s :=
-  rfl
-#align approximates_linear_on.to_local_homeomorph_target ApproximatesLinearOn.toLocalHomeomorph_target
-
 theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
     (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
     closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toLocalHomeomorph f s hc hs).target :=
   (hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse ε0 hε).mono hε
-    (Subset.refl _)
+    Subset.rfl
 #align approximates_linear_on.closed_ball_subset_target ApproximatesLinearOn.closedBall_subset_target
 
 end ApproximatesLinearOn
@@ -565,8 +563,7 @@ theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[
     ∃ s : Set E, a ∈ s ∧ IsOpen s ∧
       ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) := by
   simp only [← and_assoc]
-  refine' ((nhds_basis_opens a).exists_iff _).1 _
-  exact fun s t => ApproximatesLinearOn.mono_set
+  refine ((nhds_basis_opens a).exists_iff fun s t => ApproximatesLinearOn.mono_set).1 ?_
   exact
     hf.approximates_deriv_on_nhds <|
       f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
@@ -657,7 +654,7 @@ theorem localInverse_unique (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {
 #align has_strict_fderiv_at.local_inverse_unique HasStrictFDerivAt.localInverse_unique
 
 /-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
-then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -/
+then the inverse function `hf.localInverse f` has derivative `f'.symm` at `f a`. -/
 theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
     HasStrictFDerivAt (hf.localInverse f f' a) (f'.symm : F →L[𝕜] E) (f a) :=
   (hf.toLocalHomeomorph f).hasStrictFDerivAt_symm hf.image_mem_toLocalHomeomorph_target <| by

Also rename dimH_image_le_of_locally_lipschitz_on to dimH_image_le_of_locally_lipschitzOn.

@@ -482,7 +482,7 @@ theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSp
   have fg : EqOn f g s := fun x hx => by simp_rw [← uf hx, Pi.sub_apply, add_sub_cancel'_right]
   have hg : ApproximatesLinearOn g (f' : E →L[ℝ] F) univ (lipschitzExtensionConstant F * c) := by
     apply LipschitzOnWith.approximatesLinearOn
-    rw [lipschitz_on_univ]
+    rw [lipschitzOn_univ]
     convert hu
     ext x
     simp only [add_sub_cancel', ContinuousLinearEquiv.coe_coe, Pi.sub_apply]

@@ -150,8 +150,8 @@ theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜
   simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
 #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith
 
-alias approximatesLinearOn_iff_lipschitzOnWith ↔
-  lipschitzOnWith _root_.LipschitzOnWith.approximatesLinearOn
+alias ⟨lipschitzOnWith, _root_.LipschitzOnWith.approximatesLinearOn⟩ :=
+  approximatesLinearOn_iff_lipschitzOnWith
 #align approximates_linear_on.lipschitz_on_with ApproximatesLinearOn.lipschitzOnWith
 #align lipschitz_on_with.approximates_linear_on LipschitzOnWith.approximatesLinearOn
 

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

This has nice performance benefits.

@@ -64,15 +64,15 @@ open scoped Topology Classical NNReal
 
 noncomputable section
 
-variable {𝕜 : Type _} [NontriviallyNormedField 𝕜]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
 
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
-variable {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
-variable {G : Type _} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
 
-variable {G' : Type _} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
+variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
 
 variable {ε : ℝ}
 
@@ -467,8 +467,8 @@ def toHomeomorph (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
 
 /-- In a real vector space, a function `f` that approximates a linear equivalence on a subset `s`
 can be extended to a homeomorphism of the whole space. -/
-theorem exists_homeomorph_extension {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {F : Type _} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {s : Set E}
+theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {s : Set E}
     {f : E → F} {f' : E ≃L[ℝ] F} {c : ℝ≥0} (hf : ApproximatesLinearOn f (f' : E →L[ℝ] F) s c)
     (hc : Subsingleton E ∨ lipschitzExtensionConstant F * c < ‖(f'.symm : F →L[ℝ] E)‖₊⁻¹) :
     ∃ g : E ≃ₜ F, EqOn f g s := by
@@ -734,11 +734,11 @@ theorem open_map_of_strict_deriv [CompleteSpace 𝕜] {f f' : 𝕜 → 𝕜}
 
 namespace ContDiffAt
 
-variable {𝕂 : Type _} [IsROrC 𝕂]
+variable {𝕂 : Type*} [IsROrC 𝕂]
 
-variable {E' : Type _} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
+variable {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
 
-variable {F' : Type _} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
+variable {F' : Type*} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
 
 variable [CompleteSpace E'] (f : E' → F') {f' : E' ≃L[𝕂] F'} {a : E'}
 

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.calculus.inverse
-! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.ContDiff
 import Mathlib.Analysis.NormedSpace.Banach
 
+#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
+
 /-!
 # Inverse function theorem
 

@@ -221,7 +221,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     control. Therefore, the bound can be checked at the next step, and so on inductively.
     -/
   set g := fun x => x + f'symm (y - f x) with hg
-  set u := fun n : ℕ => (g^[n]) b with hu
+  set u := fun n : ℕ => g^[n] b with hu
   have usucc : ∀ n, u (n + 1) = g (u n) := by simp [hu, ← iterate_succ_apply' g _ b]
   -- First bound: if `f z` is close to `y`, then `g z` is close to `z` (i.e., almost a fixed point).
   have A : ∀ z, dist (g z) z ≤ f'symm.nnnorm * dist (f z) y := by

Following on from #4702, another hundred sample uses of the gcongr tactic.

@@ -201,10 +201,10 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
   · refine' ⟨b, by simp [ε0], _⟩
     have : dist y (f b) ≤ 0 :=
       (mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
-    simp only [dist_le_zero] at this 
+    simp only [dist_le_zero] at this
     rw [this]
   have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
-  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc 
+  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
   have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
   /- We have to show that `y` can be written as `f x` for some `x ∈ closed_ball b ε`.
@@ -214,7 +214,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     is nonlinear, this map is not contracting (it is not even continuous), but still the proof of
     the contraction theorem holds: `uₙ = gⁿ b` is a Cauchy sequence, converging exponentially fast
     to the desired point `x`. Instead of appealing to general results, we check this by hand.
-  
+
     The main point is that `f (u n)` becomes exponentially close to `y`, and therefore
     `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive
     bound on `dist (u n) b`, from which one checks that `u n` stays in the ball on which one has a
@@ -244,7 +244,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
           sub_add_cancel]
       _ ≤ c * ‖z + v - z‖ := (hf _ (hε hgz) _ (hε hz))
       _ ≤ c * (f'symm.nnnorm * dist (f z) y) := by
-        apply mul_le_mul_of_nonneg_left _ (NNReal.coe_nonneg c)
+        gcongr
         simpa [dist_eq_norm'] using f'symm.bound (y - f z)
       _ = c * f'symm.nnnorm * dist (f z) y := by ring
   -- Third bound: a complicated bound on `dist w b` (that will show up in the induction) is enough
@@ -259,14 +259,15 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
           f'symm.nnnorm * dist (f b) y * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) :=
         by ring
       _ ≤ f'symm.nnnorm * dist (f b) y * 1 := by
-        apply mul_le_mul_of_nonneg_left _ (mul_nonneg (NNReal.coe_nonneg _) dist_nonneg)
+        gcongr
         rw [sub_le_self_iff]
-        exact pow_nonneg (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _)) _
+        positivity
       _ ≤ f'symm.nnnorm * (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε) := by
         rw [mul_one]
-        exact mul_le_mul_of_nonneg_left (mem_closedBall'.1 hy) (NNReal.coe_nonneg _)
+        gcongr
+        exact mem_closedBall'.1 hy
       _ = ε * (1 - c * f'symm.nnnorm) := by field_simp; ring
-      
+
   /- Main inductive control: `f (u n)` becomes exponentially close to `y`, and therefore
     `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive
     bound on `dist (u n) b`, from which one checks that `u n` remains in the ball on which we
@@ -283,8 +284,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
         _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := (add_le_add (A _) le_rfl)
         _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) +
               f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) *
-                dist (f b) y :=
-          (add_le_add (mul_le_mul_of_nonneg_left IH.1 (NNReal.coe_nonneg _)) IH.2)
+                dist (f b) y := by gcongr; exact IH.1; exact IH.2
         _ = f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n.succ) /
               (1 - (c : ℝ) * f'symm.nnnorm) * dist (f b) y := by
           field_simp [Jcf', pow_succ]; ring
@@ -292,8 +292,9 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     calc
       dist (f (g (u n))) y ≤ c * f'symm.nnnorm * dist (f (u n)) y :=
         B _ (C n _ IH.2) (C n.succ _ Ign)
-      _ ≤ (c : ℝ) * f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) :=
-        (mul_le_mul_of_nonneg_left IH.1 (mul_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _)))
+      _ ≤ (c : ℝ) * f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) := by
+        gcongr
+        apply IH.1
       _ = ((c : ℝ) * f'symm.nnnorm) ^ n.succ * dist (f b) y := by simp only [pow_succ']; ring
   -- Deduce from the inductive bound that `uₙ` is a Cauchy sequence, therefore converging.
   have : CauchySeq u
@@ -301,8 +302,9 @@ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f
     calc
       dist (u n) (u (n + 1)) = dist (g (u n)) (u n) := by rw [usucc, dist_comm]
       _ ≤ f'symm.nnnorm * dist (f (u n)) y := (A _)
-      _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) :=
-        (mul_le_mul_of_nonneg_left (D n).1 (NNReal.coe_nonneg _))
+      _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) := by
+        gcongr
+        exact (D n).1
       _ = f'symm.nnnorm * dist (f b) y * ((c : ℝ) * f'symm.nnnorm) ^ n := by ring
   obtain ⟨x, hx⟩ : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete this
   -- As all the `uₙ` belong to the ball `closed_ball b ε`, so does their limit `x`.
@@ -434,7 +436,7 @@ theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Sub
       abel
     _ ≤ N * (c * ‖y' - x'‖) := (mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _))
     _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by
-      apply_rules [mul_le_mul_of_nonneg_left, NNReal.coe_nonneg]
+      gcongr
       rw [← dist_eq_norm, ← dist_eq_norm]
       exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩
     _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by
@@ -542,7 +544,7 @@ theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
   · refine' ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => _⟩
     simp [@Subsingleton.elim E hE x y]
   have := hf.def hc
-  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this 
+  rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
   rcases this with ⟨s, has, hs⟩
   exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
 #align has_strict_fderiv_at.approximates_deriv_on_nhds HasStrictFDerivAt.approximates_deriv_on_nhds
@@ -797,4 +799,3 @@ theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
 #align cont_diff_at.to_local_inverse ContDiffAt.to_localInverse
 
 end ContDiffAt
-