analysis.complex.open_mapping ⟷ Mathlib.Analysis.Complex.OpenMapping

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

@@ -160,7 +160,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz
     replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by
       simpa only [mem_ball_zero_iff, norm_eq_abs, abs_of_real, abs_norm]
-    simpa only [gray, ray, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel'_right,
+    simpa only [gray, ray, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel,
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result
@@ -176,7 +176,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (s «expr ⊆ » U) -/
 #print AnalyticOn.is_constant_or_isOpen /-
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the

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

@@ -164,11 +164,11 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result
-    push_neg at h 
+    push_neg at h
     obtain ⟨z, hz, hrz⟩ := h
     specialize h1 z hz 0 (mem_ball_self hr)
     have h7 := h1.eventually_constant_or_nhds_le_map_nhds_aux.resolve_left hrz
-    rw [show gray z 0 = g z₀ by simp [gray, ray], ← map_compose] at h7 
+    rw [show gray z 0 = g z₀ by simp [gray, ray], ← map_compose] at h7
     refine' h7.trans (map_mono _)
     have h10 : Continuous fun t : ℂ => z₀ + t • z :=
       continuous_const.add (continuous_id'.smul continuous_const)
@@ -187,7 +187,7 @@ theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreco
   by_cases ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
     exact Or.inl ⟨g z₀, hg.eq_on_of_preconnected_of_eventually_eq analyticOn_const hU hz₀ h⟩
-  · push_neg at h 
+  · push_neg at h
     refine' Or.inr fun s hs1 hs2 => is_open_iff_mem_nhds.mpr _
     rintro z ⟨w, hw1, rfl⟩
     exact

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

@@ -3,9 +3,9 @@ Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 -/
-import Mathbin.Analysis.Analytic.IsolatedZeros
-import Mathbin.Analysis.Complex.CauchyIntegral
-import Mathbin.Analysis.Complex.AbsMax
+import Analysis.Analytic.IsolatedZeros
+import Analysis.Complex.CauchyIntegral
+import Analysis.Complex.AbsMax
 
 #align_import analysis.complex.open_mapping from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
 
@@ -176,7 +176,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (s «expr ⊆ » U) -/
 #print AnalyticOn.is_constant_or_isOpen /-
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the

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

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
-
-! This file was ported from Lean 3 source module analysis.complex.open_mapping
-! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Analytic.IsolatedZeros
 import Mathbin.Analysis.Complex.CauchyIntegral
 import Mathbin.Analysis.Complex.AbsMax
 
+#align_import analysis.complex.open_mapping from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
+
 /-!
 # The open mapping theorem for holomorphic functions
 
@@ -179,7 +176,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » U) -/
 #print AnalyticOn.is_constant_or_isOpen /-
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the

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

@@ -45,6 +45,7 @@ open scoped Topology
 variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {f : ℂ → ℂ} {g : E → ℂ}
   {z₀ w : ℂ} {ε r m : ℝ}
 
+#print DiffContOnCl.ball_subset_image_closedBall /-
 /-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
 contains a disk of radius `ε / 2`. -/
 theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
@@ -76,7 +77,9 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
   have h10 : f z = f z₀ := (h9 (mem_ball_self hr)).symm
   exact not_eventually.mpr hz₀ (mem_of_superset (ball_mem_nhds z₀ hr) (h10 ▸ h9))
 #align diff_cont_on_cl.ball_subset_image_closed_ball DiffContOnCl.ball_subset_image_closedBall
+-/
 
+#print AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux /-
 /-- A function `f : ℂ → ℂ` which is analytic at a point `z₀` is either constant in a neighborhood
 of `z₀`, or behaves locally like an open function (in the sense that the image of every neighborhood
 of `z₀` is a neighborhood of `f z₀`, as in `is_open_map_iff_nhds_le`). For a function `f : E → ℂ`
@@ -116,7 +119,9 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt
     (h6.ball_subset_image_closed_ball hr (fun z hz => hfx z hz) (not_eventually.mp h)).trans
       (image_subset f (closed_ball_subset_closed_ball inf_le_right))
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds_aux AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux
+-/
 
+#print AnalyticAt.eventually_constant_or_nhds_le_map_nhds /-
 /-- The *open mapping theorem* for holomorphic functions, local version: is a function `g : E → ℂ`
 is analytic at a point `z₀`, then either it is constant in a neighborhood of `z₀`, or it maps every
 neighborhood of `z₀` to a neighborhood of `z₀`. For the particular case of a holomorphic function on
@@ -172,8 +177,10 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       continuous_const.add (continuous_id'.smul continuous_const)
     simpa using h10.tendsto 0
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+#print AnalyticOn.is_constant_or_isOpen /-
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/
@@ -190,4 +197,5 @@ theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreco
       (hg w (hs1 hw1)).eventually_constant_or_nhds_le_map_nhds.resolve_left (h w (hs1 hw1))
         (image_mem_map (hs2.mem_nhds hw1))
 #align analytic_on.is_constant_or_is_open AnalyticOn.is_constant_or_isOpen
+-/
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 
 ! This file was ported from Lean 3 source module analysis.complex.open_mapping
-! leanprover-community/mathlib commit f9dd3204df14a0749cd456fac1e6849dfe7d2b88
+! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Analysis.Complex.AbsMax
 /-!
 # The open mapping theorem for holomorphic functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves the open mapping theorem for holomorphic functions, namely that an analytic
 function on a preconnected set of the complex plane is either constant or open. The main step is to
 show a local version of the theorem that states that if `f` is analytic at a point `z₀`, then either

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

@@ -170,7 +170,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     simpa using h10.tendsto 0
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (s «expr ⊆ » U) -/
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/

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

@@ -66,7 +66,7 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
   refine' ⟨z, ball_subset_closed_ball hz1, sub_eq_zero.mp _⟩
   have h6 := h1.differentiable_on.eventually_differentiable_at (is_open_ball.mem_nhds hz1)
   refine' (eventually_eq_or_eq_zero_of_is_local_min_norm h6 hz2).resolve_left fun key => _
-  have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by filter_upwards [key]with h <;> field_simp
+  have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by filter_upwards [key] with h <;> field_simp
   replace h7 : ∃ᶠ w in 𝓝[≠] z, f w = f z := (h7.filter_mono nhdsWithin_le_nhds).Frequently
   have h8 : IsPreconnected (ball z₀ r) := (convex_ball z₀ r).IsPreconnected
   have h9 := h3.eq_on_of_preconnected_of_frequently_eq analyticOn_const h8 hz1 h7
@@ -159,7 +159,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result
-    push_neg  at h 
+    push_neg at h 
     obtain ⟨z, hz, hrz⟩ := h
     specialize h1 z hz 0 (mem_ball_self hr)
     have h7 := h1.eventually_constant_or_nhds_le_map_nhds_aux.resolve_left hrz
@@ -180,7 +180,7 @@ theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreco
   by_cases ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
     exact Or.inl ⟨g z₀, hg.eq_on_of_preconnected_of_eventually_eq analyticOn_const hU hz₀ h⟩
-  · push_neg  at h 
+  · push_neg at h 
     refine' Or.inr fun s hs1 hs2 => is_open_iff_mem_nhds.mpr _
     rintro z ⟨w, hw1, rfl⟩
     exact

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

@@ -159,11 +159,11 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result
-    push_neg  at h
+    push_neg  at h 
     obtain ⟨z, hz, hrz⟩ := h
     specialize h1 z hz 0 (mem_ball_self hr)
     have h7 := h1.eventually_constant_or_nhds_le_map_nhds_aux.resolve_left hrz
-    rw [show gray z 0 = g z₀ by simp [gray, ray], ← map_compose] at h7
+    rw [show gray z 0 = g z₀ by simp [gray, ray], ← map_compose] at h7 
     refine' h7.trans (map_mono _)
     have h10 : Continuous fun t : ℂ => z₀ + t • z :=
       continuous_const.add (continuous_id'.smul continuous_const)
@@ -180,7 +180,7 @@ theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreco
   by_cases ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
     exact Or.inl ⟨g z₀, hg.eq_on_of_preconnected_of_eventually_eq analyticOn_const hU hz₀ h⟩
-  · push_neg  at h
+  · push_neg  at h 
     refine' Or.inr fun s hs1 hs2 => is_open_iff_mem_nhds.mpr _
     rintro z ⟨w, hw1, rfl⟩
     exact

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

@@ -37,7 +37,7 @@ second step is implemented in `diff_cont_on_cl.ball_subset_image_closed_ball`.
 
 open Set Filter Metric Complex
 
-open Topology
+open scoped Topology
 
 variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {f : ℂ → ℂ} {g : E → ℂ}
   {z₀ w : ℂ} {ε r m : ℝ}

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 
 ! This file was ported from Lean 3 source module analysis.complex.open_mapping
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit f9dd3204df14a0749cd456fac1e6849dfe7d2b88
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -154,7 +154,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       simpa [gray, ray] using h w e2
     have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz
     replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by
-      simpa only [mem_ball_zero_iff, norm_eq_abs, abs_of_real, abs_norm_eq_norm]
+      simpa only [mem_ball_zero_iff, norm_eq_abs, abs_of_real, abs_norm]
     simpa only [gray, ray, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel'_right,
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right

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

@@ -170,7 +170,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     simpa using h10.tendsto 0
 #align analytic_at.eventually_constant_or_nhds_le_map_nhds AnalyticAt.eventually_constant_or_nhds_le_map_nhds
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (s «expr ⊆ » U) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (s «expr ⊆ » U) -/
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/

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

@@ -131,7 +131,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     -- If g is eventually constant along every direction, then it is eventually constant
     refine eventually_of_mem (ball_mem_nhds z₀ hr) fun z hz => ?_
     refine (eq_or_ne z z₀).casesOn (congr_arg g) fun h' => ?_
-    replace h' : ‖z - z₀‖ ≠ 0 := by simpa only [Ne.def, norm_eq_zero, sub_eq_zero]
+    replace h' : ‖z - z₀‖ ≠ 0 := by simpa only [Ne, norm_eq_zero, sub_eq_zero]
     let w : E := ‖z - z₀‖⁻¹ • (z - z₀)
     have h3 : ∀ t ∈ ball (0 : ℂ) r, gray w t = g z₀ := by
       have e1 : IsPreconnected (ball (0 : ℂ) r) := (convex_ball 0 r).isPreconnected

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 | |

@@ -142,7 +142,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz
     replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by
       simpa only [mem_ball_zero_iff, norm_eq_abs, abs_ofReal, abs_norm]
-    simpa only [ray, gray, w, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel'_right,
+    simpa only [ray, gray, w, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel,
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -57,8 +57,8 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
     linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
       norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
   have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
-  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
-  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
+  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z :=
+    exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
   refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
   have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
   refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_

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

@@ -123,7 +123,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
   obtain ⟨r, hr, hgr⟩ := isOpen_iff.mp (isOpen_analyticAt ℂ g) z₀ hg
   have h1 : ∀ z ∈ sphere (0 : E) 1, AnalyticOn ℂ (gray z) (ball 0 r) := by
     refine fun z hz t ht => AnalyticAt.comp ?_ ?_
-    · exact hgr (by simpa [norm_smul, mem_sphere_zero_iff_norm.mp hz] using ht)
+    · exact hgr (by simpa [ray, norm_smul, mem_sphere_zero_iff_norm.mp hz] using ht)
     · exact analyticAt_const.add
         ((ContinuousLinearMap.smulRight (ContinuousLinearMap.id ℂ ℂ) z).analyticAt t)
   by_cases h : ∀ z ∈ sphere (0 : E) 1, ∀ᶠ t in 𝓝 0, gray z t = gray z 0
@@ -135,14 +135,14 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     let w : E := ‖z - z₀‖⁻¹ • (z - z₀)
     have h3 : ∀ t ∈ ball (0 : ℂ) r, gray w t = g z₀ := by
       have e1 : IsPreconnected (ball (0 : ℂ) r) := (convex_ball 0 r).isPreconnected
-      have e2 : w ∈ sphere (0 : E) 1 := by simp [norm_smul, inv_mul_cancel h']
+      have e2 : w ∈ sphere (0 : E) 1 := by simp [w, norm_smul, inv_mul_cancel h']
       specialize h1 w e2
       apply h1.eqOn_of_preconnected_of_eventuallyEq analyticOn_const e1 (mem_ball_self hr)
-      simpa using h w e2
+      simpa [ray, gray] using h w e2
     have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz
     replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by
       simpa only [mem_ball_zero_iff, norm_eq_abs, abs_ofReal, abs_norm]
-    simpa only [smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel'_right,
+    simpa only [ray, gray, w, smul_smul, mul_inv_cancel h', one_smul, add_sub_cancel'_right,
       Function.comp_apply, coe_smul] using h3 (↑‖z - z₀‖) h4
   · right
     -- Otherwise, it is open along at least one direction and that implies the result
@@ -150,7 +150,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     obtain ⟨z, hz, hrz⟩ := h
     specialize h1 z hz 0 (mem_ball_self hr)
     have h7 := h1.eventually_constant_or_nhds_le_map_nhds_aux.resolve_left hrz
-    rw [show gray z 0 = g z₀ by simp, ← map_compose] at h7
+    rw [show gray z 0 = g z₀ by simp [gray, ray], ← map_compose] at h7
     refine h7.trans (map_mono ?_)
     have h10 : Continuous fun t : ℂ => z₀ + t • z :=
       continuous_const.add (continuous_id'.smul continuous_const)

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -21,8 +21,8 @@ its image `f z₀`. The results extend in higher dimension to `g : E → ℂ`.
 The proof of the local version on `ℂ` goes through two main steps: first, assuming that the function
 is not constant around `z₀`, use the isolated zero principle to show that `‖f z‖` is bounded below
 on a small `sphere z₀ r` around `z₀`, and then use the maximum principle applied to the auxiliary
-function `(λ z, ‖f z - v‖)` to show that any `v` close enough to `f z₀` is in `f '' ball z₀ r`. That
-second step is implemented in `DiffContOnCl.ball_subset_image_closedBall`.
+function `(fun z ↦ ‖f z - v‖)` to show that any `v` close enough to `f z₀` is in `f '' ball z₀ r`.
+That second step is implemented in `DiffContOnCl.ball_subset_image_closedBall`.
 
 ## Main results
 
@@ -45,7 +45,7 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
     (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
     ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
   /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
-    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
+    the function `fun z ↦ ‖f z - v‖`: it is bounded below on the circle, and takes a small value
     at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
     hence that `v` is in the range of `f`. -/
   rintro v hv

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>

@@ -88,9 +88,9 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt
     ∃ ρ > 0, AnalyticOn ℂ f (closedBall z₀ ρ) ∧ ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀ := by
     simpa only [setOf_and, subset_inter_iff] using
       nhds_basis_closedBall.mem_iff.mp (h2.and (eventually_nhdsWithin_iff.mp h1))
-  replace h3 : DiffContOnCl ℂ f (ball z₀ ρ)
-  exact ⟨h3.differentiableOn.mono ball_subset_closedBall,
-    (closure_ball z₀ hρ.lt.ne.symm).symm ▸ h3.continuousOn⟩
+  replace h3 : DiffContOnCl ℂ f (ball z₀ ρ) :=
+    ⟨h3.differentiableOn.mono ball_subset_closedBall,
+      (closure_ball z₀ hρ.lt.ne.symm).symm ▸ h3.continuousOn⟩
   let r := ρ ⊓ R
   have hr : 0 < r := lt_inf_iff.mpr ⟨hρ, hR⟩
   have h5 : closedBall z₀ r ⊆ closedBall z₀ ρ := closedBall_subset_closedBall inf_le_left

Follow-up #9184

@@ -161,7 +161,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/
 theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreconnected U) :
-    (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ (s) (_ : s ⊆ U), IsOpen s → IsOpen (g '' s) := by
+    (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s) := by
   by_cases h : ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
     exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOn_const hU hz₀ h⟩

I've also got a change to make this required, but I'd like to land this first.

@@ -126,7 +126,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
     · exact hgr (by simpa [norm_smul, mem_sphere_zero_iff_norm.mp hz] using ht)
     · exact analyticAt_const.add
         ((ContinuousLinearMap.smulRight (ContinuousLinearMap.id ℂ ℂ) z).analyticAt t)
-  by_cases ∀ z ∈ sphere (0 : E) 1, ∀ᶠ t in 𝓝 0, gray z t = gray z 0
+  by_cases h : ∀ z ∈ sphere (0 : E) 1, ∀ᶠ t in 𝓝 0, gray z t = gray z 0
   · left
     -- If g is eventually constant along every direction, then it is eventually constant
     refine eventually_of_mem (ball_mem_nhds z₀ hr) fun z hz => ?_
@@ -162,7 +162,7 @@ is analytic on a connected set `U`, then either it is constant on `U`, or it is
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/
 theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreconnected U) :
     (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ (s) (_ : s ⊆ U), IsOpen s → IsOpen (g '' s) := by
-  by_cases ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
+  by_cases h : ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
     exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOn_const hU hz₀ h⟩
   · push_neg at h

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

This has nice performance benefits.

@@ -36,7 +36,7 @@ open Set Filter Metric Complex
 
 open scoped Topology
 
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {f : ℂ → ℂ} {g : E → ℂ}
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set E} {f : ℂ → ℂ} {g : E → ℂ}
   {z₀ w : ℂ} {ε r m : ℝ}
 
 /-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
-
-! This file was ported from Lean 3 source module analysis.complex.open_mapping
-! leanprover-community/mathlib commit f9dd3204df14a0749cd456fac1e6849dfe7d2b88
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Analytic.IsolatedZeros
 import Mathlib.Analysis.Complex.CauchyIntegral
 import Mathlib.Analysis.Complex.AbsMax
 
+#align_import analysis.complex.open_mapping from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
+
 /-!
 # The open mapping theorem for holomorphic functions
 

Also rename 2 lemmas

• IsCompact.exists_local_min_mem_open -> IsCompact.exists_isLocalMin_mem_open;
• Metric.exists_local_min_mem_ball -> Metric.exists_isLocal_min_mem_ball.

@@ -61,7 +61,7 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
       norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
   have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
   obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
-  exact exists_local_min_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
+  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
   refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
   have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
   refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_