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

@@ -42,7 +42,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)
   target := Ioi (0 : ℝ) ×ˢ Ioo (-π) π
   map_target' := by
     rintro ⟨r, θ⟩ ⟨hr, hθ⟩
-    dsimp at hr hθ 
+    dsimp at hr hθ
     rcases eq_or_ne θ 0 with (rfl | h'θ)
     · simpa using hr
     · right
@@ -54,14 +54,14 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       true_and_iff, Complex.arg_lt_pi_iff]
     constructor
     · cases hxy
-      · dsimp at hxy ; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
+      · dsimp at hxy; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
       · linarith [sq_nonneg x, sq_pos_of_ne_zero _ hxy]
     · cases hxy
       · exact Or.inl (le_of_lt hxy)
       · exact Or.inr hxy
   right_inv' := by
     rintro ⟨r, θ⟩ ⟨hr, hθ⟩
-    dsimp at hr hθ 
+    dsimp at hr hθ
     simp only [Prod.mk.inj_iff]
     constructor
     · conv_rhs => rw [← sqrt_sq (le_of_lt hr), ← one_mul (r ^ 2), ← sin_sq_add_cos_sq θ]
@@ -79,7 +79,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)
         Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←
-      mul_assoc] at Z 
+      mul_assoc] at Z
     simpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z
   open_target := isOpen_Ioi.Prod isOpen_Ioo
   open_source :=
@@ -125,7 +125,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
     by
     intro x hx
     simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_set_of_eq, not_lt,
-      Classical.not_not] at hx 
+      Classical.not_not] at hx
     exact hx.2
   have B : volume ((LinearMap.snd ℝ ℝ ℝ).ker : Set (ℝ × ℝ)) = 0 :=
     by

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

@@ -34,7 +34,7 @@ open scoped Real Topology
 /-- The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
 @[simps]
-def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
+def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     where
   toFun q := (Real.sqrt (q.1 ^ 2 + q.2 ^ 2), Complex.arg (Complex.equivRealProd.symm q))
   invFun p := (p.1 * cos p.2, p.1 * sin p.2)

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.SpecialFunctions.Trigonometric.Deriv
-import Mathbin.MeasureTheory.Function.Jacobian
+import Analysis.SpecialFunctions.Trigonometric.Deriv
+import MeasureTheory.Function.Jacobian
 
 #align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.special_functions.polar_coord
-! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecialFunctions.Trigonometric.Deriv
 import Mathbin.MeasureTheory.Function.Jacobian
 
+#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
+
 /-!
 # Polar coordinates
 

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

@@ -33,6 +33,7 @@ open Real Set MeasureTheory
 open scoped Real Topology
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print polarCoord /-
 /-- The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
 @[simps]
@@ -101,7 +102,9 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     · exact (Complex.continuousAt_arg hz).ContinuousWithinAt
     · exact complex.equiv_real_prod_clm.symm.continuous.continuous_on
 #align polar_coord polarCoord
+-/
 
+#print hasFDerivAt_polarCoord_symm /-
 theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
     HasFDerivAt polarCoord.symm
       (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
@@ -116,6 +119,7 @@ theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
       2 <;>
     simp only [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]
 #align has_fderiv_at_polar_coord_symm hasFDerivAt_polarCoord_symm
+-/
 
 #print polarCoord_source_ae_eq_univ /-
 theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -141,7 +141,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
 #print integral_comp_polarCoord_symm /-
 theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
-    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p :=
+    ∫ p in polarCoord.target, p.1 • f (polarCoord.symm p) = ∫ p, f p :=
   by
   set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
     (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
@@ -157,7 +157,7 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
     ring
   symm
   calc
-    (∫ p, f p) = ∫ p in polar_coord.source, f p :=
+    ∫ p, f p = ∫ p in polar_coord.source, f p :=
       by
       rw [← integral_univ]
       apply set_integral_congr_set_ae

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.special_functions.polar_coord
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
+! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.MeasureTheory.Function.Jacobian
 /-!
 # Polar coordinates
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define polar coordinates, as a local homeomorphism in `ℝ^2` between `ℝ^2 - (-∞, 0]` and
 `(0, +∞) × (-π, π)`. Its inverse is given by `(r, θ) ↦ (r cos θ, r sin θ)`.
 

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

@@ -166,7 +166,6 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
       apply set_integral_congr polar_coord.open_target.measurable_set fun x hx => _
       rw [B_det, abs_of_pos]
       exact hx.1
-    
 #align integral_comp_polar_coord_symm integral_comp_polarCoord_symm
 -/
 

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

@@ -114,6 +114,7 @@ theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
     simp only [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]
 #align has_fderiv_at_polar_coord_symm hasFDerivAt_polarCoord_symm
 
+#print polarCoord_source_ae_eq_univ /-
 theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
   by
   have A : polar_coord.sourceᶜ ⊆ (LinearMap.snd ℝ ℝ ℝ).ker :=
@@ -132,7 +133,9 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
   simp only [ae_eq_univ]
   exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
+-/
 
+#print integral_comp_polarCoord_symm /-
 theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p :=
@@ -165,4 +168,5 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
       exact hx.1
     
 #align integral_comp_polar_coord_symm integral_comp_polarCoord_symm
+-/
 

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

@@ -37,7 +37,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     where
   toFun q := (Real.sqrt (q.1 ^ 2 + q.2 ^ 2), Complex.arg (Complex.equivRealProd.symm q))
   invFun p := (p.1 * cos p.2, p.1 * sin p.2)
-  source := { q | 0 < q.1 } ∪ { q | q.2 ≠ 0 }
+  source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
   target := Ioi (0 : ℝ) ×ˢ Ioo (-π) π
   map_target' := by
     rintro ⟨r, θ⟩ ⟨hr, hθ⟩
@@ -91,8 +91,8 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     by
     apply ((continuous_fst.pow 2).add (continuous_snd.pow 2)).sqrt.ContinuousOn.Prod
     have A :
-      maps_to complex.equiv_real_prod.symm ({ q : ℝ × ℝ | 0 < q.1 } ∪ { q : ℝ × ℝ | q.2 ≠ 0 })
-        { z | 0 < z.re ∨ z.im ≠ 0 } :=
+      maps_to complex.equiv_real_prod.symm ({q : ℝ × ℝ | 0 < q.1} ∪ {q : ℝ × ℝ | q.2 ≠ 0})
+        {z | 0 < z.re ∨ z.im ≠ 0} :=
       by rintro ⟨x, y⟩ hxy; simpa only using hxy
     apply ContinuousOn.comp (fun z hz => _) _ A
     · exact (Complex.continuousAt_arg hz).ContinuousWithinAt
@@ -106,7 +106,8 @@ theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
       p :=
   by
   rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
-  convert HasFDerivAt.prod
+  convert
+      HasFDerivAt.prod
         (has_fderiv_at_fst.mul ((has_deriv_at_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
         (has_fderiv_at_fst.mul ((has_deriv_at_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using
       2 <;>

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

@@ -41,7 +41,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
   target := Ioi (0 : ℝ) ×ˢ Ioo (-π) π
   map_target' := by
     rintro ⟨r, θ⟩ ⟨hr, hθ⟩
-    dsimp at hr hθ
+    dsimp at hr hθ 
     rcases eq_or_ne θ 0 with (rfl | h'θ)
     · simpa using hr
     · right
@@ -53,14 +53,14 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       true_and_iff, Complex.arg_lt_pi_iff]
     constructor
     · cases hxy
-      · dsimp at hxy; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
+      · dsimp at hxy ; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
       · linarith [sq_nonneg x, sq_pos_of_ne_zero _ hxy]
     · cases hxy
       · exact Or.inl (le_of_lt hxy)
       · exact Or.inr hxy
   right_inv' := by
     rintro ⟨r, θ⟩ ⟨hr, hθ⟩
-    dsimp at hr hθ
+    dsimp at hr hθ 
     simp only [Prod.mk.inj_iff]
     constructor
     · conv_rhs => rw [← sqrt_sq (le_of_lt hr), ← one_mul (r ^ 2), ← sin_sq_add_cos_sq θ]
@@ -78,7 +78,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
         Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←
-      mul_assoc] at Z
+      mul_assoc] at Z 
     simpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z
   open_target := isOpen_Ioi.Prod isOpen_Ioo
   open_source :=
@@ -119,7 +119,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
     by
     intro x hx
     simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_set_of_eq, not_lt,
-      Classical.not_not] at hx
+      Classical.not_not] at hx 
     exact hx.2
   have B : volume ((LinearMap.snd ℝ ℝ ℝ).ker : Set (ℝ × ℝ)) = 0 :=
     by

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

@@ -27,7 +27,7 @@ noncomputable section
 
 open Real Set MeasureTheory
 
-open Real Topology
+open scoped Real Topology
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.

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

@@ -53,8 +53,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       true_and_iff, Complex.arg_lt_pi_iff]
     constructor
     · cases hxy
-      · dsimp at hxy
-        linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
+      · dsimp at hxy; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
       · linarith [sq_nonneg x, sq_pos_of_ne_zero _ hxy]
     · cases hxy
       · exact Or.inl (le_of_lt hxy)
@@ -94,9 +93,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     have A :
       maps_to complex.equiv_real_prod.symm ({ q : ℝ × ℝ | 0 < q.1 } ∪ { q : ℝ × ℝ | q.2 ≠ 0 })
         { z | 0 < z.re ∨ z.im ≠ 0 } :=
-      by
-      rintro ⟨x, y⟩ hxy
-      simpa only using hxy
+      by rintro ⟨x, y⟩ hxy; simpa only using hxy
     apply ContinuousOn.comp (fun z hz => _) _ A
     · exact (Complex.continuousAt_arg hz).ContinuousWithinAt
     · exact complex.equiv_real_prod_clm.symm.continuous.continuous_on

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

@@ -102,19 +102,19 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     · exact complex.equiv_real_prod_clm.symm.continuous.continuous_on
 #align polar_coord polarCoord
 
-theorem hasFderivAt_polarCoord_symm (p : ℝ × ℝ) :
-    HasFderivAt polarCoord.symm
+theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
+    HasFDerivAt polarCoord.symm
       (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
           !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2]).toContinuousLinearMap
       p :=
   by
   rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
-  convert HasFderivAt.prod
-        (has_fderiv_at_fst.mul ((has_deriv_at_cos p.2).comp_hasFderivAt p hasFderivAt_snd))
-        (has_fderiv_at_fst.mul ((has_deriv_at_sin p.2).comp_hasFderivAt p hasFderivAt_snd)) using
+  convert HasFDerivAt.prod
+        (has_fderiv_at_fst.mul ((has_deriv_at_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
+        (has_fderiv_at_fst.mul ((has_deriv_at_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using
       2 <;>
     simp only [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]
-#align has_fderiv_at_polar_coord_symm hasFderivAt_polarCoord_symm
+#align has_fderiv_at_polar_coord_symm hasFDerivAt_polarCoord_symm
 
 theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
   by
@@ -143,8 +143,8 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
     (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
         !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2]).toContinuousLinearMap with
     hB
-  have A : ∀ p ∈ polar_coord.symm.source, HasFderivAt polar_coord.symm (B p) p := fun p hp =>
-    hasFderivAt_polarCoord_symm p
+  have A : ∀ p ∈ polar_coord.symm.source, HasFDerivAt polar_coord.symm (B p) p := fun p hp =>
+    hasFDerivAt_polarCoord_symm p
   have B_det : ∀ p, (B p).det = p.1 := by
     intro p
     conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module analysis.special_functions.polar_coord
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Analysis.SpecialFunctions.Trigonometric.Deriv
 import Mathbin.MeasureTheory.Function.Jacobian
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -66,7 +66,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     · conv_rhs => rw [← sqrt_sq (le_of_lt hr), ← one_mul (r ^ 2), ← sin_sq_add_cos_sq θ]
       congr 1
       ring
-    · convert Complex.arg_mul_cos_add_sin_mul_i hr ⟨hθ.1, hθ.2.le⟩
+    · convert Complex.arg_mul_cos_add_sin_mul_I hr ⟨hθ.1, hθ.2.le⟩
       simp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.ofReal_cos,
         Complex.ofReal_sin]
       ring
@@ -76,7 +76,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       simp only [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
         Complex.ofReal_re, Complex.mul_re, Complex.I_re, MulZeroClass.mul_zero, Complex.ofReal_im,
         Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
-    have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.I)
+    have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z
     simpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

@@ -101,7 +101,6 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     · exact complex.equiv_real_prod_clm.symm.continuous.continuous_on
 #align polar_coord polarCoord
 
-/- ./././Mathport/Syntax/Translate/Tactic/Basic.lean:31:4: unsupported: too many args: matrix.notation ... #[[]] -/
 theorem hasFderivAt_polarCoord_symm (p : ℝ × ℝ) :
     HasFderivAt polarCoord.symm
       (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
@@ -135,7 +134,6 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
   exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
 
-/- ./././Mathport/Syntax/Translate/Tactic/Basic.lean:31:4: unsupported: too many args: matrix.notation ... #[[]] -/
 theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

@@ -101,13 +101,11 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     · exact complex.equiv_real_prod_clm.symm.continuous.continuous_on
 #align polar_coord polarCoord
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `«expr!![ » -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation -/
+/- ./././Mathport/Syntax/Translate/Tactic/Basic.lean:31:4: unsupported: too many args: matrix.notation ... #[[]] -/
 theorem hasFderivAt_polarCoord_symm (p : ℝ × ℝ) :
     HasFderivAt polarCoord.symm
       (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
-          («expr!![ »
-            "./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation")).toContinuousLinearMap
+          !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2]).toContinuousLinearMap
       p :=
   by
   rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
@@ -137,16 +135,14 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
   exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `«expr!![ » -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation -/
+/- ./././Mathport/Syntax/Translate/Tactic/Basic.lean:31:4: unsupported: too many args: matrix.notation ... #[[]] -/
 theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p :=
   by
   set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
     (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
-        («expr!![ »
-          "./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation")).toContinuousLinearMap with
+        !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2]).toContinuousLinearMap with
     hB
   have A : ∀ p ∈ polar_coord.symm.source, HasFderivAt polar_coord.symm (B p) p := fun p hp =>
     hasFderivAt_polarCoord_symm p

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

@@ -111,8 +111,7 @@ theorem hasFderivAt_polarCoord_symm (p : ℝ × ℝ) :
       p :=
   by
   rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
-  convert
-      HasFderivAt.prod
+  convert HasFderivAt.prod
         (has_fderiv_at_fst.mul ((has_deriv_at_cos p.2).comp_hasFderivAt p hasFderivAt_snd))
         (has_fderiv_at_fst.mul ((has_deriv_at_sin p.2).comp_hasFderivAt p hasFderivAt_snd)) using
       2 <;>

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

@@ -67,8 +67,8 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       congr 1
       ring
     · convert Complex.arg_mul_cos_add_sin_mul_i hr ⟨hθ.1, hθ.2.le⟩
-      simp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.of_real_cos,
-        Complex.of_real_sin]
+      simp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.ofReal_cos,
+        Complex.ofReal_sin]
       ring
   left_inv' := by
     rintro ⟨x, y⟩ hxy
@@ -77,9 +77,9 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
         Complex.ofReal_re, Complex.mul_re, Complex.I_re, MulZeroClass.mul_zero, Complex.ofReal_im,
         Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.I)
-    simp only [← Complex.of_real_cos, ← Complex.of_real_sin, mul_add, ← Complex.ofReal_mul, ←
+    simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z
-    simpa [A, -Complex.of_real_cos, -Complex.of_real_sin] using Complex.ext_iff.1 Z
+    simpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z
   open_target := isOpen_Ioi.Prod isOpen_Ioo
   open_source :=
     (isOpen_lt continuous_const continuous_fst).union

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

@@ -72,10 +72,10 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       ring
   left_inv' := by
     rintro ⟨x, y⟩ hxy
-    have A : sqrt (x ^ 2 + y ^ 2) = Complex.AbsTheory.Complex.abs (x + y * Complex.I) := by
+    have A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by
       simp only [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
-        Complex.ofReal_re, Complex.mul_re, Complex.i_re, MulZeroClass.mul_zero, Complex.ofReal_im,
-        Complex.i_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
+        Complex.ofReal_re, Complex.mul_re, Complex.I_re, MulZeroClass.mul_zero, Complex.ofReal_im,
+        Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.I)
     simp only [← Complex.of_real_cos, ← Complex.of_real_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z

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

@@ -74,8 +74,8 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
     rintro ⟨x, y⟩ hxy
     have A : sqrt (x ^ 2 + y ^ 2) = Complex.AbsTheory.Complex.abs (x + y * Complex.I) := by
       simp only [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
-        Complex.ofReal_re, Complex.mul_re, Complex.i_re, mul_zero, Complex.ofReal_im, Complex.i_im,
-        sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
+        Complex.ofReal_re, Complex.mul_re, Complex.i_re, MulZeroClass.mul_zero, Complex.ofReal_im,
+        Complex.i_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.I)
     simp only [← Complex.of_real_cos, ← Complex.of_real_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z

mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba

@@ -67,17 +67,17 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
       congr 1
       ring
     · convert Complex.arg_mul_cos_add_sin_mul_i hr ⟨hθ.1, hθ.2.le⟩
-      simp only [Complex.equivRealProd_symm_apply, Complex.of_real_mul, Complex.of_real_cos,
+      simp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.of_real_cos,
         Complex.of_real_sin]
       ring
   left_inv' := by
     rintro ⟨x, y⟩ hxy
-    have A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.i) := by
+    have A : sqrt (x ^ 2 + y ^ 2) = Complex.AbsTheory.Complex.abs (x + y * Complex.I) := by
       simp only [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
-        Complex.of_real_re, Complex.mul_re, Complex.i_re, mul_zero, Complex.of_real_im,
-        Complex.i_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
-    have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.i)
-    simp only [← Complex.of_real_cos, ← Complex.of_real_sin, mul_add, ← Complex.of_real_mul, ←
+        Complex.ofReal_re, Complex.mul_re, Complex.i_re, mul_zero, Complex.ofReal_im, Complex.i_im,
+        sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
+    have Z := Complex.abs_mul_cos_add_sin_mul_i (x + y * Complex.I)
+    simp only [← Complex.of_real_cos, ← Complex.of_real_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z
     simpa [A, -Complex.of_real_cos, -Complex.of_real_sin] using Complex.ext_iff.1 Z
   open_target := isOpen_Ioi.Prod isOpen_Ioo

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

@@ -102,12 +102,12 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ)
 #align polar_coord polarCoord
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `«expr!![ » -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:391:14: unsupported user notation matrix.notation -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation -/
 theorem hasFderivAt_polarCoord_symm (p : ℝ × ℝ) :
     HasFderivAt polarCoord.symm
       (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
           («expr!![ »
-            "./././Mathport/Syntax/Translate/Expr.lean:391:14: unsupported user notation matrix.notation")).toContinuousLinearMap
+            "./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation")).toContinuousLinearMap
       p :=
   by
   rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
@@ -139,7 +139,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ :=
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `«expr!![ » -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:391:14: unsupported user notation matrix.notation -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation -/
 theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p :=
@@ -147,7 +147,7 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
   set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
     (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
         («expr!![ »
-          "./././Mathport/Syntax/Translate/Expr.lean:391:14: unsupported user notation matrix.notation")).toContinuousLinearMap with
+          "./././Mathport/Syntax/Translate/Expr.lean:387:14: unsupported user notation matrix.notation")).toContinuousLinearMap with
     hB
   have A : ∀ p ∈ polar_coord.symm.source, HasFderivAt polar_coord.symm (B p) p := fun p hp =>
     hasFderivAt_polarCoord_symm p

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

This matches our general policy and zpow_two_pos_of_ne_zero.

Also define sq_pos_of_ne_zero as an alias.

@@ -49,8 +49,8 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
       true_and_iff, Complex.arg_lt_pi_iff]
     constructor
     · cases' hxy with hxy hxy
-      · dsimp at hxy; linarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]
-      · linarith [sq_nonneg x, sq_pos_of_ne_zero _ hxy]
+      · dsimp at hxy; linarith [sq_pos_of_ne_zero hxy.ne', sq_nonneg y]
+      · linarith [sq_nonneg x, sq_pos_of_ne_zero hxy]
     · cases' hxy with hxy hxy
       · exact Or.inl (le_of_lt hxy)
       · exact Or.inr hxy

Done with a global search and replace, and then (to fix the #align lines), replace (#align \S*)setIntegral with $1set_integral.

@@ -141,12 +141,12 @@ theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [Normed
   calc
     ∫ p, f p = ∫ p in polarCoord.source, f p := by
       rw [← integral_univ]
-      apply set_integral_congr_set_ae
+      apply setIntegral_congr_set_ae
       exact polarCoord_source_ae_eq_univ.symm
     _ = ∫ p in polarCoord.target, abs (B p).det • f (polarCoord.symm p) := by
       apply integral_target_eq_integral_abs_det_fderiv_smul volume A
     _ = ∫ p in polarCoord.target, p.1 • f (polarCoord.symm p) := by
-      apply set_integral_congr polarCoord.open_target.measurableSet fun x hx => ?_
+      apply setIntegral_congr polarCoord.open_target.measurableSet fun x hx => ?_
       rw [B_det, abs_of_pos]
       exact hx.1
 #align integral_comp_polar_coord_symm integral_comp_polarCoord_symm

This adds the notation √r for Real.sqrt r. The precedence is such that √x⁻¹ is parsed as √(x⁻¹); not because this is particularly desirable, but because it's the default and the choice doesn't really matter.

This is extracted from #7907, which adds a more general nth root typeclass. The idea is to perform all the boring substitutions downstream quickly, so that we can play around with custom elaborators with a much slower rate of code-rot. This PR also won't rot as quickly, as it does not forbid writing x.sqrt as that PR does.

While perhaps claiming √ for Real.sqrt is greedy; it:

• Is far more common thatn NNReal.sqrt and Nat.sqrt
• Is far more interesting to mathlib than sqrt on Float
• Can be overloaded anyway, so this does not prevent downstream code using the notation on their own types.
• Will be replaced by a more general typeclass in a future PR.

Zulip

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

@@ -30,7 +30,7 @@ open scoped Real Topology
 It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
 @[simps]
 def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
-  toFun q := (Real.sqrt (q.1 ^ 2 + q.2 ^ 2), Complex.arg (Complex.equivRealProd.symm q))
+  toFun q := (√(q.1 ^ 2 + q.2 ^ 2), Complex.arg (Complex.equivRealProd.symm q))
   invFun p := (p.1 * cos p.2, p.1 * sin p.2)
   source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
   target := Ioi (0 : ℝ) ×ˢ Ioo (-π) π
@@ -68,7 +68,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
       ring
   left_inv' := by
     rintro ⟨x, y⟩ _
-    have A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by
+    have A : √(x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by
       rw [Complex.abs_apply, Complex.normSq_add_mul_I]
     have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←

@@ -41,7 +41,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
     · simpa using hr
     · right
       simp at hr
-      simpa only [ne_of_gt hr, Ne.def, mem_setOf_eq, mul_eq_zero, false_or_iff,
+      simpa only [ne_of_gt hr, Ne, mem_setOf_eq, mul_eq_zero, false_or_iff,
         sin_eq_zero_iff_of_lt_of_lt hθ.1 hθ.2] using h'θ
   map_source' := by
     rintro ⟨x, y⟩ hxy
@@ -115,7 +115,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
     exact hx.2
   have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
     apply Measure.addHaar_submodule
-    rw [Ne.def, LinearMap.ker_eq_top]
+    rw [Ne, LinearMap.ker_eq_top]
     intro h
     have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]
     simp at this

We want to avoid making Lean unfold smul during unification. A separate instance does helps at the cost of some elaboration failures.

@@ -100,7 +100,7 @@ theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
   convert HasFDerivAt.prod (𝕜 := ℝ)
     (hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
     (hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
-  simp [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]
+  simp [smul_smul, add_comm, neg_mul, smul_neg, neg_smul _ (ContinuousLinearMap.snd ℝ ℝ ℝ)]
 #align has_fderiv_at_polar_coord_symm hasFDerivAt_polarCoord_symm
 
 -- Porting note: this instance is needed but not automatically synthesised

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

@@ -134,7 +134,7 @@ theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [Normed
   have B_det : ∀ p, (B p).det = p.1 := by
     intro p
     conv_rhs => rw [← one_mul p.1, ← cos_sq_add_sin_sq p.2]
-    simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
+    simp only [B, neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,
       Matrix.det_fin_two_of, sub_neg_eq_add]
     ring
   symm

Rename

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

and related theorems.

@@ -89,7 +89,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
     refine' ContinuousOn.comp (f := Complex.equivRealProd.symm)
       (g := Complex.arg) (fun z hz => _) _ A
     · exact (Complex.continuousAt_arg hz).continuousWithinAt
-    · exact Complex.equivRealProdClm.symm.continuous.continuousOn
+    · exact Complex.equivRealProdCLM.symm.continuous.continuousOn
 #align polar_coord polarCoord
 
 theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
@@ -162,7 +162,7 @@ open scoped Real
 /-- The polar coordinates partial homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
 protected noncomputable def polarCoord : PartialHomeomorph ℂ (ℝ × ℝ) :=
-  equivRealProdClm.toHomeomorph.transPartialHomeomorph polarCoord
+  equivRealProdCLM.toHomeomorph.transPartialHomeomorph polarCoord
 
 protected theorem polarCoord_apply (a : ℂ) :
     Complex.polarCoord a = (Complex.abs a, Complex.arg a) := by
@@ -177,7 +177,7 @@ protected theorem polarCoord_target :
 @[simp]
 protected theorem polarCoord_symm_apply (p : ℝ × ℝ) :
     Complex.polarCoord.symm p = p.1 * (Real.cos p.2 + Real.sin p.2 * Complex.I) := by
-  simp [Complex.polarCoord, equivRealProdClm_symm_apply, mul_add, mul_assoc]
+  simp [Complex.polarCoord, equivRealProdCLM_symm_apply, mul_add, mul_assoc]
 
 theorem polardCoord_symm_abs (p : ℝ × ℝ) :
     Complex.abs (Complex.polarCoord.symm p) = |p.1| := by simp

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

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

@@ -5,6 +5,7 @@ Authors: Sébastien Gouëzel
 -/
 import Mathlib.MeasureTheory.Function.Jacobian
 import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
 
 #align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
 

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.

@@ -11,7 +11,7 @@ import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
 /-!
 # Polar coordinates
 
-We define polar coordinates, as a local homeomorphism in `ℝ^2` between `ℝ^2 - (-∞, 0]` and
+We define polar coordinates, as a partial homeomorphism in `ℝ^2` between `ℝ^2 - (-∞, 0]` and
 `(0, +∞) × (-π, π)`. Its inverse is given by `(r, θ) ↦ (r cos θ, r sin θ)`.
 
 It satisfies the following change of variables formula (see `integral_comp_polarCoord_symm`):
@@ -25,7 +25,7 @@ open Real Set MeasureTheory
 
 open scoped Real Topology
 
-/-- The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
+/-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
 @[simps]
 def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
@@ -158,7 +158,7 @@ namespace Complex
 
 open scoped Real
 
-/-- The polar coordinates local homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
+/-- The polar coordinates partial homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
 protected noncomputable def polarCoord : PartialHomeomorph ℂ (ℝ × ℝ) :=
   equivRealProdClm.toHomeomorph.transPartialHomeomorph polarCoord

In preparation of future PRs dealing with estimates of the complex logarithm and its Taylor series, this introduces Complex.slitPlane for the set of complex numbers not on the closed negative real axis (in Analysis.SpecialFunctions.Complex.Arg), adds a bunch of API lemmas, and replaces hypotheses of the form 0 < x.re ∨ x.im ≠ 0 by x ∈ slitPlane in several other files.

(We do not introduce a new file for that to avoid circular imports with Analysis.SpecialFunctions.Complex.Arg.)

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

@@ -83,7 +83,7 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
   continuousOn_toFun := by
     apply ((continuous_fst.pow 2).add (continuous_snd.pow 2)).sqrt.continuousOn.prod
     have A : MapsTo Complex.equivRealProd.symm ({q : ℝ × ℝ | 0 < q.1} ∪ {q : ℝ × ℝ | q.2 ≠ 0})
-        {z | 0 < z.re ∨ z.im ≠ 0} := by
+        Complex.slitPlane := by
       rintro ⟨x, y⟩ hxy; simpa only using hxy
     refine' ContinuousOn.comp (f := Complex.equivRealProd.symm)
       (g := Complex.arg) (fun z hz => _) _ A
@@ -161,18 +161,17 @@ open scoped Real
 /-- The polar coordinates local homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
 protected noncomputable def polarCoord : PartialHomeomorph ℂ (ℝ × ℝ) :=
-  equivRealProdClm.toHomeomorph.toPartialHomeomorph.trans polarCoord
+  equivRealProdClm.toHomeomorph.transPartialHomeomorph polarCoord
 
 protected theorem polarCoord_apply (a : ℂ) :
     Complex.polarCoord a = (Complex.abs a, Complex.arg a) := by
   simp_rw [Complex.abs_def, Complex.normSq_apply, ← pow_two]
   rfl
 
-protected theorem polarCoord_source :
-    Complex.polarCoord.source = {a | 0 < a.re} ∪ {a | a.im ≠ 0} := by simp [Complex.polarCoord]
+protected theorem polarCoord_source : Complex.polarCoord.source = slitPlane := rfl
 
 protected theorem polarCoord_target :
-    Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π := by simp [Complex.polarCoord]
+    Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π := rfl
 
 @[simp]
 protected theorem polarCoord_symm_apply (p : ℝ × ℝ) :

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

@@ -28,7 +28,7 @@ open scoped Real Topology
 /-- The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
 @[simps]
-def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
+def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
   toFun q := (Real.sqrt (q.1 ^ 2 + q.2 ^ 2), Complex.arg (Complex.equivRealProd.symm q))
   invFun p := (p.1 * cos p.2, p.1 * sin p.2)
   source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
@@ -160,8 +160,8 @@ open scoped Real
 
 /-- The polar coordinates local homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
 It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
-protected noncomputable def polarCoord : LocalHomeomorph ℂ (ℝ × ℝ) :=
-  equivRealProdClm.toHomeomorph.toLocalHomeomorph.trans polarCoord
+protected noncomputable def polarCoord : PartialHomeomorph ℂ (ℝ × ℝ) :=
+  equivRealProdClm.toHomeomorph.toPartialHomeomorph.trans polarCoord
 
 protected theorem polarCoord_apply (a : ℂ) :
     Complex.polarCoord a = (Complex.abs a, Complex.arg a) := by

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

@@ -77,10 +77,10 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
   open_source :=
     (isOpen_lt continuous_const continuous_fst).union
       (isOpen_ne_fun continuous_snd continuous_const)
-  continuous_invFun :=
+  continuousOn_invFun :=
     ((continuous_fst.mul (continuous_cos.comp continuous_snd)).prod_mk
         (continuous_fst.mul (continuous_sin.comp continuous_snd))).continuousOn
-  continuous_toFun := by
+  continuousOn_toFun := by
     apply ((continuous_fst.pow 2).add (continuous_snd.pow 2)).sqrt.continuousOn.prod
     have A : MapsTo Complex.equivRealProd.symm ({q : ℝ × ℝ | 0 < q.1} ∪ {q : ℝ × ℝ | q.2 ≠ 0})
         {z | 0 < z.re ∨ z.im ≠ 0} := by

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -19,8 +19,6 @@ It satisfies the following change of variables formula (see `integral_comp_polar
 
 -/
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
 noncomputable section Real
 
 open Real Set MeasureTheory

@@ -3,8 +3,8 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
 import Mathlib.MeasureTheory.Function.Jacobian
+import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
 
 #align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
 
@@ -19,11 +19,10 @@ It satisfies the following change of variables formula (see `integral_comp_polar
 
 -/
 
-
-noncomputable section
-
 local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
+noncomputable section Real
+
 open Real Set MeasureTheory
 
 open scoped Real Topology
@@ -152,3 +151,44 @@ theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [Normed
       rw [B_det, abs_of_pos]
       exact hx.1
 #align integral_comp_polar_coord_symm integral_comp_polarCoord_symm
+
+end Real
+
+noncomputable section Complex
+
+namespace Complex
+
+open scoped Real
+
+/-- The polar coordinates local homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
+It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
+protected noncomputable def polarCoord : LocalHomeomorph ℂ (ℝ × ℝ) :=
+  equivRealProdClm.toHomeomorph.toLocalHomeomorph.trans polarCoord
+
+protected theorem polarCoord_apply (a : ℂ) :
+    Complex.polarCoord a = (Complex.abs a, Complex.arg a) := by
+  simp_rw [Complex.abs_def, Complex.normSq_apply, ← pow_two]
+  rfl
+
+protected theorem polarCoord_source :
+    Complex.polarCoord.source = {a | 0 < a.re} ∪ {a | a.im ≠ 0} := by simp [Complex.polarCoord]
+
+protected theorem polarCoord_target :
+    Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π := by simp [Complex.polarCoord]
+
+@[simp]
+protected theorem polarCoord_symm_apply (p : ℝ × ℝ) :
+    Complex.polarCoord.symm p = p.1 * (Real.cos p.2 + Real.sin p.2 * Complex.I) := by
+  simp [Complex.polarCoord, equivRealProdClm_symm_apply, mul_add, mul_assoc]
+
+theorem polardCoord_symm_abs (p : ℝ × ℝ) :
+    Complex.abs (Complex.polarCoord.symm p) = |p.1| := by simp
+
+protected theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E]
+    [NormedSpace ℝ E] (f : ℂ → E) :
+    (∫ p in polarCoord.target, p.1 • f (Complex.polarCoord.symm p)) = ∫ p, f p := by
+  rw [← (Complex.volume_preserving_equiv_real_prod.symm).integral_comp
+    measurableEquivRealProd.symm.measurableEmbedding, ← integral_comp_polarCoord_symm]
+  rfl
+
+end Complex

Also add formula for cos (x / 2), sin (x / 2) and golf some proofs.

@@ -71,13 +71,11 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
   left_inv' := by
     rintro ⟨x, y⟩ _
     have A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by
-      simp [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
-        Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im,
-        Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
+      rw [Complex.abs_apply, Complex.normSq_add_mul_I]
     have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←
       mul_assoc] at Z
-    simpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z
+    simp [A]
   open_target := isOpen_Ioi.prod isOpen_Ioo
   open_source :=
     (isOpen_lt continuous_const continuous_fst).union

Also add completeSpace_prod, integrable_prod.

@@ -128,7 +128,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
 
 theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] (f : ℝ × ℝ → E) :
+    (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
   set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>
     LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)

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

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

@@ -72,7 +72,7 @@ def polarCoord : LocalHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
     rintro ⟨x, y⟩ _
     have A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by
       simp [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re,
-        Complex.ofReal_re, Complex.mul_re, Complex.I_re, MulZeroClass.mul_zero, Complex.ofReal_im,
+        Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im,
         Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add]
     have Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)
     simp only [← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ←

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

This has nice performance benefits.

@@ -127,7 +127,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
   exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le
 #align polar_coord_source_ae_eq_univ polarCoord_source_ae_eq_univ
 
-theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [CompleteSpace E] (f : ℝ × ℝ → E) :
     (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p := by
   set B : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ := fun p =>

@@ -22,7 +22,7 @@ It satisfies the following change of variables formula (see `integral_comp_polar
 
 noncomputable section
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
 open Real Set MeasureTheory
 

• 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) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.special_functions.polar_coord
-! leanprover-community/mathlib commit 8f9fea08977f7e450770933ee6abb20733b47c92
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
 import Mathlib.MeasureTheory.Function.Jacobian
 
+#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
+
 /-!
 # Polar coordinates
 

This is supposed to mean "an additive Haar measure", not adding something to Haar, so it should be one word and not two.

@@ -121,7 +121,7 @@ theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by
       Classical.not_not] at hx
     exact hx.2
   have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
-    apply Measure.add_haar_submodule
+    apply Measure.addHaar_submodule
     rw [Ne.def, LinearMap.ker_eq_top]
     intro h
     have : (LinearMap.snd ℝ ℝ ℝ) (0, 1) = (0 : ℝ × ℝ →ₗ[ℝ] ℝ) (0, 1) := by rw [h]

@@ -146,7 +146,7 @@ theorem integral_comp_polarCoord_symm {E : Type _} [NormedAddCommGroup E] [Norme
     ring
   symm
   calc
-    (∫ p, f p) = ∫ p in polarCoord.source, f p := by
+    ∫ p, f p = ∫ p in polarCoord.source, f p := by
       rw [← integral_univ]
       apply set_integral_congr_set_ae
       exact polarCoord_source_ae_eq_univ.symm