analysis.complex.upper_half_plane.metric ⟷ Mathlib.Analysis.Complex.UpperHalfPlane.Metric

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

@@ -5,7 +5,7 @@ Authors: Yury Kudryashov
 -/
 import Analysis.Complex.UpperHalfPlane.Topology
 import Analysis.SpecialFunctions.Arsinh
-import Geometry.Euclidean.Inversion
+import Geometry.Euclidean.Inversion.Basic
 
 #align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
 
@@ -50,7 +50,7 @@ theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqr
 #print UpperHalfPlane.sinh_half_dist /-
 theorem sinh_half_dist (z w : ℍ) :
     sinh (dist z w / 2) = dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) := by
-  rw [dist_eq, mul_div_cancel_left (arsinh _) two_ne_zero, sinh_arsinh]
+  rw [dist_eq, mul_div_cancel_left₀ (arsinh _) two_ne_zero, sinh_arsinh]
 #align upper_half_plane.sinh_half_dist UpperHalfPlane.sinh_half_dist
 -/
 
@@ -223,7 +223,7 @@ theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) :
   by
   have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
   simp only [Complex.dist_eq, Complex.sq_abs, norm_sq_apply, coe_re, coe_im, center_re, center_im,
-    cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
+    cosh_dist', mul_div_cancel₀ _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
     sq]
   ring
 #align upper_half_plane.dist_coe_center_sq UpperHalfPlane.dist_coe_center_sq

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

@@ -469,7 +469,7 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a :
   by
   refine' Isometry.of_dist_eq fun y₁ y₂ => _
   simp only [dist_eq, coe_pos_real_smul, pos_real_im]; congr 2
-  rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
+  rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (hMul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
     Real.norm_eq_abs, mul_left_comm]
   exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
 #align upper_half_plane.isometry_pos_mul UpperHalfPlane.isometry_pos_mul

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

@@ -249,7 +249,7 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
   have hr₀' : 0 ≤ w.im * sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
   have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
   simp only [← cosh_strict_mono_on.cmp_map_eq dist_nonneg hr₀, ←
-    (@strictMonoOn_pow ℝ _ _ two_pos).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
+    (@pow_left_strictMonoOn ℝ _ _ two_pos).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
   rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
 #align upper_half_plane.cmp_dist_eq_cmp_dist_coe_center UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center
 -/

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

@@ -3,9 +3,9 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Analysis.Complex.UpperHalfPlane.Topology
-import Mathbin.Analysis.SpecialFunctions.Arsinh
-import Mathbin.Geometry.Euclidean.Inversion
+import Analysis.Complex.UpperHalfPlane.Topology
+import Analysis.SpecialFunctions.Arsinh
+import Geometry.Euclidean.Inversion
 
 #align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.metric
-! leanprover-community/mathlib commit 9240e8be927a0955b9a82c6c85ef499ee3a626b8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Complex.UpperHalfPlane.Topology
 import Mathbin.Analysis.SpecialFunctions.Arsinh
 import Mathbin.Geometry.Euclidean.Inversion
 
+#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"9240e8be927a0955b9a82c6c85ef499ee3a626b8"
+
 /-!
 # Metric on the upper half-plane
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.metric
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
+! leanprover-community/mathlib commit 9240e8be927a0955b9a82c6c85ef499ee3a626b8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Geometry.Euclidean.Inversion
 /-!
 # Metric on the upper half-plane
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define a `metric_space` structure on the `upper_half_plane`. We use hyperbolic
 (Poincaré) distance given by
 `dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * real.sqrt (z.im * w.im)))` instead of the induced

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

@@ -41,15 +41,20 @@ namespace UpperHalfPlane
 instance : Dist ℍ :=
   ⟨fun z w => 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im)))⟩
 
+#print UpperHalfPlane.dist_eq /-
 theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im))) :=
   rfl
 #align upper_half_plane.dist_eq UpperHalfPlane.dist_eq
+-/
 
+#print UpperHalfPlane.sinh_half_dist /-
 theorem sinh_half_dist (z w : ℍ) :
     sinh (dist z w / 2) = dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) := by
   rw [dist_eq, mul_div_cancel_left (arsinh _) two_ne_zero, sinh_arsinh]
 #align upper_half_plane.sinh_half_dist UpperHalfPlane.sinh_half_dist
+-/
 
+#print UpperHalfPlane.cosh_half_dist /-
 theorem cosh_half_dist (z w : ℍ) :
     cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * sqrt (z.im * w.im)) :=
   by
@@ -63,24 +68,32 @@ theorem cosh_half_dist (z w : ℍ) :
     Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
   ring
 #align upper_half_plane.cosh_half_dist UpperHalfPlane.cosh_half_dist
+-/
 
+#print UpperHalfPlane.tanh_half_dist /-
 theorem tanh_half_dist (z w : ℍ) : tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) :=
   by
   rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
   exact (mul_pos (zero_lt_two' ℝ) (sqrt_pos.2 <| mul_pos z.im_pos w.im_pos)).ne'
 #align upper_half_plane.tanh_half_dist UpperHalfPlane.tanh_half_dist
+-/
 
+#print UpperHalfPlane.exp_half_dist /-
 theorem exp_half_dist (z w : ℍ) :
     exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * sqrt (z.im * w.im)) := by
   rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]
 #align upper_half_plane.exp_half_dist UpperHalfPlane.exp_half_dist
+-/
 
+#print UpperHalfPlane.cosh_dist /-
 theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) := by
   rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
     sq_sqrt (mul_pos z.im_pos w.im_pos).le, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc,
     mul_div_mul_left _ _ (two_ne_zero' ℝ)]
 #align upper_half_plane.cosh_dist UpperHalfPlane.cosh_dist
+-/
 
+#print UpperHalfPlane.sinh_half_dist_add_dist /-
 theorem sinh_half_dist_add_dist (a b c : ℍ) :
     sinh ((dist a b + dist b c) / 2) =
       (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
@@ -94,21 +107,29 @@ theorem sinh_half_dist_add_dist (a b c : ℍ) :
       mul_comm] <;>
     exact (im_pos _).le
 #align upper_half_plane.sinh_half_dist_add_dist UpperHalfPlane.sinh_half_dist_add_dist
+-/
 
+#print UpperHalfPlane.dist_comm /-
 protected theorem dist_comm (z w : ℍ) : dist z w = dist w z := by
   simp only [dist_eq, dist_comm (z : ℂ), mul_comm]
 #align upper_half_plane.dist_comm UpperHalfPlane.dist_comm
+-/
 
+#print UpperHalfPlane.dist_le_iff_le_sinh /-
 theorem dist_le_iff_le_sinh :
     dist z w ≤ r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) ≤ sinh (r / 2) := by
   rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
 #align upper_half_plane.dist_le_iff_le_sinh UpperHalfPlane.dist_le_iff_le_sinh
+-/
 
+#print UpperHalfPlane.dist_eq_iff_eq_sinh /-
 theorem dist_eq_iff_eq_sinh :
     dist z w = r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) = sinh (r / 2) := by
   rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist]
 #align upper_half_plane.dist_eq_iff_eq_sinh UpperHalfPlane.dist_eq_iff_eq_sinh
+-/
 
+#print UpperHalfPlane.dist_eq_iff_eq_sq_sinh /-
 theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
     dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 :=
   by
@@ -118,7 +139,9 @@ theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
   · exact div_nonneg dist_nonneg (mul_nonneg zero_le_two <| sqrt_nonneg _)
   · exact sinh_nonneg_iff.2 (div_nonneg hr zero_le_two)
 #align upper_half_plane.dist_eq_iff_eq_sq_sinh UpperHalfPlane.dist_eq_iff_eq_sq_sinh
+-/
 
+#print UpperHalfPlane.dist_triangle /-
 protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c :=
   by
   rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
@@ -130,13 +153,17 @@ protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c
   · rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
     exact b.im_ne_zero
 #align upper_half_plane.dist_triangle UpperHalfPlane.dist_triangle
+-/
 
+#print UpperHalfPlane.dist_le_dist_coe_div_sqrt /-
 theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) :=
   by
   rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
   exact div_nonneg dist_nonneg (mul_nonneg zero_le_two (sqrt_nonneg _))
 #align upper_half_plane.dist_le_dist_coe_div_sqrt UpperHalfPlane.dist_le_dist_coe_div_sqrt
+-/
 
+#print UpperHalfPlane.metricSpaceAux /-
 /-- An auxiliary `metric_space` instance on the upper half-plane. This instance has bad projection
 to `topological_space`. We replace it later. -/
 def metricSpaceAux : MetricSpace ℍ where
@@ -147,9 +174,11 @@ def metricSpaceAux : MetricSpace ℍ where
   eq_of_dist_eq_zero z w h := by
     simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, Subtype.coe_inj] using h
 #align upper_half_plane.metric_space_aux UpperHalfPlane.metricSpaceAux
+-/
 
 open Complex
 
+#print UpperHalfPlane.cosh_dist' /-
 theorem cosh_dist' (z w : ℍ) :
     Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) :=
   by
@@ -157,27 +186,37 @@ theorem cosh_dist' (z w : ℍ) :
   field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, norm_sq_apply, H, H.ne']
   ring
 #align upper_half_plane.cosh_dist' UpperHalfPlane.cosh_dist'
+-/
 
+#print UpperHalfPlane.center /-
 /-- Euclidean center of the circle with center `z` and radius `r` in the hyperbolic metric. -/
 def center (z : ℍ) (r : ℝ) : ℍ :=
   ⟨⟨z.re, z.im * cosh r⟩, mul_pos z.im_pos (cosh_pos _)⟩
 #align upper_half_plane.center UpperHalfPlane.center
+-/
 
+#print UpperHalfPlane.center_re /-
 @[simp]
 theorem center_re (z r) : (center z r).re = z.re :=
   rfl
 #align upper_half_plane.center_re UpperHalfPlane.center_re
+-/
 
+#print UpperHalfPlane.center_im /-
 @[simp]
 theorem center_im (z r) : (center z r).im = z.im * cosh r :=
   rfl
 #align upper_half_plane.center_im UpperHalfPlane.center_im
+-/
 
+#print UpperHalfPlane.center_zero /-
 @[simp]
 theorem center_zero (z : ℍ) : center z 0 = z :=
   Subtype.ext <| ext rfl <| by rw [coe_im, coe_im, center_im, Real.cosh_zero, mul_one]
 #align upper_half_plane.center_zero UpperHalfPlane.center_zero
+-/
 
+#print UpperHalfPlane.dist_coe_center_sq /-
 theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) :
     dist (z : ℂ) (w.center r) ^ 2 =
       2 * z.im * w.im * (cosh (dist z w) - cosh r) + (w.im * sinh r) ^ 2 :=
@@ -188,13 +227,17 @@ theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) :
     sq]
   ring
 #align upper_half_plane.dist_coe_center_sq UpperHalfPlane.dist_coe_center_sq
+-/
 
+#print UpperHalfPlane.dist_coe_center /-
 theorem dist_coe_center (z w : ℍ) (r : ℝ) :
     dist (z : ℂ) (w.center r) =
       sqrt (2 * z.im * w.im * (cosh (dist z w) - cosh r) + (w.im * sinh r) ^ 2) :=
   by rw [← sqrt_sq dist_nonneg, dist_coe_center_sq]
 #align upper_half_plane.dist_coe_center UpperHalfPlane.dist_coe_center
+-/
 
+#print UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center /-
 theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
     cmp (dist z w) r = cmp (dist (z : ℂ) (w.center r)) (w.im * sinh r) :=
   by
@@ -209,39 +252,55 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
     (@strictMonoOn_pow ℝ _ _ two_pos).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
   rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
 #align upper_half_plane.cmp_dist_eq_cmp_dist_coe_center UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center
+-/
 
+#print UpperHalfPlane.dist_eq_iff_dist_coe_center_eq /-
 theorem dist_eq_iff_dist_coe_center_eq : dist z w = r ↔ dist (z : ℂ) (w.center r) = w.im * sinh r :=
   eq_iff_eq_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r)
 #align upper_half_plane.dist_eq_iff_dist_coe_center_eq UpperHalfPlane.dist_eq_iff_dist_coe_center_eq
+-/
 
+#print UpperHalfPlane.dist_self_center /-
 @[simp]
 theorem dist_self_center (z : ℍ) (r : ℝ) : dist (z : ℂ) (z.center r) = z.im * (cosh r - 1) :=
   by
   rw [dist_of_re_eq (z.center_re r).symm, dist_comm, Real.dist_eq, mul_sub, mul_one]
   exact abs_of_nonneg (sub_nonneg.2 <| le_mul_of_one_le_right z.im_pos.le (one_le_cosh _))
 #align upper_half_plane.dist_self_center UpperHalfPlane.dist_self_center
+-/
 
+#print UpperHalfPlane.dist_center_dist /-
 @[simp]
 theorem dist_center_dist (z w : ℍ) : dist (z : ℂ) (w.center (dist z w)) = w.im * sinh (dist z w) :=
   dist_eq_iff_dist_coe_center_eq.1 rfl
 #align upper_half_plane.dist_center_dist UpperHalfPlane.dist_center_dist
+-/
 
+#print UpperHalfPlane.dist_lt_iff_dist_coe_center_lt /-
 theorem dist_lt_iff_dist_coe_center_lt : dist z w < r ↔ dist (z : ℂ) (w.center r) < w.im * sinh r :=
   lt_iff_lt_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r)
 #align upper_half_plane.dist_lt_iff_dist_coe_center_lt UpperHalfPlane.dist_lt_iff_dist_coe_center_lt
+-/
 
+#print UpperHalfPlane.lt_dist_iff_lt_dist_coe_center /-
 theorem lt_dist_iff_lt_dist_coe_center : r < dist z w ↔ w.im * sinh r < dist (z : ℂ) (w.center r) :=
   lt_iff_lt_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 <| cmp_dist_eq_cmp_dist_coe_center z w r)
 #align upper_half_plane.lt_dist_iff_lt_dist_coe_center UpperHalfPlane.lt_dist_iff_lt_dist_coe_center
+-/
 
+#print UpperHalfPlane.dist_le_iff_dist_coe_center_le /-
 theorem dist_le_iff_dist_coe_center_le : dist z w ≤ r ↔ dist (z : ℂ) (w.center r) ≤ w.im * sinh r :=
   le_iff_le_of_cmp_eq_cmp (cmp_dist_eq_cmp_dist_coe_center z w r)
 #align upper_half_plane.dist_le_iff_dist_coe_center_le UpperHalfPlane.dist_le_iff_dist_coe_center_le
+-/
 
+#print UpperHalfPlane.le_dist_iff_le_dist_coe_center /-
 theorem le_dist_iff_le_dist_coe_center : r < dist z w ↔ w.im * sinh r < dist (z : ℂ) (w.center r) :=
   lt_iff_lt_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 <| cmp_dist_eq_cmp_dist_coe_center z w r)
 #align upper_half_plane.le_dist_iff_le_dist_coe_center UpperHalfPlane.le_dist_iff_le_dist_coe_center
+-/
 
+#print UpperHalfPlane.dist_of_re_eq /-
 /-- For two points on the same vertical line, the distance is equal to the distance between the
 logarithms of their imaginary parts. -/
 theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) :=
@@ -256,7 +315,9 @@ theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im)
   field_simp [z.im_pos, w.im_pos, z.im_ne_zero, w.im_ne_zero]
   ring
 #align upper_half_plane.dist_of_re_eq UpperHalfPlane.dist_of_re_eq
+-/
 
+#print UpperHalfPlane.dist_log_im_le /-
 /-- Hyperbolic distance between two points is greater than or equal to the distance between the
 logarithms of their imaginary parts. -/
 theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
@@ -270,17 +331,23 @@ theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
             simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w))
         zero_le_two
 #align upper_half_plane.dist_log_im_le UpperHalfPlane.dist_log_im_le
+-/
 
+#print UpperHalfPlane.im_le_im_mul_exp_dist /-
 theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * exp (dist z w) :=
   by
   rw [← div_le_iff' w.im_pos, ← exp_log z.im_pos, ← exp_log w.im_pos, ← Real.exp_sub, exp_le_exp]
   exact (le_abs_self _).trans (dist_log_im_le z w)
 #align upper_half_plane.im_le_im_mul_exp_dist UpperHalfPlane.im_le_im_mul_exp_dist
+-/
 
+#print UpperHalfPlane.im_div_exp_dist_le /-
 theorem im_div_exp_dist_le (z w : ℍ) : z.im / exp (dist z w) ≤ w.im :=
   (div_le_iff (exp_pos _)).2 (im_le_im_mul_exp_dist z w)
 #align upper_half_plane.im_div_exp_dist_le UpperHalfPlane.im_div_exp_dist_le
+-/
 
+#print UpperHalfPlane.dist_coe_le /-
 /-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
 and the imaginary part of one of the points. -/
 theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (exp (dist z w) - 1) :=
@@ -290,7 +357,9 @@ theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (exp (dist z w) -
     _ = w.im * (exp (dist z w) - 1) := by
       rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, Real.sinh_add_cosh]
 #align upper_half_plane.dist_coe_le UpperHalfPlane.dist_coe_le
+-/
 
+#print UpperHalfPlane.le_dist_coe /-
 /-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
 and the imaginary part of one of the points. -/
 theorem le_dist_coe (z w : ℍ) : w.im * (1 - exp (-dist z w)) ≤ dist (z : ℂ) w :=
@@ -300,6 +369,7 @@ theorem le_dist_coe (z w : ℍ) : w.im * (1 - exp (-dist z w)) ≤ dist (z : ℂ
       by rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; ring
     _ ≤ dist (z : ℂ) w := sub_le_iff_le_add.2 <| dist_triangle _ _ _
 #align upper_half_plane.le_dist_coe UpperHalfPlane.le_dist_coe
+-/
 
 /-- The hyperbolic metric on the upper half plane. We ensure that the projection to
 `topological_space` is definitionally equal to the subtype topology. -/
@@ -324,6 +394,7 @@ instance : MetricSpace ℍ :=
       refine' fun w hw => (dist_coe_le w z).trans_lt _
       rwa [← lt_div_iff' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
 
+#print UpperHalfPlane.im_pos_of_dist_center_le /-
 theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w (center z r) ≤ z.im * sinh r) :
     0 < w.im :=
   calc
@@ -332,7 +403,9 @@ theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w (cent
     _ ≤ (z.center r).im - dist (z.center r : ℂ) w := (sub_le_sub_left (by rwa [dist_comm]) _)
     _ ≤ w.im := sub_le_comm.1 <| (le_abs_self _).trans (abs_im_le_abs <| z.center r - w)
 #align upper_half_plane.im_pos_of_dist_center_le UpperHalfPlane.im_pos_of_dist_center_le
+-/
 
+#print UpperHalfPlane.image_coe_closedBall /-
 theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
     (coe : ℍ → ℂ) '' closedBall z r = closedBall (z.center r) (z.im * sinh r) :=
   by
@@ -343,7 +416,9 @@ theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
     lift w to ℍ using im_pos_of_dist_center_le hw
     exact mem_image_of_mem _ (dist_le_iff_dist_coe_center_le.2 hw)
 #align upper_half_plane.image_coe_closed_ball UpperHalfPlane.image_coe_closedBall
+-/
 
+#print UpperHalfPlane.image_coe_ball /-
 theorem image_coe_ball (z : ℍ) (r : ℝ) :
     (coe : ℍ → ℂ) '' ball z r = ball (z.center r) (z.im * sinh r) :=
   by
@@ -354,7 +429,9 @@ theorem image_coe_ball (z : ℍ) (r : ℝ) :
     lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closed_ball hw)
     exact mem_image_of_mem _ (dist_lt_iff_dist_coe_center_lt.2 hw)
 #align upper_half_plane.image_coe_ball UpperHalfPlane.image_coe_ball
+-/
 
+#print UpperHalfPlane.image_coe_sphere /-
 theorem image_coe_sphere (z : ℍ) (r : ℝ) :
     (coe : ℍ → ℂ) '' sphere z r = sphere (z.center r) (z.im * sinh r) :=
   by
@@ -365,23 +442,29 @@ theorem image_coe_sphere (z : ℍ) (r : ℝ) :
     lift w to ℍ using im_pos_of_dist_center_le (sphere_subset_closed_ball hw)
     exact mem_image_of_mem _ (dist_eq_iff_dist_coe_center_eq.2 hw)
 #align upper_half_plane.image_coe_sphere UpperHalfPlane.image_coe_sphere
+-/
 
 instance : ProperSpace ℍ := by
   refine' ⟨fun z r => _⟩
   rw [← inducing_coe.is_compact_iff, image_coe_closed_ball]
   apply is_compact_closed_ball
 
+#print UpperHalfPlane.isometry_vertical_line /-
 theorem isometry_vertical_line (a : ℝ) : Isometry fun y => mk ⟨a, exp y⟩ (exp_pos y) :=
   by
   refine' Isometry.of_dist_eq fun y₁ y₂ => _
   rw [dist_of_re_eq]
   exacts [congr_arg₂ _ (log_exp _) (log_exp _), rfl]
 #align upper_half_plane.isometry_vertical_line UpperHalfPlane.isometry_vertical_line
+-/
 
+#print UpperHalfPlane.isometry_real_vadd /-
 theorem isometry_real_vadd (a : ℝ) : Isometry ((· +ᵥ ·) a : ℍ → ℍ) :=
   Isometry.of_dist_eq fun y₁ y₂ => by simp only [dist_eq, coe_vadd, vadd_im, dist_add_left]
 #align upper_half_plane.isometry_real_vadd UpperHalfPlane.isometry_real_vadd
+-/
 
+#print UpperHalfPlane.isometry_pos_mul /-
 theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a : ℍ → ℍ) :=
   by
   refine' Isometry.of_dist_eq fun y₁ y₂ => _
@@ -390,6 +473,7 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a :
     Real.norm_eq_abs, mul_left_comm]
   exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
 #align upper_half_plane.isometry_pos_mul UpperHalfPlane.isometry_pos_mul
+-/
 
 /-- `SL(2, ℝ)` acts on the upper half plane as an isometry.-/
 instance : IsometricSMul SL(2, ℝ) ℍ :=

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

@@ -403,9 +403,9 @@ instance : IsometricSMul SL(2, ℝ) ℍ :=
         simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im,
           div_one, Complex.inv_im, Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg,
           mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero, one_ne_zero, or_false_iff,
-          dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ← map_mul, ←
-          Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply, mul_div (2 : ℝ),
-          div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
+          dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ←
+          AbsoluteValue.map_mul, ← Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply,
+          mul_div (2 : ℝ), div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
     by_cases hc : g 1 0 = 0
     · obtain ⟨u, v, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_eq_zero g hc
       rw [h]

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

@@ -269,7 +269,6 @@ theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
           div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _)) <| by
             simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w))
         zero_le_two
-    
 #align upper_half_plane.dist_log_im_le UpperHalfPlane.dist_log_im_le
 
 theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * exp (dist z w) :=
@@ -290,7 +289,6 @@ theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (exp (dist z w) -
       dist_triangle_right _ _ _
     _ = w.im * (exp (dist z w) - 1) := by
       rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, Real.sinh_add_cosh]
-    
 #align upper_half_plane.dist_coe_le UpperHalfPlane.dist_coe_le
 
 /-- An upper estimate on the complex distance between two points in terms of the hyperbolic distance
@@ -301,7 +299,6 @@ theorem le_dist_coe (z w : ℍ) : w.im * (1 - exp (-dist z w)) ≤ dist (z : ℂ
         dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) :=
       by rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; ring
     _ ≤ dist (z : ℂ) w := sub_le_iff_le_add.2 <| dist_triangle _ _ _
-    
 #align upper_half_plane.le_dist_coe UpperHalfPlane.le_dist_coe
 
 /-- The hyperbolic metric on the upper half plane. We ensure that the projection to
@@ -334,7 +331,6 @@ theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w (cent
     _ = (z.center r).im - z.im * sinh r := (mul_sub _ _ _)
     _ ≤ (z.center r).im - dist (z.center r : ℂ) w := (sub_le_sub_left (by rwa [dist_comm]) _)
     _ ≤ w.im := sub_le_comm.1 <| (le_abs_self _).trans (abs_im_le_abs <| z.center r - w)
-    
 #align upper_half_plane.im_pos_of_dist_center_le UpperHalfPlane.im_pos_of_dist_center_le
 
 theorem image_coe_closedBall (z : ℍ) (r : ℝ) :

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

@@ -201,7 +201,7 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
   letI := metric_space_aux
   cases' lt_or_le r 0 with hr₀ hr₀
   · trans Ordering.gt
-    exacts[(hr₀.trans_le dist_nonneg).cmp_eq_gt,
+    exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
       ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
   have hr₀' : 0 ≤ w.im * sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
   have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
@@ -249,7 +249,7 @@ theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im)
   have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
   rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
       sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
-    [skip;exact h]
+    [skip; exact h]
   nth_rw 4 [← abs_of_pos w.im_pos]
   simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
   congr 1
@@ -379,7 +379,7 @@ theorem isometry_vertical_line (a : ℝ) : Isometry fun y => mk ⟨a, exp y⟩ (
   by
   refine' Isometry.of_dist_eq fun y₁ y₂ => _
   rw [dist_of_re_eq]
-  exacts[congr_arg₂ _ (log_exp _) (log_exp _), rfl]
+  exacts [congr_arg₂ _ (log_exp _) (log_exp _), rfl]
 #align upper_half_plane.isometry_vertical_line UpperHalfPlane.isometry_vertical_line
 
 theorem isometry_real_vadd (a : ℝ) : Isometry ((· +ᵥ ·) a : ℍ → ℍ) :=

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

@@ -30,7 +30,7 @@ ball/sphere with another center and radius.
 
 noncomputable section
 
-open UpperHalfPlane ComplexConjugate NNReal Topology MatrixGroups
+open scoped UpperHalfPlane ComplexConjugate NNReal Topology MatrixGroups
 
 open Set Metric Filter Real
 

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

@@ -299,9 +299,7 @@ theorem le_dist_coe (z w : ℍ) : w.im * (1 - exp (-dist z w)) ≤ dist (z : ℂ
   calc
     w.im * (1 - exp (-dist z w)) =
         dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) :=
-      by
-      rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]
-      ring
+      by rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; ring
     _ ≤ dist (z : ℂ) w := sub_le_iff_le_add.2 <| dist_triangle _ _ _
     
 #align upper_half_plane.le_dist_coe UpperHalfPlane.le_dist_coe

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -249,7 +249,7 @@ theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im)
   have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
   rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
       sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
-    [skip, exact h]
+    [skip;exact h]
   nth_rw 4 [← abs_of_pos w.im_pos]
   simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
   congr 1

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -401,7 +401,7 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a :
 instance : IsometricSMul SL(2, ℝ) ℍ :=
   ⟨fun g =>
     by
-    have h₀ : Isometry (fun z => ModularGroup.s • z : ℍ → ℍ) :=
+    have h₀ : Isometry (fun z => ModularGroup.S • z : ℍ → ℍ) :=
       Isometry.of_dist_eq fun y₁ y₂ =>
         by
         have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le

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: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.metric
-! leanprover-community/mathlib commit f06058e64b7e8397234455038f3f8aec83aaba5a
+! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -127,7 +127,7 @@ protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c
     exact
       div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
         (EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj ↑b))
-  · rw [dist_comm, dist_pos, Ne.def, Complex.eq_conj_iff_im]
+  · rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
     exact b.im_ne_zero
 #align upper_half_plane.dist_triangle UpperHalfPlane.dist_triangle
 

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

@@ -405,7 +405,7 @@ instance : IsometricSMul SL(2, ℝ) ℍ :=
       Isometry.of_dist_eq fun y₁ y₂ =>
         by
         have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le
-        have h₂ : Complex.AbsTheory.Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
+        have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
         simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im,
           div_one, Complex.inv_im, Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg,
           mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero, one_ne_zero, or_false_iff,

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

@@ -141,7 +141,7 @@ theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w /
 to `topological_space`. We replace it later. -/
 def metricSpaceAux : MetricSpace ℍ where
   dist := dist
-  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
+  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, MulZeroClass.mul_zero]
   dist_comm := UpperHalfPlane.dist_comm
   dist_triangle := UpperHalfPlane.dist_triangle
   eq_of_dist_eq_zero z w h := by

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

@@ -405,7 +405,7 @@ instance : IsometricSMul SL(2, ℝ) ℍ :=
       Isometry.of_dist_eq fun y₁ y₂ =>
         by
         have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le
-        have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
+        have h₂ : Complex.AbsTheory.Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
         simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im,
           div_one, Complex.inv_im, Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg,
           mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero, one_ne_zero, or_false_iff,

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -398,7 +398,7 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a :
 #align upper_half_plane.isometry_pos_mul UpperHalfPlane.isometry_pos_mul
 
 /-- `SL(2, ℝ)` acts on the upper half plane as an isometry.-/
-instance : HasIsometricSmul SL(2, ℝ) ℍ :=
+instance : IsometricSMul SL(2, ℝ) ℍ :=
   ⟨fun g =>
     by
     have h₀ : Isometry (fun z => ModularGroup.s • z : ℍ → ℍ) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/62e8311c791f02c47451bf14aa2501048e7c2f33

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.metric
-! leanprover-community/mathlib commit 832a8ba8f10f11fea99367c469ff802e69a5b8ec
+! leanprover-community/mathlib commit f06058e64b7e8397234455038f3f8aec83aaba5a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -30,7 +30,7 @@ ball/sphere with another center and radius.
 
 noncomputable section
 
-open UpperHalfPlane ComplexConjugate NNReal Topology
+open UpperHalfPlane ComplexConjugate NNReal Topology MatrixGroups
 
 open Set Metric Filter Real
 
@@ -397,5 +397,29 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a :
   exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
 #align upper_half_plane.isometry_pos_mul UpperHalfPlane.isometry_pos_mul
 
+/-- `SL(2, ℝ)` acts on the upper half plane as an isometry.-/
+instance : HasIsometricSmul SL(2, ℝ) ℍ :=
+  ⟨fun g =>
+    by
+    have h₀ : Isometry (fun z => ModularGroup.s • z : ℍ → ℍ) :=
+      Isometry.of_dist_eq fun y₁ y₂ =>
+        by
+        have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le
+        have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
+        simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im,
+          div_one, Complex.inv_im, Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg,
+          mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero, one_ne_zero, or_false_iff,
+          dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ← map_mul, ←
+          Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply, mul_div (2 : ℝ),
+          div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
+    by_cases hc : g 1 0 = 0
+    · obtain ⟨u, v, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_eq_zero g hc
+      rw [h]
+      exact (isometry_real_vadd v).comp (isometry_pos_mul u)
+    · obtain ⟨u, v, w, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_ne_zero g hc
+      rw [h]
+      exact
+        (isometry_real_vadd w).comp (h₀.comp <| (isometry_real_vadd v).comp <| isometry_pos_mul u)⟩
+
 end UpperHalfPlane
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -38,7 +38,7 @@ variable {z w : ℍ} {r R : ℝ}
 
 namespace UpperHalfPlane
 
-instance : HasDist ℍ :=
+instance : Dist ℍ :=
   ⟨fun z w => 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im)))⟩
 
 theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im))) :=

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

@@ -333,8 +333,8 @@ theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ} (h : dist w (cent
     0 < w.im :=
   calc
     0 < z.im * (cosh r - sinh r) := mul_pos z.im_pos (sub_pos.2 <| sinh_lt_cosh _)
-    _ = (z.center r).im - z.im * sinh r := mul_sub _ _ _
-    _ ≤ (z.center r).im - dist (z.center r : ℂ) w := sub_le_sub_left (by rwa [dist_comm]) _
+    _ = (z.center r).im - z.im * sinh r := (mul_sub _ _ _)
+    _ ≤ (z.center r).im - dist (z.center r : ℂ) w := (sub_le_sub_left (by rwa [dist_comm]) _)
     _ ≤ w.im := sub_le_comm.1 <| (le_abs_self _).trans (abs_im_le_abs <| z.center r - w)
     
 #align upper_half_plane.im_pos_of_dist_center_le UpperHalfPlane.im_pos_of_dist_center_le

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

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

@@ -305,8 +305,8 @@ theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ}
     (h : dist w (center z r) ≤ z.im * Real.sinh r) : 0 < w.im :=
   calc
     0 < z.im * (Real.cosh r - Real.sinh r) := mul_pos z.im_pos (sub_pos.2 <| sinh_lt_cosh _)
-    _ = (z.center r).im - z.im * Real.sinh r := (mul_sub _ _ _)
-    _ ≤ (z.center r).im - dist (z.center r : ℂ) w := (sub_le_sub_left (by rwa [dist_comm]) _)
+    _ = (z.center r).im - z.im * Real.sinh r := mul_sub _ _ _
+    _ ≤ (z.center r).im - dist (z.center r : ℂ) w := sub_le_sub_left (by rwa [dist_comm]) _
     _ ≤ w.im := sub_le_comm.1 <| (le_abs_self _).trans (abs_im_le_abs <| z.center r - w)
 #align upper_half_plane.im_pos_of_dist_center_le UpperHalfPlane.im_pos_of_dist_center_le
 

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>

@@ -14,7 +14,7 @@ import Mathlib.Geometry.Euclidean.Inversion.Basic
 
 In this file we define a `MetricSpace` structure on the `UpperHalfPlane`. We use hyperbolic
 (Poincaré) distance given by
-`dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * Real.sqrt (z.im * w.im)))` instead of the induced
+`dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * √(z.im * w.im)))` instead of the induced
 Euclidean distance because the hyperbolic distance is invariant under holomorphic automorphisms of
 the upper half-plane. However, we ensure that the projection to `TopologicalSpace` is
 definitionally equal to the induced topological space structure.
@@ -36,19 +36,19 @@ variable {z w : ℍ} {r R : ℝ}
 namespace UpperHalfPlane
 
 instance : Dist ℍ :=
-  ⟨fun z w => 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im)))⟩
+  ⟨fun z w => 2 * arsinh (dist (z : ℂ) w / (2 * √(z.im * w.im)))⟩
 
-theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqrt (z.im * w.im))) :=
+theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * √(z.im * w.im))) :=
   rfl
 #align upper_half_plane.dist_eq UpperHalfPlane.dist_eq
 
 theorem sinh_half_dist (z w : ℍ) :
-    sinh (dist z w / 2) = dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) := by
+    sinh (dist z w / 2) = dist (z : ℂ) w / (2 * √(z.im * w.im)) := by
   rw [dist_eq, mul_div_cancel_left₀ (arsinh _) two_ne_zero, sinh_arsinh]
 #align upper_half_plane.sinh_half_dist UpperHalfPlane.sinh_half_dist
 
 theorem cosh_half_dist (z w : ℍ) :
-    cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * sqrt (z.im * w.im)) := by
+    cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) := by
   rw [← sq_eq_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
   · congr 1
     simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
@@ -64,7 +64,7 @@ theorem tanh_half_dist (z w : ℍ) :
 #align upper_half_plane.tanh_half_dist UpperHalfPlane.tanh_half_dist
 
 theorem exp_half_dist (z w : ℍ) :
-    exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * sqrt (z.im * w.im)) := by
+    exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * √(z.im * w.im)) := by
   rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]
 #align upper_half_plane.exp_half_dist UpperHalfPlane.exp_half_dist
 
@@ -75,12 +75,12 @@ theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2
 
 theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
     (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
-      (2 * sqrt (a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by
+      (2 * √(a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by
   simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
   rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _),
     dist_comm (b : ℂ), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
   congr 2
-  rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (sqrt a.im), mul_mul_mul_comm, mul_self_sqrt,
+  rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (√a.im), mul_mul_mul_comm, mul_self_sqrt,
       mul_comm] <;> exact (im_pos _).le
 #align upper_half_plane.sinh_half_dist_add_dist UpperHalfPlane.sinh_half_dist_add_dist
 
@@ -89,12 +89,12 @@ protected theorem dist_comm (z w : ℍ) : dist z w = dist w z := by
 #align upper_half_plane.dist_comm UpperHalfPlane.dist_comm
 
 theorem dist_le_iff_le_sinh :
-    dist z w ≤ r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) ≤ sinh (r / 2) := by
+    dist z w ≤ r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) ≤ sinh (r / 2) := by
   rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
 #align upper_half_plane.dist_le_iff_le_sinh UpperHalfPlane.dist_le_iff_le_sinh
 
 theorem dist_eq_iff_eq_sinh :
-    dist z w = r ↔ dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) = sinh (r / 2) := by
+    dist z w = r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) = sinh (r / 2) := by
   rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist]
 #align upper_half_plane.dist_eq_iff_eq_sinh UpperHalfPlane.dist_eq_iff_eq_sinh
 
@@ -114,7 +114,7 @@ protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c
     exact b.im_ne_zero
 #align upper_half_plane.dist_triangle UpperHalfPlane.dist_triangle
 
-theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) := by
+theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / √(z.im * w.im) := by
   rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
   positivity
 #align upper_half_plane.dist_le_dist_coe_div_sqrt UpperHalfPlane.dist_le_dist_coe_div_sqrt
@@ -169,7 +169,7 @@ theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^
 #align upper_half_plane.dist_coe_center_sq UpperHalfPlane.dist_coe_center_sq
 
 theorem dist_coe_center (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) =
-    sqrt (2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2) := by
+    √(2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2) := by
   rw [← sqrt_sq dist_nonneg, dist_coe_center_sq]
 #align upper_half_plane.dist_coe_center UpperHalfPlane.dist_coe_center
 
@@ -238,7 +238,6 @@ nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log
   field_simp
   ring
 #align upper_half_plane.dist_of_re_eq UpperHalfPlane.dist_of_re_eq
-
 /-- Hyperbolic distance between two points is greater than or equal to the distance between the
 logarithms of their imaginary parts. -/
 theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
@@ -247,9 +246,9 @@ theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
       Eq.symm <| dist_of_re_eq rfl
     _ ≤ dist z w := by
       simp_rw [dist_eq]
+      dsimp only [coe_mk, mk_im]
       gcongr
-      · simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
-      · simp
+      simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
 #align upper_half_plane.dist_log_im_le UpperHalfPlane.dist_log_im_le
 
 theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * Real.exp (dist z w) := by
@@ -287,7 +286,7 @@ instance : MetricSpace ℍ :=
   metricSpaceAux.replaceTopology <| by
     refine' le_antisymm (continuous_id_iff_le.1 _) _
     · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
-      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x => by positivity
+      have : ∀ x : ℍ × ℍ, 2 * √(x.1.im * x.2.im) ≠ 0 := fun x => by positivity
       -- `continuity` fails to apply `Continuous.div`
       apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
         Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,

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

@@ -364,7 +364,7 @@ theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry (a • · : ℍ 
   exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
 #align upper_half_plane.isometry_pos_mul UpperHalfPlane.isometry_pos_mul
 
-/-- `SL(2, ℝ)` acts on the upper half plane as an isometry.-/
+/-- `SL(2, ℝ)` acts on the upper half plane as an isometry. -/
 instance : IsometricSMul SL(2, ℝ) ℍ :=
   ⟨fun g => by
     have h₀ : Isometry (fun z => ModularGroup.S • z : ℍ → ℍ) :=

@@ -110,7 +110,7 @@ protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c
     div_mul_eq_mul_div]
   · gcongr
     exact EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ))
-  · rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
+  · rw [dist_comm, dist_pos, Ne, Complex.conj_eq_iff_im]
     exact b.im_ne_zero
 #align upper_half_plane.dist_triangle UpperHalfPlane.dist_triangle
 

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 | |

@@ -44,7 +44,7 @@ theorem dist_eq (z w : ℍ) : dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * sqr
 
 theorem sinh_half_dist (z w : ℍ) :
     sinh (dist z w / 2) = dist (z : ℂ) w / (2 * sqrt (z.im * w.im)) := by
-  rw [dist_eq, mul_div_cancel_left (arsinh _) two_ne_zero, sinh_arsinh]
+  rw [dist_eq, mul_div_cancel_left₀ (arsinh _) two_ne_zero, sinh_arsinh]
 #align upper_half_plane.sinh_half_dist UpperHalfPlane.sinh_half_dist
 
 theorem cosh_half_dist (z w : ℍ) :
@@ -163,7 +163,7 @@ theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^
     2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
   have H : 2 * z.im * w.im ≠ 0 := by positivity
   simp only [Complex.dist_eq, Complex.sq_abs, normSq_apply, coe_re, coe_im, center_re, center_im,
-    cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
+    cosh_dist', mul_div_cancel₀ _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
     sq]
   ring
 #align upper_half_plane.dist_coe_center_sq UpperHalfPlane.dist_coe_center_sq

The new names and argument orders match the corresponding * lemmas, which I already took care of in a previous PR.

From LeanAPAP

@@ -245,9 +245,11 @@ theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
   calc
     dist (log z.im) (log w.im) = dist (mk ⟨0, z.im⟩ z.im_pos) (mk ⟨0, w.im⟩ w.im_pos) :=
       Eq.symm <| dist_of_re_eq rfl
-    _ ≤ dist z w := mul_le_mul_of_nonneg_left (arsinh_le_arsinh.2 <|
-      div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _)) <| by
-        simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)) zero_le_two
+    _ ≤ dist z w := by
+      simp_rw [dist_eq]
+      gcongr
+      · simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
+      · simp
 #align upper_half_plane.dist_log_im_le UpperHalfPlane.dist_log_im_le
 
 theorem im_le_im_mul_exp_dist (z w : ℍ) : z.im ≤ w.im * Real.exp (dist z w) := by

@@ -49,21 +49,18 @@ theorem sinh_half_dist (z w : ℍ) :
 
 theorem cosh_half_dist (z w : ℍ) :
     cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * sqrt (z.im * w.im)) := by
-  have H₁ : (2 ^ 2 : ℝ) = 4 := by norm_num1
-  have H₂ : 0 < z.im * w.im := mul_pos z.im_pos w.im_pos
-  have H₃ : 0 < 2 * sqrt (z.im * w.im) := mul_pos two_pos (sqrt_pos.2 H₂)
-  rw [← sq_eq_sq (cosh_pos _).le (div_nonneg dist_nonneg H₃.le), cosh_sq', sinh_half_dist, div_pow,
-    div_pow, one_add_div (pow_ne_zero 2 H₃.ne'), mul_pow, sq_sqrt H₂.le, H₁]
-  congr 1
-  simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
-    Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
-  ring
+  rw [← sq_eq_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
+  · congr 1
+    simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
+      Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
+    ring
+  all_goals positivity
 #align upper_half_plane.cosh_half_dist UpperHalfPlane.cosh_half_dist
 
 theorem tanh_half_dist (z w : ℍ) :
     tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) := by
   rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
-  exact (mul_pos (zero_lt_two' ℝ) (sqrt_pos.2 <| mul_pos z.im_pos w.im_pos)).ne'
+  positivity
 #align upper_half_plane.tanh_half_dist UpperHalfPlane.tanh_half_dist
 
 theorem exp_half_dist (z w : ℍ) :
@@ -73,8 +70,7 @@ theorem exp_half_dist (z w : ℍ) :
 
 theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) := by
   rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
-    sq_sqrt (mul_pos z.im_pos w.im_pos).le, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc,
-    mul_div_mul_left _ _ (two_ne_zero' ℝ)]
+    sq_sqrt, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc, mul_div_mul_left] <;> positivity
 #align upper_half_plane.cosh_dist UpperHalfPlane.cosh_dist
 
 theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
@@ -106,23 +102,21 @@ theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
     dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by
   rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
   · norm_num
-  · exact (mul_pos z.im_pos w.im_pos).le
-  · exact div_nonneg dist_nonneg (mul_nonneg zero_le_two <| sqrt_nonneg _)
-  · exact sinh_nonneg_iff.2 (div_nonneg hr zero_le_two)
+  all_goals positivity
 #align upper_half_plane.dist_eq_iff_eq_sq_sinh UpperHalfPlane.dist_eq_iff_eq_sq_sinh
 
 protected theorem dist_triangle (a b c : ℍ) : dist a c ≤ dist a b + dist b c := by
   rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,
     div_mul_eq_mul_div]
-  · exact div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
-      (EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ)))
+  · gcongr
+    exact EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ))
   · rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
     exact b.im_ne_zero
 #align upper_half_plane.dist_triangle UpperHalfPlane.dist_triangle
 
 theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / sqrt (z.im * w.im) := by
   rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
-  exact div_nonneg dist_nonneg (mul_nonneg zero_le_two (sqrt_nonneg _))
+  positivity
 #align upper_half_plane.dist_le_dist_coe_div_sqrt UpperHalfPlane.dist_le_dist_coe_div_sqrt
 
 /-- An auxiliary `MetricSpace` instance on the upper half-plane. This instance has bad projection
@@ -141,14 +135,13 @@ open Complex
 
 theorem cosh_dist' (z w : ℍ) :
     Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) := by
-  have H : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
-  field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply, H, H.ne']
+  field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply]
   ring
 #align upper_half_plane.cosh_dist' UpperHalfPlane.cosh_dist'
 
 /-- Euclidean center of the circle with center `z` and radius `r` in the hyperbolic metric. -/
 def center (z : ℍ) (r : ℝ) : ℍ :=
-  ⟨⟨z.re, z.im * Real.cosh r⟩, mul_pos z.im_pos (cosh_pos _)⟩
+  ⟨⟨z.re, z.im * Real.cosh r⟩, by positivity⟩
 #align upper_half_plane.center UpperHalfPlane.center
 
 @[simp]
@@ -168,7 +161,7 @@ theorem center_zero (z : ℍ) : center z 0 = z :=
 
 theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
     2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2 := by
-  have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
+  have H : 2 * z.im * w.im ≠ 0 := by positivity
   simp only [Complex.dist_eq, Complex.sq_abs, normSq_apply, coe_re, coe_im, center_re, center_im,
     cosh_dist', mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ←
     sq]
@@ -187,8 +180,8 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
   · trans Ordering.gt
     exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
       ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
-  have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
-  have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
+  have hr₀' : 0 ≤ w.im * Real.sinh r := by positivity
+  have hzw₀ : 0 < 2 * z.im * w.im := by positivity
   simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀, ←
     (pow_left_strictMonoOn two_ne_zero).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
   rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
@@ -235,14 +228,14 @@ theorem le_dist_iff_le_dist_coe_center :
 /-- For two points on the same vertical line, the distance is equal to the distance between the
 logarithms of their imaginary parts. -/
 nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log w.im) := by
-  have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
+  have h₀ : 0 < z.im / w.im := by positivity
   rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero,
     sinh_log h₀, dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
   [skip; exact h]
   nth_rw 4 [← abs_of_pos w.im_pos]
   simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
   congr 1
-  field_simp [z.im_pos, w.im_pos, z.im_ne_zero, w.im_ne_zero]
+  field_simp
   ring
 #align upper_half_plane.dist_of_re_eq UpperHalfPlane.dist_of_re_eq
 
@@ -292,8 +285,7 @@ instance : MetricSpace ℍ :=
   metricSpaceAux.replaceTopology <| by
     refine' le_antisymm (continuous_id_iff_le.1 _) _
     · refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
-      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
-        mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
+      have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x => by positivity
       -- `continuity` fails to apply `Continuous.div`
       apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh,
         Continuous.dist, continuous_coe.comp, continuous_fst, continuous_snd,

The names for lemmas about monotonicity of (a ^ ·) and (· ^ n) were a mess. This PR tidies up everything related by following the naming convention for (a * ·) and (· * b). Namely, (a ^ ·) is pow_right and (· ^ n) is pow_left in lemma names. All lemma renames follow the corresponding multiplication lemma names closely.

Renames

Algebra.GroupPower.Order

• pow_mono → pow_right_mono
• pow_le_pow → pow_le_pow_right
• pow_le_pow_of_le_left → pow_le_pow_left
• pow_lt_pow_of_lt_left → pow_lt_pow_left
• strictMonoOn_pow → pow_left_strictMonoOn
• pow_strictMono_right → pow_right_strictMono
• pow_lt_pow → pow_lt_pow_right
• pow_lt_pow_iff → pow_lt_pow_iff_right
• pow_le_pow_iff → pow_le_pow_iff_right
• self_lt_pow → lt_self_pow
• strictAnti_pow → pow_right_strictAnti
• pow_lt_pow_iff_of_lt_one → pow_lt_pow_iff_right_of_lt_one
• pow_lt_pow_of_lt_one → pow_lt_pow_right_of_lt_one
• lt_of_pow_lt_pow → lt_of_pow_lt_pow_left
• le_of_pow_le_pow → le_of_pow_le_pow_left
• pow_lt_pow₀ → pow_lt_pow_right₀

Algebra.GroupPower.CovariantClass

• pow_le_pow_of_le_left' → pow_le_pow_left'
• nsmul_le_nsmul_of_le_right → nsmul_le_nsmul_right
• pow_lt_pow' → pow_lt_pow_right'
• nsmul_lt_nsmul → nsmul_lt_nsmul_left
• pow_strictMono_left → pow_right_strictMono'
• nsmul_strictMono_right → nsmul_left_strictMono
• StrictMono.pow_right' → StrictMono.pow_const
• StrictMono.nsmul_left → StrictMono.const_nsmul
• pow_strictMono_right' → pow_left_strictMono
• nsmul_strictMono_left → nsmul_right_strictMono
• Monotone.pow_right → Monotone.pow_const
• Monotone.nsmul_left → Monotone.const_nsmul
• lt_of_pow_lt_pow' → lt_of_pow_lt_pow_left'
• lt_of_nsmul_lt_nsmul → lt_of_nsmul_lt_nsmul_right
• pow_le_pow' → pow_le_pow_right'
• nsmul_le_nsmul → nsmul_le_nsmul_left
• pow_le_pow_of_le_one' → pow_le_pow_right_of_le_one'
• nsmul_le_nsmul_of_nonpos → nsmul_le_nsmul_left_of_nonpos
• le_of_pow_le_pow' → le_of_pow_le_pow_left'
• le_of_nsmul_le_nsmul' → le_of_nsmul_le_nsmul_right'
• pow_le_pow_iff' → pow_le_pow_iff_right'
• nsmul_le_nsmul_iff → nsmul_le_nsmul_iff_left
• pow_lt_pow_iff' → pow_lt_pow_iff_right'
• nsmul_lt_nsmul_iff → nsmul_lt_nsmul_iff_left

Data.Nat.Pow

• Nat.pow_lt_pow_of_lt_left → Nat.pow_lt_pow_left
• Nat.pow_le_iff_le_left → Nat.pow_le_pow_iff_left
• Nat.pow_lt_iff_lt_left → Nat.pow_lt_pow_iff_left

Lemmas added

• pow_le_pow_iff_left
• pow_lt_pow_iff_left
• pow_right_injective
• pow_right_inj
• Nat.pow_le_pow_left to have the correct name since Nat.pow_le_pow_of_le_left is in Std.
• Nat.pow_le_pow_right to have the correct name since Nat.pow_le_pow_of_le_right is in Std.

Lemmas removed

• self_le_pow was a duplicate of le_self_pow.
• Nat.pow_lt_pow_of_lt_right is defeq to pow_lt_pow_right.
• Nat.pow_right_strictMono is defeq to pow_right_strictMono.
• Nat.pow_le_iff_le_right is defeq to pow_le_pow_iff_right.
• Nat.pow_lt_iff_lt_right is defeq to pow_lt_pow_iff_right.

Other changes

• A bunch of proofs have been golfed.
• Some lemma assumptions have been turned from 0 < n or 1 ≤ n to n ≠ 0.
• A few Nat lemmas have been protected.
• One docstring has been fixed.

@@ -190,7 +190,7 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
   have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
   have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
   simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀, ←
-    (@strictMonoOn_pow ℝ _ _ two_pos).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
+    (pow_left_strictMonoOn two_ne_zero).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
   rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
 #align upper_half_plane.cmp_dist_eq_cmp_dist_coe_center UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center
 

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -358,11 +358,11 @@ theorem isometry_vertical_line (a : ℝ) : Isometry fun y => mk ⟨a, exp y⟩ (
   exacts [congr_arg₂ _ (log_exp _) (log_exp _), rfl]
 #align upper_half_plane.isometry_vertical_line UpperHalfPlane.isometry_vertical_line
 
-theorem isometry_real_vadd (a : ℝ) : Isometry ((· +ᵥ ·) a : ℍ → ℍ) :=
+theorem isometry_real_vadd (a : ℝ) : Isometry (a +ᵥ · : ℍ → ℍ) :=
   Isometry.of_dist_eq fun y₁ y₂ => by simp only [dist_eq, coe_vadd, vadd_im, dist_add_left]
 #align upper_half_plane.isometry_real_vadd UpperHalfPlane.isometry_real_vadd
 
-theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry ((· • ·) a : ℍ → ℍ) := by
+theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry (a • · : ℍ → ℍ) := by
   refine' Isometry.of_dist_eq fun y₁ y₂ => _
   simp only [dist_eq, coe_pos_real_smul, pos_real_im]; congr 2
   rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
@@ -385,11 +385,9 @@ instance : IsometricSMul SL(2, ℝ) ℍ :=
           mul_div (2 : ℝ), div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
     by_cases hc : g 1 0 = 0
     · obtain ⟨u, v, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_eq_zero g hc
-      dsimp only at h
       rw [h]
       exact (isometry_real_vadd v).comp (isometry_pos_mul u)
     · obtain ⟨u, v, w, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_ne_zero g hc
-      dsimp only at h
       rw [h]
       exact
         (isometry_real_vadd w).comp (h₀.comp <| (isometry_real_vadd v).comp <| isometry_pos_mul u)⟩

Define sigma-compact subsets of a topological space and show their basic properties.

• compact sets are sigma-compact
• countable unions of (sigma-)compact sets are sigma-compact
• closed subsets of sigma-compact sets are sigma-compact.

Relate them to sigma-compact space: a set is sigma-compact iff it is a sigma-compact space (w.r.t. the subspace topology).

In a later PR, we'll show that sigma-compact measure zero sets are nowhere dense.

• depends on: #7528

Co-authored-by: David Loeffler <d.loeffler.01@cantab.net> Co-authored-by: grunweg <grunweg@posteo.de>

@@ -349,7 +349,7 @@ theorem image_coe_sphere (z : ℍ) (r : ℝ) :
 
 instance : ProperSpace ℍ := by
   refine' ⟨fun z r => _⟩
-  rw [← inducing_subtype_val.isCompact_iff (f := ((↑) : ℍ → ℂ)), image_coe_closedBall]
+  rw [inducing_subtype_val.isCompact_iff (f := ((↑) : ℍ → ℂ)), image_coe_closedBall]
   apply isCompact_closedBall
 
 theorem isometry_vertical_line (a : ℝ) : Isometry fun y => mk ⟨a, exp y⟩ (exp_pos y) := by

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

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

@@ -129,7 +129,7 @@ theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w /
 to `TopologicalSpace`. We replace it later. -/
 def metricSpaceAux : MetricSpace ℍ where
   dist := dist
-  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, MulZeroClass.mul_zero]
+  dist_self z := by rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
   dist_comm := UpperHalfPlane.dist_comm
   dist_triangle := UpperHalfPlane.dist_triangle
   eq_of_dist_eq_zero {z w} h := by

• 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 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.metric
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
 import Mathlib.Analysis.SpecialFunctions.Arsinh
 import Mathlib.Geometry.Euclidean.Inversion.Basic
 
+#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
+
 /-!
 # Metric on the upper half-plane
 

Move theorems about image of a sphere passing through the center to a new file.

@@ -10,7 +10,7 @@ Authors: Yury Kudryashov
 -/
 import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
 import Mathlib.Analysis.SpecialFunctions.Arsinh
-import Mathlib.Geometry.Euclidean.Inversion
+import Mathlib.Geometry.Euclidean.Inversion.Basic
 
 /-!
 # Metric on the upper half-plane

Also define UpperHalfPlane.coe so that type synonym doesn't leak through API.

@@ -136,7 +136,7 @@ def metricSpaceAux : MetricSpace ℍ where
   dist_comm := UpperHalfPlane.dist_comm
   dist_triangle := UpperHalfPlane.dist_triangle
   eq_of_dist_eq_zero {z w} h := by
-    simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, Subtype.coe_inj] using h
+    simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, ext_iff] using h
   edist_dist _ _ := by exact ENNReal.coe_nnreal_eq _
 #align upper_half_plane.metric_space_aux UpperHalfPlane.metricSpaceAux
 
@@ -166,7 +166,7 @@ theorem center_im (z r) : (center z r).im = z.im * Real.cosh r :=
 
 @[simp]
 theorem center_zero (z : ℍ) : center z 0 = z :=
-  Subtype.ext <| Complex.ext rfl <| by rw [coe_im, coe_im, center_im, Real.cosh_zero, mul_one]
+  ext' rfl <| by rw [center_im, Real.cosh_zero, mul_one]
 #align upper_half_plane.center_zero UpperHalfPlane.center_zero
 
 theorem dist_coe_center_sq (z w : ℍ) (r : ℝ) : dist (z : ℂ) (w.center r) ^ 2 =
@@ -253,8 +253,8 @@ nonrec theorem dist_of_re_eq (h : z.re = w.re) : dist z w = dist (log z.im) (log
 logarithms of their imaginary parts. -/
 theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
   calc
-    dist (log z.im) (log w.im) = @dist ℍ _ ⟨⟨0, z.im⟩, z.im_pos⟩ ⟨⟨0, w.im⟩, w.im_pos⟩ :=
-      Eq.symm <| @dist_of_re_eq ⟨⟨0, z.im⟩, z.im_pos⟩ ⟨⟨0, w.im⟩, w.im_pos⟩ rfl
+    dist (log z.im) (log w.im) = dist (mk ⟨0, z.im⟩ z.im_pos) (mk ⟨0, w.im⟩ w.im_pos) :=
+      Eq.symm <| dist_of_re_eq rfl
     _ ≤ dist z w := mul_le_mul_of_nonneg_left (arsinh_le_arsinh.2 <|
       div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _)) <| by
         simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)) zero_le_two