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

@@ -141,16 +141,16 @@ theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_
 #align complex.cpow_neg_one Complex.cpow_neg_one
 -/
 
-#print Complex.cpow_nat_cast /-
+#print Complex.cpow_natCast /-
 @[simp, norm_cast]
-theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
+theorem cpow_natCast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
   | n + 1 =>
     if hx : x = 0 then by
       simp only [hx, pow_succ', Complex.zero_cpow (Nat.cast_ne_zero.2 (Nat.succ_ne_zero _)),
         MulZeroClass.zero_mul]
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast n]
-#align complex.cpow_nat_cast Complex.cpow_nat_cast
+#align complex.cpow_nat_cast Complex.cpow_natCast
 -/
 
 #print Complex.cpow_two /-
@@ -160,15 +160,15 @@ theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 := by rw [← cpow_nat_cast];
 #align complex.cpow_two Complex.cpow_two
 -/
 
-#print Complex.cpow_int_cast /-
+#print Complex.cpow_intCast /-
 @[simp, norm_cast]
-theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
+theorem cpow_intCast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
   | -[n+1] => by
     rw [zpow_negSucc] <;>
       simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_natCast,
         cpow_nat_cast]
-#align complex.cpow_int_cast Complex.cpow_int_cast
+#align complex.cpow_int_cast Complex.cpow_intCast
 -/
 
 #print Complex.cpow_nat_inv_pow /-
@@ -276,11 +276,11 @@ section Tactics
 namespace NormNum
 
 theorem cpow_pos (a b : ℂ) (b' : ℕ) (c : ℂ) (hb : b = b') (h : a ^ b' = c) : a ^ b = c := by
-  rw [← h, hb, Complex.cpow_nat_cast]
+  rw [← h, hb, Complex.cpow_natCast]
 #align norm_num.cpow_pos NormNum.cpow_pos
 
 theorem cpow_neg (a b : ℂ) (b' : ℕ) (c c' : ℂ) (hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') :
-    a ^ (-b) = c' := by rw [← hc, ← h, hb, Complex.cpow_neg, Complex.cpow_nat_cast]
+    a ^ (-b) = c' := by rw [← hc, ← h, hb, Complex.cpow_neg, Complex.cpow_natCast]
 #align norm_num.cpow_neg NormNum.cpow_neg
 
 open Tactic

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

@@ -166,7 +166,7 @@ theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
   | -[n+1] => by
     rw [zpow_negSucc] <;>
-      simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_ofNat,
+      simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_natCast,
         cpow_nat_cast]
 #align complex.cpow_int_cast Complex.cpow_int_cast
 -/

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

@@ -147,7 +147,7 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
   | n + 1 =>
     if hx : x = 0 then by
-      simp only [hx, pow_succ, Complex.zero_cpow (Nat.cast_ne_zero.2 (Nat.succ_ne_zero _)),
+      simp only [hx, pow_succ', Complex.zero_cpow (Nat.cast_ne_zero.2 (Nat.succ_ne_zero _)),
         MulZeroClass.zero_mul]
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast n]
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
@@ -218,7 +218,7 @@ theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ :=
   by
   simp_rw [Complex.cpow_def, log_inv_eq_ite, inv_eq_zero, map_eq_zero, ite_mul, neg_mul,
-    IsROrC.conj_inv, apply_ite conj, apply_ite NormedSpace.exp, apply_ite Inv.inv, map_zero,
+    RCLike.conj_inv, apply_ite conj, apply_ite NormedSpace.exp, apply_ite Inv.inv, map_zero,
     map_one, NormedSpace.exp_neg, inv_one, inv_zero, ← NormedSpace.exp_conj, map_mul, conj_conj]
   split_ifs with hx hn ha ha <;> rfl
 #align complex.inv_cpow_eq_ite Complex.inv_cpow_eq_ite

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

@@ -78,10 +78,10 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 
   by
   constructor
   · intro hyp
-    simp only [cpow_def, eq_self_iff_true, if_true] at hyp 
+    simp only [cpow_def, eq_self_iff_true, if_true] at hyp
     by_cases x = 0
-    · subst h; simp only [if_true, eq_self_iff_true] at hyp ; right; exact ⟨rfl, hyp.symm⟩
-    · rw [if_neg h] at hyp ; left; exact ⟨h, hyp.symm⟩
+    · subst h; simp only [if_true, eq_self_iff_true] at hyp; right; exact ⟨rfl, hyp.symm⟩
+    · rw [if_neg h] at hyp; left; exact ⟨h, hyp.symm⟩
   · rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
     · exact zero_cpow h
     · exact cpow_zero _
@@ -179,7 +179,7 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
     rw [← cpow_nat_cast, ← cpow_mul _ this.1 this.2, inv_mul_cancel, cpow_one]
     exact_mod_cast hn
   rw [mul_comm, ← of_real_nat_cast, ← of_real_inv, of_real_mul_im, ← div_eq_inv_mul]
-  rw [← pos_iff_ne_zero] at hn 
+  rw [← pos_iff_ne_zero] at hn
   have hn' : 0 < (n : ℝ) := by assumption_mod_cast
   have hn1 : 1 ≤ (n : ℝ) := by exact_mod_cast Nat.succ_le_iff.2 hn
   constructor

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

@@ -111,7 +111,8 @@ theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by
 
 #print Complex.cpow_add /-
 theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by
-  simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] <;> simp_all [exp_add, mul_add]
+  simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] <;>
+    simp_all [NormedSpace.exp_add, mul_add]
 #align complex.cpow_add Complex.cpow_add
 -/
 
@@ -125,7 +126,7 @@ theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (lo
 
 #print Complex.cpow_neg /-
 theorem cpow_neg (x y : ℂ) : x ^ (-y) = (x ^ y)⁻¹ := by
-  simp only [cpow_def, neg_eq_zero, mul_neg] <;> split_ifs <;> simp [exp_neg]
+  simp only [cpow_def, neg_eq_zero, mul_neg] <;> split_ifs <;> simp [NormedSpace.exp_neg]
 #align complex.cpow_neg Complex.cpow_neg
 -/
 
@@ -208,7 +209,7 @@ theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
   have ha'' : (a : ℂ) ≠ 0 := of_real_ne_zero.mpr ha'.ne'
   have hb'' : (b : ℂ) ≠ 0 := of_real_ne_zero.mpr hb'.ne'
   rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_of_real_mul ha' hb'', of_real_log ha,
-    add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
+    add_mul, NormedSpace.exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
 #align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
 -/
 
@@ -217,8 +218,8 @@ theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ :=
   by
   simp_rw [Complex.cpow_def, log_inv_eq_ite, inv_eq_zero, map_eq_zero, ite_mul, neg_mul,
-    IsROrC.conj_inv, apply_ite conj, apply_ite exp, apply_ite Inv.inv, map_zero, map_one, exp_neg,
-    inv_one, inv_zero, ← exp_conj, map_mul, conj_conj]
+    IsROrC.conj_inv, apply_ite conj, apply_ite NormedSpace.exp, apply_ite Inv.inv, map_zero,
+    map_one, NormedSpace.exp_neg, inv_one, inv_zero, ← NormedSpace.exp_conj, map_mul, conj_conj]
   split_ifs with hx hn ha ha <;> rfl
 #align complex.inv_cpow_eq_ite Complex.inv_cpow_eq_ite
 -/
@@ -245,8 +246,8 @@ theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
 theorem conj_cpow_eq_ite (x : ℂ) (n : ℂ) :
     conj x ^ n = if x.arg = π then x ^ n else conj (x ^ conj n) :=
   by
-  simp_rw [cpow_def, map_eq_zero, apply_ite conj, map_one, map_zero, ← exp_conj, map_mul, conj_conj,
-    log_conj_eq_ite]
+  simp_rw [cpow_def, map_eq_zero, apply_ite conj, map_one, map_zero, ← NormedSpace.exp_conj,
+    map_mul, conj_conj, log_conj_eq_ite]
   split_ifs with hcx hn hx <;> rfl
 #align complex.conj_cpow_eq_ite Complex.conj_cpow_eq_ite
 -/

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
   Rémy Degenne, David Loeffler
 -/
-import Mathbin.Analysis.SpecialFunctions.Complex.Log
+import Analysis.SpecialFunctions.Complex.Log
 
 #align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"33c67ae661dd8988516ff7f247b0be3018cdd952"
 

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

@@ -3,14 +3,11 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
   Rémy Degenne, David Loeffler
-
-! This file was ported from Lean 3 source module analysis.special_functions.pow.complex
-! leanprover-community/mathlib commit 33c67ae661dd8988516ff7f247b0be3018cdd952
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecialFunctions.Complex.Log
 
+#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"33c67ae661dd8988516ff7f247b0be3018cdd952"
+
 /-! # Power function on `ℂ`
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

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

@@ -45,26 +45,36 @@ theorem cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y :=
 #align complex.cpow_eq_pow Complex.cpow_eq_pow
 -/
 
+#print Complex.cpow_def /-
 theorem cpow_def (x y : ℂ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) :=
   rfl
 #align complex.cpow_def Complex.cpow_def
+-/
 
+#print Complex.cpow_def_of_ne_zero /-
 theorem cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) :=
   if_neg hx
 #align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zero
+-/
 
+#print Complex.cpow_zero /-
 @[simp]
 theorem cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def]
 #align complex.cpow_zero Complex.cpow_zero
+-/
 
+#print Complex.cpow_eq_zero_iff /-
 @[simp]
 theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by simp only [cpow_def];
   split_ifs <;> simp [*, exp_ne_zero]
 #align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff
+-/
 
+#print Complex.zero_cpow /-
 @[simp]
 theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
 #align complex.zero_cpow Complex.zero_cpow
+-/
 
 #print Complex.zero_cpow_eq_iff /-
 theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 :=
@@ -87,11 +97,13 @@ theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 
 #align complex.eq_zero_cpow_iff Complex.eq_zero_cpow_iff
 -/
 
+#print Complex.cpow_one /-
 @[simp]
 theorem cpow_one (x : ℂ) : x ^ (1 : ℂ) = x :=
   if hx : x = 0 then by simp [hx, cpow_def]
   else by rw [cpow_def, if_neg (one_ne_zero : (1 : ℂ) ≠ 0), if_neg hx, mul_one, exp_log hx]
 #align complex.cpow_one Complex.cpow_one
+-/
 
 #print Complex.one_cpow /-
 @[simp]
@@ -100,27 +112,38 @@ theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by
 #align complex.one_cpow Complex.one_cpow
 -/
 
+#print Complex.cpow_add /-
 theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by
   simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] <;> simp_all [exp_add, mul_add]
 #align complex.cpow_add Complex.cpow_add
+-/
 
+#print Complex.cpow_mul /-
 theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
     x ^ (y * z) = (x ^ y) ^ z := by
   simp only [cpow_def]
   split_ifs <;> simp_all [exp_ne_zero, log_exp h₁ h₂, mul_assoc]
 #align complex.cpow_mul Complex.cpow_mul
+-/
 
+#print Complex.cpow_neg /-
 theorem cpow_neg (x y : ℂ) : x ^ (-y) = (x ^ y)⁻¹ := by
   simp only [cpow_def, neg_eq_zero, mul_neg] <;> split_ifs <;> simp [exp_neg]
 #align complex.cpow_neg Complex.cpow_neg
+-/
 
+#print Complex.cpow_sub /-
 theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by
   rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]
 #align complex.cpow_sub Complex.cpow_sub
+-/
 
+#print Complex.cpow_neg_one /-
 theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_neg x 1
 #align complex.cpow_neg_one Complex.cpow_neg_one
+-/
 
+#print Complex.cpow_nat_cast /-
 @[simp, norm_cast]
 theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
@@ -130,12 +153,16 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
         MulZeroClass.zero_mul]
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast n]
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
+-/
 
+#print Complex.cpow_two /-
 @[simp]
 theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 := by rw [← cpow_nat_cast];
   simp only [Nat.cast_bit0, Nat.cast_one]
 #align complex.cpow_two Complex.cpow_two
+-/
 
+#print Complex.cpow_int_cast /-
 @[simp, norm_cast]
 theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
@@ -144,7 +171,9 @@ theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
       simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_ofNat,
         cpow_nat_cast]
 #align complex.cpow_int_cast Complex.cpow_int_cast
+-/
 
+#print Complex.cpow_nat_inv_pow /-
 theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x :=
   by
   suffices im (log x * n⁻¹) ∈ Ioc (-π) π
@@ -167,7 +196,9 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
       _ = π * 1 := (mul_one π).symm
       _ ≤ π * n := mul_le_mul_of_nonneg_left hn1 real.pi_pos.le
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
+-/
 
+#print Complex.mul_cpow_ofReal_nonneg /-
 theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r :=
   by
@@ -182,6 +213,7 @@ theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
   rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_of_real_mul ha' hb'', of_real_log ha,
     add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
 #align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
+-/
 
 #print Complex.inv_cpow_eq_ite /-
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
@@ -200,6 +232,7 @@ theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)
 #align complex.inv_cpow Complex.inv_cpow
 -/
 
+#print Complex.inv_cpow_eq_ite' /-
 /-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
     (x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n :=
@@ -209,6 +242,7 @@ theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
   · rfl
   · rw [inv_cpow _ _ h]
 #align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'
+-/
 
 #print Complex.conj_cpow_eq_ite /-
 theorem conj_cpow_eq_ite (x : ℂ) (n : ℂ) :
@@ -226,9 +260,11 @@ theorem conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x
 #align complex.conj_cpow Complex.conj_cpow
 -/
 
+#print Complex.cpow_conj /-
 theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) := by
   rw [conj_cpow _ _ hx, conj_conj]
 #align complex.cpow_conj Complex.cpow_conj
+-/
 
 end Complex
 

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

@@ -168,7 +168,7 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
       _ ≤ π * n := mul_le_mul_of_nonneg_left hn1 real.pi_pos.le
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
-theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
+theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r :=
   by
   rcases eq_or_ne r 0 with (rfl | hr)
@@ -181,7 +181,7 @@ theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
   have hb'' : (b : ℂ) ≠ 0 := of_real_ne_zero.mpr hb'.ne'
   rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_of_real_mul ha' hb'', of_real_log ha,
     add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
-#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_of_real_nonneg
+#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
 
 #print Complex.inv_cpow_eq_ite /-
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :

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

@@ -161,13 +161,11 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
       -π * n ≤ -π * 1 := mul_le_mul_of_nonpos_left hn1 (neg_nonpos.2 real.pi_pos.le)
       _ = -π := (mul_one _)
       _ < im (log x) := neg_pi_lt_log_im _
-      
   · rw [div_le_iff hn']
     calc
       im (log x) ≤ π := log_im_le_pi _
       _ = π * 1 := (mul_one π).symm
       _ ≤ π * n := mul_le_mul_of_nonneg_left hn1 real.pi_pos.le
-      
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
 theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :

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

@@ -71,10 +71,10 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 
   by
   constructor
   · intro hyp
-    simp only [cpow_def, eq_self_iff_true, if_true] at hyp
+    simp only [cpow_def, eq_self_iff_true, if_true] at hyp 
     by_cases x = 0
-    · subst h; simp only [if_true, eq_self_iff_true] at hyp; right; exact ⟨rfl, hyp.symm⟩
-    · rw [if_neg h] at hyp; left; exact ⟨h, hyp.symm⟩
+    · subst h; simp only [if_true, eq_self_iff_true] at hyp ; right; exact ⟨rfl, hyp.symm⟩
+    · rw [if_neg h] at hyp ; left; exact ⟨h, hyp.symm⟩
   · rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
     · exact zero_cpow h
     · exact cpow_zero _
@@ -152,7 +152,7 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
     rw [← cpow_nat_cast, ← cpow_mul _ this.1 this.2, inv_mul_cancel, cpow_one]
     exact_mod_cast hn
   rw [mul_comm, ← of_real_nat_cast, ← of_real_inv, of_real_mul_im, ← div_eq_inv_mul]
-  rw [← pos_iff_ne_zero] at hn
+  rw [← pos_iff_ne_zero] at hn 
   have hn' : 0 < (n : ℝ) := by assumption_mod_cast
   have hn1 : 1 ≤ (n : ℝ) := by exact_mod_cast Nat.succ_le_iff.2 hn
   constructor

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

@@ -20,7 +20,7 @@ We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.
 -/
 
 
-open Classical Real Topology Filter ComplexConjugate
+open scoped Classical Real Topology Filter ComplexConjugate
 
 open Filter Finset Set
 

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

@@ -45,53 +45,23 @@ theorem cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y :=
 #align complex.cpow_eq_pow Complex.cpow_eq_pow
 -/
 
-/- warning: complex.cpow_def -> Complex.cpow_def is a dubious translation:
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-but is expected to have type
-  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (ite.{1} Complex (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.instDecidableEqComplex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (ite.{1} Complex (Eq.{1} Complex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.instDecidableEqComplex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex)) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y)))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_def Complex.cpow_defₓ'. -/
 theorem cpow_def (x y : ℂ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) :=
   rfl
 #align complex.cpow_def Complex.cpow_def
 
-/- warning: complex.cpow_def_of_ne_zero -> Complex.cpow_def_of_ne_zero is a dubious translation:
-lean 3 declaration is
-  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (forall (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (Complex.log x) y)))
-but is expected to have type
-  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) -> (forall (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y)))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zeroₓ'. -/
 theorem cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) :=
   if_neg hx
 #align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zero
 
-/- warning: complex.cpow_zero -> Complex.cpow_zero is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne)))
-but is expected to have type
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_zero Complex.cpow_zeroₓ'. -/
 @[simp]
 theorem cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def]
 #align complex.cpow_zero Complex.cpow_zero
 
-/- warning: complex.cpow_eq_zero_iff -> Complex.cpow_eq_zero_iff is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex) (y : Complex), Iff (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (And (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (Ne.{1} Complex y (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))))
-but is expected to have type
-  forall (x : Complex) (y : Complex), Iff (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (And (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Ne.{1} Complex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iffₓ'. -/
 @[simp]
 theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by simp only [cpow_def];
   split_ifs <;> simp [*, exp_ne_zero]
 #align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff
 
-/- warning: complex.zero_cpow -> Complex.zero_cpow is a dubious translation:
-lean 3 declaration is
-  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero))) x) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complex.zero_cpow Complex.zero_cpowₓ'. -/
 @[simp]
 theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
 #align complex.zero_cpow Complex.zero_cpow
@@ -117,12 +87,6 @@ theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 
 #align complex.eq_zero_cpow_iff Complex.eq_zero_cpow_iff
 -/
 
-/- warning: complex.cpow_one -> Complex.cpow_one is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne)))) x
-but is expected to have type
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex))) x
-Case conversion may be inaccurate. Consider using '#align complex.cpow_one Complex.cpow_oneₓ'. -/
 @[simp]
 theorem cpow_one (x : ℂ) : x ^ (1 : ℂ) = x :=
   if hx : x = 0 then by simp [hx, cpow_def]
@@ -136,63 +100,27 @@ theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by
 #align complex.one_cpow Complex.one_cpow
 -/
 
-/- warning: complex.cpow_add -> Complex.cpow_add is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complex.cpow_add Complex.cpow_addₓ'. -/
 theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by
   simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] <;> simp_all [exp_add, mul_add]
 #align complex.cpow_add Complex.cpow_add
 
-/- warning: complex.cpow_mul -> Complex.cpow_mul is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complex.cpow_mul Complex.cpow_mulₓ'. -/
 theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
     x ^ (y * z) = (x ^ y) ^ z := by
   simp only [cpow_def]
   split_ifs <;> simp_all [exp_ne_zero, log_exp h₁ h₂, mul_assoc]
 #align complex.cpow_mul Complex.cpow_mul
 
-/- warning: complex.cpow_neg -> Complex.cpow_neg is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Neg.neg.{0} Complex Complex.instNegComplex y)) (Inv.inv.{0} Complex Complex.instInvComplex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_neg Complex.cpow_negₓ'. -/
 theorem cpow_neg (x y : ℂ) : x ^ (-y) = (x ^ y)⁻¹ := by
   simp only [cpow_def, neg_eq_zero, mul_neg] <;> split_ifs <;> simp [exp_neg]
 #align complex.cpow_neg Complex.cpow_neg
 
-/- warning: complex.cpow_sub -> Complex.cpow_sub is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complex.cpow_sub Complex.cpow_subₓ'. -/
 theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by
   rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]
 #align complex.cpow_sub Complex.cpow_sub
 
-/- warning: complex.cpow_neg_one -> Complex.cpow_neg_one is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (Neg.neg.{0} Complex Complex.hasNeg (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne))))) (Inv.inv.{0} Complex Complex.hasInv x)
-but is expected to have type
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Neg.neg.{0} Complex Complex.instNegComplex (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex)))) (Inv.inv.{0} Complex Complex.instInvComplex x)
-Case conversion may be inaccurate. Consider using '#align complex.cpow_neg_one Complex.cpow_neg_oneₓ'. -/
 theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_neg x 1
 #align complex.cpow_neg_one Complex.cpow_neg_one
 
-/- warning: complex.cpow_nat_cast -> Complex.cpow_nat_cast is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align complex.cpow_nat_cast Complex.cpow_nat_castₓ'. -/
 @[simp, norm_cast]
 theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
@@ -203,23 +131,11 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast n]
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
 
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-but is expected to have type
-  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 2 (instOfNat.{0} Complex 2 (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
-Case conversion may be inaccurate. Consider using '#align complex.cpow_two Complex.cpow_twoₓ'. -/
 @[simp]
 theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 := by rw [← cpow_nat_cast];
   simp only [Nat.cast_bit0, Nat.cast_one]
 #align complex.cpow_two Complex.cpow_two
 
-/- warning: complex.cpow_int_cast -> Complex.cpow_int_cast is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex) (n : Int), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Int Complex (HasLiftT.mk.{1, 1} Int Complex (CoeTCₓ.coe.{1, 1} Int Complex (Int.castCoe.{0} Complex (AddGroupWithOne.toHasIntCast.{0} Complex Complex.addGroupWithOne)))) n)) (HPow.hPow.{0, 0, 0} Complex Int Complex (instHPow.{0, 0} Complex Int (DivInvMonoid.Pow.{0} Complex (DivisionRing.toDivInvMonoid.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.normedField))))) x n)
-but is expected to have type
-  forall (x : Complex) (n : Int), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Int.cast.{0} Complex (Ring.toIntCast.{0} Complex Complex.instRingComplex) n)) (HPow.hPow.{0, 0, 0} Complex Int Complex (instHPow.{0, 0} Complex Int (DivInvMonoid.Pow.{0} Complex (DivisionRing.toDivInvMonoid.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.instNormedFieldComplex))))) x n)
-Case conversion may be inaccurate. Consider using '#align complex.cpow_int_cast Complex.cpow_int_castₓ'. -/
 @[simp, norm_cast]
 theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
@@ -229,12 +145,6 @@ theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
         cpow_nat_cast]
 #align complex.cpow_int_cast Complex.cpow_int_cast
 
-/- warning: complex.cpow_nat_inv_pow -> Complex.cpow_nat_inv_pow is a dubious translation:
-lean 3 declaration is
-  forall (x : Complex) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (Ring.toMonoid.{0} Complex Complex.ring))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (Inv.inv.{0} Complex Complex.hasInv ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Complex (HasLiftT.mk.{1, 1} Nat Complex (CoeTCₓ.coe.{1, 1} Nat Complex (Nat.castCoe.{0} Complex (AddMonoidWithOne.toNatCast.{0} Complex (AddGroupWithOne.toAddMonoidWithOne.{0} Complex Complex.addGroupWithOne))))) n))) n) x)
-but is expected to have type
-  forall (x : Complex) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Inv.inv.{0} Complex Complex.instInvComplex (Nat.cast.{0} Complex (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) n))) n) x)
-Case conversion may be inaccurate. Consider using '#align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_powₓ'. -/
 theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x :=
   by
   suffices im (log x * n⁻¹) ∈ Ioc (-π) π
@@ -260,12 +170,6 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
       
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
-/- warning: complex.mul_cpow_of_real_nonneg -> Complex.mul_cpow_of_real_nonneg is a dubious translation:
-lean 3 declaration is
-  forall {a : Real} {b : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) b) -> (forall (r : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) a) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) b)) r) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) a) r) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) b) r)))
-but is expected to have type
-  forall {a : Real} {b : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) b) -> (forall (r : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.ofReal' a) (Complex.ofReal' b)) r) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Complex.ofReal' a) r) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Complex.ofReal' b) r)))
-Case conversion may be inaccurate. Consider using '#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_of_real_nonnegₓ'. -/
 theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r :=
   by
@@ -298,9 +202,6 @@ theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)
 #align complex.inv_cpow Complex.inv_cpow
 -/
 
-/- warning: complex.inv_cpow_eq_ite' -> Complex.inv_cpow_eq_ite' is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'ₓ'. -/
 /-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
     (x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n :=
@@ -327,9 +228,6 @@ theorem conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x
 #align complex.conj_cpow Complex.conj_cpow
 -/
 
-/- warning: complex.cpow_conj -> Complex.cpow_conj is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.cpow_conj Complex.cpow_conjₓ'. -/
 theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) := by
   rw [conj_cpow _ _ hx, conj_conj]
 #align complex.cpow_conj Complex.cpow_conj

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

@@ -82,9 +82,7 @@ but is expected to have type
   forall (x : Complex) (y : Complex), Iff (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (And (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Ne.{1} Complex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))))
 Case conversion may be inaccurate. Consider using '#align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iffₓ'. -/
 @[simp]
-theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 :=
-  by
-  simp only [cpow_def]
+theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by simp only [cpow_def];
   split_ifs <;> simp [*, exp_ne_zero]
 #align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff
 
@@ -105,13 +103,8 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 
   · intro hyp
     simp only [cpow_def, eq_self_iff_true, if_true] at hyp
     by_cases x = 0
-    · subst h
-      simp only [if_true, eq_self_iff_true] at hyp
-      right
-      exact ⟨rfl, hyp.symm⟩
-    · rw [if_neg h] at hyp
-      left
-      exact ⟨h, hyp.symm⟩
+    · subst h; simp only [if_true, eq_self_iff_true] at hyp; right; exact ⟨rfl, hyp.symm⟩
+    · rw [if_neg h] at hyp; left; exact ⟨h, hyp.symm⟩
   · rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
     · exact zero_cpow h
     · exact cpow_zero _
@@ -217,9 +210,7 @@ but is expected to have type
   forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 2 (instOfNat.{0} Complex 2 (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
 Case conversion may be inaccurate. Consider using '#align complex.cpow_two Complex.cpow_twoₓ'. -/
 @[simp]
-theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 :=
-  by
-  rw [← cpow_nat_cast]
+theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 := by rw [← cpow_nat_cast];
   simp only [Nat.cast_bit0, Nat.cast_one]
 #align complex.cpow_two Complex.cpow_two
 

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

@@ -308,10 +308,7 @@ theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'ₓ'. -/
 /-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
@@ -340,10 +337,7 @@ theorem conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x
 -/
 
 /- warning: complex.cpow_conj -> Complex.cpow_conj is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align complex.cpow_conj Complex.cpow_conjₓ'. -/
 theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) := by
   rw [conj_cpow _ _ hx, conj_conj]

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

@@ -5,7 +5,7 @@ Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébasti
   Rémy Degenne, David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.special_functions.pow.complex
-! leanprover-community/mathlib commit 4fa54b337f7d52805480306db1b1439c741848c8
+! leanprover-community/mathlib commit 33c67ae661dd8988516ff7f247b0be3018cdd952
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.SpecialFunctions.Complex.Log
 
 /-! # Power function on `ℂ`
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.
 -/
 

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

@@ -308,7 +308,7 @@ theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)
 lean 3 declaration is
   forall (x : Complex) (n : Complex), Eq.{1} Complex (Inv.inv.{0} Complex Complex.hasInv (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x n)) (ite.{1} Complex (Eq.{1} Real (Complex.arg x) Real.pi) (Real.decidableEq (Complex.arg x) Real.pi) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (Inv.inv.{0} Complex Complex.hasInv x) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) n))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (Inv.inv.{0} Complex Complex.hasInv x) n))
 but is expected to have type
-  forall (x : Complex) (n : Complex), Eq.{1} Complex (Inv.inv.{0} Complex Complex.instInvComplex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x n)) (ite.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (a : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) a) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n))) (Eq.{1} Real (Complex.arg x) Real.pi) (Real.decidableEq (Complex.arg x) Real.pi) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) n))
+  forall (x : Complex) (n : Complex), Eq.{1} Complex (Inv.inv.{0} Complex Complex.instInvComplex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x n)) (ite.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (a : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) a) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n))) (Eq.{1} Real (Complex.arg x) Real.pi) (Real.decidableEq (Complex.arg x) Real.pi) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) n))
 Case conversion may be inaccurate. Consider using '#align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'ₓ'. -/
 /-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
@@ -340,7 +340,7 @@ theorem conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x
 lean 3 declaration is
   forall (x : Complex) (n : Complex), (Ne.{1} Real (Complex.arg x) Real.pi) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) n)) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) x) n)))
 but is expected to have type
-  forall (x : Complex) (n : Complex), (Ne.{1} Real (Complex.arg x) Real.pi) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex.instPowComplex) x (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n)) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (fun (_x : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (NonUnitalNonAssocSemiring.toMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)))))) (starRingEnd.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (instHPow.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex Complex.instPowComplex) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) x) n)))
+  forall (x : Complex) (n : Complex), (Ne.{1} Real (Complex.arg x) Real.pi) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) n) Complex.instPowComplex) x (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n)) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (fun (_x : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (NonUnitalNonAssocSemiring.toMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex)))))) (starRingEnd.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) (instHPow.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) x) Complex Complex.instPowComplex) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) x) n)))
 Case conversion may be inaccurate. Consider using '#align complex.cpow_conj Complex.cpow_conjₓ'. -/
 theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) := by
   rw [conj_cpow _ _ hx, conj_conj]

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

@@ -23,33 +23,61 @@ open Filter Finset Set
 
 namespace Complex
 
+#print Complex.cpow /-
 /-- The complex power function `x ^ y`, given by `x ^ y = exp(y log x)` (where `log` is the
 principal determination of the logarithm), unless `x = 0` where one sets `0 ^ 0 = 1` and
 `0 ^ y = 0` for `y ≠ 0`. -/
 noncomputable def cpow (x y : ℂ) : ℂ :=
   if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)
 #align complex.cpow Complex.cpow
+-/
 
 noncomputable instance : Pow ℂ ℂ :=
   ⟨cpow⟩
 
+#print Complex.cpow_eq_pow /-
 @[simp]
 theorem cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y :=
   rfl
 #align complex.cpow_eq_pow Complex.cpow_eq_pow
+-/
 
+/- warning: complex.cpow_def -> Complex.cpow_def is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (ite.{1} Complex (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (Complex.decidableEq x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (ite.{1} Complex (Eq.{1} Complex y (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (Complex.decidableEq y (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne))) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (Complex.log x) y)))
+but is expected to have type
+  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (ite.{1} Complex (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.instDecidableEqComplex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (ite.{1} Complex (Eq.{1} Complex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.instDecidableEqComplex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex)) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_def Complex.cpow_defₓ'. -/
 theorem cpow_def (x y : ℂ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) :=
   rfl
 #align complex.cpow_def Complex.cpow_def
 
+/- warning: complex.cpow_def_of_ne_zero -> Complex.cpow_def_of_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (forall (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (Complex.log x) y)))
+but is expected to have type
+  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) -> (forall (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zeroₓ'. -/
 theorem cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) :=
   if_neg hx
 #align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zero
 
+/- warning: complex.cpow_zero -> Complex.cpow_zero is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne)))
+but is expected to have type
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_zero Complex.cpow_zeroₓ'. -/
 @[simp]
 theorem cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def]
 #align complex.cpow_zero Complex.cpow_zero
 
+/- warning: complex.cpow_eq_zero_iff -> Complex.cpow_eq_zero_iff is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (y : Complex), Iff (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (And (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) (Ne.{1} Complex y (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))))
+but is expected to have type
+  forall (x : Complex) (y : Complex), Iff (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (And (Eq.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) (Ne.{1} Complex y (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iffₓ'. -/
 @[simp]
 theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 :=
   by
@@ -57,10 +85,17 @@ theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 :=
   split_ifs <;> simp [*, exp_ne_zero]
 #align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff
 
+/- warning: complex.zero_cpow -> Complex.zero_cpow is a dubious translation:
+lean 3 declaration is
+  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero))) x) (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero))))
+but is expected to have type
+  forall {x : Complex}, (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex)) x) (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex)))
+Case conversion may be inaccurate. Consider using '#align complex.zero_cpow Complex.zero_cpowₓ'. -/
 @[simp]
 theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
 #align complex.zero_cpow Complex.zero_cpow
 
+#print Complex.zero_cpow_eq_iff /-
 theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 :=
   by
   constructor
@@ -78,43 +113,90 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 
     · exact zero_cpow h
     · exact cpow_zero _
 #align complex.zero_cpow_eq_iff Complex.zero_cpow_eq_iff
+-/
 
+#print Complex.eq_zero_cpow_iff /-
 theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
   rw [← zero_cpow_eq_iff, eq_comm]
 #align complex.eq_zero_cpow_iff Complex.eq_zero_cpow_iff
+-/
 
+/- warning: complex.cpow_one -> Complex.cpow_one is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne)))) x
+but is expected to have type
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex))) x
+Case conversion may be inaccurate. Consider using '#align complex.cpow_one Complex.cpow_oneₓ'. -/
 @[simp]
 theorem cpow_one (x : ℂ) : x ^ (1 : ℂ) = x :=
   if hx : x = 0 then by simp [hx, cpow_def]
   else by rw [cpow_def, if_neg (one_ne_zero : (1 : ℂ) ≠ 0), if_neg hx, mul_one, exp_log hx]
 #align complex.cpow_one Complex.cpow_one
 
+#print Complex.one_cpow /-
 @[simp]
 theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by
   rw [cpow_def] <;> split_ifs <;> simp_all [one_ne_zero]
 #align complex.one_cpow Complex.one_cpow
+-/
 
+/- warning: complex.cpow_add -> Complex.cpow_add is a dubious translation:
+lean 3 declaration is
+  forall {x : Complex} (y : Complex) (z : Complex), (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.hasAdd) y z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x z)))
+but is expected to have type
+  forall {x : Complex} (y : Complex) (z : Complex), (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAddComplex) y z)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x z)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_add Complex.cpow_addₓ'. -/
 theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by
   simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] <;> simp_all [exp_add, mul_add]
 #align complex.cpow_add Complex.cpow_add
 
+/- warning: complex.cpow_mul -> Complex.cpow_mul is a dubious translation:
+lean 3 declaration is
+  forall {x : Complex} {y : Complex} (z : Complex), (LT.lt.{0} Real Real.hasLt (Neg.neg.{0} Real Real.hasNeg Real.pi) (Complex.im (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (Complex.log x) y))) -> (LE.le.{0} Real Real.hasLe (Complex.im (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (Complex.log x) y)) Real.pi) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) y z)) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) z))
+but is expected to have type
+  forall {x : Complex} {y : Complex} (z : Complex), (LT.lt.{0} Real Real.instLTReal (Neg.neg.{0} Real Real.instNegReal Real.pi) (Complex.im (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y))) -> (LE.le.{0} Real Real.instLEReal (Complex.im (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.log x) y)) Real.pi) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) y z)) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) z))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_mul Complex.cpow_mulₓ'. -/
 theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
     x ^ (y * z) = (x ^ y) ^ z := by
   simp only [cpow_def]
   split_ifs <;> simp_all [exp_ne_zero, log_exp h₁ h₂, mul_assoc]
 #align complex.cpow_mul Complex.cpow_mul
 
+/- warning: complex.cpow_neg -> Complex.cpow_neg is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (Neg.neg.{0} Complex Complex.hasNeg y)) (Inv.inv.{0} Complex Complex.hasInv (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y))
+but is expected to have type
+  forall (x : Complex) (y : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Neg.neg.{0} Complex Complex.instNegComplex y)) (Inv.inv.{0} Complex Complex.instInvComplex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_neg Complex.cpow_negₓ'. -/
 theorem cpow_neg (x y : ℂ) : x ^ (-y) = (x ^ y)⁻¹ := by
   simp only [cpow_def, neg_eq_zero, mul_neg] <;> split_ifs <;> simp [exp_neg]
 #align complex.cpow_neg Complex.cpow_neg
 
+/- warning: complex.cpow_sub -> Complex.cpow_sub is a dubious translation:
+lean 3 declaration is
+  forall {x : Complex} (y : Complex) (z : Complex), (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (OfNat.mk.{0} Complex 0 (Zero.zero.{0} Complex Complex.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.hasSub) y z)) (HDiv.hDiv.{0, 0, 0} Complex Complex Complex (instHDiv.{0} Complex (DivInvMonoid.toHasDiv.{0} Complex (DivisionRing.toDivInvMonoid.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.normedField))))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x y) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x z)))
+but is expected to have type
+  forall {x : Complex} (y : Complex) (z : Complex), (Ne.{1} Complex x (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZeroComplex))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSubComplex) y z)) (HDiv.hDiv.{0, 0, 0} Complex Complex Complex (instHDiv.{0} Complex (Field.toDiv.{0} Complex Complex.instFieldComplex)) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x y) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x z)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_sub Complex.cpow_subₓ'. -/
 theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by
   rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]
 #align complex.cpow_sub Complex.cpow_sub
 
+/- warning: complex.cpow_neg_one -> Complex.cpow_neg_one is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (Neg.neg.{0} Complex Complex.hasNeg (OfNat.ofNat.{0} Complex 1 (OfNat.mk.{0} Complex 1 (One.one.{0} Complex Complex.hasOne))))) (Inv.inv.{0} Complex Complex.hasInv x)
+but is expected to have type
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Neg.neg.{0} Complex Complex.instNegComplex (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOneComplex)))) (Inv.inv.{0} Complex Complex.instInvComplex x)
+Case conversion may be inaccurate. Consider using '#align complex.cpow_neg_one Complex.cpow_neg_oneₓ'. -/
 theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_neg x 1
 #align complex.cpow_neg_one Complex.cpow_neg_one
 
+/- warning: complex.cpow_nat_cast -> Complex.cpow_nat_cast is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (n : Nat), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Complex (HasLiftT.mk.{1, 1} Nat Complex (CoeTCₓ.coe.{1, 1} Nat Complex (Nat.castCoe.{0} Complex (AddMonoidWithOne.toNatCast.{0} Complex (AddGroupWithOne.toAddMonoidWithOne.{0} Complex Complex.addGroupWithOne))))) n)) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (Ring.toMonoid.{0} Complex Complex.ring))) x n)
+but is expected to have type
+  forall (x : Complex) (n : Nat), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Nat.cast.{0} Complex (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) n)) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) x n)
+Case conversion may be inaccurate. Consider using '#align complex.cpow_nat_cast Complex.cpow_nat_castₓ'. -/
 @[simp, norm_cast]
 theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
@@ -125,6 +207,12 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast n]
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
 
+/- warning: complex.cpow_two -> Complex.cpow_two is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (OfNat.ofNat.{0} Complex 2 (OfNat.mk.{0} Complex 2 (bit0.{0} Complex Complex.hasAdd (One.one.{0} Complex Complex.hasOne))))) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (Ring.toMonoid.{0} Complex Complex.ring))) x (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
+but is expected to have type
+  forall (x : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (OfNat.ofNat.{0} Complex 2 (instOfNat.{0} Complex 2 (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_two Complex.cpow_twoₓ'. -/
 @[simp]
 theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 :=
   by
@@ -132,6 +220,12 @@ theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ 2 :=
   simp only [Nat.cast_bit0, Nat.cast_one]
 #align complex.cpow_two Complex.cpow_two
 
+/- warning: complex.cpow_int_cast -> Complex.cpow_int_cast is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (n : Int), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Int Complex (HasLiftT.mk.{1, 1} Int Complex (CoeTCₓ.coe.{1, 1} Int Complex (Int.castCoe.{0} Complex (AddGroupWithOne.toHasIntCast.{0} Complex Complex.addGroupWithOne)))) n)) (HPow.hPow.{0, 0, 0} Complex Int Complex (instHPow.{0, 0} Complex Int (DivInvMonoid.Pow.{0} Complex (DivisionRing.toDivInvMonoid.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.normedField))))) x n)
+but is expected to have type
+  forall (x : Complex) (n : Int), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Int.cast.{0} Complex (Ring.toIntCast.{0} Complex Complex.instRingComplex) n)) (HPow.hPow.{0, 0, 0} Complex Int Complex (instHPow.{0, 0} Complex Int (DivInvMonoid.Pow.{0} Complex (DivisionRing.toDivInvMonoid.{0} Complex (NormedDivisionRing.toDivisionRing.{0} Complex (NormedField.toNormedDivisionRing.{0} Complex Complex.instNormedFieldComplex))))) x n)
+Case conversion may be inaccurate. Consider using '#align complex.cpow_int_cast Complex.cpow_int_castₓ'. -/
 @[simp, norm_cast]
 theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
@@ -141,6 +235,12 @@ theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
         cpow_nat_cast]
 #align complex.cpow_int_cast Complex.cpow_int_cast
 
+/- warning: complex.cpow_nat_inv_pow -> Complex.cpow_nat_inv_pow is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (Ring.toMonoid.{0} Complex Complex.ring))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x (Inv.inv.{0} Complex Complex.hasInv ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Complex (HasLiftT.mk.{1, 1} Nat Complex (CoeTCₓ.coe.{1, 1} Nat Complex (Nat.castCoe.{0} Complex (AddMonoidWithOne.toNatCast.{0} Complex (AddGroupWithOne.toAddMonoidWithOne.{0} Complex Complex.addGroupWithOne))))) n))) n) x)
+but is expected to have type
+  forall (x : Complex) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.Pow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiringComplex)))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x (Inv.inv.{0} Complex Complex.instInvComplex (Nat.cast.{0} Complex (Semiring.toNatCast.{0} Complex Complex.instSemiringComplex) n))) n) x)
+Case conversion may be inaccurate. Consider using '#align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_powₓ'. -/
 theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x :=
   by
   suffices im (log x * n⁻¹) ∈ Ioc (-π) π
@@ -166,6 +266,12 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
       
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
+/- warning: complex.mul_cpow_of_real_nonneg -> Complex.mul_cpow_of_real_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {a : Real} {b : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) b) -> (forall (r : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) a) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) b)) r) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.hasMul) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) a) r) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Real Complex (HasLiftT.mk.{1, 1} Real Complex (CoeTCₓ.coe.{1, 1} Real Complex (coeBase.{1, 1} Real Complex Complex.hasCoe))) b) r)))
+but is expected to have type
+  forall {a : Real} {b : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) b) -> (forall (r : Complex), Eq.{1} Complex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (Complex.ofReal' a) (Complex.ofReal' b)) r) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMulComplex) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Complex.ofReal' a) r) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Complex.ofReal' b) r)))
+Case conversion may be inaccurate. Consider using '#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_of_real_nonnegₓ'. -/
 theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r :=
   by
@@ -181,6 +287,7 @@ theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
     add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
 #align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_of_real_nonneg
 
+#print Complex.inv_cpow_eq_ite /-
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ :=
   by
@@ -189,11 +296,20 @@ theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     inv_one, inv_zero, ← exp_conj, map_mul, conj_conj]
   split_ifs with hx hn ha ha <;> rfl
 #align complex.inv_cpow_eq_ite Complex.inv_cpow_eq_ite
+-/
 
+#print Complex.inv_cpow /-
 theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)⁻¹ := by
   rw [inv_cpow_eq_ite, if_neg hx]
 #align complex.inv_cpow Complex.inv_cpow
+-/
 
+/- warning: complex.inv_cpow_eq_ite' -> Complex.inv_cpow_eq_ite' is a dubious translation:
+lean 3 declaration is
+  forall (x : Complex) (n : Complex), Eq.{1} Complex (Inv.inv.{0} Complex Complex.hasInv (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) x n)) (ite.{1} Complex (Eq.{1} Real (Complex.arg x) Real.pi) (Real.decidableEq (Complex.arg x) Real.pi) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (Inv.inv.{0} Complex Complex.hasInv x) (coeFn.{1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (fun (_x : RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) => Complex -> Complex) (RingHom.hasCoeToFun.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.commSemiring))) (starRingEnd.{0} Complex Complex.commSemiring Complex.starRing) n))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.hasPow) (Inv.inv.{0} Complex Complex.hasInv x) n))
+but is expected to have type
+  forall (x : Complex) (n : Complex), Eq.{1} Complex (Inv.inv.{0} Complex Complex.instInvComplex (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) x n)) (ite.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (a : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) a) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) 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(CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex (instHPow.{0, 0} Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) n) Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} 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(CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) n))) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPowComplex) (Inv.inv.{0} Complex Complex.instInvComplex x) n))
+Case conversion may be inaccurate. Consider using '#align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'ₓ'. -/
 /-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
     (x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n :=
@@ -204,6 +320,7 @@ theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
   · rw [inv_cpow _ _ h]
 #align complex.inv_cpow_eq_ite' Complex.inv_cpow_eq_ite'
 
+#print Complex.conj_cpow_eq_ite /-
 theorem conj_cpow_eq_ite (x : ℂ) (n : ℂ) :
     conj x ^ n = if x.arg = π then x ^ n else conj (x ^ conj n) :=
   by
@@ -211,11 +328,20 @@ theorem conj_cpow_eq_ite (x : ℂ) (n : ℂ) :
     log_conj_eq_ite]
   split_ifs with hcx hn hx <;> rfl
 #align complex.conj_cpow_eq_ite Complex.conj_cpow_eq_ite
+-/
 
+#print Complex.conj_cpow /-
 theorem conj_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : conj x ^ n = conj (x ^ conj n) := by
   rw [conj_cpow_eq_ite, if_neg hx]
 #align complex.conj_cpow Complex.conj_cpow
+-/
 
+/- warning: complex.cpow_conj -> Complex.cpow_conj is a dubious translation:
+lean 3 declaration is
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Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (CommSemiring.toSemiring.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex)))))) (starRingEnd.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) (HPow.hPow.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) (instHPow.{0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) x) Complex Complex.instPowComplex) (FunLike.coe.{1, 1, 1} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex (fun (_x : Complex) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Complex) => Complex) _x) (MulHomClass.toFunLike.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalNonAssocSemiring.toMul.{0} Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))) (NonUnitalRingHomClass.toMulHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) (RingHomClass.toNonUnitalRingHomClass.{0, 0, 0} (RingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex))) Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (RingHom.instRingHomClassRingHom.{0, 0} Complex Complex (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)) (Semiring.toNonAssocSemiring.{0} Complex (CommSemiring.toSemiring.{0} Complex Complex.instCommSemiringComplex)))))) (starRingEnd.{0} Complex Complex.instCommSemiringComplex Complex.instStarRingComplexToNonUnitalSemiringToNonUnitalCommSemiringToNonUnitalCommRingCommRing) x) n)))
+Case conversion may be inaccurate. Consider using '#align complex.cpow_conj Complex.cpow_conjₓ'. -/
 theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (conj x ^ n) := by
   rw [conj_cpow _ _ hx, conj_conj]
 #align complex.cpow_conj Complex.cpow_conj

mathlib commit https://github.com/leanprover-community/mathlib/commit/f8c79b0a623404854a2902b836eac32156fd7712

@@ -5,7 +5,7 @@ Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébasti
   Rémy Degenne, David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.special_functions.pow.complex
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
+! leanprover-community/mathlib commit 4fa54b337f7d52805480306db1b1439c741848c8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -222,3 +222,61 @@ theorem cpow_conj (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x ^ conj n = conj (c
 
 end Complex
 
+section Tactics
+
+/-!
+## Tactic extensions for complex powers
+-/
+
+
+namespace NormNum
+
+theorem cpow_pos (a b : ℂ) (b' : ℕ) (c : ℂ) (hb : b = b') (h : a ^ b' = c) : a ^ b = c := by
+  rw [← h, hb, Complex.cpow_nat_cast]
+#align norm_num.cpow_pos NormNum.cpow_pos
+
+theorem cpow_neg (a b : ℂ) (b' : ℕ) (c c' : ℂ) (hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') :
+    a ^ (-b) = c' := by rw [← hc, ← h, hb, Complex.cpow_neg, Complex.cpow_nat_cast]
+#align norm_num.cpow_neg NormNum.cpow_neg
+
+open Tactic
+
+/-- Generalized version of `prove_cpow`, `prove_nnrpow`, `prove_ennrpow`. -/
+unsafe def prove_rpow' (pos neg zero : Name) (α β one a b : expr) : tactic (expr × expr) := do
+  let na ← a.to_rat
+  let icα ← mk_instance_cache α
+  let icβ ← mk_instance_cache β
+  match match_sign b with
+    | Sum.inl b => do
+      let nc ← mk_instance_cache q(ℕ)
+      let (icβ, nc, b', hb) ← prove_nat_uncast icβ nc b
+      let (icα, c, h) ← prove_pow a na icα b'
+      let cr ← c
+      let (icα, c', hc) ← prove_inv icα c cr
+      pure (c', (expr.const neg []).mk_app [a, b, b', c, c', hb, h, hc])
+    | Sum.inr ff => pure (one, expr.const zero [] a)
+    | Sum.inr tt => do
+      let nc ← mk_instance_cache q(ℕ)
+      let (icβ, nc, b', hb) ← prove_nat_uncast icβ nc b
+      let (icα, c, h) ← prove_pow a na icα b'
+      pure (c, (expr.const Pos []).mk_app [a, b, b', c, hb, h])
+#align norm_num.prove_rpow' norm_num.prove_rpow'
+
+/-- Evaluate `complex.cpow a b` where `a` is a rational numeral and `b` is an integer. -/
+unsafe def prove_cpow : expr → expr → tactic (expr × expr) :=
+  prove_rpow' `` cpow_pos `` cpow_neg `` Complex.cpow_zero q(ℂ) q(ℂ) q((1 : ℂ))
+#align norm_num.prove_cpow norm_num.prove_cpow
+
+/-- Evaluates expressions of the form `cpow a b` and `a ^ b` in the special case where
+`b` is an integer and `a` is a positive rational (so it's really just a rational power). -/
+@[norm_num]
+unsafe def eval_cpow : expr → tactic (expr × expr)
+  | q(@Pow.pow _ _ Complex.hasPow $(a) $(b)) => b.to_int >> prove_cpow a b
+  | q(Complex.cpow $(a) $(b)) => b.to_int >> prove_cpow a b
+  | _ => tactic.failed
+#align norm_num.eval_cpow norm_num.eval_cpow
+
+end NormNum
+
+end Tactics
+

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -138,21 +138,21 @@ lemma cpow_mul_ofNat (x y : ℂ) (n : ℕ) [n.AtLeastTwo] :
   cpow_mul_nat x y n
 
 @[simp, norm_cast]
-theorem cpow_nat_cast (x : ℂ) (n : ℕ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_nat_mul x n 1
-#align complex.cpow_nat_cast Complex.cpow_nat_cast
+theorem cpow_natCast (x : ℂ) (n : ℕ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_nat_mul x n 1
+#align complex.cpow_nat_cast Complex.cpow_natCast
 
 /-- See Note [no_index around OfNat.ofNat] -/
 @[simp]
 lemma cpow_ofNat (x : ℂ) (n : ℕ) [n.AtLeastTwo] :
     x ^ (no_index (OfNat.ofNat n) : ℂ) = x ^ (OfNat.ofNat n : ℕ) :=
-  cpow_nat_cast x n
+  cpow_natCast x n
 
 theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ (2 : ℕ) := cpow_ofNat x 2
 #align complex.cpow_two Complex.cpow_two
 
 @[simp, norm_cast]
-theorem cpow_int_cast (x : ℂ) (n : ℤ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_int_mul x n 1
-#align complex.cpow_int_cast Complex.cpow_int_cast
+theorem cpow_intCast (x : ℂ) (n : ℤ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_int_mul x n 1
+#align complex.cpow_int_cast Complex.cpow_intCast
 
 @[simp]
 theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x := by
@@ -173,7 +173,7 @@ because the equality fails, e.g., for `x = -I`, `n = 2`, `y = 1/2`. -/
 lemma cpow_int_mul' {x : ℂ} {n : ℤ} (hlt : -π < n * x.arg) (hle : n * x.arg ≤ π) (y : ℂ) :
     x ^ (n * y) = (x ^ n) ^ y := by
   rw [mul_comm] at hlt hle
-  rw [cpow_mul, cpow_int_cast] <;> simpa [log_im]
+  rw [cpow_mul, cpow_intCast] <;> simpa [log_im]
 
 /-- A version of `Complex.cpow_nat_mul` with RHS that matches `Complex.cpow_mul`.
 
@@ -219,7 +219,7 @@ theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
 #align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
 
 lemma natCast_mul_natCast_cpow (m n : ℕ) (s : ℂ) : (m * n : ℂ) ^ s = m ^ s * n ^ s :=
-  ofReal_nat_cast m ▸ ofReal_nat_cast n ▸ mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg s
+  ofReal_natCast m ▸ ofReal_natCast n ▸ mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg s
 
 lemma natCast_cpow_natCast_mul (n m : ℕ) (z : ℂ) : (n : ℂ) ^ (m * z) = ((n : ℂ) ^ m) ^ z := by
   refine cpow_nat_mul' (x := n) (n := m) ?_ ?_ z
@@ -274,11 +274,11 @@ end Complex
 -- namespace NormNum
 
 -- theorem cpow_pos (a b : ℂ) (b' : ℕ) (c : ℂ) (hb : b = b') (h : a ^ b' = c) : a ^ b = c := by
---   rw [← h, hb, Complex.cpow_nat_cast]
+--   rw [← h, hb, Complex.cpow_natCast]
 -- #align norm_num.cpow_pos NormNum.cpow_pos
 
 -- theorem cpow_neg (a b : ℂ) (b' : ℕ) (c c' : ℂ) (hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') :
---     a ^ (-b) = c' := by rw [← hc, ← h, hb, Complex.cpow_neg, Complex.cpow_nat_cast]
+--     a ^ (-b) = c' := by rw [← hc, ← h, hb, Complex.cpow_neg, Complex.cpow_natCast]
 -- #align norm_num.cpow_neg NormNum.cpow_neg
 
 -- open Tactic

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -229,7 +229,7 @@ lemma natCast_cpow_natCast_mul (n m : ℕ) (z : ℂ) : (n : ℂ) ^ (m * z) = ((n
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ := by
   simp_rw [Complex.cpow_def, log_inv_eq_ite, inv_eq_zero, map_eq_zero, ite_mul, neg_mul,
-    IsROrC.conj_inv, apply_ite conj, apply_ite exp, apply_ite Inv.inv, map_zero, map_one, exp_neg,
+    RCLike.conj_inv, apply_ite conj, apply_ite exp, apply_ite Inv.inv, map_zero, map_one, exp_neg,
     inv_one, inv_zero, ← exp_conj, map_mul, conj_conj]
   split_ifs with hx hn ha ha <;> rfl
 #align complex.inv_cpow_eq_ite Complex.inv_cpow_eq_ite

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -13,7 +13,8 @@ import Mathlib.Analysis.SpecialFunctions.Complex.Log
 We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.
 -/
 
-open Classical Real Topology Filter ComplexConjugate Finset Set
+open scoped Classical
+open Real Topology Filter ComplexConjugate Finset Set
 
 namespace Complex
 

This PR cleans up some lemmas where we're taking named instance parameters [h : Nat.AtLeastTwo n] and manually proving inequalities like 0 < n from h by adding an instance of NeZero n from Nat.AtLeastTwo n as well as a couple of other helper lemmas.

This removes Nat.AtLeastTwo.ne_zero, as NeZero.ne_zero replaces it.

@@ -161,9 +161,9 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ
 
 /-- See Note [no_index around OfNat.ofNat] -/
 @[simp]
-lemma cpow_ofNat_inv_pow (x : ℂ) (n : ℕ) [h : n.AtLeastTwo] :
+lemma cpow_ofNat_inv_pow (x : ℂ) (n : ℕ) [n.AtLeastTwo] :
     (x ^ ((no_index (OfNat.ofNat n) : ℂ)⁻¹)) ^ (no_index (OfNat.ofNat n) : ℕ) = x :=
-  cpow_nat_inv_pow _ (two_pos.trans_le h.1).ne'
+  cpow_nat_inv_pow _ (NeZero.ne n)
 
 /-- A version of `Complex.cpow_int_mul` with RHS that matches `Complex.cpow_mul`.
 
@@ -193,10 +193,10 @@ lemma pow_cpow_nat_inv {x : ℂ} {n : ℕ} (h₀ : n ≠ 0) (hlt : -(π / n) < x
   · rwa [← div_lt_iff' (Nat.cast_pos.2 h₀.bot_lt), neg_div]
   · rwa [← le_div_iff' (Nat.cast_pos.2 h₀.bot_lt)]
 
-lemma pow_cpow_ofNat_inv {x : ℂ} {n : ℕ} [h : n.AtLeastTwo] (hlt : -(π / OfNat.ofNat n) < x.arg)
+lemma pow_cpow_ofNat_inv {x : ℂ} {n : ℕ} [n.AtLeastTwo] (hlt : -(π / OfNat.ofNat n) < x.arg)
     (hle : x.arg ≤ π / OfNat.ofNat n) :
     (x ^ (OfNat.ofNat n : ℕ)) ^ ((OfNat.ofNat n : ℂ)⁻¹) = x :=
-  pow_cpow_nat_inv (two_pos.trans_le h.1).ne' hlt hle
+  pow_cpow_nat_inv (NeZero.ne n) hlt hle
 
 /-- See also `Complex.pow_cpow_ofNat_inv` for a version that also works for `x * I`, `0 ≤ x`. -/
 lemma sq_cpow_two_inv {x : ℂ} (hx : 0 < x.re) : (x ^ (2 : ℕ)) ^ (2⁻¹ : ℂ) = x :=

This is the fourth PR in a sequence that adds auxiliary lemmas from the EulerProducts project to Mathlib.

It adds some lemmas on functions of complex numbers applied to natural numbers:

lemma Complex.natCast_log {n : ℕ} : Real.log n = log n

lemma Complex.natCast_arg {n : ℕ} : arg n = 0

lemma Complex.natCast_mul_natCast_cpow (m n : ℕ) (s : ℂ) : (m * n : ℂ) ^ s = m ^ s * n ^ s

lemma Complex.natCast_cpow_natCast_mul (n m : ℕ) (z : ℂ) : (n : ℂ) ^ (m * z) = ((n : ℂ) ^ m) ^ z

lemma Complex.norm_log_natCast_le_rpow_div (n : ℕ) {ε : ℝ} (hε : 0 < ε) : ‖log n‖ ≤ n ^ ε / ε

(and ofNat versions of the first two).

@@ -217,6 +217,14 @@ theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
     add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
 #align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
 
+lemma natCast_mul_natCast_cpow (m n : ℕ) (s : ℂ) : (m * n : ℂ) ^ s = m ^ s * n ^ s :=
+  ofReal_nat_cast m ▸ ofReal_nat_cast n ▸ mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg s
+
+lemma natCast_cpow_natCast_mul (n m : ℕ) (z : ℂ) : (n : ℂ) ^ (m * z) = ((n : ℂ) ^ m) ^ z := by
+  refine cpow_nat_mul' (x := n) (n := m) ?_ ?_ z
+  · simp only [natCast_arg, mul_zero, Left.neg_neg_iff, pi_pos]
+  · simp only [natCast_arg, mul_zero, pi_pos.le]
+
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ := by
   simp_rw [Complex.cpow_def, log_inv_eq_ite, inv_eq_zero, map_eq_zero, ite_mul, neg_mul,

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

@@ -58,7 +58,7 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧
   constructor
   · intro hyp
     simp only [cpow_def, eq_self_iff_true, if_true] at hyp
-    by_cases x = 0
+    by_cases h : x = 0
     · subst h
       simp only [if_true, eq_self_iff_true] at hyp
       right

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -120,8 +120,8 @@ lemma cpow_int_mul (x : ℂ) (n : ℤ) (y : ℂ) : x ^ (n * y) = (x ^ y) ^ n :=
 
 lemma cpow_mul_int (x y : ℂ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n := by rw [mul_comm, cpow_int_mul]
 
-lemma cpow_nat_mul (x : ℂ) (n : ℕ) (y : ℂ) : x ^ (n * y) = (x ^ y) ^ n := by
-  exact_mod_cast cpow_int_mul x n y
+lemma cpow_nat_mul (x : ℂ) (n : ℕ) (y : ℂ) : x ^ (n * y) = (x ^ y) ^ n :=
+  mod_cast cpow_int_mul x n y
 
 /-- See Note [no_index around OfNat.ofNat] -/
 lemma cpow_ofNat_mul (x : ℂ) (n : ℕ) [n.AtLeastTwo] (y : ℂ) :

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>

@@ -13,8 +13,6 @@ import Mathlib.Analysis.SpecialFunctions.Complex.Log
 We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.
 -/
 
-local macro_rules | `($x ^ $y)   => `(HPow.hPow $x $y) -- Porting note: See issue #2220
-
 open Classical Real Topology Filter ComplexConjugate Finset Set
 
 namespace Complex

• Add Complex.cpow_int_mul, Complex.cpow_mul_int, Complex.cpow_nat_mul, and Complex.cpow_mul_nat.
• Generalize Complex.cpow_two to Complex.cpow_ofNat.
• Golf some proofs using new lemmas.

Motivated by a lemma in the Mandelbrot set connectedness project.

Co-Authored-By: @girving

@@ -112,55 +112,99 @@ theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x
 theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_neg x 1
 #align complex.cpow_neg_one Complex.cpow_neg_one
 
+/-- See also `Complex.cpow_int_mul'`. -/
+lemma cpow_int_mul (x : ℂ) (n : ℤ) (y : ℂ) : x ^ (n * y) = (x ^ y) ^ n := by
+  rcases eq_or_ne x 0 with rfl | hx
+  · rcases eq_or_ne n 0 with rfl | hn
+    · simp
+    · rcases eq_or_ne y 0 with rfl | hy <;> simp [*, zero_zpow]
+  · rw [cpow_def_of_ne_zero hx, cpow_def_of_ne_zero hx, mul_left_comm, exp_int_mul]
+
+lemma cpow_mul_int (x y : ℂ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n := by rw [mul_comm, cpow_int_mul]
+
+lemma cpow_nat_mul (x : ℂ) (n : ℕ) (y : ℂ) : x ^ (n * y) = (x ^ y) ^ n := by
+  exact_mod_cast cpow_int_mul x n y
+
+/-- See Note [no_index around OfNat.ofNat] -/
+lemma cpow_ofNat_mul (x : ℂ) (n : ℕ) [n.AtLeastTwo] (y : ℂ) :
+    x ^ (no_index (OfNat.ofNat n) * y) = (x ^ y) ^ (OfNat.ofNat n : ℕ) :=
+  cpow_nat_mul x n y
+
+lemma cpow_mul_nat (x y : ℂ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n := by
+  rw [mul_comm, cpow_nat_mul]
+
+/-- See Note [no_index around OfNat.ofNat] -/
+lemma cpow_mul_ofNat (x y : ℂ) (n : ℕ) [n.AtLeastTwo] :
+    x ^ (y * no_index (OfNat.ofNat n)) = (x ^ y) ^ (OfNat.ofNat n : ℕ) :=
+  cpow_mul_nat x y n
+
 @[simp, norm_cast]
-theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
-  | 0 => by simp
-  | n + 1 =>
-    if hx : x = 0 then by
-      simp only [hx, pow_succ, Complex.zero_cpow (Nat.cast_ne_zero.2 (Nat.succ_ne_zero _)),
-        zero_mul]
-    else by simp [cpow_add, hx, pow_add, cpow_nat_cast x n]
+theorem cpow_nat_cast (x : ℂ) (n : ℕ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_nat_mul x n 1
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
 
+/-- See Note [no_index around OfNat.ofNat] -/
 @[simp]
-theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ (2 : ℕ) := by
-  rw [← cpow_nat_cast]
-  simp only [Nat.cast_ofNat]
+lemma cpow_ofNat (x : ℂ) (n : ℕ) [n.AtLeastTwo] :
+    x ^ (no_index (OfNat.ofNat n) : ℂ) = x ^ (OfNat.ofNat n : ℕ) :=
+  cpow_nat_cast x n
+
+theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ (2 : ℕ) := cpow_ofNat x 2
 #align complex.cpow_two Complex.cpow_two
 
-open Int in
 @[simp, norm_cast]
-theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
-  | (n : ℕ) => by simp
-  | -[n+1] => by
-    rw [zpow_negSucc]
-    simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_ofNat,
-      cpow_nat_cast]
+theorem cpow_int_cast (x : ℂ) (n : ℤ) : x ^ (n : ℂ) = x ^ n := by simpa using cpow_int_mul x n 1
 #align complex.cpow_int_cast Complex.cpow_int_cast
 
+@[simp]
 theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x := by
-  suffices im (log x * (n⁻¹ : ℂ)) ∈ Ioc (-π) π by
-    rw [← cpow_nat_cast, ← cpow_mul _ this.1 this.2, inv_mul_cancel, cpow_one]
-    exact_mod_cast hn
-  rw [mul_comm, ← ofReal_nat_cast, ← ofReal_inv, ofReal_mul_im, ← div_eq_inv_mul]
-  rw [← pos_iff_ne_zero] at hn
-  have hn' : 0 < (n : ℝ) := by assumption_mod_cast
-  have hn1 : 1 ≤ (n : ℝ) := by exact_mod_cast Nat.succ_le_iff.2 hn
-  constructor
-  · rw [lt_div_iff hn']
-    calc
-      -π * n ≤ -π * 1 := mul_le_mul_of_nonpos_left hn1 (neg_nonpos.2 Real.pi_pos.le)
-      _ = -π := (mul_one _)
-      _ < im (log x) := neg_pi_lt_log_im _
-
-  · rw [div_le_iff hn']
-    calc
-      im (log x) ≤ π := log_im_le_pi _
-      _ = π * 1 := (mul_one π).symm
-      _ ≤ π * n := mul_le_mul_of_nonneg_left hn1 Real.pi_pos.le
-
+  rw [← cpow_nat_mul, mul_inv_cancel, cpow_one]
+  assumption_mod_cast
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
+/-- See Note [no_index around OfNat.ofNat] -/
+@[simp]
+lemma cpow_ofNat_inv_pow (x : ℂ) (n : ℕ) [h : n.AtLeastTwo] :
+    (x ^ ((no_index (OfNat.ofNat n) : ℂ)⁻¹)) ^ (no_index (OfNat.ofNat n) : ℕ) = x :=
+  cpow_nat_inv_pow _ (two_pos.trans_le h.1).ne'
+
+/-- A version of `Complex.cpow_int_mul` with RHS that matches `Complex.cpow_mul`.
+
+The assumptions on the arguments are needed
+because the equality fails, e.g., for `x = -I`, `n = 2`, `y = 1/2`. -/
+lemma cpow_int_mul' {x : ℂ} {n : ℤ} (hlt : -π < n * x.arg) (hle : n * x.arg ≤ π) (y : ℂ) :
+    x ^ (n * y) = (x ^ n) ^ y := by
+  rw [mul_comm] at hlt hle
+  rw [cpow_mul, cpow_int_cast] <;> simpa [log_im]
+
+/-- A version of `Complex.cpow_nat_mul` with RHS that matches `Complex.cpow_mul`.
+
+The assumptions on the arguments are needed
+because the equality fails, e.g., for `x = -I`, `n = 2`, `y = 1/2`. -/
+lemma cpow_nat_mul' {x : ℂ} {n : ℕ} (hlt : -π < n * x.arg) (hle : n * x.arg ≤ π) (y : ℂ) :
+    x ^ (n * y) = (x ^ n) ^ y :=
+  cpow_int_mul' hlt hle y
+
+lemma cpow_ofNat_mul' {x : ℂ} {n : ℕ} [n.AtLeastTwo] (hlt : -π < OfNat.ofNat n * x.arg)
+    (hle : OfNat.ofNat n * x.arg ≤ π) (y : ℂ) :
+    x ^ (OfNat.ofNat n * y) = (x ^ (OfNat.ofNat n : ℕ)) ^ y :=
+  cpow_nat_mul' hlt hle y
+
+lemma pow_cpow_nat_inv {x : ℂ} {n : ℕ} (h₀ : n ≠ 0) (hlt : -(π / n) < x.arg) (hle : x.arg ≤ π / n) :
+    (x ^ n) ^ (n⁻¹ : ℂ) = x := by
+  rw [← cpow_nat_mul', mul_inv_cancel (Nat.cast_ne_zero.2 h₀), cpow_one]
+  · rwa [← div_lt_iff' (Nat.cast_pos.2 h₀.bot_lt), neg_div]
+  · rwa [← le_div_iff' (Nat.cast_pos.2 h₀.bot_lt)]
+
+lemma pow_cpow_ofNat_inv {x : ℂ} {n : ℕ} [h : n.AtLeastTwo] (hlt : -(π / OfNat.ofNat n) < x.arg)
+    (hle : x.arg ≤ π / OfNat.ofNat n) :
+    (x ^ (OfNat.ofNat n : ℕ)) ^ ((OfNat.ofNat n : ℂ)⁻¹) = x :=
+  pow_cpow_nat_inv (two_pos.trans_le h.1).ne' hlt hle
+
+/-- See also `Complex.pow_cpow_ofNat_inv` for a version that also works for `x * I`, `0 ≤ x`. -/
+lemma sq_cpow_two_inv {x : ℂ} (hx : 0 < x.re) : (x ^ (2 : ℕ)) ^ (2⁻¹ : ℂ) = x :=
+  pow_cpow_ofNat_inv (neg_pi_div_two_lt_arg_iff.2 <| .inl hx)
+    (arg_le_pi_div_two_iff.2 <| .inl hx.le)
+
 theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r := by
   rcases eq_or_ne r 0 with (rfl | hr)

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).

@@ -118,7 +118,7 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | n + 1 =>
     if hx : x = 0 then by
       simp only [hx, pow_succ, Complex.zero_cpow (Nat.cast_ne_zero.2 (Nat.succ_ne_zero _)),
-        MulZeroClass.zero_mul]
+        zero_mul]
     else by simp [cpow_add, hx, pow_add, cpow_nat_cast x n]
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
 
@@ -166,9 +166,9 @@ theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
   rcases eq_or_ne r 0 with (rfl | hr)
   · simp only [cpow_zero, mul_one]
   rcases eq_or_lt_of_le ha with (rfl | ha')
-  · rw [ofReal_zero, MulZeroClass.zero_mul, zero_cpow hr, MulZeroClass.zero_mul]
+  · rw [ofReal_zero, zero_mul, zero_cpow hr, zero_mul]
   rcases eq_or_lt_of_le hb with (rfl | hb')
-  · rw [ofReal_zero, MulZeroClass.mul_zero, zero_cpow hr, MulZeroClass.mul_zero]
+  · rw [ofReal_zero, mul_zero, zero_cpow hr, mul_zero]
   have ha'' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha'.ne'
   have hb'' : (b : ℂ) ≠ 0 := ofReal_ne_zero.mpr hb'.ne'
   rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_ofReal_mul ha' hb'', ofReal_log ha,

@@ -13,6 +13,7 @@ import Mathlib.Analysis.SpecialFunctions.Complex.Log
 We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.
 -/
 
+local macro_rules | `($x ^ $y)   => `(HPow.hPow $x $y) -- Porting note: See issue #2220
 
 open Classical Real Topology Filter ComplexConjugate Finset Set
 
@@ -55,7 +56,7 @@ theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
 theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
 #align complex.zero_cpow Complex.zero_cpow
 
-theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
+theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
   constructor
   · intro hyp
     simp only [cpow_def, eq_self_iff_true, if_true] at hyp
@@ -72,7 +73,7 @@ theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 
     · exact cpow_zero _
 #align complex.zero_cpow_eq_iff Complex.zero_cpow_eq_iff
 
-theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
+theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = (0 : ℂ) ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by
   rw [← zero_cpow_eq_iff, eq_comm]
 #align complex.eq_zero_cpow_iff Complex.eq_zero_cpow_iff
 
@@ -111,9 +112,8 @@ theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x
 theorem cpow_neg_one (x : ℂ) : x ^ (-1 : ℂ) = x⁻¹ := by simpa using cpow_neg x 1
 #align complex.cpow_neg_one Complex.cpow_neg_one
 
--- Porting note: couldn't find a way to use `^` for the RHS
 @[simp, norm_cast]
-theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = HPow.hPow x (n : ℕ)
+theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = x ^ n
   | 0 => by simp
   | n + 1 =>
     if hx : x = 0 then by
@@ -123,14 +123,14 @@ theorem cpow_nat_cast (x : ℂ) : ∀ n : ℕ, x ^ (n : ℂ) = HPow.hPow x (n :
 #align complex.cpow_nat_cast Complex.cpow_nat_cast
 
 @[simp]
-theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = HPow.hPow x (2 : ℕ) := by
+theorem cpow_two (x : ℂ) : x ^ (2 : ℂ) = x ^ (2 : ℕ) := by
   rw [← cpow_nat_cast]
   simp only [Nat.cast_ofNat]
 #align complex.cpow_two Complex.cpow_two
 
 open Int in
 @[simp, norm_cast]
-theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = HPow.hPow x n
+theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = x ^ n
   | (n : ℕ) => by simp
   | -[n+1] => by
     rw [zpow_negSucc]
@@ -138,7 +138,7 @@ theorem cpow_int_cast (x : ℂ) : ∀ n : ℤ, x ^ (n : ℂ) = HPow.hPow x n
       cpow_nat_cast]
 #align complex.cpow_int_cast Complex.cpow_int_cast
 
-theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : HPow.hPow (x ^ (n⁻¹ : ℂ)) n = x := by
+theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℂ)) ^ n = x := by
   suffices im (log x * (n⁻¹ : ℂ)) ∈ Ioc (-π) π by
     rw [← cpow_nat_cast, ← cpow_mul _ this.1 this.2, inv_mul_cancel, cpow_one]
     exact_mod_cast hn

• depends on: #5966

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

@@ -3,14 +3,11 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
   Rémy Degenne, David Loeffler
-
-! This file was ported from Lean 3 source module analysis.special_functions.pow.complex
-! leanprover-community/mathlib commit 4fa54b337f7d52805480306db1b1439c741848c8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Complex.Log
 
+#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
+
 /-! # Power function on `ℂ`
 
 We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.

@@ -164,9 +164,7 @@ theorem cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : n ≠ 0) : HPow.hPow (x ^ (n
 
 #align complex.cpow_nat_inv_pow Complex.cpow_nat_inv_pow
 
--- TODO: should log_of_real_mul and of_real_log use ofReal in their names?
-
-theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
+theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
     ((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r := by
   rcases eq_or_ne r 0 with (rfl | hr)
   · simp only [cpow_zero, mul_one]
@@ -176,9 +174,9 @@ theorem mul_cpow_of_real_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : 
   · rw [ofReal_zero, MulZeroClass.mul_zero, zero_cpow hr, MulZeroClass.mul_zero]
   have ha'' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha'.ne'
   have hb'' : (b : ℂ) ≠ 0 := ofReal_ne_zero.mpr hb'.ne'
-  rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_of_real_mul ha' hb'', of_real_log ha,
+  rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_ofReal_mul ha' hb'', ofReal_log ha,
     add_mul, exp_add, ← cpow_def_of_ne_zero ha'', ← cpow_def_of_ne_zero hb'']
-#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_of_real_nonneg
+#align complex.mul_cpow_of_real_nonneg Complex.mul_cpow_ofReal_nonneg
 
 theorem inv_cpow_eq_ite (x : ℂ) (n : ℂ) :
     x⁻¹ ^ n = if x.arg = π then conj (x ^ conj n)⁻¹ else (x ^ n)⁻¹ := by

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -192,7 +192,7 @@ theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)
   rw [inv_cpow_eq_ite, if_neg hx]
 #align complex.inv_cpow Complex.inv_cpow
 
-/-- `complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
+/-- `Complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
 theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
     (x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n := by
   rw [inv_cpow_eq_ite, apply_ite conj, conj_conj, conj_conj]
@@ -258,7 +258,7 @@ end Complex
 --       pure (c, (expr.const Pos []).mk_app [a, b, b', c, hb, h])
 -- #align norm_num.prove_rpow' norm_num.prove_rpow'
 
--- /-- Evaluate `complex.cpow a b` where `a` is a rational numeral and `b` is an integer. -/
+-- /-- Evaluate `Complex.cpow a b` where `a` is a rational numeral and `b` is an integer. -/
 -- unsafe def prove_cpow : expr → expr → tactic (expr × expr) :=
 --   prove_rpow' `` cpow_pos `` cpow_neg `` Complex.cpow_zero q(ℂ) q(ℂ) q((1 : ℂ))
 -- #align norm_num.prove_cpow norm_num.prove_cpow

Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com>