ring_theory.localization.away.basic ⟷ Mathlib.RingTheory.Localization.Away.Basic

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

@@ -239,11 +239,11 @@ theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMa
 #align self_zpow_of_nonneg selfZPow_of_nonneg
 -/
 
-#print selfZPow_coe_nat /-
+#print selfZPow_natCast /-
 @[simp]
-theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
+theorem selfZPow_natCast (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
   selfZPow_of_nonneg _ _ (Int.natCast_nonneg d)
-#align self_zpow_coe_nat selfZPow_coe_nat
+#align self_zpow_coe_nat selfZPow_natCast
 -/
 
 #print selfZPow_zero /-
@@ -269,26 +269,25 @@ theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
 #align self_zpow_of_nonpos selfZPow_of_nonpos
 -/
 
-#print selfZPow_neg_coe_nat /-
+#print selfZPow_neg_natCast /-
 @[simp]
-theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
+theorem selfZPow_neg_natCast (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
   simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.natCast_nonneg d))]
-#align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
+#align self_zpow_neg_coe_nat selfZPow_neg_natCast
 -/
 
-#print selfZPow_sub_cast_nat /-
+#print selfZPow_sub_natCast /-
 @[simp]
-theorem selfZPow_sub_cast_nat {n m : ℕ} :
-    selfZPow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
+theorem selfZPow_sub_natCast {n m : ℕ} : selfZPow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
   by
   by_cases h : m ≤ n
   ·
     rw [IsLocalization.eq_mk'_iff_mul_eq, Submonoid.pow_apply, Subtype.coe_mk, ← Int.ofNat_sub h,
-      selfZPow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
-  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_coe_nat,
+      selfZPow_natCast, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
+  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_natCast,
       IsLocalization.mk'_eq_iff_eq]
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
-#align self_zpow_sub_cast_nat selfZPow_sub_cast_nat
+#align self_zpow_sub_cast_nat selfZPow_sub_natCast
 -/
 
 #print selfZPow_add /-
@@ -301,11 +300,11 @@ theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZ
       selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_of_nonneg hn hm, pow_add]
   · have : n + m = n.nat_abs - m.nat_abs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
-    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_natCast,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow]
   · have : n + m = m.nat_abs - n.nat_abs := by
       rw [Int.natAbs_of_nonneg hm, Int.ofNat_natAbs_of_nonpos hn.le, sub_neg_eq_add, add_comm]
-    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_natCast,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
   · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
       Int.natAbs_add_of_nonpos hn hm, ← mk'_mul, one_mul]
@@ -380,7 +379,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
   simp only [← Subtype.val_eq_coe, ← hy] at H
   obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
-  rw [← mk'_one B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat, ← map_pow _ _ d,
+  rw [← mk'_one B, selfZPow_pow_sub, selfZPow_natCast, selfZPow_natCast, ← map_pow _ _ d,
     mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
   exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'

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

@@ -242,7 +242,7 @@ theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMa
 #print selfZPow_coe_nat /-
 @[simp]
 theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
-  selfZPow_of_nonneg _ _ (Int.coe_nat_nonneg d)
+  selfZPow_of_nonneg _ _ (Int.natCast_nonneg d)
 #align self_zpow_coe_nat selfZPow_coe_nat
 -/
 
@@ -272,7 +272,7 @@ theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
 #print selfZPow_neg_coe_nat /-
 @[simp]
 theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
-  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
+  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.natCast_nonneg d))]
 #align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
 -/
 
@@ -298,7 +298,7 @@ theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZ
   cases' le_or_lt 0 n with hn hn <;> cases' le_or_lt 0 m with hm hm
   ·
     rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_nonneg _ _ hm,
-      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
+      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_of_nonneg hn hm, pow_add]
   · have : n + m = n.nat_abs - m.nat_abs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
     rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
@@ -308,7 +308,7 @@ theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZ
     rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
   · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
-      Int.natAbs_add_neg hn hm, ← mk'_mul, one_mul]
+      Int.natAbs_add_of_nonpos hn hm, ← mk'_mul, one_mul]
     congr
     ext
     simp [pow_add]

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

@@ -138,7 +138,7 @@ noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
   · intro x y hxy
     obtain ⟨c, eq⟩ := (IsLocalization.eq_iff_exists M S).mp hxy
     obtain ⟨u, hu⟩ := H c
-    rwa [← hu, Units.mul_right_inj] at eq 
+    rwa [← hu, Units.mul_right_inj] at eq
   · intro y
     obtain ⟨⟨x, s⟩, eq⟩ := IsLocalization.surj M y
     obtain ⟨u, hu⟩ := H s
@@ -346,12 +346,12 @@ theorem selfZPow_pow_sub (a : R) (b : B) (m d : ℤ) :
   constructor
   · intro h
     have := congr_arg (fun s : B => s * selfZPow x B d) h
-    simp only at this 
-    rwa [mul_assoc, mul_assoc, selfZPow_neg_mul, mul_one, mul_comm b _] at this 
+    simp only at this
+    rwa [mul_assoc, mul_assoc, selfZPow_neg_mul, mul_one, mul_comm b _] at this
   · intro h
     have := congr_arg (fun s : B => s * selfZPow x B (-d)) h
-    simp only at this 
-    rwa [mul_comm _ b, mul_assoc b _ _, selfZPow_mul_neg, mul_one] at this 
+    simp only at this
+    rwa [mul_comm _ b, mul_assoc b _ _, selfZPow_mul_neg, mul_one] at this
 #align self_zpow_pow_sub selfZPow_pow_sub
 -/
 
@@ -368,7 +368,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     haveI :=
       @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
         (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-    simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
+    simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H
     apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
     rw [← H]
     apply mul_ne_zero hb (pow_ne_zero _ _)
@@ -377,7 +377,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
         (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
         (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
     exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-  simp only [← Subtype.val_eq_coe, ← hy] at H 
+  simp only [← Subtype.val_eq_coe, ← hy] at H
   obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
   rw [← mk'_one B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat, ← map_pow _ _ d,

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

@@ -359,7 +359,30 @@ variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 #print exists_reduced_fraction' /-
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
-    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by classical
+    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by
+  classical
+  obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
+  obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
+  have ha₀ : a₀ ≠ 0 :=
+    by
+    haveI :=
+      @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
+        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+    simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
+    apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
+    rw [← H]
+    apply mul_ne_zero hb (pow_ne_zero _ _)
+    exact
+      IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
+        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+        (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
+    exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+  simp only [← Subtype.val_eq_coe, ← hy] at H 
+  obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
+  refine' ⟨a, m - d, _⟩
+  rw [← mk'_one B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat, ← map_pow _ _ d,
+    mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
+  exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'
 -/
 

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

@@ -359,30 +359,7 @@ variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 #print exists_reduced_fraction' /-
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
-    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by
-  classical
-  obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
-  obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
-  have ha₀ : a₀ ≠ 0 :=
-    by
-    haveI :=
-      @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
-        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-    simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
-    apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
-    rw [← H]
-    apply mul_ne_zero hb (pow_ne_zero _ _)
-    exact
-      IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
-        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-        (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
-    exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-  simp only [← Subtype.val_eq_coe, ← hy] at H 
-  obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
-  refine' ⟨a, m - d, _⟩
-  rw [← mk'_one B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat, ← map_pow _ _ d,
-    mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
-  exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
+    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by classical
 #align exists_reduced_fraction' exists_reduced_fraction'
 -/
 

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

@@ -226,140 +226,140 @@ variable (x : R)
 
 variable (B : Type _) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
-#print selfZpow /-
+#print selfZPow /-
 /-- `self_zpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
-noncomputable def selfZpow (m : ℤ) : B :=
+noncomputable def selfZPow (m : ℤ) : B :=
   if hm : 0 ≤ m then algebraMap _ _ x ^ m.natAbs else mk' _ (1 : R) (Submonoid.pow x m.natAbs)
-#align self_zpow selfZpow
+#align self_zpow selfZPow
 -/
 
-#print selfZpow_of_nonneg /-
-theorem selfZpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZpow x B n = algebraMap R B x ^ n.natAbs :=
+#print selfZPow_of_nonneg /-
+theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMap R B x ^ n.natAbs :=
   dif_pos hn
-#align self_zpow_of_nonneg selfZpow_of_nonneg
+#align self_zpow_of_nonneg selfZPow_of_nonneg
 -/
 
-#print selfZpow_coe_nat /-
+#print selfZPow_coe_nat /-
 @[simp]
-theorem selfZpow_coe_nat (d : ℕ) : selfZpow x B d = algebraMap R B x ^ d :=
-  selfZpow_of_nonneg _ _ (Int.coe_nat_nonneg d)
-#align self_zpow_coe_nat selfZpow_coe_nat
+theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
+  selfZPow_of_nonneg _ _ (Int.coe_nat_nonneg d)
+#align self_zpow_coe_nat selfZPow_coe_nat
 -/
 
-#print selfZpow_zero /-
+#print selfZPow_zero /-
 @[simp]
-theorem selfZpow_zero : selfZpow x B 0 = 1 := by simp [selfZpow_of_nonneg _ _ le_rfl]
-#align self_zpow_zero selfZpow_zero
+theorem selfZPow_zero : selfZPow x B 0 = 1 := by simp [selfZPow_of_nonneg _ _ le_rfl]
+#align self_zpow_zero selfZPow_zero
 -/
 
-#print selfZpow_of_neg /-
-theorem selfZpow_of_neg {n : ℤ} (hn : n < 0) :
-    selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
+#print selfZPow_of_neg /-
+theorem selfZPow_of_neg {n : ℤ} (hn : n < 0) :
+    selfZPow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   dif_neg hn.not_le
-#align self_zpow_of_neg selfZpow_of_neg
+#align self_zpow_of_neg selfZPow_of_neg
 -/
 
-#print selfZpow_of_nonpos /-
-theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
-    selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
+#print selfZPow_of_nonpos /-
+theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
+    selfZPow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   by
   by_cases hn0 : n = 0
-  · simp [hn0, selfZpow_zero, Submonoid.pow_apply]
-  · simp [selfZpow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
-#align self_zpow_of_nonpos selfZpow_of_nonpos
+  · simp [hn0, selfZPow_zero, Submonoid.pow_apply]
+  · simp [selfZPow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
+#align self_zpow_of_nonpos selfZPow_of_nonpos
 -/
 
-#print selfZpow_neg_coe_nat /-
+#print selfZPow_neg_coe_nat /-
 @[simp]
-theorem selfZpow_neg_coe_nat (d : ℕ) : selfZpow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
-  simp [selfZpow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
-#align self_zpow_neg_coe_nat selfZpow_neg_coe_nat
+theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
+  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
+#align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
 -/
 
-#print selfZpow_sub_cast_nat /-
+#print selfZPow_sub_cast_nat /-
 @[simp]
-theorem selfZpow_sub_cast_nat {n m : ℕ} :
-    selfZpow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
+theorem selfZPow_sub_cast_nat {n m : ℕ} :
+    selfZPow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
   by
   by_cases h : m ≤ n
   ·
     rw [IsLocalization.eq_mk'_iff_mul_eq, Submonoid.pow_apply, Subtype.coe_mk, ← Int.ofNat_sub h,
-      selfZpow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
-  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZpow_neg_coe_nat,
+      selfZPow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
+  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_coe_nat,
       IsLocalization.mk'_eq_iff_eq]
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
-#align self_zpow_sub_cast_nat selfZpow_sub_cast_nat
+#align self_zpow_sub_cast_nat selfZPow_sub_cast_nat
 -/
 
-#print selfZpow_add /-
+#print selfZPow_add /-
 @[simp]
-theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZpow x B m :=
+theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m :=
   by
   cases' le_or_lt 0 n with hn hn <;> cases' le_or_lt 0 m with hm hm
   ·
-    rw [selfZpow_of_nonneg _ _ hn, selfZpow_of_nonneg _ _ hm,
-      selfZpow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
+    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_nonneg _ _ hm,
+      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
   · have : n + m = n.nat_abs - m.nat_abs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
-    rw [selfZpow_of_nonneg _ _ hn, selfZpow_of_neg _ _ hm, this, selfZpow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow]
   · have : n + m = m.nat_abs - n.nat_abs := by
       rw [Int.natAbs_of_nonneg hm, Int.ofNat_natAbs_of_nonpos hn.le, sub_neg_eq_add, add_comm]
-    rw [selfZpow_of_nonneg _ _ hm, selfZpow_of_neg _ _ hn, this, selfZpow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
-  · rw [selfZpow_of_neg _ _ hn, selfZpow_of_neg _ _ hm, selfZpow_of_neg _ _ (add_neg hn hm),
+  · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
       Int.natAbs_add_neg hn hm, ← mk'_mul, one_mul]
     congr
     ext
     simp [pow_add]
-#align self_zpow_add selfZpow_add
+#align self_zpow_add selfZPow_add
 -/
 
-#print selfZpow_mul_neg /-
-theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
+#print selfZPow_mul_neg /-
+theorem selfZPow_mul_neg (d : ℤ) : selfZPow x B d * selfZPow x B (-d) = 1 :=
   by
   by_cases hd : d ≤ 0
-  · erw [selfZpow_of_nonpos x B hd, selfZpow_of_nonneg, ← map_pow, Int.natAbs_neg,
+  · erw [selfZPow_of_nonpos x B hd, selfZPow_of_nonneg, ← map_pow, Int.natAbs_neg,
       IsLocalization.mk'_spec, map_one]
     apply nonneg_of_neg_nonpos
     rwa [neg_neg]
-  · erw [selfZpow_of_nonneg x B (le_of_not_le hd), selfZpow_of_nonpos, ← map_pow, Int.natAbs_neg,
+  · erw [selfZPow_of_nonneg x B (le_of_not_le hd), selfZPow_of_nonpos, ← map_pow, Int.natAbs_neg,
       @IsLocalization.mk'_spec' R _ (Submonoid.powers x) B _ _ _ 1 (Submonoid.pow x d.nat_abs),
       map_one]
     refine' nonpos_of_neg_nonneg (le_of_lt _)
     rwa [neg_neg, ← not_le]
-#align self_zpow_mul_neg selfZpow_mul_neg
+#align self_zpow_mul_neg selfZPow_mul_neg
 -/
 
-#print selfZpow_neg_mul /-
-theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := by
-  rw [mul_comm, selfZpow_mul_neg x B d]
-#align self_zpow_neg_mul selfZpow_neg_mul
+#print selfZPow_neg_mul /-
+theorem selfZPow_neg_mul (d : ℤ) : selfZPow x B (-d) * selfZPow x B d = 1 := by
+  rw [mul_comm, selfZPow_mul_neg x B d]
+#align self_zpow_neg_mul selfZPow_neg_mul
 -/
 
-#print selfZpow_pow_sub /-
-theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
-    selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
-      selfZpow x B m * mk' B a (1 : Submonoid.powers x) = selfZpow x B d * b :=
+#print selfZPow_pow_sub /-
+theorem selfZPow_pow_sub (a : R) (b : B) (m d : ℤ) :
+    selfZPow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
+      selfZPow x B m * mk' B a (1 : Submonoid.powers x) = selfZPow x B d * b :=
   by
-  rw [sub_eq_add_neg, selfZpow_add, mul_assoc, mul_comm _ (mk' B a 1), ← mul_assoc]
+  rw [sub_eq_add_neg, selfZPow_add, mul_assoc, mul_comm _ (mk' B a 1), ← mul_assoc]
   constructor
   · intro h
-    have := congr_arg (fun s : B => s * selfZpow x B d) h
+    have := congr_arg (fun s : B => s * selfZPow x B d) h
     simp only at this 
-    rwa [mul_assoc, mul_assoc, selfZpow_neg_mul, mul_one, mul_comm b _] at this 
+    rwa [mul_assoc, mul_assoc, selfZPow_neg_mul, mul_one, mul_comm b _] at this 
   · intro h
-    have := congr_arg (fun s : B => s * selfZpow x B (-d)) h
+    have := congr_arg (fun s : B => s * selfZPow x B (-d)) h
     simp only at this 
-    rwa [mul_comm _ b, mul_assoc b _ _, selfZpow_mul_neg, mul_one] at this 
-#align self_zpow_pow_sub selfZpow_pow_sub
+    rwa [mul_comm _ b, mul_assoc b _ _, selfZPow_mul_neg, mul_one] at this 
+#align self_zpow_pow_sub selfZPow_pow_sub
 -/
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 #print exists_reduced_fraction' /-
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
-    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
+    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by
   classical
   obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
   obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
@@ -380,7 +380,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
   simp only [← Subtype.val_eq_coe, ← hy] at H 
   obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
-  rw [← mk'_one B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat, ← map_pow _ _ d,
+  rw [← mk'_one B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat, ← map_pow _ _ d,
     mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
   exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'

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

@@ -3,8 +3,8 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 -/
-import Mathbin.RingTheory.UniqueFactorizationDomain
-import Mathbin.RingTheory.Localization.Basic
+import RingTheory.UniqueFactorizationDomain
+import RingTheory.Localization.Basic
 
 #align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-
-! This file was ported from Lean 3 source module ring_theory.localization.away.basic
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.RingTheory.UniqueFactorizationDomain
 import Mathbin.RingTheory.Localization.Basic
 
+#align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
 /-!
 # Localizations away from an element
 

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

@@ -65,13 +65,16 @@ noncomputable def invSelf : S :=
 #align is_localization.away.inv_self IsLocalization.Away.invSelf
 -/
 
+#print IsLocalization.Away.mul_invSelf /-
 @[simp]
 theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by
   convert IsLocalization.mk'_mul_mk'_eq_one _ 1; symm; apply IsLocalization.mk'_one
 #align is_localization.away.mul_inv_self IsLocalization.Away.mul_invSelf
+-/
 
 variable {g : R →+* P}
 
+#print IsLocalization.Away.lift /-
 /-- Given `x : R`, a localization map `F : R →+* S` away from `x`, and a map of `comm_semiring`s
 `g : R →+* P` such that `g x` is invertible, the homomorphism induced from `S` to `P` sending
 `z : S` to `g y * (g x)⁻ⁿ`, where `y : R, n : ℕ` are such that `z = F y * (F x)⁻ⁿ`. -/
@@ -82,17 +85,23 @@ noncomputable def lift (hg : IsUnit (g x)) : S →+* P :=
       rw [← hn, g.map_pow]
       exact IsUnit.map (powMonoidHom n : P →* P) hg
 #align is_localization.away.lift IsLocalization.Away.lift
+-/
 
+#print IsLocalization.Away.AwayMap.lift_eq /-
 @[simp]
 theorem AwayMap.lift_eq (hg : IsUnit (g x)) (a : R) : lift x hg ((algebraMap R S) a) = g a :=
   lift_eq _ _
 #align is_localization.away.away_map.lift_eq IsLocalization.Away.AwayMap.lift_eq
+-/
 
+#print IsLocalization.Away.AwayMap.lift_comp /-
 @[simp]
 theorem AwayMap.lift_comp (hg : IsUnit (g x)) : (lift x hg).comp (algebraMap R S) = g :=
   lift_comp _
 #align is_localization.away.away_map.lift_comp IsLocalization.Away.AwayMap.lift_comp
+-/
 
+#print IsLocalization.Away.awayToAwayRight /-
 /-- Given `x y : R` and localizations `S`, `P` away from `x` and `x * y`
 respectively, the homomorphism induced from `S` to `P`. -/
 noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x * y) P] : S →+* P :=
@@ -101,15 +110,18 @@ noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x
       isUnit_of_mul_eq_one ((algebraMap R P) x) (mk' P y ⟨x * y, Submonoid.mem_powers _⟩) <| by
         rw [mul_mk'_eq_mk'_of_mul, mk'_self]
 #align is_localization.away.away_to_away_right IsLocalization.Away.awayToAwayRight
+-/
 
 variable (S) (Q : Type _) [CommSemiring Q] [Algebra P Q]
 
+#print IsLocalization.Away.map /-
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
     [IsLocalization.Away (f r) Q] : S →+* Q :=
   IsLocalization.map Q f
     (show Submonoid.powers r ≤ (Submonoid.powers (f r)).comap f by rintro x ⟨n, rfl⟩; use n; simp)
 #align is_localization.away.map IsLocalization.Away.map
+-/
 
 end Away
 
@@ -121,6 +133,7 @@ section AtUnits
 
 variable (R) (S) (M)
 
+#print IsLocalization.atUnits /-
 /-- The localization at a module of units is isomorphic to the ring -/
 noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
   by
@@ -137,6 +150,7 @@ noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
     rw [RingHom.map_mul, ← Eq, ← hu, mul_assoc, ← RingHom.map_mul]
     simp
 #align is_localization.at_units IsLocalization.atUnits
+-/
 
 #print IsLocalization.atUnit /-
 /-- The localization away from a unit is isomorphic to the ring -/
@@ -158,6 +172,7 @@ noncomputable def atOne [IsLocalization.Away (1 : R) S] : R ≃ₐ[R] S :=
 #align is_localization.at_one IsLocalization.atOne
 -/
 
+#print IsLocalization.away_of_isUnit_of_bijective /-
 theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [CommRing S]
     [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
     IsLocalization.Away r S :=
@@ -169,6 +184,7 @@ theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [Comm
       · rintro rfl; exact ⟨1, rfl⟩
       · rintro ⟨⟨_, n, rfl⟩, e⟩; exact (hr.pow _).mul_right_inj.mp e }
 #align is_localization.away_of_is_unit_of_bijective IsLocalization.away_of_isUnit_of_bijective
+-/
 
 end AtUnits
 
@@ -180,18 +196,22 @@ open IsLocalization
 
 variable {M}
 
+#print Localization.awayLift /-
 /-- Given a map `f : R →+* S` and an element `r : R`, such that `f r` is invertible,
   we may construct a map `Rᵣ →+* S`. -/
 noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
     Localization.Away r →+* P :=
   IsLocalization.Away.lift r hr
 #align localization.away_lift Localization.awayLift
+-/
 
+#print Localization.awayMap /-
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable abbrev awayMap (f : R →+* P) (r : R) :
     Localization.Away r →+* Localization.Away (f r) :=
   IsLocalization.Away.map _ _ f r
 #align localization.away_map Localization.awayMap
+-/
 
 end Localization
 
@@ -216,24 +236,33 @@ noncomputable def selfZpow (m : ℤ) : B :=
 #align self_zpow selfZpow
 -/
 
+#print selfZpow_of_nonneg /-
 theorem selfZpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZpow x B n = algebraMap R B x ^ n.natAbs :=
   dif_pos hn
 #align self_zpow_of_nonneg selfZpow_of_nonneg
+-/
 
+#print selfZpow_coe_nat /-
 @[simp]
 theorem selfZpow_coe_nat (d : ℕ) : selfZpow x B d = algebraMap R B x ^ d :=
   selfZpow_of_nonneg _ _ (Int.coe_nat_nonneg d)
 #align self_zpow_coe_nat selfZpow_coe_nat
+-/
 
+#print selfZpow_zero /-
 @[simp]
 theorem selfZpow_zero : selfZpow x B 0 = 1 := by simp [selfZpow_of_nonneg _ _ le_rfl]
 #align self_zpow_zero selfZpow_zero
+-/
 
+#print selfZpow_of_neg /-
 theorem selfZpow_of_neg {n : ℤ} (hn : n < 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   dif_neg hn.not_le
 #align self_zpow_of_neg selfZpow_of_neg
+-/
 
+#print selfZpow_of_nonpos /-
 theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   by
@@ -241,12 +270,16 @@ theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
   · simp [hn0, selfZpow_zero, Submonoid.pow_apply]
   · simp [selfZpow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
 #align self_zpow_of_nonpos selfZpow_of_nonpos
+-/
 
+#print selfZpow_neg_coe_nat /-
 @[simp]
 theorem selfZpow_neg_coe_nat (d : ℕ) : selfZpow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
   simp [selfZpow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
 #align self_zpow_neg_coe_nat selfZpow_neg_coe_nat
+-/
 
+#print selfZpow_sub_cast_nat /-
 @[simp]
 theorem selfZpow_sub_cast_nat {n m : ℕ} :
     selfZpow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
@@ -259,7 +292,9 @@ theorem selfZpow_sub_cast_nat {n m : ℕ} :
       IsLocalization.mk'_eq_iff_eq]
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
 #align self_zpow_sub_cast_nat selfZpow_sub_cast_nat
+-/
 
+#print selfZpow_add /-
 @[simp]
 theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZpow x B m :=
   by
@@ -281,7 +316,9 @@ theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZ
     ext
     simp [pow_add]
 #align self_zpow_add selfZpow_add
+-/
 
+#print selfZpow_mul_neg /-
 theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
   by
   by_cases hd : d ≤ 0
@@ -295,11 +332,15 @@ theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
     refine' nonpos_of_neg_nonneg (le_of_lt _)
     rwa [neg_neg, ← not_le]
 #align self_zpow_mul_neg selfZpow_mul_neg
+-/
 
+#print selfZpow_neg_mul /-
 theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := by
   rw [mul_comm, selfZpow_mul_neg x B d]
 #align self_zpow_neg_mul selfZpow_neg_mul
+-/
 
+#print selfZpow_pow_sub /-
 theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
     selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
       selfZpow x B m * mk' B a (1 : Submonoid.powers x) = selfZpow x B d * b :=
@@ -315,9 +356,11 @@ theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
     simp only at this 
     rwa [mul_comm _ b, mul_assoc b _ _, selfZpow_mul_neg, mul_one] at this 
 #align self_zpow_pow_sub selfZpow_pow_sub
+-/
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
+#print exists_reduced_fraction' /-
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
   classical
@@ -344,6 +387,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
   exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'
+-/
 
 end NumDenom
 

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

@@ -321,28 +321,28 @@ variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
   classical
-    obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
-    obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
-    have ha₀ : a₀ ≠ 0 :=
-      by
-      haveI :=
-        @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
-          (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-      simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
-      apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
-      rw [← H]
-      apply mul_ne_zero hb (pow_ne_zero _ _)
-      exact
-        IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
-          (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-          (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
-      exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-    simp only [← Subtype.val_eq_coe, ← hy] at H 
-    obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
-    refine' ⟨a, m - d, _⟩
-    rw [← mk'_one B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat, ← map_pow _ _ d,
-      mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
-    exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
+  obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
+  obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
+  have ha₀ : a₀ ≠ 0 :=
+    by
+    haveI :=
+      @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
+        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+    simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
+    apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
+    rw [← H]
+    apply mul_ne_zero hb (pow_ne_zero _ _)
+    exact
+      IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
+        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+        (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
+    exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+  simp only [← Subtype.val_eq_coe, ← hy] at H 
+  obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
+  refine' ⟨a, m - d, _⟩
+  rw [← mk'_one B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat, ← map_pow _ _ d,
+    mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
+  exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'
 
 end NumDenom

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

@@ -128,7 +128,7 @@ noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
   · intro x y hxy
     obtain ⟨c, eq⟩ := (IsLocalization.eq_iff_exists M S).mp hxy
     obtain ⟨u, hu⟩ := H c
-    rwa [← hu, Units.mul_right_inj] at eq
+    rwa [← hu, Units.mul_right_inj] at eq 
   · intro y
     obtain ⟨⟨x, s⟩, eq⟩ := IsLocalization.surj M y
     obtain ⟨u, hu⟩ := H s
@@ -308,18 +308,18 @@ theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
   constructor
   · intro h
     have := congr_arg (fun s : B => s * selfZpow x B d) h
-    simp only at this
-    rwa [mul_assoc, mul_assoc, selfZpow_neg_mul, mul_one, mul_comm b _] at this
+    simp only at this 
+    rwa [mul_assoc, mul_assoc, selfZpow_neg_mul, mul_one, mul_comm b _] at this 
   · intro h
     have := congr_arg (fun s : B => s * selfZpow x B (-d)) h
-    simp only at this
-    rwa [mul_comm _ b, mul_assoc b _ _, selfZpow_mul_neg, mul_one] at this
+    simp only at this 
+    rwa [mul_comm _ b, mul_assoc b _ _, selfZpow_mul_neg, mul_one] at this 
 #align self_zpow_pow_sub selfZpow_pow_sub
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
-    ∃ (a : R)(n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
+    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
   classical
     obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
     obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
@@ -328,7 +328,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
       haveI :=
         @is_domain_of_le_non_zero_divisors B _ R _ _ _ (Submonoid.powers x) _
           (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-      simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H
+      simp only [map_zero, ← Subtype.val_eq_coe, ← hy, map_pow] at H 
       apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
       rw [← H]
       apply mul_ne_zero hb (pow_ne_zero _ _)
@@ -337,7 +337,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
           (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
           (mem_non_zero_divisors_iff_ne_zero.mpr hx.ne_zero)
       exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-    simp only [← Subtype.val_eq_coe, ← hy] at H
+    simp only [← Subtype.val_eq_coe, ← hy] at H 
     obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
     refine' ⟨a, m - d, _⟩
     rw [← mk'_one B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat, ← map_pow _ _ d,

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

@@ -65,12 +65,6 @@ noncomputable def invSelf : S :=
 #align is_localization.away.inv_self IsLocalization.Away.invSelf
 -/
 
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 @[simp]
 theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by
   convert IsLocalization.mk'_mul_mk'_eq_one _ 1; symm; apply IsLocalization.mk'_one
@@ -78,12 +72,6 @@ theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by
 
 variable {g : R →+* P}
 
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 /-- Given `x : R`, a localization map `F : R →+* S` away from `x`, and a map of `comm_semiring`s
 `g : R →+* P` such that `g x` is invertible, the homomorphism induced from `S` to `P` sending
 `z : S` to `g y * (g x)⁻ⁿ`, where `y : R, n : ℕ` are such that `z = F y * (F x)⁻ⁿ`. -/
@@ -95,31 +83,16 @@ noncomputable def lift (hg : IsUnit (g x)) : S →+* P :=
       exact IsUnit.map (powMonoidHom n : P →* P) hg
 #align is_localization.away.lift IsLocalization.Away.lift
 
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 @[simp]
 theorem AwayMap.lift_eq (hg : IsUnit (g x)) (a : R) : lift x hg ((algebraMap R S) a) = g a :=
   lift_eq _ _
 #align is_localization.away.away_map.lift_eq IsLocalization.Away.AwayMap.lift_eq
 
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 @[simp]
 theorem AwayMap.lift_comp (hg : IsUnit (g x)) : (lift x hg).comp (algebraMap R S) = g :=
   lift_comp _
 #align is_localization.away.away_map.lift_comp IsLocalization.Away.AwayMap.lift_comp
 
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 /-- Given `x y : R` and localizations `S`, `P` away from `x` and `x * y`
 respectively, the homomorphism induced from `S` to `P`. -/
 noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x * y) P] : S →+* P :=
@@ -131,9 +104,6 @@ noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x
 
 variable (S) (Q : Type _) [CommSemiring Q] [Algebra P Q]
 
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 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
     [IsLocalization.Away (f r) Q] : S →+* Q :=
@@ -151,12 +121,6 @@ section AtUnits
 
 variable (R) (S) (M)
 
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 /-- The localization at a module of units is isomorphic to the ring -/
 noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
   by
@@ -194,12 +158,6 @@ noncomputable def atOne [IsLocalization.Away (1 : R) S] : R ≃ₐ[R] S :=
 #align is_localization.at_one IsLocalization.atOne
 -/
 
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 theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [CommRing S]
     [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
     IsLocalization.Away r S :=
@@ -222,12 +180,6 @@ open IsLocalization
 
 variable {M}
 
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 /-- Given a map `f : R →+* S` and an element `r : R`, such that `f r` is invertible,
   we may construct a map `Rᵣ →+* S`. -/
 noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
@@ -235,9 +187,6 @@ noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
   IsLocalization.Away.lift r hr
 #align localization.away_lift Localization.awayLift
 
-/- warning: localization.away_map -> Localization.awayMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align localization.away_map Localization.awayMapₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable abbrev awayMap (f : R →+* P) (r : R) :
     Localization.Away r →+* Localization.Away (f r) :=
@@ -267,54 +216,24 @@ noncomputable def selfZpow (m : ℤ) : B :=
 #align self_zpow selfZpow
 -/
 
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 theorem selfZpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZpow x B n = algebraMap R B x ^ n.natAbs :=
   dif_pos hn
 #align self_zpow_of_nonneg selfZpow_of_nonneg
 
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 @[simp]
 theorem selfZpow_coe_nat (d : ℕ) : selfZpow x B d = algebraMap R B x ^ d :=
   selfZpow_of_nonneg _ _ (Int.coe_nat_nonneg d)
 #align self_zpow_coe_nat selfZpow_coe_nat
 
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 @[simp]
 theorem selfZpow_zero : selfZpow x B 0 = 1 := by simp [selfZpow_of_nonneg _ _ le_rfl]
 #align self_zpow_zero selfZpow_zero
 
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 theorem selfZpow_of_neg {n : ℤ} (hn : n < 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   dif_neg hn.not_le
 #align self_zpow_of_neg selfZpow_of_neg
 
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 theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   by
@@ -323,23 +242,11 @@ theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
   · simp [selfZpow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
 #align self_zpow_of_nonpos selfZpow_of_nonpos
 
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 @[simp]
 theorem selfZpow_neg_coe_nat (d : ℕ) : selfZpow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
   simp [selfZpow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
 #align self_zpow_neg_coe_nat selfZpow_neg_coe_nat
 
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-Case conversion may be inaccurate. Consider using '#align self_zpow_sub_cast_nat selfZpow_sub_cast_natₓ'. -/
 @[simp]
 theorem selfZpow_sub_cast_nat {n m : ℕ} :
     selfZpow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
@@ -353,12 +260,6 @@ theorem selfZpow_sub_cast_nat {n m : ℕ} :
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
 #align self_zpow_sub_cast_nat selfZpow_sub_cast_nat
 
-/- warning: self_zpow_add -> selfZpow_add is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align self_zpow_add selfZpow_addₓ'. -/
 @[simp]
 theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZpow x B m :=
   by
@@ -381,12 +282,6 @@ theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZ
     simp [pow_add]
 #align self_zpow_add selfZpow_add
 
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-Case conversion may be inaccurate. Consider using '#align self_zpow_mul_neg selfZpow_mul_negₓ'. -/
 theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
   by
   by_cases hd : d ≤ 0
@@ -401,19 +296,10 @@ theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
     rwa [neg_neg, ← not_le]
 #align self_zpow_mul_neg selfZpow_mul_neg
 
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-Case conversion may be inaccurate. Consider using '#align self_zpow_neg_mul selfZpow_neg_mulₓ'. -/
 theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := by
   rw [mul_comm, selfZpow_mul_neg x B d]
 #align self_zpow_neg_mul selfZpow_neg_mul
 
-/- warning: self_zpow_pow_sub -> selfZpow_pow_sub is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align self_zpow_pow_sub selfZpow_pow_subₓ'. -/
 theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
     selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
       selfZpow x B m * mk' B a (1 : Submonoid.powers x) = selfZpow x B d * b :=
@@ -432,9 +318,6 @@ theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
-/- warning: exists_reduced_fraction' -> exists_reduced_fraction' is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align exists_reduced_fraction' exists_reduced_fraction'ₓ'. -/
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R)(n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
   classical

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

@@ -72,11 +72,8 @@ but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align is_localization.away.mul_inv_self IsLocalization.Away.mul_invSelfₓ'. -/
 @[simp]
-theorem mul_invSelf : algebraMap R S x * invSelf x = 1 :=
-  by
-  convert IsLocalization.mk'_mul_mk'_eq_one _ 1
-  symm
-  apply IsLocalization.mk'_one
+theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by
+  convert IsLocalization.mk'_mul_mk'_eq_one _ 1; symm; apply IsLocalization.mk'_one
 #align is_localization.away.mul_inv_self IsLocalization.Away.mul_invSelf
 
 variable {g : R →+* P}
@@ -141,11 +138,7 @@ Case conversion may be inaccurate. Consider using '#align is_localization.away.m
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
     [IsLocalization.Away (f r) Q] : S →+* Q :=
   IsLocalization.map Q f
-    (show Submonoid.powers r ≤ (Submonoid.powers (f r)).comap f
-      by
-      rintro x ⟨n, rfl⟩
-      use n
-      simp)
+    (show Submonoid.powers r ≤ (Submonoid.powers (f r)).comap f by rintro x ⟨n, rfl⟩; use n; simp)
 #align is_localization.away.map IsLocalization.Away.map
 
 end Away
@@ -210,19 +203,13 @@ Case conversion may be inaccurate. Consider using '#align is_localization.away_o
 theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [CommRing S]
     [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
     IsLocalization.Away r S :=
-  { map_units := by
-      rintro ⟨_, n, rfl⟩
-      exact (algebraMap R S).isUnit_map (hr.pow _)
-    surj := fun z => by
-      obtain ⟨z', rfl⟩ := H.2 z
-      exact ⟨⟨z', 1⟩, by simp⟩
+  { map_units := by rintro ⟨_, n, rfl⟩; exact (algebraMap R S).isUnit_map (hr.pow _)
+    surj := fun z => by obtain ⟨z', rfl⟩ := H.2 z; exact ⟨⟨z', 1⟩, by simp⟩
     eq_iff_exists := fun x y => by
       erw [H.1.eq_iff]
       constructor
-      · rintro rfl
-        exact ⟨1, rfl⟩
-      · rintro ⟨⟨_, n, rfl⟩, e⟩
-        exact (hr.pow _).mul_right_inj.mp e }
+      · rintro rfl; exact ⟨1, rfl⟩
+      · rintro ⟨⟨_, n, rfl⟩, e⟩; exact (hr.pow _).mul_right_inj.mp e }
 #align is_localization.away_of_is_unit_of_bijective IsLocalization.away_of_isUnit_of_bijective
 
 end AtUnits

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

@@ -99,10 +99,7 @@ noncomputable def lift (hg : IsUnit (g x)) : S →+* P :=
 #align is_localization.away.lift IsLocalization.Away.lift
 
 /- warning: is_localization.away.away_map.lift_eq -> IsLocalization.Away.AwayMap.lift_eq is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.away.away_map.lift_eq IsLocalization.Away.AwayMap.lift_eqₓ'. -/
 @[simp]
 theorem AwayMap.lift_eq (hg : IsUnit (g x)) (a : R) : lift x hg ((algebraMap R S) a) = g a :=
@@ -138,10 +135,7 @@ noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x
 variable (S) (Q : Type _) [CommSemiring Q] [Algebra P Q]
 
 /- warning: is_localization.away.map -> IsLocalization.Away.map is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.away.map IsLocalization.Away.mapₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
@@ -255,10 +249,7 @@ noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
 #align localization.away_lift Localization.awayLift
 
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 Case conversion may be inaccurate. Consider using '#align localization.away_map Localization.awayMapₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable abbrev awayMap (f : R →+* P) (r : R) :
@@ -434,10 +425,7 @@ theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := b
 #align self_zpow_neg_mul selfZpow_neg_mul
 
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 Case conversion may be inaccurate. Consider using '#align self_zpow_pow_sub selfZpow_pow_subₓ'. -/
 theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
     selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
@@ -458,10 +446,7 @@ theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 /- warning: exists_reduced_fraction' -> exists_reduced_fraction' is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align exists_reduced_fraction' exists_reduced_fraction'ₓ'. -/
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R)(n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 
 ! This file was ported from Lean 3 source module ring_theory.localization.away.basic
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.RingTheory.Localization.Basic
 /-!
 # Localizations away from an element
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main definitions
 
  * `is_localization.away (x : R) S` expresses that `S` is a localization away from `x`, as an

mathlib commit https://github.com/leanprover-community/mathlib/commit/1b0a28e1c93409dbf6d69526863cd9984ef652ce

@@ -43,21 +43,31 @@ section Away
 
 variable (x : R)
 
+#print IsLocalization.Away /-
 /-- Given `x : R`, the typeclass `is_localization.away x S` states that `S` is
 isomorphic to the localization of `R` at the submonoid generated by `x`. -/
 abbrev Away (S : Type _) [CommSemiring S] [Algebra R S] :=
   IsLocalization (Submonoid.powers x) S
 #align is_localization.away IsLocalization.Away
+-/
 
 namespace Away
 
 variable [IsLocalization.Away x S]
 
+#print IsLocalization.Away.invSelf /-
 /-- Given `x : R` and a localization map `F : R →+* S` away from `x`, `inv_self` is `(F x)⁻¹`. -/
 noncomputable def invSelf : S :=
   mk' S (1 : R) ⟨x, Submonoid.mem_powers _⟩
 #align is_localization.away.inv_self IsLocalization.Away.invSelf
+-/
 
+/- warning: is_localization.away.mul_inv_self -> IsLocalization.Away.mul_invSelf is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_localization.away.mul_inv_self IsLocalization.Away.mul_invSelfₓ'. -/
 @[simp]
 theorem mul_invSelf : algebraMap R S x * invSelf x = 1 :=
   by
@@ -68,6 +78,12 @@ theorem mul_invSelf : algebraMap R S x * invSelf x = 1 :=
 
 variable {g : R →+* P}
 
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+Case conversion may be inaccurate. Consider using '#align is_localization.away.lift IsLocalization.Away.liftₓ'. -/
 /-- Given `x : R`, a localization map `F : R →+* S` away from `x`, and a map of `comm_semiring`s
 `g : R →+* P` such that `g x` is invertible, the homomorphism induced from `S` to `P` sending
 `z : S` to `g y * (g x)⁻ⁿ`, where `y : R, n : ℕ` are such that `z = F y * (F x)⁻ⁿ`. -/
@@ -79,16 +95,34 @@ noncomputable def lift (hg : IsUnit (g x)) : S →+* P :=
       exact IsUnit.map (powMonoidHom n : P →* P) hg
 #align is_localization.away.lift IsLocalization.Away.lift
 
+/- warning: is_localization.away.away_map.lift_eq -> IsLocalization.Away.AwayMap.lift_eq is a dubious translation:
+lean 3 declaration is
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_inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u3} R P (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4)))))) g a)
+Case conversion may be inaccurate. Consider using '#align is_localization.away.away_map.lift_eq IsLocalization.Away.AwayMap.lift_eqₓ'. -/
 @[simp]
 theorem AwayMap.lift_eq (hg : IsUnit (g x)) (a : R) : lift x hg ((algebraMap R S) a) = g a :=
   lift_eq _ _
 #align is_localization.away.away_map.lift_eq IsLocalization.Away.AwayMap.lift_eq
 
+/- warning: is_localization.away.away_map.lift_comp -> IsLocalization.Away.AwayMap.lift_comp 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 is_localization.away.away_map.lift_comp IsLocalization.Away.AwayMap.lift_compₓ'. -/
 @[simp]
 theorem AwayMap.lift_comp (hg : IsUnit (g x)) : (lift x hg).comp (algebraMap R S) = g :=
   lift_comp _
 #align is_localization.away.away_map.lift_comp IsLocalization.Away.AwayMap.lift_comp
 
+/- warning: is_localization.away.away_to_away_right -> IsLocalization.Away.awayToAwayRight is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {S : Type.{u2}} [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] {P : Type.{u3}} [_inst_4 : CommSemiring.{u3} P] (x : R) [_inst_5 : IsLocalization.Away.{u1, u2} R _inst_1 x S _inst_2 _inst_3] (y : R) [_inst_6 : Algebra.{u1, u3} R P _inst_1 (CommSemiring.toSemiring.{u3} P _inst_4)] [_inst_7 : IsLocalization.Away.{u1, u3} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) P _inst_4 _inst_6], RingHom.{u2, u3} S P (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {S : Type.{u2}} [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] {P : Type.{u3}} [_inst_4 : CommSemiring.{u3} P] (x : R) [_inst_5 : IsLocalization.Away.{u1, u2} R _inst_1 x S _inst_2 _inst_3] (y : R) [_inst_6 : Algebra.{u1, u3} R P _inst_1 (CommSemiring.toSemiring.{u3} P _inst_4)] [_inst_7 : IsLocalization.Away.{u1, u3} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x y) P _inst_4 _inst_6], RingHom.{u2, u3} S P (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))
+Case conversion may be inaccurate. Consider using '#align is_localization.away.away_to_away_right IsLocalization.Away.awayToAwayRightₓ'. -/
 /-- Given `x y : R` and localizations `S`, `P` away from `x` and `x * y`
 respectively, the homomorphism induced from `S` to `P`. -/
 noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x * y) P] : S →+* P :=
@@ -100,6 +134,12 @@ noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x
 
 variable (S) (Q : Type _) [CommSemiring Q] [Algebra P Q]
 
+/- warning: is_localization.away.map -> IsLocalization.Away.map is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (S : Type.{u2}) [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] {P : Type.{u3}} [_inst_4 : CommSemiring.{u3} P] (Q : Type.{u4}) [_inst_6 : CommSemiring.{u4} Q] [_inst_7 : Algebra.{u3, u4} P Q _inst_4 (CommSemiring.toSemiring.{u4} Q _inst_6)] (f : RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) (r : R) [_inst_8 : IsLocalization.Away.{u1, u2} R _inst_1 r S _inst_2 _inst_3] [_inst_9 : IsLocalization.Away.{u3, u4} P _inst_4 (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) (fun (_x : RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) => R -> P) (RingHom.hasCoeToFun.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) f r) Q _inst_6 _inst_7], RingHom.{u2, u4} S Q (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u4} Q (CommSemiring.toSemiring.{u4} Q _inst_6))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (S : Type.{u2}) [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] {P : Type.{u3}} [_inst_4 : CommSemiring.{u3} P] (Q : Type.{u4}) [_inst_6 : CommSemiring.{u4} Q] [_inst_7 : Algebra.{u3, u4} P Q _inst_4 (CommSemiring.toSemiring.{u4} Q _inst_6)] (f : RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) (r : R) [_inst_8 : IsLocalization.Away.{u1, u2} R _inst_1 r S _inst_2 _inst_3] [_inst_9 : IsLocalization.Away.{u3, u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => P) r) _inst_4 (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => P) _x) (MulHomClass.toFunLike.{max u1 u3, u1, u3} (RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) R P (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u3} P (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} P (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u3, u1, u3} (RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) R P (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} P (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u3, u1, u3} (RingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4))) R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4)) (RingHom.instRingHomClassRingHom.{u1, u3} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} P (CommSemiring.toSemiring.{u3} P _inst_4)))))) f r) Q _inst_6 _inst_7], RingHom.{u2, u4} S Q (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u4} Q (CommSemiring.toSemiring.{u4} Q _inst_6))
+Case conversion may be inaccurate. Consider using '#align is_localization.away.map IsLocalization.Away.mapₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
     [IsLocalization.Away (f r) Q] : S →+* Q :=
@@ -121,6 +161,12 @@ section AtUnits
 
 variable (R) (S) (M)
 
+/- warning: is_localization.at_units -> IsLocalization.atUnits is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_1 : CommSemiring.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (S : Type.{u2}) [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : IsLocalization.{u1, u2} R _inst_1 M S _inst_2 _inst_3], (forall (x : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M), IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M) R (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M) R (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M) R (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) M) R (coeSubtype.{succ u1} R (fun (x : R) => Membership.Mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x M))))) x)) -> (AlgEquiv.{u1, u1, u2} R R S _inst_1 (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S _inst_2) (Algebra.id.{u1} R _inst_1) _inst_3)
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_1 : CommSemiring.{u1} R] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (S : Type.{u2}) [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : IsLocalization.{u1, u2} R _inst_1 M S _inst_2 _inst_3], (forall (x : Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x M)), IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Subtype.val.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x (SetLike.coe.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) M)) x)) -> (AlgEquiv.{u1, u1, u2} R R S _inst_1 (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S _inst_2) (Algebra.id.{u1} R _inst_1) _inst_3)
+Case conversion may be inaccurate. Consider using '#align is_localization.at_units IsLocalization.atUnitsₓ'. -/
 /-- The localization at a module of units is isomorphic to the ring -/
 noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
   by
@@ -138,6 +184,7 @@ noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S :=
     simp
 #align is_localization.at_units IsLocalization.atUnits
 
+#print IsLocalization.atUnit /-
 /-- The localization away from a unit is isomorphic to the ring -/
 noncomputable def atUnit (x : R) (e : IsUnit x) [IsLocalization.Away x S] : R ≃ₐ[R] S :=
   by
@@ -148,12 +195,21 @@ noncomputable def atUnit (x : R) (e : IsUnit x) [IsLocalization.Away x S] : R 
   use u.inv ^ n
   simp [← hxn, ← hu, ← mul_pow]
 #align is_localization.at_unit IsLocalization.atUnit
+-/
 
+#print IsLocalization.atOne /-
 /-- The localization at one is isomorphic to the ring. -/
 noncomputable def atOne [IsLocalization.Away (1 : R) S] : R ≃ₐ[R] S :=
   @atUnit R _ S _ _ (1 : R) isUnit_one _
 #align is_localization.at_one IsLocalization.atOne
+-/
 
+/- warning: is_localization.away_of_is_unit_of_bijective -> IsLocalization.away_of_isUnit_of_bijective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_6 : CommRing.{u1} R] [_inst_7 : CommRing.{u2} S] [_inst_8 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_6) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_7))] {r : R}, (IsUnit.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_6)) r) -> (Function.Bijective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_6))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_7)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_6))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_7)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_6))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_7)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_6) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_7)) _inst_8))) -> (IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_6) r S (CommRing.toCommSemiring.{u2} S _inst_7) _inst_8)
+but is expected to have type
+  forall {R : Type.{u2}} (S : Type.{u1}) [_inst_6 : CommRing.{u2} R] [_inst_7 : CommRing.{u1} S] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_6) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7))] {r : R}, (IsUnit.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6)))) r) -> (Function.Bijective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_6))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_6) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_7)) _inst_8))) -> (IsLocalization.Away.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_6) r S (CommRing.toCommSemiring.{u1} S _inst_7) _inst_8)
+Case conversion may be inaccurate. Consider using '#align is_localization.away_of_is_unit_of_bijective IsLocalization.away_of_isUnit_of_bijectiveₓ'. -/
 theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [CommRing S]
     [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
     IsLocalization.Away r S :=
@@ -182,6 +238,12 @@ open IsLocalization
 
 variable {M}
 
+/- warning: localization.away_lift -> Localization.awayLift is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {P : Type.{u2}} [_inst_4 : CommSemiring.{u2} P] (f : RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (r : R), (IsUnit.{u2} P (MonoidWithZero.toMonoid.{u2} P (Semiring.toMonoidWithZero.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) (fun (_x : RingHom.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) => R -> P) (RingHom.hasCoeToFun.{u1, u2} R P (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4))) f r)) -> (RingHom.{u1, u2} (Localization.Away.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) r) P (Semiring.toNonAssocSemiring.{u1} (Localization.Away.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) r) (CommSemiring.toSemiring.{u1} (Localization.Away.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) r) (Localization.commSemiring.{u1} R _inst_1 (Submonoid.powers.{u1} R (CommMonoid.toMonoid.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1)) r)))) (Semiring.toNonAssocSemiring.{u2} P (CommSemiring.toSemiring.{u2} P _inst_4)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align localization.away_lift Localization.awayLiftₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, such that `f r` is invertible,
   we may construct a map `Rᵣ →+* S`. -/
 noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
@@ -189,6 +251,12 @@ noncomputable abbrev awayLift (f : R →+* P) (r : R) (hr : IsUnit (f r)) :
   IsLocalization.Away.lift r hr
 #align localization.away_lift Localization.awayLift
 
+/- warning: localization.away_map -> Localization.awayMap is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align localization.away_map Localization.awayMapₓ'. -/
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable abbrev awayMap (f : R →+* P) (r : R) :
     Localization.Away r →+* Localization.Away (f r) :=
@@ -211,29 +279,61 @@ variable (x : R)
 
 variable (B : Type _) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
+#print selfZpow /-
 /-- `self_zpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
 noncomputable def selfZpow (m : ℤ) : B :=
   if hm : 0 ≤ m then algebraMap _ _ x ^ m.natAbs else mk' _ (1 : R) (Submonoid.pow x m.natAbs)
 #align self_zpow selfZpow
+-/
 
+/- warning: self_zpow_of_nonneg -> selfZpow_of_nonneg is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_of_nonneg selfZpow_of_nonnegₓ'. -/
 theorem selfZpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZpow x B n = algebraMap R B x ^ n.natAbs :=
   dif_pos hn
 #align self_zpow_of_nonneg selfZpow_of_nonneg
 
+/- warning: self_zpow_coe_nat -> selfZpow_coe_nat is a dubious translation:
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 @[simp]
 theorem selfZpow_coe_nat (d : ℕ) : selfZpow x B d = algebraMap R B x ^ d :=
   selfZpow_of_nonneg _ _ (Int.coe_nat_nonneg d)
 #align self_zpow_coe_nat selfZpow_coe_nat
 
+/- warning: self_zpow_zero -> selfZpow_zero is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (B : Type.{u2}) [_inst_2 : CommRing.{u2} B] [_inst_3 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))] [_inst_4 : IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) x B (CommRing.toCommSemiring.{u2} B _inst_2) _inst_3], Eq.{succ u2} B (selfZpow.{u1, u2} R _inst_1 x B _inst_2 _inst_3 _inst_4 (OfNat.ofNat.{0} Int 0 (instOfNatInt 0))) (OfNat.ofNat.{u2} B 1 (One.toOfNat1.{u2} B (Semiring.toOne.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))))
+Case conversion may be inaccurate. Consider using '#align self_zpow_zero selfZpow_zeroₓ'. -/
 @[simp]
 theorem selfZpow_zero : selfZpow x B 0 = 1 := by simp [selfZpow_of_nonneg _ _ le_rfl]
 #align self_zpow_zero selfZpow_zero
 
+/- warning: self_zpow_of_neg -> selfZpow_of_neg is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (B : Type.{u2}) [_inst_2 : CommRing.{u2} B] [_inst_3 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))] [_inst_4 : IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) x B (CommRing.toCommSemiring.{u2} B _inst_2) _inst_3] {n : Int}, (LT.lt.{0} Int Int.instLTInt n (OfNat.ofNat.{0} Int 0 (instOfNatInt 0))) -> (Eq.{succ u2} B (selfZpow.{u1, u2} R _inst_1 x B _inst_2 _inst_3 _inst_4 n) (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) (Submonoid.powers.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) x) B (CommRing.toCommSemiring.{u2} B _inst_2) _inst_3 _inst_4 (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Submonoid.pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) x (Int.natAbs n))))
+Case conversion may be inaccurate. Consider using '#align self_zpow_of_neg selfZpow_of_negₓ'. -/
 theorem selfZpow_of_neg {n : ℤ} (hn : n < 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   dif_neg hn.not_le
 #align self_zpow_of_neg selfZpow_of_neg
 
+/- warning: self_zpow_of_nonpos -> selfZpow_of_nonpos is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align self_zpow_of_nonpos selfZpow_of_nonposₓ'. -/
 theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
     selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   by
@@ -242,11 +342,23 @@ theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
   · simp [selfZpow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
 #align self_zpow_of_nonpos selfZpow_of_nonpos
 
+/- warning: self_zpow_neg_coe_nat -> selfZpow_neg_coe_nat is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_neg_coe_nat selfZpow_neg_coe_natₓ'. -/
 @[simp]
 theorem selfZpow_neg_coe_nat (d : ℕ) : selfZpow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
   simp [selfZpow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
 #align self_zpow_neg_coe_nat selfZpow_neg_coe_nat
 
+/- warning: self_zpow_sub_cast_nat -> selfZpow_sub_cast_nat is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_sub_cast_nat selfZpow_sub_cast_natₓ'. -/
 @[simp]
 theorem selfZpow_sub_cast_nat {n m : ℕ} :
     selfZpow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) :=
@@ -260,6 +372,12 @@ theorem selfZpow_sub_cast_nat {n m : ℕ} :
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
 #align self_zpow_sub_cast_nat selfZpow_sub_cast_nat
 
+/- warning: self_zpow_add -> selfZpow_add is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_add selfZpow_addₓ'. -/
 @[simp]
 theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZpow x B m :=
   by
@@ -282,6 +400,12 @@ theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZ
     simp [pow_add]
 #align self_zpow_add selfZpow_add
 
+/- warning: self_zpow_mul_neg -> selfZpow_mul_neg is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_mul_neg selfZpow_mul_negₓ'. -/
 theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
   by
   by_cases hd : d ≤ 0
@@ -296,10 +420,22 @@ theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 :=
     rwa [neg_neg, ← not_le]
 #align self_zpow_mul_neg selfZpow_mul_neg
 
+/- warning: self_zpow_neg_mul -> selfZpow_neg_mul is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align self_zpow_neg_mul selfZpow_neg_mulₓ'. -/
 theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := by
   rw [mul_comm, selfZpow_mul_neg x B d]
 #align self_zpow_neg_mul selfZpow_neg_mul
 
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+Case conversion may be inaccurate. Consider using '#align self_zpow_pow_sub selfZpow_pow_subₓ'. -/
 theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
     selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
       selfZpow x B m * mk' B a (1 : Submonoid.powers x) = selfZpow x B d * b :=
@@ -318,6 +454,12 @@ theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
+/- warning: exists_reduced_fraction' -> exists_reduced_fraction' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (B : Type.{u2}) [_inst_2 : CommRing.{u2} B] [_inst_3 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_2))] [_inst_4 : IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) x B (CommRing.toCommSemiring.{u2} B _inst_2) _inst_3] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_6 : NormalizationMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_5)] [_inst_7 : UniqueFactorizationMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_5)] {b : B}, (Ne.{succ u2} B b (OfNat.ofNat.{u2} B 0 (OfNat.mk.{u2} B 0 (Zero.zero.{u2} B (MulZeroClass.toHasZero.{u2} B (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_2)))))))))) -> (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) x) -> (Exists.{succ u1} R (fun (a : R) => Exists.{1} Int (fun (n : Int) => And (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x a)) (Eq.{succ u2} B (HMul.hMul.{u2, u2, u2} B B B (instHMul.{u2} B (Distrib.toHasMul.{u2} B (Ring.toDistrib.{u2} B (CommRing.toRing.{u2} B _inst_2)))) (selfZpow.{u1, u2} R _inst_1 x B _inst_2 _inst_3 _inst_4 n) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_2)))) (fun (_x : RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_2)))) => R -> B) (RingHom.hasCoeToFun.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_2)))) (algebraMap.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_2)) _inst_3) a)) b))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : R) (B : Type.{u2}) [_inst_2 : CommRing.{u2} B] [_inst_3 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))] [_inst_4 : IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) x B (CommRing.toCommSemiring.{u2} B _inst_2) _inst_3] [_inst_5 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_6 : NormalizationMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_5)] [_inst_7 : UniqueFactorizationMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_5)] {b : B}, (Ne.{succ u2} B b (OfNat.ofNat.{u2} B 0 (Zero.toOfNat0.{u2} B (CommMonoidWithZero.toZero.{u2} B (CommSemiring.toCommMonoidWithZero.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))))) -> (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) x) -> (Exists.{succ u1} R (fun (a : R) => Exists.{1} Int (fun (n : Int) => And (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x a)) (Eq.{succ u2} B (HMul.hMul.{u2, u2, u2} B ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) a) B (instHMul.{u2} B (NonUnitalNonAssocRing.toMul.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_2))))) (selfZpow.{u1, u2} R _inst_1 x B _inst_2 _inst_3 _inst_4 n) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))) R B (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))) R B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)))) R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2))))))) (algebraMap.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_2)) _inst_3) a)) b))))
+Case conversion may be inaccurate. Consider using '#align exists_reduced_fraction' exists_reduced_fraction'ₓ'. -/
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R)(n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
   classical

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -3,12 +3,12 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 
-! This file was ported from Lean 3 source module ring_theory.localization.away
-! leanprover-community/mathlib commit 570e9f4877079b3a923135b3027ac3be8695ab8c
+! This file was ported from Lean 3 source module ring_theory.localization.away.basic
+! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.RingTheory.AdjoinRoot
+import Mathbin.RingTheory.UniqueFactorizationDomain
 import Mathbin.RingTheory.Localization.Basic
 
 /-!
@@ -199,38 +199,13 @@ end Localization
 
 end CommSemiring
 
-open Polynomial AdjoinRoot Localization
+open Localization
 
 variable {R : Type _} [CommRing R]
 
-attribute [local instance] IsLocalization.algHom_subsingleton AdjoinRoot.algHom_subsingleton
-
-/-- The `R`-`alg_equiv` between the localization of `R` away from `r` and
-    `R` with an inverse of `r` adjoined. -/
-noncomputable def Localization.awayEquivAdjoin (r : R) : Away r ≃ₐ[R] AdjoinRoot (C r * X - 1) :=
-  AlgEquiv.ofAlgHom
-    { awayLift _ r _ with
-      commutes' :=
-        IsLocalization.Away.AwayMap.lift_eq r (isUnit_of_mul_eq_one _ _ <| root_is_inv r) }
-    (liftHom _ (IsLocalization.Away.invSelf r) <| by
-      simp only [map_sub, map_mul, aeval_C, aeval_X, IsLocalization.Away.mul_invSelf, aeval_one,
-        sub_self])
-    (Subsingleton.elim _ _) (Subsingleton.elim _ _)
-#align localization.away_equiv_adjoin Localization.awayEquivAdjoin
-
-theorem IsLocalization.adjoin_inv (r : R) : IsLocalization.Away r (AdjoinRoot <| C r * X - 1) :=
-  IsLocalization.isLocalization_of_algEquiv _ (Localization.awayEquivAdjoin r)
-#align is_localization.adjoin_inv IsLocalization.adjoin_inv
-
-theorem IsLocalization.Away.finitePresentation (r : R) {S} [CommRing S] [Algebra R S]
-    [IsLocalization.Away r S] : Algebra.FinitePresentation R S :=
-  (AdjoinRoot.finitePresentation _).Equiv <|
-    (Localization.awayEquivAdjoin r).symm.trans <| IsLocalization.algEquiv (Submonoid.powers r) _ _
-#align is_localization.away.finite_presentation IsLocalization.Away.finitePresentation
-
 section NumDenom
 
-open multiplicity UniqueFactorizationMonoid IsLocalization
+open UniqueFactorizationMonoid IsLocalization
 
 variable (x : R)
 

A PR analogous to #12338 and #12361: reformatting proofs following the multiple goals linter of #12339.

@@ -295,13 +295,13 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
         (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
     simp only [map_zero, ← hy, map_pow] at H
     apply ((injective_iff_map_eq_zero' (algebraMap R B)).mp _ a₀).mpr.mt
-    rw [← H]
-    apply mul_ne_zero hb (pow_ne_zero _ _)
-    exact
-      IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
-        (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
-        (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)
-    exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+    · rw [← H]
+      apply mul_ne_zero hb (pow_ne_zero _ _)
+      exact
+        IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors B
+          (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
+          (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)
+    · exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
   simp only [← hy] at H
   obtain ⟨m, a, hyp1, hyp2⟩ := WfDvdMonoid.max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -189,9 +189,9 @@ theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMa
 #align self_zpow_of_nonneg selfZPow_of_nonneg
 
 @[simp]
-theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
+theorem selfZPow_natCast (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
   selfZPow_of_nonneg _ _ (Int.natCast_nonneg d)
-#align self_zpow_coe_nat selfZPow_coe_nat
+#align self_zpow_coe_nat selfZPow_natCast
 
 @[simp]
 theorem selfZPow_zero : selfZPow x B 0 = 1 := by
@@ -211,20 +211,25 @@ theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
 #align self_zpow_of_nonpos selfZPow_of_nonpos
 
 @[simp]
-theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
+theorem selfZPow_neg_natCast (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
   simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.natCast_nonneg d))]
-#align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
+#align self_zpow_neg_coe_nat selfZPow_neg_natCast
 
 @[simp]
-theorem selfZPow_sub_cast_nat {n m : ℕ} :
+theorem selfZPow_sub_natCast {n m : ℕ} :
     selfZPow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) := by
   by_cases h : m ≤ n
   · rw [IsLocalization.eq_mk'_iff_mul_eq, Submonoid.pow_apply, Subtype.coe_mk, ← Int.ofNat_sub h,
-      selfZPow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
-  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_coe_nat,
+      selfZPow_natCast, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
+  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_natCast,
       IsLocalization.mk'_eq_iff_eq]
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
-#align self_zpow_sub_cast_nat selfZPow_sub_cast_nat
+#align self_zpow_sub_cast_nat selfZPow_sub_natCast
+
+-- 2024-04-05
+@[deprecated] alias selfZPow_coe_nat := selfZPow_natCast
+@[deprecated] alias selfZPow_neg_coe_nat := selfZPow_neg_natCast
+@[deprecated] alias selfZPow_sub_cast_nat := selfZPow_sub_natCast
 
 @[simp]
 theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m := by
@@ -233,11 +238,11 @@ theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZ
       selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_of_nonneg hn hm, pow_add]
   · have : n + m = n.natAbs - m.natAbs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
-    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_natCast,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow]
   · have : n + m = m.natAbs - n.natAbs := by
       rw [Int.natAbs_of_nonneg hm, Int.ofNat_natAbs_of_nonpos hn.le, sub_neg_eq_add, add_comm]
-    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_natCast,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
   · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
       Int.natAbs_add_of_nonpos hn.le hm.le, ← mk'_mul, one_mul]
@@ -300,7 +305,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
   simp only [← hy] at H
   obtain ⟨m, a, hyp1, hyp2⟩ := WfDvdMonoid.max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
-  rw [← mk'_one (M := Submonoid.powers x) B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat,
+  rw [← mk'_one (M := Submonoid.powers x) B, selfZPow_pow_sub, selfZPow_natCast, selfZPow_natCast,
     ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
   exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'

Most lemmas are too specific to be reused so I deleted them. They must have been used for tactics in Lean 3.

I moved the few lemmas that were generally useful (although I could only see one use, in RingTheory.Localization.Away.Basic!) to Data.Int.Defs.

@@ -3,7 +3,6 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 -/
-import Mathlib.Init.Data.Int.CompLemmas
 import Mathlib.RingTheory.UniqueFactorizationDomain
 import Mathlib.RingTheory.Localization.Basic
 
@@ -231,7 +230,7 @@ theorem selfZPow_sub_cast_nat {n m : ℕ} :
 theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m := by
   rcases le_or_lt 0 n with hn | hn <;> rcases le_or_lt 0 m with hm | hm
   · rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_nonneg _ _ hm,
-      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
+      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_of_nonneg hn hm, pow_add]
   · have : n + m = n.natAbs - m.natAbs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
     rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
@@ -241,7 +240,7 @@ theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZ
     rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
   · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
-      Int.natAbs_add_neg hn hm, ← mk'_mul, one_mul]
+      Int.natAbs_add_of_nonpos hn.le hm.le, ← mk'_mul, one_mul]
     congr
     ext
     simp [pow_add]

Reduce the diff of #11499

Renames

All in the Int namespace:

• ofNat_eq_cast → ofNat_eq_natCast
• cast_eq_cast_iff_Nat → natCast_inj
• natCast_eq_ofNat → ofNat_eq_natCast
• coe_nat_sub → natCast_sub
• coe_nat_nonneg → natCast_nonneg
• sign_coe_add_one → sign_natCast_add_one
• nat_succ_eq_int_succ → natCast_succ
• succ_neg_nat_succ → succ_neg_natCast_succ
• coe_pred_of_pos → natCast_pred_of_pos
• coe_nat_div → natCast_div
• coe_nat_ediv → natCast_ediv
• sign_coe_nat_of_nonzero → sign_natCast_of_ne_zero
• toNat_coe_nat → toNat_natCast
• toNat_coe_nat_add_one → toNat_natCast_add_one
• coe_nat_dvd → natCast_dvd_natCast
• coe_nat_dvd_left → natCast_dvd
• coe_nat_dvd_right → dvd_natCast
• le_coe_nat_sub → le_natCast_sub
• succ_coe_nat_pos → succ_natCast_pos
• coe_nat_modEq_iff → natCast_modEq_iff
• coe_natAbs → natCast_natAbs
• coe_nat_eq_zero → natCast_eq_zero
• coe_nat_ne_zero → natCast_ne_zero
• coe_nat_ne_zero_iff_pos → natCast_ne_zero_iff_pos
• abs_coe_nat → abs_natCast
• coe_nat_nonpos_iff → natCast_nonpos_iff

Also rename Nat.coe_nat_dvd to Nat.cast_dvd_cast

@@ -191,7 +191,7 @@ theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMa
 
 @[simp]
 theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
-  selfZPow_of_nonneg _ _ (Int.coe_nat_nonneg d)
+  selfZPow_of_nonneg _ _ (Int.natCast_nonneg d)
 #align self_zpow_coe_nat selfZPow_coe_nat
 
 @[simp]
@@ -213,7 +213,7 @@ theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
 
 @[simp]
 theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
-  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
+  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.natCast_nonneg d))]
 #align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
 
 @[simp]

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -32,7 +32,6 @@ commutative ring, field of fractions
 section CommSemiring
 
 variable {R : Type*} [CommSemiring R] (M : Submonoid R) {S : Type*} [CommSemiring S]
-
 variable [Algebra R S] {P : Type*} [CommSemiring P]
 
 namespace IsLocalization
@@ -179,7 +178,6 @@ section NumDen
 open IsLocalization
 
 variable (x : R)
-
 variable (B : Type*) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
 /-- `selfZPow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/

• makes UniqueFactorizationMonoid.max_power_factor obsolete
• makes the proof of multiplicity.finite_prime_left trivial
• relax the condition of exists_reduced_fraction'

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -176,7 +176,7 @@ variable {R : Type*} [CommRing R]
 
 section NumDen
 
-open UniqueFactorizationMonoid IsLocalization
+open IsLocalization
 
 variable (x : R)
 
@@ -281,7 +281,7 @@ theorem selfZPow_pow_sub (a : R) (b : B) (m d : ℤ) :
     rwa [mul_comm _ b, mul_assoc b _ _, selfZPow_mul_neg, mul_one] at this
 #align self_zpow_pow_sub selfZPow_pow_sub
 
-variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
+variable [IsDomain R] [WfDvdMonoid R]
 
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
     ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by
@@ -301,8 +301,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
         (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)
     exact IsLocalization.injective B (powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
   simp only [← hy] at H
-  classical
-  obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
+  obtain ⟨m, a, hyp1, hyp2⟩ := WfDvdMonoid.max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
   rw [← mk'_one (M := Submonoid.powers x) B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat,
     ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -231,7 +231,7 @@ theorem selfZPow_sub_cast_nat {n m : ℕ} :
 
 @[simp]
 theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m := by
-  cases' le_or_lt 0 n with hn hn <;> cases' le_or_lt 0 m with hm hm
+  rcases le_or_lt 0 n with hn | hn <;> rcases le_or_lt 0 m with hm | hm
   · rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_nonneg _ _ hm,
       selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
   · have : n + m = n.natAbs - m.natAbs := by

Normalising to naming convention rule number 6.

@@ -182,109 +182,109 @@ variable (x : R)
 
 variable (B : Type*) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
-/-- `selfZpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
-noncomputable def selfZpow (m : ℤ) : B :=
+/-- `selfZPow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
+noncomputable def selfZPow (m : ℤ) : B :=
   if _ : 0 ≤ m then algebraMap _ _ x ^ m.natAbs else mk' _ (1 : R) (Submonoid.pow x m.natAbs)
-#align self_zpow selfZpow
+#align self_zpow selfZPow
 
-theorem selfZpow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZpow x B n = algebraMap R B x ^ n.natAbs :=
+theorem selfZPow_of_nonneg {n : ℤ} (hn : 0 ≤ n) : selfZPow x B n = algebraMap R B x ^ n.natAbs :=
   dif_pos hn
-#align self_zpow_of_nonneg selfZpow_of_nonneg
+#align self_zpow_of_nonneg selfZPow_of_nonneg
 
 @[simp]
-theorem selfZpow_coe_nat (d : ℕ) : selfZpow x B d = algebraMap R B x ^ d :=
-  selfZpow_of_nonneg _ _ (Int.coe_nat_nonneg d)
-#align self_zpow_coe_nat selfZpow_coe_nat
+theorem selfZPow_coe_nat (d : ℕ) : selfZPow x B d = algebraMap R B x ^ d :=
+  selfZPow_of_nonneg _ _ (Int.coe_nat_nonneg d)
+#align self_zpow_coe_nat selfZPow_coe_nat
 
 @[simp]
-theorem selfZpow_zero : selfZpow x B 0 = 1 := by
-  simp [selfZpow_of_nonneg _ _ le_rfl]
-#align self_zpow_zero selfZpow_zero
+theorem selfZPow_zero : selfZPow x B 0 = 1 := by
+  simp [selfZPow_of_nonneg _ _ le_rfl]
+#align self_zpow_zero selfZPow_zero
 
-theorem selfZpow_of_neg {n : ℤ} (hn : n < 0) :
-    selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
+theorem selfZPow_of_neg {n : ℤ} (hn : n < 0) :
+    selfZPow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) :=
   dif_neg hn.not_le
-#align self_zpow_of_neg selfZpow_of_neg
+#align self_zpow_of_neg selfZPow_of_neg
 
-theorem selfZpow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
-    selfZpow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) := by
+theorem selfZPow_of_nonpos {n : ℤ} (hn : n ≤ 0) :
+    selfZPow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs) := by
   by_cases hn0 : n = 0
-  · simp [hn0, selfZpow_zero, Submonoid.pow_apply]
-  · simp [selfZpow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
-#align self_zpow_of_nonpos selfZpow_of_nonpos
+  · simp [hn0, selfZPow_zero, Submonoid.pow_apply]
+  · simp [selfZPow_of_neg _ _ (lt_of_le_of_ne hn hn0)]
+#align self_zpow_of_nonpos selfZPow_of_nonpos
 
 @[simp]
-theorem selfZpow_neg_coe_nat (d : ℕ) : selfZpow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
-  simp [selfZpow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
-#align self_zpow_neg_coe_nat selfZpow_neg_coe_nat
+theorem selfZPow_neg_coe_nat (d : ℕ) : selfZPow x B (-d) = mk' _ (1 : R) (Submonoid.pow x d) := by
+  simp [selfZPow_of_nonpos _ _ (neg_nonpos.mpr (Int.coe_nat_nonneg d))]
+#align self_zpow_neg_coe_nat selfZPow_neg_coe_nat
 
 @[simp]
-theorem selfZpow_sub_cast_nat {n m : ℕ} :
-    selfZpow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) := by
+theorem selfZPow_sub_cast_nat {n m : ℕ} :
+    selfZPow x B (n - m) = mk' _ (x ^ n) (Submonoid.pow x m) := by
   by_cases h : m ≤ n
   · rw [IsLocalization.eq_mk'_iff_mul_eq, Submonoid.pow_apply, Subtype.coe_mk, ← Int.ofNat_sub h,
-      selfZpow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
-  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZpow_neg_coe_nat,
+      selfZPow_coe_nat, ← map_pow, ← map_mul, ← pow_add, Nat.sub_add_cancel h]
+  · rw [← neg_sub, ← Int.ofNat_sub (le_of_not_le h), selfZPow_neg_coe_nat,
       IsLocalization.mk'_eq_iff_eq]
     simp [Submonoid.pow_apply, ← pow_add, Nat.sub_add_cancel (le_of_not_le h)]
-#align self_zpow_sub_cast_nat selfZpow_sub_cast_nat
+#align self_zpow_sub_cast_nat selfZPow_sub_cast_nat
 
 @[simp]
-theorem selfZpow_add {n m : ℤ} : selfZpow x B (n + m) = selfZpow x B n * selfZpow x B m := by
+theorem selfZPow_add {n m : ℤ} : selfZPow x B (n + m) = selfZPow x B n * selfZPow x B m := by
   cases' le_or_lt 0 n with hn hn <;> cases' le_or_lt 0 m with hm hm
-  · rw [selfZpow_of_nonneg _ _ hn, selfZpow_of_nonneg _ _ hm,
-      selfZpow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
+  · rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_nonneg _ _ hm,
+      selfZPow_of_nonneg _ _ (add_nonneg hn hm), Int.natAbs_add_nonneg hn hm, pow_add]
   · have : n + m = n.natAbs - m.natAbs := by
       rw [Int.natAbs_of_nonneg hn, Int.ofNat_natAbs_of_nonpos hm.le, sub_neg_eq_add]
-    rw [selfZpow_of_nonneg _ _ hn, selfZpow_of_neg _ _ hm, this, selfZpow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hn, selfZPow_of_neg _ _ hm, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow]
   · have : n + m = m.natAbs - n.natAbs := by
       rw [Int.natAbs_of_nonneg hm, Int.ofNat_natAbs_of_nonpos hn.le, sub_neg_eq_add, add_comm]
-    rw [selfZpow_of_nonneg _ _ hm, selfZpow_of_neg _ _ hn, this, selfZpow_sub_cast_nat,
+    rw [selfZPow_of_nonneg _ _ hm, selfZPow_of_neg _ _ hn, this, selfZPow_sub_cast_nat,
       IsLocalization.mk'_eq_mul_mk'_one, map_pow, mul_comm]
-  · rw [selfZpow_of_neg _ _ hn, selfZpow_of_neg _ _ hm, selfZpow_of_neg _ _ (add_neg hn hm),
+  · rw [selfZPow_of_neg _ _ hn, selfZPow_of_neg _ _ hm, selfZPow_of_neg _ _ (add_neg hn hm),
       Int.natAbs_add_neg hn hm, ← mk'_mul, one_mul]
     congr
     ext
     simp [pow_add]
-#align self_zpow_add selfZpow_add
+#align self_zpow_add selfZPow_add
 
-theorem selfZpow_mul_neg (d : ℤ) : selfZpow x B d * selfZpow x B (-d) = 1 := by
+theorem selfZPow_mul_neg (d : ℤ) : selfZPow x B d * selfZPow x B (-d) = 1 := by
   by_cases hd : d ≤ 0
-  · erw [selfZpow_of_nonpos x B hd, selfZpow_of_nonneg, ← map_pow, Int.natAbs_neg,
+  · erw [selfZPow_of_nonpos x B hd, selfZPow_of_nonneg, ← map_pow, Int.natAbs_neg,
       IsLocalization.mk'_spec, map_one]
     apply nonneg_of_neg_nonpos
     rwa [neg_neg]
-  · erw [selfZpow_of_nonneg x B (le_of_not_le hd), selfZpow_of_nonpos, ← map_pow, Int.natAbs_neg,
+  · erw [selfZPow_of_nonneg x B (le_of_not_le hd), selfZPow_of_nonpos, ← map_pow, Int.natAbs_neg,
       @IsLocalization.mk'_spec' R _ (Submonoid.powers x) B _ _ _ 1 (Submonoid.pow x d.natAbs),
       map_one]
     refine' nonpos_of_neg_nonneg (le_of_lt _)
     rwa [neg_neg, ← not_le]
-#align self_zpow_mul_neg selfZpow_mul_neg
+#align self_zpow_mul_neg selfZPow_mul_neg
 
-theorem selfZpow_neg_mul (d : ℤ) : selfZpow x B (-d) * selfZpow x B d = 1 := by
-  rw [mul_comm, selfZpow_mul_neg x B d]
-#align self_zpow_neg_mul selfZpow_neg_mul
+theorem selfZPow_neg_mul (d : ℤ) : selfZPow x B (-d) * selfZPow x B d = 1 := by
+  rw [mul_comm, selfZPow_mul_neg x B d]
+#align self_zpow_neg_mul selfZPow_neg_mul
 
-theorem selfZpow_pow_sub (a : R) (b : B) (m d : ℤ) :
-    selfZpow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
-      selfZpow x B m * mk' B a (1 : Submonoid.powers x) = selfZpow x B d * b := by
-  rw [sub_eq_add_neg, selfZpow_add, mul_assoc, mul_comm _ (mk' B a 1), ← mul_assoc]
+theorem selfZPow_pow_sub (a : R) (b : B) (m d : ℤ) :
+    selfZPow x B (m - d) * mk' B a (1 : Submonoid.powers x) = b ↔
+      selfZPow x B m * mk' B a (1 : Submonoid.powers x) = selfZPow x B d * b := by
+  rw [sub_eq_add_neg, selfZPow_add, mul_assoc, mul_comm _ (mk' B a 1), ← mul_assoc]
   constructor
   · intro h
-    have := congr_arg (fun s : B => s * selfZpow x B d) h
+    have := congr_arg (fun s : B => s * selfZPow x B d) h
     simp only at this
-    rwa [mul_assoc, mul_assoc, selfZpow_neg_mul, mul_one, mul_comm b _] at this
+    rwa [mul_assoc, mul_assoc, selfZPow_neg_mul, mul_one, mul_comm b _] at this
   · intro h
-    have := congr_arg (fun s : B => s * selfZpow x B (-d)) h
+    have := congr_arg (fun s : B => s * selfZPow x B (-d)) h
     simp only at this
-    rwa [mul_comm _ b, mul_assoc b _ _, selfZpow_mul_neg, mul_one] at this
-#align self_zpow_pow_sub selfZpow_pow_sub
+    rwa [mul_comm _ b, mul_assoc b _ _, selfZPow_mul_neg, mul_one] at this
+#align self_zpow_pow_sub selfZPow_pow_sub
 
 variable [IsDomain R] [NormalizationMonoid R] [UniqueFactorizationMonoid R]
 
 theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
-    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZpow x B n * algebraMap R B a = b := by
+    ∃ (a : R) (n : ℤ), ¬x ∣ a ∧ selfZPow x B n * algebraMap R B a = b := by
   obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
   obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
   have ha₀ : a₀ ≠ 0 := by
@@ -304,7 +304,7 @@ theorem exists_reduced_fraction' {b : B} (hb : b ≠ 0) (hx : Irreducible x) :
   classical
   obtain ⟨m, a, hyp1, hyp2⟩ := max_power_factor ha₀ hx
   refine' ⟨a, m - d, _⟩
-  rw [← mk'_one (M := Submonoid.powers x) B, selfZpow_pow_sub, selfZpow_coe_nat, selfZpow_coe_nat,
+  rw [← mk'_one (M := Submonoid.powers x) B, selfZPow_pow_sub, selfZPow_coe_nat, selfZPow_coe_nat,
     ← map_pow _ _ d, mul_comm _ b, H, hyp2, map_mul, map_pow _ _ m]
   exact ⟨hyp1, congr_arg _ (IsLocalization.mk'_one _ _)⟩
 #align exists_reduced_fraction' exists_reduced_fraction'

@@ -117,30 +117,10 @@ section AtUnits
 
 variable (R) (S)
 
-/-- The localization at a module of units is isomorphic to the ring. -/
-noncomputable def atUnits (H : ∀ x : M, IsUnit (x : R)) : R ≃ₐ[R] S := by
-  refine' AlgEquiv.ofBijective (Algebra.ofId R S) ⟨_, _⟩
-  · intro x y hxy
-    obtain ⟨c, eq⟩ := (IsLocalization.eq_iff_exists M S).mp hxy
-    obtain ⟨u, hu⟩ := H c
-    rwa [← hu, Units.mul_right_inj] at eq
-  · intro y
-    obtain ⟨⟨x, s⟩, eq⟩ := IsLocalization.surj M y
-    obtain ⟨u, hu⟩ := H s
-    use x * u.inv
-    dsimp [Algebra.ofId, RingHom.toFun_eq_coe, AlgHom.coe_mks]
-    rw [RingHom.map_mul, ← eq, ← hu, mul_assoc, ← RingHom.map_mul]
-    simp
-#align is_localization.at_units IsLocalization.atUnits
-
 /-- The localization away from a unit is isomorphic to the ring. -/
-noncomputable def atUnit (x : R) (e : IsUnit x) [IsLocalization.Away x S] : R ≃ₐ[R] S := by
-  apply atUnits R (Submonoid.powers x)
-  rintro ⟨xn, n, hxn⟩
-  obtain ⟨u, hu⟩ := e
-  rw [isUnit_iff_exists_inv]
-  use u.inv ^ n
-  simp [← hxn, ← hu, ← mul_pow]
+noncomputable def atUnit (x : R) (e : IsUnit x) [IsLocalization.Away x S] : R ≃ₐ[R] S :=
+  atUnits R (Submonoid.powers x)
+    (by rwa [Submonoid.powers_eq_closure, Submonoid.closure_le, Set.singleton_subset_iff])
 #align is_localization.at_unit IsLocalization.atUnit
 
 /-- The localization at one is isomorphic to the ring. -/

The other direction is a consequence of IsLocalization.map_units.

Also do the same for LocalizationMap and IsLocalizedModule.

This means we have one less fact to prove when constructing an IsLocalization (etc.) instance (thus many proofs are golfed), but once we construct it we still have access to the eq_iff_exists lemmas (without the prime) so the API doesn't get less powerful.

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -157,13 +157,10 @@ theorem away_of_isUnit_of_bijective {R : Type*} (S : Type*) [CommRing R] [CommRi
     surj' := fun z => by
       obtain ⟨z', rfl⟩ := H.2 z
       exact ⟨⟨z', 1⟩, by simp⟩
-    eq_iff_exists' := fun {x y} => by
+    exists_of_eq := fun {x y} => by
       erw [H.1.eq_iff]
-      constructor
-      · rintro rfl
-        exact ⟨1, rfl⟩
-      · rintro ⟨⟨_, n, rfl⟩, e⟩
-        exact (hr.pow _).mul_right_inj.mp e }
+      rintro rfl
+      exact ⟨1, rfl⟩ }
 #align is_localization.away_of_is_unit_of_bijective IsLocalization.away_of_isUnit_of_bijective
 
 end AtUnits

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

This has nice performance benefits.

@@ -31,9 +31,9 @@ commutative ring, field of fractions
 
 section CommSemiring
 
-variable {R : Type _} [CommSemiring R] (M : Submonoid R) {S : Type _} [CommSemiring S]
+variable {R : Type*} [CommSemiring R] (M : Submonoid R) {S : Type*} [CommSemiring S]
 
-variable [Algebra R S] {P : Type _} [CommSemiring P]
+variable [Algebra R S] {P : Type*} [CommSemiring P]
 
 namespace IsLocalization
 
@@ -43,7 +43,7 @@ variable (x : R)
 
 /-- Given `x : R`, the typeclass `IsLocalization.Away x S` states that `S` is
 isomorphic to the localization of `R` at the submonoid generated by `x`. -/
-abbrev Away (S : Type _) [CommSemiring S] [Algebra R S] :=
+abbrev Away (S : Type*) [CommSemiring S] [Algebra R S] :=
   IsLocalization (Submonoid.powers x) S
 #align is_localization.away IsLocalization.Away
 
@@ -95,7 +95,7 @@ noncomputable def awayToAwayRight (y : R) [Algebra R P] [IsLocalization.Away (x
         rw [mul_mk'_eq_mk'_of_mul, mk'_self]
 #align is_localization.away.away_to_away_right IsLocalization.Away.awayToAwayRight
 
-variable (S) (Q : Type _) [CommSemiring Q] [Algebra P Q]
+variable (S) (Q : Type*) [CommSemiring Q] [Algebra P Q]
 
 /-- Given a map `f : R →+* S` and an element `r : R`, we may construct a map `Rᵣ →+* Sᵣ`. -/
 noncomputable def map (f : R →+* P) (r : R) [IsLocalization.Away r S]
@@ -148,7 +148,7 @@ noncomputable def atOne [IsLocalization.Away (1 : R) S] : R ≃ₐ[R] S :=
   @atUnit R _ S _ _ (1 : R) isUnit_one _
 #align is_localization.at_one IsLocalization.atOne
 
-theorem away_of_isUnit_of_bijective {R : Type _} (S : Type _) [CommRing R] [CommRing S]
+theorem away_of_isUnit_of_bijective {R : Type*} (S : Type*) [CommRing R] [CommRing S]
     [Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
     IsLocalization.Away r S :=
   { map_units' := by
@@ -195,7 +195,7 @@ end CommSemiring
 
 open Localization
 
-variable {R : Type _} [CommRing R]
+variable {R : Type*} [CommRing R]
 
 section NumDen
 
@@ -203,7 +203,7 @@ open UniqueFactorizationMonoid IsLocalization
 
 variable (x : R)
 
-variable (B : Type _) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
+variable (B : Type*) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
 /-- `selfZpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
 noncomputable def selfZpow (m : ℤ) : B :=

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-
-! This file was ported from Lean 3 source module ring_theory.localization.away.basic
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Init.Data.Int.CompLemmas
 import Mathlib.RingTheory.UniqueFactorizationDomain
 import Mathlib.RingTheory.Localization.Basic
 
+#align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
+
 /-!
 # Localizations away from an element
 

@@ -8,6 +8,7 @@ Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baan
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathlib.Init.Data.Int.CompLemmas
 import Mathlib.RingTheory.UniqueFactorizationDomain
 import Mathlib.RingTheory.Localization.Basic
 

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

@@ -207,7 +207,7 @@ variable (x : R)
 
 variable (B : Type _) [CommRing B] [Algebra R B] [IsLocalization.Away x B]
 
-/-- `self_zpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
+/-- `selfZpow x (m : ℤ)` is `x ^ m` as an element of the localization away from `x`. -/
 noncomputable def selfZpow (m : ℤ) : B :=
   if _ : 0 ≤ m then algebraMap _ _ x ^ m.natAbs else mk' _ (1 : R) (Submonoid.pow x m.natAbs)
 #align self_zpow selfZpow

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