fix Data.Int.Basic: fix zsmul on ℤ (#1217)

This way two instances for `m • n`, `m n : ℤ`, are defeq. Also use defeq to golf some proofs.

Mathlib/Algebra/GroupPower/Lemmas.lean

@@ -1218,33 +1218,29 @@ open Multiplicative
 
 -- Porting note: the proof became a little roundabout while porting.
 @[simp]
-theorem Nat.to_add_pow (a : Multiplicative ℕ) (b : ℕ) : toAdd (a ^ b) = toAdd a * b := by
-  induction' b with b ihs
-  · erw [_root_.pow_zero, toAdd_one, mul_zero]
-  · simp [*, _root_.pow_succ, add_comm, Nat.mul_succ]
-#align nat.to_add_pow Nat.to_add_pow
+theorem Nat.toAdd_pow (a : Multiplicative ℕ) (b : ℕ) : toAdd (a ^ b) = toAdd a * b :=
+  mul_comm _ _
+#align nat.to_add_pow Nat.toAdd_pow
 
 @[simp]
-theorem Nat.of_add_mul (a b : ℕ) : ofAdd (a * b) = ofAdd a ^ b :=
-  (Nat.to_add_pow _ _).symm
-#align nat.of_add_mul Nat.of_add_mul
+theorem Nat.ofAdd_mul (a b : ℕ) : ofAdd (a * b) = ofAdd a ^ b :=
+  (Nat.toAdd_pow _ _).symm
+#align nat.of_add_mul Nat.ofAdd_mul
 
 @[simp]
-theorem Int.to_add_pow (a : Multiplicative ℤ) (b : ℕ) : toAdd (a ^ b) = toAdd a * b := by
-  induction b <;> simp [*, mul_add, pow_succ, add_comm]
-#align int.to_add_pow Int.to_add_pow
+theorem Int.toAdd_pow (a : Multiplicative ℤ) (b : ℕ) : toAdd (a ^ b) = toAdd a * b :=
+  mul_comm _ _
+#align int.to_add_pow Int.toAdd_pow
 
 @[simp]
-theorem Int.to_add_zpow (a : Multiplicative ℤ) (b : ℤ) : toAdd (a ^ b) = toAdd a * b :=
-  Int.induction_on b (by simp) (by simp (config := { contextual := true }) [zpow_add, mul_add])
-    (by
-      simp (config := { contextual := true }) [zpow_add, mul_add, sub_eq_add_neg])
-#align int.to_add_zpow Int.to_add_zpow
+theorem Int.toAdd_zpow (a : Multiplicative ℤ) (b : ℤ) : toAdd (a ^ b) = toAdd a * b :=
+  mul_comm _ _
+#align int.to_add_zpow Int.toAdd_zpow
 
 @[simp]
-theorem Int.of_add_mul (a b : ℤ) : ofAdd (a * b) = ofAdd a ^ b :=
-  (Int.to_add_zpow _ _).symm
-#align int.of_add_mul Int.of_add_mul
+theorem Int.ofAdd_mul (a b : ℤ) : ofAdd (a * b) = ofAdd a ^ b :=
+  (Int.toAdd_zpow _ _).symm
+#align int.of_add_mul Int.ofAdd_mul
 
 end Multiplicative
 

Mathlib/Data/Int/Basic.lean

@@ -48,10 +48,14 @@ instance : CommRing ℤ where
   add_left_neg := Int.add_left_neg
   nsmul := (·*·)
   nsmul_zero := Int.zero_mul
-  nsmul_succ n x := by
-    show ofNat (Nat.succ n) * x = x + ofNat n * x
-    simp only [ofNat_eq_coe]
-    rw [Int.ofNat_succ, Int.add_mul, Int.add_comm, Int.one_mul]
+  nsmul_succ n x :=
+    show (n + 1 : ℤ) * x = x + n * x
+    by rw [Int.add_mul, Int.add_comm, Int.one_mul]
+  zsmul := (·*·)
+  zsmul_zero' := Int.zero_mul
+  zsmul_succ' m n := by
+    simp only [ofNat_eq_coe, ofNat_succ, Int.add_mul, Int.add_comm, Int.one_mul]
+  zsmul_neg' m n := by simp only [negSucc_coe, ofNat_succ, Int.neg_mul]
   sub_eq_add_neg _ _ := Int.sub_eq_add_neg
   natCast := (·)
   natCast_zero := rfl