fix: definition of Lean.Grind.Field (#9520)

This PR corrects the changes to `Lean.Grind.Field` made in #9500. 

(The lack of examples of fields in the core repository is a problem! I
guess it is likely that for interval arithmetic we will at least need
`Rat` soon.)

src/Init/Grind/Ring/Field.lean

@@ -27,10 +27,10 @@ class Field (α : Type u) extends CommRing α, Inv α, Div α where
   mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1
   /-- The zeroth power of any element is one. -/
   zpow_zero : ∀ a : α, a ^ (0 : Int) = 1
-  /-- The first power of any element is the element itself. -/
-  zpow_one : ∀ a : α, a ^ (1 : Int) = a
-  /-- Power to a sum is the product of the powers. -/
-  zpow_add : ∀ a : α, ∀ n m : Int, a ^ (n + m) = a ^ n * a ^ m
+  /-- The (n+1)-st power of any element is the element multiplied by the n-th power. -/
+  zpow_succ : ∀ (a : α) (n : Nat), a ^ (n + 1 : Int) = a ^ (n : Int) * a
+  /-- Raising to a negative power is the inverse of raising to the positive power. -/
+  zpow_neg : ∀ (a : α) (n : Int), a ^ (-n) = (a ^ n)⁻¹
 
 attribute [instance 100] Field.toInv Field.toDiv Field.zpow
 
@@ -61,6 +61,13 @@ theorem inv_inv (a : α) : a⁻¹⁻¹ = a := by
     apply eq_inv_of_mul_eq_one
     exact mul_inv_cancel h
 
+theorem inv_eq_iff_eq_iff (a b : α) : a⁻¹ = b ↔ a = b⁻¹ := by
+  constructor
+  · intro h
+    rw [← h, inv_inv]
+  · intro h
+    rw [h, inv_inv]
+
 theorem inv_neg (a : α) : (-a)⁻¹ = -a⁻¹ := by
   by_cases h : a = 0
   · subst h
@@ -99,7 +106,7 @@ theorem of_mul_eq_zero {a b : α} : a*b = 0 → a = 0 ∨ b = 0 := by
   have := Field.zero_ne_one (α := α)
   simp [h] at this
 
-theorem mul_inv (a b : α) : (a*b)⁻¹ = a⁻¹*b⁻¹ := by
+theorem inv_mul (a b : α) : (a*b)⁻¹ = a⁻¹*b⁻¹ := by
   cases (Classical.em (a = 0)); simp [*, Semiring.zero_mul, Field.inv_zero]
   cases (Classical.em (b = 0)); simp [*, Semiring.mul_zero, Field.inv_zero]
   cases (Classical.em (a*b = 0)); simp [*, Field.inv_zero]
@@ -126,14 +133,41 @@ theorem of_pow_eq_zero (a : α) (n : Nat) : a^n = 0 → a = 0 := by
     have := ih h
     contradiction
 
-theorem zpow_neg (a : α) (n : Int) : a ^ (-n) = (a ^ n)⁻¹ := by
-  apply eq_inv_of_mul_eq_one
-  rw [← zpow_add, Int.add_left_neg, zpow_zero]
-
 theorem zpow_natCast (a : α) (n : Nat) : a ^ (n : Int) = a ^ n := by
   induction n
   next => simp [zpow_zero, Semiring.pow_zero]
-  next n ih => rw [Int.natCast_add_one, zpow_add, ih, zpow_one, Semiring.pow_succ]
+  next n ih => rw [Int.natCast_add_one, zpow_succ, ih, Semiring.pow_succ]
+
+theorem zpow_one (a : α) : a ^ (1 : Int) = a := by
+  have : (1 : Int) = 0 + 1 := rfl
+  rw [this, ← Int.natCast_zero, zpow_succ, Int.natCast_zero, zpow_zero, Semiring.one_mul]
+
+theorem zpow_neg_one (a : α) : a ^ (-1) = a⁻¹ := by
+  rw [zpow_neg, zpow_one]
+
+theorem zpow_add_one {a : α} (h : a ≠ 0) (n : Int) : a ^ (n + 1) = a ^ n * a := by
+  match n with
+  | (n : Nat) => rw [zpow_succ, zpow_natCast]
+  | -(n + 1 : Nat) =>
+    rw [zpow_neg, Int.natCast_add, Int.cast_ofNat_Int, Int.neg_add, Int.neg_add_cancel_right,
+      zpow_neg, zpow_succ, inv_mul, Semiring.mul_assoc, inv_mul_cancel h, Semiring.mul_one]
+
+theorem zpow_sub_one {a : α} (h : a ≠ 0) (n : Int) : a ^ (n - 1) = a ^ n * a⁻¹ := by
+  have h₁ := zpow_add_one h (-n)
+  have h₂ : -n + 1 = - (n - 1) := by omega
+  rw [h₂, zpow_neg, inv_eq_iff_eq_iff] at h₁
+  rw [h₁, inv_mul, zpow_neg, inv_inv]
+
+theorem zpow_add {a : α} (h : a ≠ 0) (m n : Int) : a ^ (m + n) = a ^ m * a ^ n := by
+  match n with
+  | (n : Nat) =>
+    induction n with
+    | zero => simp [zpow_zero, Semiring.mul_one]
+    | succ n ih => rw [Int.natCast_add_one, zpow_succ, ← Int.add_assoc, zpow_add_one h, ih, Semiring.mul_assoc]
+  | -(n + 1 : Nat) =>
+    induction n with
+    | zero => simp [Int.add_neg_one, zpow_sub_one h, zpow_neg_one]
+    | succ n ih => rw [Int.natCast_add_one, Int.neg_add, Int.add_neg_one, ← Int.add_sub_assoc, zpow_sub_one h, zpow_sub_one h, ih, Semiring.mul_assoc]
 
 instance [IsCharP α 0] : NoNatZeroDivisors α := NoNatZeroDivisors.mk' <| by
   intro a b h w