case zero
n : ℕ
⊢ a 1 ^ Nat.zero = a ↑Nat.zero

case succ
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ a 1 ^ Nat.succ k = a ↑(Nat.succ k)
n k : ℕ
⊢ a 1 ^ k = a ↑k
induction' k with k IH
case zero
n : ℕ
⊢ a 1 ^ Nat.zero = a 0
case zero
n : ℕ
⊢ a 1 ^ Nat.zero = a ↑Nat.zero
rw [Nat.cast_zero]
no goals
case zero
n : ℕ
⊢ a 1 ^ Nat.zero = a 0
rfl
case succ
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ a (1 + ↑k) = a ↑(Nat.succ k)
case succ
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ a 1 ^ Nat.succ k = a ↑(Nat.succ k)
rw [pow_succ, IH, a_mul_a]
case succ.e_a
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ 1 + ↑k = ↑(Nat.succ k)
case succ
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ a (1 + ↑k) = a ↑(Nat.succ k)
congr 1
case succ.e_a
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ ↑(1 + k) = ↑(Nat.succ k)
case succ.e_a
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ 1 + ↑k = ↑(Nat.succ k)
norm_cast
no goals
case succ.e_a
n k : ℕ
IH : a 1 ^ k = a ↑k
⊢ ↑(1 + k) = ↑(Nat.succ k)
rw [Nat.one_add]