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

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