[GOAL]
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
n : ℕ
⊢ (↑n).fst = ↑n
[PROOFSTEP]
induction n
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
⊢ (↑Nat.zero).fst = ↑Nat.zero
[PROOFSTEP]
simp [*]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
n✝ : ℕ
n_ih✝ : (↑n✝).fst = ↑n✝
⊢ (↑(Nat.succ n✝)).fst = ↑(Nat.succ n✝)
[PROOFSTEP]
simp [*]
[GOAL]
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
n : ℕ
⊢ (↑n).snd = ↑n
[PROOFSTEP]
induction n
[GOAL]
case zero
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
⊢ (↑Nat.zero).snd = ↑Nat.zero
[PROOFSTEP]
simp [*]
[GOAL]
case succ
α : Type u_1
β : Type u_2
inst✝¹ : AddMonoidWithOne α
inst✝ : AddMonoidWithOne β
n✝ : ℕ
n_ih✝ : (↑n✝).snd = ↑n✝
⊢ (↑(Nat.succ n✝)).snd = ↑(Nat.succ n✝)
[PROOFSTEP]
simp [*]
