open import systemT
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning

module int where

_+_ : Int -> Int -> Int
n + O     = n
n + (S m) = S (n + m)

left-increment : (n m : Int) -> (S n) + m ≡ S (n + m)
left-increment  n    O  = refl
left-increment  n (S m) = cong S (left-increment n m)

double : Int -> Int
double O = O
double (S n) = S (S (double n))

sum-sym : (n : Int) (m : Int) -> n + m ≡ m + n
sum-sym    O     O  = refl
sum-sym    O  (S m) = cong S (sum-sym O m)
sum-sym (S n)    O  = cong S (sum-sym n O)
sum-sym (S n) (S m) = begin
    (S n) + (S m)  
  ≡⟨ refl ⟩
    S ((S n) + m)
  ≡⟨ cong S (left-increment n m)⟩
    S (S (n + m))
  ≡⟨ cong (\x -> S (S x)) (sum-sym n m) ⟩
    S (S (m + n))
  ≡⟨ sym (cong S (left-increment m n)) ⟩
    S ((S m) + n)
  ≡⟨ refl ⟩
    (S m) + (S n)
  ∎