-- Copyright 2020 INP Toulouse.
-- Authors : Mathieu Montin.

-- This version of Commut is provided to you free of charge. It is released under the FSF GPL license, http://www.fsf.org/licenses/gpl.html. 
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open import Data.Nat
open import Relation.Binary.PropositionalEquality

module Commut where

n≡n+0 :  {b}  b  b + 0
n≡n+0 {zero} = refl
n≡n+0 {suc b} = cong suc n≡n+0

s[b+n]≡b+s[n] :  {b n}  suc (b + n)  b + suc n
s[b+n]≡b+s[n] {zero} = refl
s[b+n]≡b+s[n] {suc n} = cong suc s[b+n]≡b+s[n]

comm :  {a b}  a + b  b + a
comm {zero} = n≡n+0
comm {suc a} = trans (cong suc (comm {a})) s[b+n]≡b+s[n]