module agda-term where

open import Data.Nat.Base
open import Relation.Binary.PropositionalEquality

+zero : {y : @$\mathbb{N}$@} @$\rightarrow$@ y + zero @$\equiv$@ y
+zero {zero} = refl
+zero {suc y} = cong (@$\lambda$@ yy @$\rightarrow$@ suc yy) (+zero {y})

+-suc : {x y : @$\mathbb{N}$@} @$\rightarrow$@ x + suc y @$\equiv$@ suc (x + y)
+-suc {zero} {y} = refl
+-suc {suc x} {y} = cong suc (+-suc {x} {y})

+-comm : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
+-comm zero y rewrite (+zero {y}) = refl
+-comm (suc x) y = let open @$\equiv$@-Reasoning in
  begin
  suc (x + y) @$\equiv$@@$\langle$@@$\rangle$@
  suc (x + y) @$\equiv$@@$\langle$@ cong suc (+-comm x y) @$\rangle$@
  suc (y + x) @$\equiv$@@$\langle$@ sym (+-suc {y} {x}) @$\rangle$@
  y + suc x @$\blacksquare$@

+-come : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
+-come zero y rewrite (+zero {y}) = refl
+-come (suc x) y
  rewrite (cong suc (+-come x y)) | sym (+-suc {y} {x}) = refl