-- The Agda standard library
-- An example of how Algebra.CommutativeMonoidSolver can be used

{-# OPTIONS --cubical-compatible --safe #-}

module Algebra.Solver.CommutativeMonoid.Example where

open import Relation.Binary.PropositionalEquality using (_≡_)

open import Data.Bool.Base using (_∨_)
open import Data.Bool.Properties using (∨-commutativeMonoid)

open import Data.Fin.Base using (zero; suc)
open import Data.Vec.Base using ([]; _∷_)

open import Algebra.Solver.CommutativeMonoid ∨-commutativeMonoid

test :  x y z  (x  y)  (x  z)  (z  y)  (x  x)
test a b c = let _∨_ = _⊕_ in
  prove 3 ((x  y)  (x  z)) ((z  y)  (x  x)) (a  b  c  [])
  x = var zero
  y = var (suc zero)
  z = var (suc (suc zero))