-- The Agda standard library
-- Automatic solvers for equations over naturals

-- See README.Tactic.RingSolver for examples of how to use this solver

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

module Data.Nat.Tactic.RingSolver where

open import Agda.Builtin.Reflection

open import Data.Maybe.Base using (just; nothing)
open import Data.Nat.Base using (zero; suc)
open import Data.Nat.Properties
open import Level using (0ℓ)
open import Data.Unit using ()
open import Relation.Binary.PropositionalEquality

import Tactic.RingSolver as Solver
import Tactic.RingSolver.Core.AlmostCommutativeRing as ACR

-- A module for automatically solving propositional equivalences
-- containing _+_ and _*_

ring : ACR.AlmostCommutativeRing 0ℓ 0ℓ
ring = ACR.fromCommutativeSemiring +-*-commutativeSemiring
  λ { zero  just refl; _  nothing }

  solve-∀ : Term  TC 
  solve-∀ = Solver.solve-∀-macro (quote ring)

  solve : Term  Term  TC 
  solve n = Solver.solve-macro n (quote ring)