------------------------------------------------------------------------
-- The Agda standard library
--
-- Some examples showing where the natural numbers and some related
-- operations and properties are defined, and how they can be used
------------------------------------------------------------------------

{-# OPTIONS --without-K #-}

module README.Nat where

-- The natural numbers and various arithmetic operations are defined
-- in Data.Nat.

open import Data.Nat

ex₁ : ℕ
ex₁ =  + 

-- Propositional equality and some related properties can be found
-- in Relation.Binary.PropositionalEquality.

open import Relation.Binary.PropositionalEquality

ex₂ :  +  ≡  * 
ex₂ = refl

-- Data.Nat.Properties contains a number of properties about natural
-- numbers.

import Data.Nat.Properties as Nat

ex₃ : ∀ m n → m * n ≡ n * m
ex₃ m n = Nat.*-comm m n

-- The module ≡-Reasoning in Relation.Binary.PropositionalEquality
-- provides some combinators for equational reasoning.

open ≡-Reasoning

ex₄ : ∀ m n → m * (n + ) ≡ n * m
ex₄ m n = begin
  m * (n + )  ≡⟨ cong (_*_ m) (Nat.+-identityʳ n) ⟩
  m * n        ≡⟨ Nat.*-comm m n ⟩
  n * m        ∎

-- The module SemiringSolver in Data.Nat.Solver contains a solver
-- for natural number equalities involving variables, constants, _+_
-- and _*_.

open import Data.Nat.Solver using (module +-*-Solver)
open +-*-Solver

ex₅ : ∀ m n → m * (n + ) ≡ n * m
ex₅ = solve  (λ m n → m :* (n :+ con )  :=  n :* m) refl