------------------------------------------------------------------------------
-- The gcd program is correct
------------------------------------------------------------------------------

{-# OPTIONS --exact-split              #-}
{-# OPTIONS --no-sized-types           #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K                #-}

-- This module proves the correctness of the gcd program using
-- the Euclid's algorithm.

-- N.B This module does not contain combined proofs, but it imports
-- modules which contain combined proofs.

module FOTC.Program.GCD.Partial.CorrectnessProofATP where

open import FOTC.Base
open import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP using ( 0∤x ; x∣S→x≤S )
open import FOTC.Data.Nat.Type
open import FOTC.Program.GCD.Partial.CommonDivisorATP using ( gcdCD )
open import FOTC.Program.GCD.Partial.Definitions using ( x≢0≢y ; gcdSpec )
open import FOTC.Program.GCD.Partial.DivisibleATP using ( gcdDivisible )
open import FOTC.Program.GCD.Partial.GCD using ( gcd )

import FOTC.Program.GCD.Partial.GreatestAnyCommonDivisor
open module GreatestAnyCommonDivisorATP =
  FOTC.Program.GCD.Partial.GreatestAnyCommonDivisor x∣S→x≤S 0∤x
  using ( gcdGACD )

open import FOTC.Program.GCD.Partial.TotalityATP using ( gcd-N )

------------------------------------------------------------------------------
-- The gcd is correct.
postulate gcdCorrect : ∀ {m n} → N m → N n → x≢0≢y m n → gcdSpec m n (gcd m n)
{-# ATP prove gcdCorrect gcdCD gcdGACD gcd-N gcdDivisible #-}