Greatest common divisor, with Bézout coefficients

module GcdBezout

  use int.Int
  use int.ComputerDivision
  use number.Gcd
  use ref.Ref

  let gcd (x y: int) : (result: int, ghost a: int, ghost b: int)
    requires { x >= 0 /\ y >= 0 }
    ensures  { result = gcd x y }
    ensures  { a*x + b*y = result }
    =
    let x = ref x in let y = ref y in
    label Pre in
    let ghost a = ref 1 in let ghost b = ref 0 in
    let ghost c = ref 0 in let ghost d = ref 1 in
    while !y > 0 do
       invariant { !x >= 0 /\ !y >= 0 }
       invariant { gcd !x !y = gcd (!x at Pre) (!y at Pre) }
       invariant { !a * (!x at Pre) + !b * (!y at Pre) = !x }
       invariant { !c * (!x at Pre) + !d * (!y at Pre) = !y }
       variant   { !y }
       let r = mod !x !y in let ghost q = div !x !y in
       x := !y; y := r;
       let ghost ta = !a in let ghost tb = !b in
       a := !c; b := !d;
       c := ta - !c * q; d := tb - !d * q;
    done;
    !x, !a, !b

end