## Greatest common divisor with Bezout coefficients

**Authors:** Jean-Christophe Filliâtre

**Topics:** Arithmetic / Ghost code

**Tools:** Why3

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# 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

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