(** ** Correctness proofs for the square root function *)

Inductive SqrtSpec : positive*mask -> positive -> Prop :=
 | SqrtExact s x : x=s*s -> SqrtSpec (s,IsNul) x
 | SqrtApprox s r x : x=s*s+r -> r <= s~0 -> SqrtSpec (s,IsPos r) x.(** ** Correctness proofs for the square root function *)

Inductive SqrtSpec : positive*mask -> positive -> Prop :=
 | SqrtExact s x : x=s*s -> SqrtSpec (s,IsNul) x
 | SqrtApprox s r x : x=s*s+r -> r <= s~0 -> SqrtSpec (s,IsPos r) x.(** ** Correctness proofs for the square root function *)

Inductive SqrtSpec : positive*mask -> positive -> Prop :=
 | SqrtExact s x : x=s*s -> SqrtSpec (s,IsNul) x
 | SqrtApprox s r x : x=s*s+r -> r <= s~0 -> SqrtSpec (s,IsPos r) x.

Lemma ggcd_gcd : forall a b, fst (ggcd a b) = gcd a b.