Module Type NZSqrtProp
(Import A : NZOrdAxiomsSig')
(Import B : NZSqrt' A)
(Import C : NZMulOrderProp A).
Local Notation "a ²" := (a*a) (at level 5, no associativity, format "a ²").
(** First, sqrt is non-negative *)
(** The spec of sqrt indeed determines it *)
(** Hence sqrt is a morphism *)
Instance sqrt_wd : Proper (eq==>eq) sqrt.
(** An alternate specification *)

Lemma sqrt_spec_alt : forall a, 0<=a -> exists r,
a == (√a)² + r /\ 0 <= r <= 2*√a.