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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Square Root Function *)
(** Interface of a sqrt function, then its specification on naturals *)
Module Type Sqrt (Import A : Typ).
Parameter Inline sqrt : t -> t.
End Sqrt.
Module Type SqrtNotation (A : Typ)(Import B : Sqrt A).
Notation "√ x" := (sqrt x) (at level 6).
End SqrtNotation.
Module Type Sqrt' (A : Typ) := Sqrt A <+ SqrtNotation A.
Module Type NZSqrtSpec (Import A : NZOrdAxiomsSig')(Import B : Sqrt' A).

(* Specifications: *)
Axiom sqrt_spec : forall a, 0<=a -> √a * √a <= a < S (√a) * S (√a).
Axiom sqrt_neg : forall a, a<0 -> √a == 0.
End NZSqrtSpec.