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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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(** Properties of Square Root Function *)
Module NSqrtProp (Import A : NAxiomsSig')(Import B : NSubProp A).
Module Import Private_NZSqrt := Nop <+ NZSqrtProp A A B.
Ltac auto' := trivial; try rewrite <- neq_0_lt_0; auto using le_0_l.
Ltac wrap l := intros; apply l; auto'.
(** We redefine NZSqrt's results, without the non-negative hyps *)

Lemma sqrt_spec' : forall a, √a*√a <= a < S (√a) * S (√a).