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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Properties of Square Root Function *)
+
+Require Import NAxioms NSub NZSqrt.
+
+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).
+Proof. wrap sqrt_spec. Qed.
+
+Definition sqrt_unique : forall a b, b*b<=a<(S b)*(S b) -> √a == b
+ := sqrt_unique.
+
+Lemma sqrt_square : forall a, √(a*a) == a.
+Proof. wrap sqrt_square. Qed.
+
+Definition sqrt_le_mono : forall a b, a<=b -> √a <= √b
+ := sqrt_le_mono.
+
+Definition sqrt_lt_cancel : forall a b, √a < √b -> a < b
+ := sqrt_lt_cancel.
+
+Lemma sqrt_le_square : forall a b, b*b<=a <-> b <= √a.
+Proof. wrap sqrt_le_square. Qed.
+
+Lemma sqrt_lt_square : forall a b, a<b*b <-> √a < b.
+Proof. wrap sqrt_lt_square. Qed.
+
+Definition sqrt_0 := sqrt_0.
+Definition sqrt_1 := sqrt_1.
+Definition sqrt_2 := sqrt_2.
+
+Definition sqrt_lt_lin : forall a, 1<a -> √a<a
+ := sqrt_lt_lin.
+
+Lemma sqrt_le_lin : forall a, √a<=a.
+Proof. wrap sqrt_le_lin. Qed.
+
+Definition sqrt_mul_below : forall a b, √a * √b <= √(a*b)
+ := sqrt_mul_below.
+
+Lemma sqrt_mul_above : forall a b, √(a*b) < S (√a) * S (√b).
+Proof. wrap sqrt_mul_above. Qed.
+
+Lemma sqrt_succ_le : forall a, √(S a) <= S (√a).
+Proof. wrap sqrt_succ_le. Qed.
+
+Lemma sqrt_succ_or : forall a, √(S a) == S (√a) \/ √(S a) == √a.
+Proof. wrap sqrt_succ_or. Qed.
+
+Definition sqrt_add_le : forall a b, √(a+b) <= √a + √b
+ := sqrt_add_le.
+
+Lemma add_sqrt_le : forall a b, √a + √b <= √(2*(a+b)).
+Proof. wrap add_sqrt_le. Qed.
+
+(** For the moment, we include stuff about [sqrt_up] with patching them. *)
+
+Include NZSqrtUpProp A A B Private_NZSqrt.
+
+End NSqrtProp.