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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \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.
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