From b03b65fdc44e3c6cfeceaf997cbc1a50a6c19e5c Mon Sep 17 00:00:00 2001
From: letouzey
Date: Tue, 19 Oct 2010 10:16:57 +0000
Subject: Add sqrt in Numbers

 As for power recently, we add a specification in NZ,N,Z,
 derived properties, implementations for nat, N, Z, BigN, BigZ.

  - For nat, this sqrt is brand new :-), cf NPeano.v

  - For Z, we rework what was in Zsqrt: same algorithm,
    no more refine but a pure function, based now on a sqrt
    for positive, from which we derive a Nsqrt and a Zsqrt.
    For the moment, the old Zsqrt.v file is kept as Zsqrt_compat.v.
    It is not loaded by default by Require ZArith.
    New definitions are now in Psqrt.v, Zsqrt_def.v and Nsqrt_def.v

  - For BigN, BigZ, we changed the specifications to refer to Zsqrt
    instead of using characteristic inequations.

 On the way, many extensions, in particular BinPos (lemmas about order),
 NZMulOrder (results about squares)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13564 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Numbers/Natural/Abstract/NAxioms.v     | 10 ++--
 theories/Numbers/Natural/Abstract/NProperties.v |  4 +-
 theories/Numbers/Natural/Abstract/NSqrt.v       | 64 +++++++++++++++++++++++++
 3 files changed, 72 insertions(+), 6 deletions(-)
 create mode 100644 theories/Numbers/Natural/Abstract/NSqrt.v

(limited to 'theories/Numbers/Natural/Abstract')

diff --git a/theories/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v
index 66ff2ded54..b360c016f3 100644
--- a/theories/Numbers/Natural/Abstract/NAxioms.v
+++ b/theories/Numbers/Natural/Abstract/NAxioms.v
@@ -8,7 +8,7 @@
 (*                      Evgeny Makarov, INRIA, 2007                     *)
 (************************************************************************)
 
-Require Export NZAxioms NZPow NZDiv.
+Require Export NZAxioms NZPow NZSqrt NZDiv.
 
 (** From [NZ], we obtain natural numbers just by stating that [pred 0] == 0 *)
 
@@ -32,7 +32,7 @@ Module Type Parity (Import N : NAxiomsMiniSig').
  Axiom odd_spec : forall n, odd n = true <-> Odd n.
 End Parity.
 
-(** Power function : NZPow is enough *)
+(** For Power and Sqrt functions : NZPow and NZSqrt are enough *)
 
 (** Division Function : we reuse NZDiv.DivMod and NZDiv.NZDivCommon,
     and add to that a N-specific constraint. *)
@@ -45,10 +45,12 @@ End NDivSpecific.
 (** We now group everything together. *)
 
 Module Type NAxiomsSig := NAxiomsMiniSig <+ HasCompare <+ Parity
-  <+ NZPow.NZPow <+ DivMod <+ NZDivCommon <+ NDivSpecific.
+  <+ NZPow.NZPow <+ NZSqrt.NZSqrt
+  <+ DivMod <+ NZDivCommon <+ NDivSpecific.
 
 Module Type NAxiomsSig' := NAxiomsMiniSig' <+ HasCompare <+ Parity
-  <+ NZPow.NZPow' <+ DivMod' <+ NZDivCommon <+ NDivSpecific.
+  <+ NZPow.NZPow' <+ NZSqrt.NZSqrt'
+  <+ DivMod' <+ NZDivCommon <+ NDivSpecific.
 
 
 (** It could also be interesting to have a constructive recursor function. *)
diff --git a/theories/Numbers/Natural/Abstract/NProperties.v b/theories/Numbers/Natural/Abstract/NProperties.v
index c1977f3533..fc8f27ddc9 100644
--- a/theories/Numbers/Natural/Abstract/NProperties.v
+++ b/theories/Numbers/Natural/Abstract/NProperties.v
@@ -7,9 +7,9 @@
 (************************************************************************)
 
 Require Export NAxioms.
-Require Import NMaxMin NParity NPow NDiv.
+Require Import NMaxMin NParity NPow NSqrt NDiv.
 
 (** This functor summarizes all known facts about N. *)
 
 Module Type NProp (N:NAxiomsSig) :=
- NMaxMinProp N <+ NParityProp N <+ NPowProp N <+ NDivProp N.
+ NMaxMinProp N <+ NParityProp N <+ NPowProp N <+ NSqrtProp N <+ NDivProp N.
diff --git a/theories/Numbers/Natural/Abstract/NSqrt.v b/theories/Numbers/Natural/Abstract/NSqrt.v
new file mode 100644
index 0000000000..d5916bdc2d
--- /dev/null
+++ b/theories/Numbers/Natural/Abstract/NSqrt.v
@@ -0,0 +1,64 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(*   \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 N : NAxiomsSig')(Import NS : NSubProp N).
+
+ Module Import NZSqrtP := NZSqrtProp N N NS.
+
+ 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.
+
+Lemma sqrt_unique : forall a b, b*b<=a<(S b)*(S b) -> √a == b.
+Proof. wrap sqrt_unique. Qed.
+
+Lemma sqrt_square : forall a, √(a*a) == a.
+Proof. wrap sqrt_square. Qed.
+
+Lemma sqrt_le_mono : forall a b, a<=b -> √a <= √b.
+Proof. wrap sqrt_le_mono. Qed.
+
+Lemma sqrt_lt_cancel : forall a b, √a < √b -> a < b.
+Proof. wrap sqrt_lt_cancel. Qed.
+
+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, 0<=a -> √a<=a.
+Proof. wrap sqrt_le_lin. Qed.
+
+Lemma sqrt_mul_below : forall a b, √a * √b <= √(a*b).
+Proof. wrap sqrt_mul_below. Qed.
+
+Lemma sqrt_mul_above : forall a b, √(a*b) < S (√a) * S (√b).
+Proof. wrap sqrt_mul_above. Qed.
+
+Lemma sqrt_add_le : forall a b, √(a+b) <= √a + √b.
+Proof. wrap sqrt_add_le. Qed.
+
+Lemma add_sqrt_le : forall a b, √a + √b <= √(2*(a+b)).
+Proof. wrap add_sqrt_le. Qed.
+
+End NSqrtProp.
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