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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZAxioms.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZAxioms.v | 102 |
1 files changed, 93 insertions, 9 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v index fd14cff0..fd20ce72 100644 --- a/theories/Numbers/Integer/Abstract/ZAxioms.v +++ b/theories/Numbers/Integer/Abstract/ZAxioms.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <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 *) @@ -8,11 +8,19 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: ZAxioms.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export NZAxioms. +Require Import Bool NZParity NZPow NZSqrt NZLog NZGcd NZDiv NZBits. + +(** We obtain integers by postulating that successor of predecessor + is identity. *) + +Module Type ZAxiom (Import Z : NZAxiomsSig'). + Axiom succ_pred : forall n, S (P n) == n. +End ZAxiom. -Set Implicit Arguments. +(** For historical reasons, ZAxiomsMiniSig isn't just NZ + ZAxiom, + we also add an [opp] function, that can be seen as a shortcut + for [sub 0]. *) Module Type Opp (Import T:Typ). Parameter Inline opp : t -> t. @@ -24,15 +32,91 @@ End OppNotation. Module Type Opp' (T:Typ) := Opp T <+ OppNotation T. -(** We obtain integers by postulating that every number has a predecessor. *) - Module Type IsOpp (Import Z : NZAxiomsSig')(Import O : Opp' Z). Declare Instance opp_wd : Proper (eq==>eq) opp. - Axiom succ_pred : forall n, S (P n) == n. Axiom opp_0 : - 0 == 0. Axiom opp_succ : forall n, - (S n) == P (- n). End IsOpp. -Module Type ZAxiomsSig := NZOrdAxiomsSig <+ Opp <+ IsOpp. -Module Type ZAxiomsSig' := NZOrdAxiomsSig' <+ Opp' <+ IsOpp. +Module Type OppCstNotation (Import A : NZAxiomsSig)(Import B : Opp A). + Notation "- 1" := (opp one). + Notation "- 2" := (opp two). +End OppCstNotation. + +Module Type ZAxiomsMiniSig := NZOrdAxiomsSig <+ ZAxiom <+ Opp <+ IsOpp. +Module Type ZAxiomsMiniSig' := NZOrdAxiomsSig' <+ ZAxiom <+ Opp' <+ IsOpp + <+ OppCstNotation. + + +(** Other functions and their specifications *) + +(** Absolute value *) + +Module Type HasAbs(Import Z : ZAxiomsMiniSig'). + Parameter Inline abs : t -> t. + Axiom abs_eq : forall n, 0<=n -> abs n == n. + Axiom abs_neq : forall n, n<=0 -> abs n == -n. +End HasAbs. + +(** A sign function *) + +Module Type HasSgn (Import Z : ZAxiomsMiniSig'). + Parameter Inline sgn : t -> t. + Axiom sgn_null : forall n, n==0 -> sgn n == 0. + Axiom sgn_pos : forall n, 0<n -> sgn n == 1. + Axiom sgn_neg : forall n, n<0 -> sgn n == -1. +End HasSgn. + +(** Divisions *) + +(** First, the usual Coq convention of Truncated-Toward-Bottom + (a.k.a Floor). We simply extend the NZ signature. *) + +Module Type ZDivSpecific (Import A:ZAxiomsMiniSig')(Import B : DivMod' A). + Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b. + Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0. +End ZDivSpecific. + +Module Type ZDiv (Z:ZAxiomsMiniSig) := NZDiv.NZDiv Z <+ ZDivSpecific Z. +Module Type ZDiv' (Z:ZAxiomsMiniSig) := NZDiv.NZDiv' Z <+ ZDivSpecific Z. + +(** Then, the Truncated-Toward-Zero convention. + For not colliding with Floor operations, we use different names +*) + +Module Type QuotRem (Import A : Typ). + Parameters Inline quot rem : t -> t -> t. +End QuotRem. + +Module Type QuotRemNotation (A : Typ)(Import B : QuotRem A). + Infix "÷" := quot (at level 40, left associativity). + Infix "rem" := rem (at level 40, no associativity). +End QuotRemNotation. + +Module Type QuotRem' (A : Typ) := QuotRem A <+ QuotRemNotation A. + +Module Type QuotRemSpec (Import A : ZAxiomsMiniSig')(Import B : QuotRem' A). + Declare Instance quot_wd : Proper (eq==>eq==>eq) quot. + Declare Instance rem_wd : Proper (eq==>eq==>eq) B.rem. + Axiom quot_rem : forall a b, b ~= 0 -> a == b*(a÷b) + (a rem b). + Axiom rem_bound_pos : forall a b, 0<=a -> 0<b -> 0 <= a rem b < b. + Axiom rem_opp_l : forall a b, b ~= 0 -> (-a) rem b == - (a rem b). + Axiom rem_opp_r : forall a b, b ~= 0 -> a rem (-b) == a rem b. +End QuotRemSpec. + +Module Type ZQuot (Z:ZAxiomsMiniSig) := QuotRem Z <+ QuotRemSpec Z. +Module Type ZQuot' (Z:ZAxiomsMiniSig) := QuotRem' Z <+ QuotRemSpec Z. + +(** For all other functions, the NZ axiomatizations are enough. *) + +(** Let's group everything *) + +Module Type ZAxiomsSig := ZAxiomsMiniSig <+ OrderFunctions + <+ HasAbs <+ HasSgn <+ NZParity.NZParity + <+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2 <+ NZGcd.NZGcd + <+ ZDiv <+ ZQuot <+ NZBits.NZBits <+ NZSquare. +Module Type ZAxiomsSig' := ZAxiomsMiniSig' <+ OrderFunctions' + <+ HasAbs <+ HasSgn <+ NZParity.NZParity + <+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2 <+ NZGcd.NZGcd' + <+ ZDiv' <+ ZQuot' <+ NZBits.NZBits' <+ NZSquare. |