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2010-11-10Integer division: quot and rem (trunc convention) in addition to div and modletouzey
(floor convention). We follow Haskell naming convention: quot and rem are for Round-Toward-Zero (a.k.a Trunc, what Ocaml, C, Asm do by default, cf. the ex-ZOdiv file), while div and mod are for Round-Toward-Bottom (a.k.a Floor, what Coq does historically in Zdiv). We use unicode ÷ for quot, and infix rem for rem (which is actually remainder in full). This way, both conventions can be used at the same time. Definitions (and proofs of specifications) for div mod quot rem are migrated in a new file Zdiv_def. Ex-ZOdiv file is now Zquot. With this new organisation, no need for functor application in Zdiv and Zquot. On the abstract side, ZAxiomsSig now provides div mod quot rem. Zproperties now contains properties of them. In NZDiv, we stop splitting specifications in Common vs. Specific parts. Instead, the NZ specification is be extended later, even if this leads to a useless mod_bound_pos, subsumed by more precise axioms. A few results in ZDivTrunc and ZDivFloor are improved (sgn stuff). A few proofs in Nnat, Znat, Zabs are reworked (no more dependency to Zmin, Zmax). A lcm (least common multiple) is derived abstractly from gcd and division (and hence available for nat N BigN Z BigZ :-). In these new files NLcm and ZLcm, we also provide some combined properties of div mod quot rem gcd. We also provide a new file Zeuclid implementing a third division convention, where the remainder is always positive. This file instanciate the abstract one ZDivEucl. Operation names are ZEuclid.div and ZEuclid.modulo. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13633 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-11-05Numbers: axiomatization, properties and implementations of gcdletouzey
- For nat, we create a brand-new gcd function, structural in the sense of Coq, even if it's Euclid algorithm. Cool... - We re-organize the Zgcd that was in Znumtheory, create out of it files Pgcd, Ngcd_def, Zgcd_def. Proofs of correctness are revised in order to be much simpler (no omega, no advanced lemmas of Znumtheory, etc). - Abstract Properties NZGcd / ZGcd / NGcd could still be completed, for the moment they contain up to Gauss thm. We could add stuff about (relative) primality, relationship between gcd and div,mod, or stuff about parity, etc etc. - Znumtheory remains as it was, apart for Zgcd and correctness proofs gone elsewhere. We could later take advantage of ZGcd in it. Someday, we'll have to switch from the current Zdivide inductive, to Zdivide' via exists. To be continued... git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13623 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-11-02Numbers : log2. Abstraction, properties and implementations.letouzey
Btw, we finally declare the original Zpower as the power on Z. We should switch to a more efficient one someday, but in the meantime BigN is proved with respect to the old one. TODO: reform Zlogarithm with respect to Zlog_def git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13606 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-11-02Numbers: specs about sqrt and pow of neg numbers, even in NZletouzey
These additional specs are useless (but trivially provable) for N. They are quite convenient when deriving properties in NZ. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13603 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-11-02Numbers: NZPowProp as a Module Type, some module variable renamingletouzey
We temporary use a hack to convert a module type into a module Module M := T is refused, so we force an include via Module M := Nop <+ T where Nop is an empty module. To be fixed later more beautifully... git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13602 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-10-19Add sqrt in Numbersletouzey
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
2010-10-14Numbers : also axiomatize constants 1 and 2.letouzey
Initially, I was using notation 1 := (S 0) and so on. But then, when implementing by NArith or ZArith, some lemmas statements were filled with Nsucc's and Zsucc's instead of 1 and 2's. Concerning BigN, things are rather complicated: zero, one, two aren't inlined during the functor application creating BigN. This is deliberate, at least for the other operations like BigN.add. And anyway, since zero, one, two are defined too early in NMake, we don't have 0%bigN in the body of BigN.zero but something complex that reduce to 0%bigN, same for one and two. Fortunately, apply or rewrite of generic lemmas seem to work, even if there's BigZ.zero on one side and 0 on the other... git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13555 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-10-14Numbers: new functions pow, even, odd + many reorganisationsletouzey
- Simplification of functor names, e.g. ZFooProp instead of ZFooPropFunct - The axiomatisations of the different fonctions are now in {N,Z}Axioms.v apart for Z division (three separate flavours in there own files). Content of {N,Z}AxiomsSig is extended, old version is {N,Z}AxiomsMiniSig. - In NAxioms, the recursion field isn't that useful, since we axiomatize other functions and not define them (apart in the toy NDefOps.v). We leave recursion there, but in a separate NAxiomsFullSig. - On Z, the pow function is specified to behave as Zpower : a^(-1)=0 - In BigN/BigZ, (power:t->N->t) is now pow_N, while pow is t->t->t These pow could be more clever (we convert 2nd arg to N and use pow_N). Default "^" is now (pow:t->t->t). BigN/BigZ ring is adapted accordingly - In BigN, is_even is now even, its spec is changed to use Zeven_bool. We add an odd. In BigZ, we add even and odd. - In ZBinary (implem of ZAxioms by ZArith), we create an efficient Zpow to implement pow. This Zpow should replace the current linear Zpower someday. - In NPeano (implem of NAxioms by Arith), we create pow, even, odd functions, and we modify the div and mod functions for them to be linear, structural, tail-recursive. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13546 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-07-24Updated all headers for 8.3 and trunkherbelin
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13323 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-04-29Remove the svn-specific $Id$ annotationsletouzey
- Many of them were broken, some of them after Pierre B's rework of mli for ocamldoc, but not only (many bad annotation, many files with no svn property about Id, etc) - Useless for those of us that work with git-svn (and a fortiori in a forthcoming git-only setting) - Even in svn, they seem to be of little interest git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12972 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-02-09Numbers: properties of min/max with respect to 0,S,P,add,sub,mulletouzey
With these properties, we can kill Arith/MinMax, NArith/Nminmax, and leave ZArith/Zminmax as a compatibility file only. Now the instanciations NPeano.Nat, NBinary.N, ZBinary.Z, BigZ, BigN contains all theses facts. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12718 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-29Division in numbers: kills some Include to avoid bad alias Zsucc = ZDiv.Z.Z'.Sletouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12704 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-08Numbers: axiomatization + generic properties of abs and sgn.letouzey
This allow to really finish files about division. An abs and sgn is added to BigZ. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12644 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-07Include can accept both Module and Module Typeletouzey
Syntax Include Type is still active, but deprecated, and triggers a warning. The syntax M <+ M' <+ M'', which performs internally an Include, also benefits from this: M, M', M'' can be independantly modules or module type. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12640 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-07Numbers: separation of funs, notations, axioms. Notations via module, ↵letouzey
without scope. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12639 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-06"by" becomes officially a reserved keyword of Coq (fixes "rewrite ... at ... ↵letouzey
by ...") Application in some proofs of Numbers's abstract division git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12630 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-05Numbers abstract layer: more Module Type, used especially for divisions.letouzey
Properties are now rather passed as functor arg instead of via Include or some inner modules. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12629 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-05Avoid declaring hints about refl/sym/trans of eq in DecidableType2letouzey
This used to be convenient in FSets, but since we now try to integrate DecidableType and OrderedType as foundation for other part of the stdlib, this should be avoided, otherwise some eauto take a _long_ time. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12626 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-05Division in Numbers: proofs with less auto (less sensitive to hints, in ↵letouzey
particular about eq) git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12625 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-05Division in Numbers: factorisation of signaturesletouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12624 85f007b7-540e-0410-9357-904b9bb8a0f7
2010-01-04Specific syntax for Instances in Module Type: Declare Instanceletouzey
NB: the grammar entry is placed in vernac:command on purpose even if it should have gone into vernac:gallina_ext. Camlp4 isn't factorising rules starting by "Declare" in a correct way otherwise... git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12623 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-12-17Division in Numbers : more properties, new filenames based on a paper by R. ↵letouzey
Boute Following R. Boute (paper "the Euclidean Definition of the Functions div and mod"): - ZDivFloor.v for Coq historical division (former ZDivCoq.v) - ZDivTrunc.v for Ocaml convention (former ZDivOcaml.v) - ZDivEucl.v for "Mathematical" convention 0<=r (former ZDivMath.v) These property functors are more or less finished (except that sign and abs stuff should be migrated to a separate file). git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12594 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-12-16Division in Numbers: more properties proved (still W.I.P.)letouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12591 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-12-15A generic euclidean division in Numbers (Still Work-In-Progress)letouzey
- For Z, we propose 3 conventions for the sign of the remainder... - Instanciation for nat in NPeano. - Beginning of instanciation in ZOdiv. Still many proofs to finish, etc, etc, but soon we will have a decent properties database for all divisions of all instances of Numbers (e.g. BigZ). git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12590 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-11-10Simplification of Numbers, mainly thanks to Includeletouzey
- No more nesting of Module and Module Type, we rather use Include. - Instead of in-name-qualification like NZeq, we use uniform short names + modular qualification like N.eq when necessary. - Many simplification of proofs, by some autorewrite for instance - In NZOrder, we instantiate an "order" tactic. - Some requirements in NZAxioms were superfluous: compatibility of le, min and max could be derived from the rest. - NMul removed, since it was containing only an ad-hoc result for ZNatPairs, that we've inlined in the proof of mul_wd there. - Zdomain removed (was already not compiled), idea of a module with eq and eqb reused in DecidableType.BooleanEqualityType. - ZBinDefs don't contain any definition now, migrate it to ZBinary. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12489 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-11-06Numbers: more (syntactic) changes toward new style of type classesletouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12475 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-11-03Numbers: start using Classes stuff, Equivalence, Proper, Instance, etcletouzey
TODO: finish removing the "Add Relation", "Add Morphism" fun_* fun2_* TODO: now that we have Include, flatten the hierarchy... git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12464 85f007b7-540e-0410-9357-904b9bb8a0f7
2009-10-08Init/Tactics.v: tactic with nicer name 'exfalso' for 'elimtype False'letouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12380 85f007b7-540e-0410-9357-904b9bb8a0f7
2008-12-12Uniformity with the rest of the StdLib : _symm --> _symletouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11675 85f007b7-540e-0410-9357-904b9bb8a0f7
2008-06-03In abstract parts of theories/Numbers, plus/times becomes add/mul, letouzey
for increased consistency with bignums parts (commit part II: names of files + additional translation minus --> sub) git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11040 85f007b7-540e-0410-9357-904b9bb8a0f7
2008-06-02In abstract parts of theories/Numbers, plus/times becomes add/mul, letouzey
for increased consistency with bignums parts (commit part I: content of files) git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11039 85f007b7-540e-0410-9357-904b9bb8a0f7
2008-05-15In practice, the new setoid rewrite (and the "at" syntax) allows to avoid letouzey
using the ad-hoc qsetoid_rewrite. Could QRewrite.v be made completely obsolete ? For the moment rewrite under fun and exists don't work. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10935 85f007b7-540e-0410-9357-904b9bb8a0f7
2008-05-15Coq headers + $ in theories/Numbers filesletouzey
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10934 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-11-22An update on Numbers. Added two files dealing with recursion, for ↵emakarov
information only. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10330 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-11-16Added theorems; created NZPlusOrder from NTimesOrder.emakarov
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10325 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-11-14Update on Numbers; renamed ZOrder.v to ZLt to remove clash with ↵emakarov
ZArith/Zorder on MacOS. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10323 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-11-07Replaced BinNat with a new version that is based on ↵emakarov
theories/Numbers/Natural/Binary/NBinDefs. Most of the entities in the new BinNat are notations for the development in Numbers. Also added min and max to the new natural numbers and integers. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10298 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-11-03An update of theories/Numbersemakarov
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10285 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-10-23Added Numbers/Natural/Abstract/NIso.v that proves that any two models of ↵emakarov
natural numbers are isomorphic. Added NatScope and IntScope for abstract developments. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10247 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-10-16Added transitivity and irreflexivity of <, as well as < -elimination for ↵emakarov
binary positive numbers. Added directory contribs/micromega with the generalization of Frédéric Besson's micromega tactic for an arbitrary ordered ring. So far no tactic has been defined. One has to apply the theorems and find the certificate, which is necessary to solve inequations, manually. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10226 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-10-01Added the compilation of theories/Numbers to Makefile.common. The following ↵emakarov
things compile: abstract natural numbers and integers with plus, times, minus, and order; Peano and binary implementations for natural numbers. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10161 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-09-25An update on theories/Numbers.emakarov
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10142 85f007b7-540e-0410-9357-904b9bb8a0f7
2007-09-21Update on theories/Numbers. Natural numbers are mostly complete,emakarov
need to make NZOrdAxiomsSig a subtype of NAxiomsSig. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10132 85f007b7-540e-0410-9357-904b9bb8a0f7
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