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path: root/theories/Numbers/NatInt/NZDiv.v
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* Bump year in headers.Gravatar Maxime Dénès2017-06-01
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* Update copyright headers.Gravatar Maxime Dénès2016-01-20
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* Fix some typos.Gravatar Guillaume Melquiond2015-12-07
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* Update headers.Gravatar Maxime Dénès2015-01-12
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* Updating headers.Gravatar herbelin2012-08-08
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15715 85f007b7-540e-0410-9357-904b9bb8a0f7
* Numbers: a particular case of div_uniqueGravatar letouzey2011-06-24
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14238 85f007b7-540e-0410-9357-904b9bb8a0f7
* Numbers and bitwise functions.Gravatar letouzey2010-12-06
| | | | | | | | | | | | | | | | | | | | | | | See NatInt/NZBits.v for the common axiomatization of bitwise functions over naturals / integers. Some specs aren't pretty, but easier to prove, see alternate statements in property functors {N,Z}Bits. Negative numbers are considered via the two's complement convention. We provide implementations for N (in Ndigits.v), for nat (quite dummy, just for completeness), for Z (new file Zdigits_def), for BigN (for the moment partly by converting to N, to be improved soon) and for BigZ. NOTA: For BigN.shiftl and BigN.shiftr, the two arguments are now in the reversed order (for consistency with the rest of the world): for instance BigN.shiftl 1 10 is 2^10. NOTA2: Zeven.Zdiv2 is _not_ doing (Zdiv _ 2), but rather (Zquot _ 2) on negative numbers. For the moment I've kept it intact, and have just added a Zdiv2' which is truly equivalent to (Zdiv _ 2). To reorganize someday ? git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13689 85f007b7-540e-0410-9357-904b9bb8a0f7
* Integer division: quot and rem (trunc convention) in addition to div and modGravatar letouzey2010-11-10
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | (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
* Numbers: NZPowProp as a Module Type, some module variable renamingGravatar letouzey2010-11-02
| | | | | | | | | 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
* Numbers : also axiomatize constants 1 and 2.Gravatar letouzey2010-10-14
| | | | | | | | | | | | | | | | | 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
* Numbers: new functions pow, even, odd + many reorganisationsGravatar letouzey2010-10-14
| | | | | | | | | | | | | | | | | | | | | | | | - 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
* Updated all headers for 8.3 and trunkGravatar herbelin2010-07-24
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13323 85f007b7-540e-0410-9357-904b9bb8a0f7
* Reverted 13293 commited mistakenly. Sorry for the noise.Gravatar herbelin2010-07-18
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13294 85f007b7-540e-0410-9357-904b9bb8a0f7
* Tentative de suppression de l'import automatique des hints et coercions.Gravatar herbelin2010-07-18
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13293 85f007b7-540e-0410-9357-904b9bb8a0f7
* Division in numbers: kills some Include to avoid bad alias Zsucc = ZDiv.Z.Z'.SGravatar letouzey2010-01-29
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12704 85f007b7-540e-0410-9357-904b9bb8a0f7
* Numbers: BigN and BigZ get instantiations of all properties about div and modGravatar letouzey2010-01-08
| | | | | | | | | NB: for declaring div and mod as a morphism, even when divisor is zero, I've slightly changed the definition of div_eucl: it now starts by a check of whether the divisor is zero. Not very nice, but this way we can say that BigN.div and BigZ.div _always_ answer like Zdiv.Zdiv. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12646 85f007b7-540e-0410-9357-904b9bb8a0f7
* Numbers: separation of funs, notations, axioms. Notations via module, ↵Gravatar letouzey2010-01-07
| | | | | | without scope. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12639 85f007b7-540e-0410-9357-904b9bb8a0f7
* "by" becomes officially a reserved keyword of Coq (fixes "rewrite ... at ... ↵Gravatar letouzey2010-01-06
| | | | | | | | 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
* Numbers abstract layer: more Module Type, used especially for divisions.Gravatar letouzey2010-01-05
| | | | | | | 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
* Avoid declaring hints about refl/sym/trans of eq in DecidableType2Gravatar letouzey2010-01-05
| | | | | | | | 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
* Division in Numbers: proofs with less auto (less sensitive to hints, in ↵Gravatar letouzey2010-01-05
| | | | | | particular about eq) git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12625 85f007b7-540e-0410-9357-904b9bb8a0f7
* Division in Numbers: factorisation of signaturesGravatar letouzey2010-01-05
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12624 85f007b7-540e-0410-9357-904b9bb8a0f7
* Specific syntax for Instances in Module Type: Declare InstanceGravatar letouzey2010-01-04
| | | | | | | | | 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
* Division in Numbers: more properties proved (still W.I.P.)Gravatar letouzey2009-12-16
| | | | git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12591 85f007b7-540e-0410-9357-904b9bb8a0f7
* A generic euclidean division in Numbers (Still Work-In-Progress)Gravatar letouzey2009-12-15
- 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
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