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This adds two experimental features to the typeclass implementation:
- Path cuts: a way to specify through regular expressions on instance names
search pathes that should be avoided (e.g. [proper_flip proper_flip]).
Regular expression matching is implemented through naïve derivatives.
- Forward hints for subclasses: e.g. [Equivalence -> Reflexive] is no
longer applied backwards, but introducing a specific [Equivalence] in the
environment register a [Reflexive] hint as well. Currently not
backwards-compatible, the next patch will allow to specify direction
of subclasses hints.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14671 85f007b7-540e-0410-9357-904b9bb8a0f7
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Instead of hard-coding in search.ml some substrings such
as "_admitted" or "_subproof" we don't want to see in results
of SearchAbout and co, we now have a user command:
Add Search Blacklist "foo".
Remove Search Blacklist "foo". (* the opposite *)
Print Table Search Blacklist. (* the current state *)
In Prelude.v, three substrings are blacklisted originally:
- "_admitted" for internal lemmas due to admit.
- "_subproof" for internal lemmas due to abstract.
- "Private_" for hiding auxiliary modules not meant for
global usage.
Note that substrings are searched in the fully qualified names
of the available lemmas (e.g. "Coq.Init.Peano.plus").
This commit also adds the prefix "Private_" to some internal modules
in Numbers, Z, N, etc.
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- Zpow_def, Zpower, Zpow_facts shortened thanks to stuff in BinInt.Z
- The alternative Zpower_alt is now in a separate file Zpow_alt.v,
not loaded by default.
- Some more injection lemmas in Znat (pow, div, mod, quot, rem)
- Btw, added a "square" function in Z, N, Pos, ... (instead of
Zpow_facts.Zsquare).
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Start of a uniform treatment of compare, eqb, leb, ltb:
- We now ensure that they are provided by N,Z,BigZ,BigN,Nat and Pos
- Some generic properties are derived in OrdersFacts.BoolOrderFacts
In BinPos, more work about sub_mask with nice implications
on compare (e.g. simplier proof of lt_trans).
In BinNat/BinPos, for uniformity, compare_antisym is now
(y ?= x) = CompOpp (x ?=y) instead of the symmetrical result.
In BigN / BigZ, eq_bool is now eqb
In BinIntDef, gtb and geb are kept for the moment, but
a comment advise to rather use ltb and leb. Z.div now uses
Z.ltb and Z.leb.
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Initial patch by Robbert Krebbers.
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We now specify testbit by some initial and recursive equations.
The previous spec (via a complex split of the number in
low and high parts) is now a derived property in {N,Z}Bits.v
This way, proofs of implementations are quite simplier.
Note that these new specs doesn't imply anymore that testbit is a
morphism, we have to add this as a extra spec (but this lead
to trivial proofs when implementing).
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For example, if we know that [f] is a morphism for [E1==>E2==>E],
then the goal [E (f x y) (f x' y')] will be transformed by [f_equiv]
into the subgoals [E1 x x'] and [E2 y y'].
This way, we can remove most of the explicit use of the morphism
instances in Numbers (lemmas foo_wd for each operator foo).
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- a ltac solve_proper which generalizes solve_predicate_wd and co
- using le_elim is nicer that (apply le_lteq; destruct ...)
- "apply ->" can now be "apply" most of the time.
Benefit: NumPrelude is now almost empty
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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 ?
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Some more results about sqrt. Similar results for sqrt_up.
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as log2
Some more results about log2. Similar results for log2_up.
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No infix notation "rem" for Zrem (that will probably become Z.rem in
a close future). This way, we avoid conflict with people already using
rem for their own need. Same for BigZ. We still use infix rem, but
only in the abstract layer of Numbers, in a way that doesn't inpact
the rest of Coq. Btw, the axiomatized function is now named rem
instead of remainder.
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(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.
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- 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...
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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
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These additional specs are useless (but trivially provable) for N.
They are quite convenient when deriving properties in NZ.
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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...
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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)
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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...
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- 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.
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- 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
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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.
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This allow to really finish files about division.
An abs and sgn is added to BigZ.
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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.
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without scope.
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by ...")
Application in some proofs of Numbers's abstract division
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Properties are now rather passed as functor arg instead of via Include or
some inner modules.
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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.
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particular about eq)
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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...
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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).
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- 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).
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- 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.
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TODO: finish removing the "Add Relation", "Add Morphism" fun_* fun2_*
TODO: now that we have Include, flatten the hierarchy...
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for increased consistency with bignums parts
(commit part II: names of files + additional translation minus --> sub)
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for increased consistency with bignums parts
(commit part I: content of files)
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using the ad-hoc qsetoid_rewrite. Could QRewrite.v be made completely obsolete ?
For the moment rewrite under fun and exists don't work.
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