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- provides the atan function
- shows that this function is equal between -1 and 1 to a function defined
with power series
- establishes the equality with the PI value as given by the alternated
series constructed with PI_tg
- provides a smarter theorem to compute approximations of PI, based on a
formula in the same family as the one used by John Machin in 1706
Dependencies between files have been rearranged so that the new theorems
are loaded with the library Reals.
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and of course, the definition of atan (the inverse of tan, from R to
(-PI/2, PI/2)
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Beware that the definition of PI changes in the process
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- tauto/intuition now works uniformly on and, prod, or, sum, False,
Empty_set, unit, True (and isomorphic copies of them), iff, ->, and
on all inhabited singleton types with a no-arguments constructor
such as "eq t t" (even though the last case goes out of
propositional logic: this features is so often used that it is
difficult to come back on it).
- New dtauto and dintuition works on all inductive types with one
constructors and no real arguments (for instance, they work on
records such as "Equivalence"), in addition to -> and eq-like types.
- Moreover, both of them no longer unfold inner negations (this is a
souce of incompatibility for intuition and evaluation of the level
of incompatibility on contribs still needs to be done).
Incidentally, and amazingly, fixing bug #2680 made that constants
InfA_compat and InfA_eqA in SetoidList.v lost one argument: old tauto
had indeed destructed a section hypothesis "@StrictOrder A ltA@
thinking it was a conjunction, making this section hypothesis
artificially necessary while it was not.
Renouncing to the unfolding of inner negations made auto/eauto
sometimes succeeding more, sometimes succeeding less. There is by the
way a (standard) problem with not in auto/eauto: even when given as an
"unfold hint", it works only in goals, not in hypotheses, so that auto
is not able to solve something like "forall P, (forall x, ~ P x) -> P
0 -> False". Should we automatically add a lemma of type "HYPS -> A ->
False" in the hint database everytime a lemma ""HYPS -> ~A" is
declared (and "unfold not" is a hint), and similarly for all unfold
hints?
At this occasion, also re-did some proofs of Znumtheory.
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same name for different statements as noticed by Adam Chlipala on
coq-club, avoiding stating the same axiom twice in distinct files.
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It relies on Z.pos_sub instead of a Pos.compare followed by Pos.sub.
Proofs seem to be quite easy to adapt, via some rewrite Z.pos_sub_spec.
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reducing matches).
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They are actually simpler, sometimes using the [rewrite at] variant.
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unification (potentially harmful)
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CompareSpec expects 3 propositions Peq Plt Pgt instead of 2 relations
eq lt and 2 points x y. For the moment, we still always use (Peq=eq x y),
(Plt=lt x y) (Pgt=lt y x), but this may not be always the case,
especially for Pgt. The former CompSpec is now defined in term of
CompareSpec. Compatibility is preserved (except maybe a rare unfold
or red to break the CompSpec definition).
Typically, CompareSpec looks nicer when we have infix notations, e.g.
forall x y, CompareSpec (x=y) (x<y) (y<x) (x?=x)
while CompSpec is shorter when we directly refer to predicates:
forall x y, CompSpec eq lt x y (compare x y)
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According to B. Gregoire, this stuff is obsolete. Fine control
on when to launch the VM in conversion problems is now provided
by VMcast. We were already almost never boxing definitions anymore
in stdlib files.
"(Un)Boxed Definition foo" will now trigger a parsing error,
same with Fixpoint. The option "(Un)Set Boxed Definitions"
aren't there anymore, but tolerated (as no-ops), since unknown
options raise a warning instead of an error by default.
Some more cleaning could be done in the vm.
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To avoid names¬ations clashs with list, Vector shouldn't be
"Import"ed but one can "Import Vector.VectorNotations." to have
notations.
SetoidVector at least remains to do.
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Initial plan was only to add shiftl/shiftr/land/... to N and
other number type, this is only partly done, but this work has
diverged into a big reorganisation and improvement session
of PArith,NArith,ZArith.
Bool/Bool: add lemmas orb_diag (a||a = a) and andb_diag (a&&a = a)
PArith/BinPos:
- added a power function Ppow
- iterator iter_pos moved from Zmisc to here + some lemmas
- added Psize_pos, which is 1+log2, used to define Nlog2/Zlog2
- more lemmas on Pcompare and succ/+/* and order, allow
to simplify a lot some old proofs elsewhere.
- new/revised results on Pminus (including some direct proof of
stuff from Pnat)
PArith/Pnat:
- more direct proofs (limit the need of stuff about Pmult_nat).
- provide nicer names for some lemmas (eg. Pplus_plus instead of
nat_of_P_plus_morphism), compatibility notations provided.
- kill some too-specific lemmas unused in stdlib + contribs
NArith/BinNat:
- N_of_nat, nat_of_N moved from Nnat to here.
- a lemma relating Npred and Nminus
- revised definitions and specification proofs of Npow and Nlog2
NArith/Nnat:
- shorter proofs.
- stuff about Z_of_N is moved to Znat. This way, NArith is
entirely independent from ZArith.
NArith/Ndigits:
- added bitwise operations Nand Nor Ndiff Nshiftl Nshiftr
- revised proofs about Nxor, still using functional bit stream
- use the same approach to prove properties of Nand Nor Ndiff
ZArith/BinInt: huge simplification of Zplus_assoc + cosmetic stuff
ZArith/Zcompare: nicer proofs of ugly things like Zcompare_Zplus_compat
ZArith/Znat: some nicer proofs and names, received stuff about Z_of_N
ZArith/Zmisc: almost empty new, only contain stuff about badly-named
iter. Should be reformed more someday.
ZArith/Zlog_def: Zlog2 is now based on Psize_pos, this factorizes
proofs and avoid slowdown due to adding 1 in Z instead of in positive
Zarith/Zpow_def: Zpower_opt is renamed more modestly Zpower_alt
as long as I dont't know why it's slower on powers of two.
Elsewhere: propagate new names + some nicer proofs
NB: Impact on compatibility is probably non-zero, but should be
really moderate. We'll see on contribs, but a few Require here
and there might be necessary.
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- NPeano isn't Exported by default anymore (contains pow for nat).
- in coq_micromega.ml, we specify more where to find the pow of R.
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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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This is a fairly large commit (around 140 files and 7000 lines of code
impacted), it will cause some troubles for sure (I've listed the know
regressions below, there is bound to be more).
At this state of developpement it brings few features to the user, as
the old tactics were
ported with no change. Changes are on the side of the developer mostly.
Here comes a list of the major changes. I will stay brief, but the code
is hopefully well documented so that it is reasonably easy to infer the
details from it.
Feature developer-side:
* Primitives for a "real" refine tactic (generating a goal for each
evar).
* Abstract type of tactics, goals and proofs
* Tactics can act on several goals (formally all the focused goals). An
interesting consequence of this is that the tactical (. ; [ . | ... ])
can be separated in two
tacticals (. ; .) and ( [ . | ... ] ) (although there is a conflict for
this particular syntax). We can also imagine a tactic to reorder the
goals.
* Possibility for a tactic to pass a value to following tactics (a
typical example is
an intro function which tells the following tactics which name it
introduced).
* backtracking primitives for tactics (it is now possible to implement a
tactical '+'
with (a+b);c equivalent to (a;c+b;c) (itself equivalent to
(a;c||b;c)). This is a valuable
tool to implement tactics like "auto" without nowing of the
implementation of tactics.
* A notion of proof modes, which allows to dynamically change the parser
for tactics. It is controlled at user level with the keywords Set
Default Proof Mode (this is the proof mode which is loaded at the start
of each proof) and Proof Mode (switches the proof mode of the current
proof) to control them.
* A new primitive Evd.fold_undefined which operates like an Evd.fold,
except it only goes through the evars whose body is Evar_empty. This is
a common operation throughout the code,
some of the fold-and-test-if-empty occurences have been replaced by
fold_undefined. For now,
it is only implemented as a fold-and-test, but we expect to have some
optimisations coming some day, as there can be a lot of evars in an
evar_map with this new implementation (I've observed a couple of
thousands), whereas there are rarely more than a dozen undefined ones.
Folding being a linear operation, this might result in a significant
speed-up.
* The declarative mode has been moved into the plugins. This is made
possible by the proof mode feature. I tried to document it so that it
can serve as a tutorial for a tactic mode plugin.
Features user-side:
* Unfocus does not go back to the root of the proof if several Focus-s
have been performed.
It only goes back to the point where it was last focused.
* experimental (non-documented) support of keywords
BeginSubproof/EndSubproof:
BeginSubproof focuses on first goal, one can unfocus only with
EndSubproof, and only
if the proof is completed for that goal.
* experimental (non-documented) support for bullets ('+', '-' and '*')
they act as hierarchical BeginSubproof/EndSubproof:
First time one uses '+' (for instance) it focuses on first goal, when
the subproof is
completed, one can use '+' again which unfocuses and focuses on next
first goal.
Meanwhile, one cas use '*' (for instance) to focus more deeply.
Known regressions:
* The xml plugin had some functions related to proof trees. As the
structure of proof changed significantly, they do not work anymore.
* I do not know how to implement info or show script in this new engine.
Actually I don't even know what they were suppose to actually mean in
earlier versions either. I wager they would require some calm thinking
before going back to work.
* Declarative mode not entirely working (in particular proofs by
induction need to be restored).
* A bug in the inversion tactic (observed in some contributions)
* A bug in Program (observed in some contributions)
* Minor change in the 'old' type of tactics causing some contributions
to fail.
* Compilation time takes about 10-15% longer for unknown reasons (I
suspect it might be linked to the fact that I don't perform any
reduction at QED-s, and also to some linear operations on evar_map-s
(see Evd.fold_undefined above)).
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This is a tradeoff: the Archimedean property is used instead through the discrete logarithm.
Thanks to P-M. Pedrot for demonstrating this proof scheme on coq-club.
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Old stuff DecidableType.v and OrderedType.v stay there and keep their
names for the moment, for compatibility.
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Leibniz part
Moreover, instantiation like MinMax are now made without redefining
generic properties (easier maintenance). We start using inner modules
for qualifying (e.g. Z.max_comm).
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with Inline)
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compatibility...)
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in */*/vo.itarget
On the way: no more -fsets (yes|no) and -reals (yes|no) option of configure
if you want a partial build, make a specific rule such as theories-light
Beware: these vo.itarget should not contain comments. Even if this is legal
for ocamlbuild, the $(shell cat ...) we do in Makefile can't accept that.
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Make setoid_rewrite-through-rewrite's selection of occurences more
robust: do not try unification with reduction if not needed.
This changes a few scripts that were using reduction in a far from
obvious way and could break more.
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We can now have a diamond-like approch to extentions of signatures,
instead of a linear-only chains as earlier...
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+ adaptation of {Nat,N,P,Z,Q,R}_as_DT for them to provide both eq_dec and eqb
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This way we get properties of Rmin / Rmax (almost) for free.
TODO: merge Rbasic_fun and Rminmax...
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Indeed, since sqrt is a total function, most lemmas are true, even for
negative inputs. For instance, sqrt_le_1 was
0 <= x -> 0 <= y -> x <= y -> sqrt x <= sqrt y
while sqrt_le_1_alt is just
x <= y -> sqrt x <= sqrt y.
Naming is done by adding _alt suffixes. In an ideal world (Coq 9.0?), these
new lemmas should just replace the original ones.
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1) compilation of Library.tex was failing on a "Ext_" in Diaconescu.v
In fact coqdoc was trying to recognize the end of a _emphasis_ and
hence inserted a bogus }. For the moment I've enclosed the phrase
with [ ], but this emphasis "feature" of coqdoc seems _really_
easy to broke. Matthieu ?
2) By the way, this Library document was made from latin1 and utf8
source file, hence bogus characters. All .v containing special
characters are converted to utf8, and their first line is now
mentionning this. (+ killed some old french comments and some
other avoidable special characters).
PLEASE: let's stick to this convention and avoid latin1, at least
in .v files.
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- Rmult_eq_compat_r, Rmult_eq_reg_r, Rplus_le_reg_r, Rmult_lt_reg_r,
Rmult_le_reg_r (mirrored variants of the existing _l lemmas);
- minus_IZR, opp_IZR (Ropp_Ropp_IZR), abs_IZR (mirrored Rabs_Zabs);
- Rle_abs (RRle_abs);
- Zpower_pos_powerRZ (signed variant of Zpower_nat_powerRZ).
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Idea: make coqtop more independant of the standard library.
In the future, we can imagine loading the syntax for numerals right
after their definition. For the moment, it is easier to stay lazy
and load the syntax plugins slightly before the definitions.
After this commit, the main (sole ?) references to theories/
from the core ml files are in Coqlib (but many parts of coqlib
are only used by plugins), and it mainly concerns Init
(+ Logic/JMeq and maybe a few others).
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- tacinterp.ml: les arguments tactiques de Tactic Notation n'etaient
pas evalues, laissant des variables libres (symptome: exc Not_found)
- reals: Open Local --> Local Open
- ListTactics: syntaxe des listes
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loadable plugins
- Any contrib foo leads to contrib/foo/foo_plugin.cmxs (and .cma for bytecode).
- Features that were available without any Require are now loaded systematically
when launching coqtop (see Coqtop.load_initial_plugins):
extraction, jprover, cc, ground, dp, recdef, xml
- The other plugins are loaded when a corresponding Require is done:
quote, ring, field, setoid_ring, omega, romega, micromega, fourier
- I experienced a crash (segfault) while turning subtac into a plugin, so this
one stays statically linked into coqtop for now
- When the ocaml version doesn't support natdynlink, or if "-natdynlink no"
is explicitely given to configure, coqtop is statically linked with all of
the above code as usual. Some messages [Ignore ML file Foo_plugin] may appear.
- How should coqdep handle a "Declare ML Module "foo"" if foo is an archive
and not a ml file ? For now, we suppose that the foo.{cmxs,cma} are at the
same location as the .v during the build, but can be moved later in any place of
the ml loadpath.
This is clearly an experimentation. Feedback most welcome...
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- si on met Rle_ge et Rge_le tous les deux en Resolve (et de même
pour Rlt_gt et Rgt_lt) alors on introduit un cycle,
- si on en met un des deux en Immediate, alors on perd la symétrie car si
un développement n'a un lemme en hints que pour Rge (resp Rle),
alors il ne sera pas utilisable si on met Rge_le (resp Rle_ge) en Immediate
(et c'est ce qui arrive notamment dans HighSchoolGeometry).
L'idéal serait d'introduire une notion de raisonnement modulo équivalence
dans auto afin que chaque lemme sur Rle (resp Rge) soit systématiquement
applicable aussi face à Rge (resp Rle) sans redondances et sans
cycle. Ainsi Rle_ge and co n'auraient pas un statut de hints mais plutôt un
statut de conversions implicites entre notions synonymes.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10762 85f007b7-540e-0410-9357-904b9bb8a0f7
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