fichiers en attendant une intégration à theories/Numbers



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Some details: 

 - ZAux.v is the only file left in Ints/Z. The few elements that remain in it
   are rather specific or compatibility oriented. Others parts and files have 
   been either deleted when unused or pushed into some place of ZArith. 
 - Ints/List/ is removed since it was not needed anymore
 - Ints/Tactic.v disappear: some of its tactic were unused, some already in 
   Tactics.v (case_eq, f_equal instead of eq_tac), and the nice contradict 
   has been added to Tactics.v
 - Znumtheory inherits lots of results about Zdivide, rel_prime, prime, Zgcd, ...
 - A new file Zpow_facts inherits lots of results about Zpower. Placing them 
   into Zpower would have been difficult with respect to compatibility 
   (import of ring)
 - A few things added to Zmax, Zabs, Znat, Zsqrt, Zeven, Zorder
 - Adequate adaptations to Ints/num/* (mainly renaming of lemmas) 
 
Now, concerning Zdiv, the behavior when dividing by a negative number is now 
fully proved. When this was possible, existing lemmas has been extended, 
either from strictly positive to non-zero divisor, or even to arbitrary 
divisor (especially when playing with Zmod). These extended lemmas are named
with the suffix _full, whereas the original restrictive lemmas are retained 
for compatibility. Several lemmas now have shorter proofs (based on unicity 
lemmas). Lemmas are now more or less organized by themes (division and order, 
division and usual operations, etc). Three possible choices of spec for 
divisions on negative numbers are presented: this Zdiv, the ocaml approach
and the remainder-always-positive approach. The ugly behavior of Zopp 
with the current choice of Zdiv/Zmod is now fully covered. A embryo of 
division "a la Ocaml" is given: Odiv and Omod. 




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present in Ints. For the moment, mainly: 
 - Q parts go in QArith
 - Some of the Zdivide & Zgcd stuff go in Znumtheory

More to come ...



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that "intros ? a ? b" behaves as expected.


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------------------------------------------------ 

See file PreOmega for more details and/or test-suite/succes/*Omega*.v

The zify tactic performs a Z-ification of your current goal,
transforming parts of type nat, N, positive, taking advantage of many
equivalences of operations, and of the positivity implied by these
types.

Integration with omega and romega:
 (r)omega : the earlier tactics, 100% compatible 
 (r)omega with * : full zify applied before the (r)omega run
 (r)omega with <types>, where <types> is a sub-list of {nat,N,positive,Z}, 
   applies only specific parts of zify (btw "with Z" means take advantage 
   of Zmax, Zmin, Zabs and Zsgn). 

As a particular consequence, "romega with nat" should now be a
close-to-perfect replacement for omega. Slightly more powerful, since
(forall x:nat, x*x>=0) is provable and also slightly less powerful: if
False is somewhere in the hypothesis, it doesn't use it. 

For the moment zify is done in a direct way in Ltac, using rewrite
when necessary, but crucial chains of rewrite may be made reflexive
some day.

Even though zify is designed to help (r)omega, I think it might be 
of interest for other tactics (micromega ?). Feel free to complete
zify if your favorite operation / type isn't handled yet. 


Side-effects: 

- additional results for ZArith, NArith, etc...
- definition of Ple, Plt, Pgt, Pge and notations for them in positive_scope
- romega now start by doing "intros". Since the conclusion will be negated, 
  and this operation will be justified by means of decidability, it helps
  to have as little as possible in the conclusion. 






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details).




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