Integration of theories/Ints/Z/* in ZArith and large cleanup and extension of Zdiv

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. 




git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10291 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 51 insertions, 0 deletions

diff --git a/theories/ZArith/Zsqrt.v b/theories/ZArith/Zsqrt.v
index 4a395e6f8..00da766d8 100644
--- a/theories/ZArith/Zsqrt.v
+++ b/theories/ZArith/Zsqrt.v
@@ -148,6 +148,7 @@ Definition Zsqrt_plain (x:Z) : Z :=
   end.
 
 (** A basic theorem about Zsqrt_plain *)
+
 Theorem Zsqrt_interval :
   forall n:Z,
     0 <= n ->
@@ -162,3 +163,53 @@ Proof.
   intros p Hle; elim Hle; auto.
 Qed.
 
+(** Positivity *)
+
+Theorem Zsqrt_plain_is_pos: forall n, 0 <= n ->  0 <= Zsqrt_plain n.
+Proof.
+  intros n m; case (Zsqrt_interval n); auto with zarith.
+  intros H1 H2; case (Zle_or_lt 0 (Zsqrt_plain n)); auto.
+  intros H3; absurd_hyp H2; auto; apply Zle_not_lt.
+  apply Zle_trans with ( 2 := H1 ).
+  replace ((Zsqrt_plain n + 1) * (Zsqrt_plain n + 1))
+     with (Zsqrt_plain n * Zsqrt_plain n + (2 * Zsqrt_plain n + 1));
+  auto with zarith.
+  ring.
+Qed.
+
+(** Direct correctness on squares. *)
+
+Theorem Zsqrt_square_id: forall a, 0 <= a ->  Zsqrt_plain (a * a) = a.
+Proof.
+  intros a H.
+  generalize (Zsqrt_plain_is_pos (a * a)); auto with zarith; intros Haa.
+  case (Zsqrt_interval (a * a)); auto with zarith.
+  intros H1 H2.
+  case (Zle_or_lt a (Zsqrt_plain (a * a))); intros H3; auto.
+  case Zle_lt_or_eq with (1:=H3); auto; clear H3; intros H3.
+  absurd_hyp H1; auto; apply Zlt_not_le; auto with zarith.
+  apply Zle_lt_trans with (a * Zsqrt_plain (a * a)); auto with zarith.
+  apply Zmult_lt_compat_r; auto with zarith.
+  absurd_hyp H2; auto; apply Zle_not_lt; auto with zarith.
+  apply Zmult_le_compat; auto with zarith.
+Qed.
+
+(** [Zsqrt_plain] is increasing *)
+
+Theorem Zsqrt_le:
+ forall p q, 0 <= p <= q  ->  Zsqrt_plain p <= Zsqrt_plain q.
+Proof.
+  intros p q [H1 H2]; case Zle_lt_or_eq with (1:=H2); clear H2; intros H2; 
+  [ | subst q; auto with zarith].
+  case (Zle_or_lt (Zsqrt_plain p) (Zsqrt_plain q)); auto; intros H3.
+  assert (Hp: (0 <= Zsqrt_plain q)).
+   apply Zsqrt_plain_is_pos; auto with zarith.
+  absurd (q <= p); auto with zarith.
+  apply Zle_trans with ((Zsqrt_plain q + 1) * (Zsqrt_plain q + 1)).
+  case (Zsqrt_interval q); auto with zarith.
+  apply Zle_trans with (Zsqrt_plain p * Zsqrt_plain p); auto with zarith.
+  apply Zmult_le_compat; auto with zarith.
+  case (Zsqrt_interval p); auto with zarith.
+Qed.
+
+