Some more revision of {P,N,Z}Arith + bitwise ops in Ndigits

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.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13651 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 14 insertions, 3 deletions

diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v
index c16a13dfb..d3d7b5835 100644
--- a/theories/Arith/Wf_nat.v
+++ b/theories/Arith/Wf_nat.v
@@ -270,7 +270,18 @@ Theorem iter_nat_plus :
   forall (n m:nat) (A:Type) (f:A -> A) (x:A),
     iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x).
 Proof.
-  simple induction n;
-    [ simpl in |- *; auto with arith
-      | intros; simpl in |- *; apply f_equal with (f := f); apply H ].
+  induction n; simpl; trivial.
+  intros. now f_equal.
+Qed.
+
+(** Preservation of invariants : if [f : A->A] preserves the invariant [Inv],
+    then the iterates of [f] also preserve it. *)
+
+Theorem iter_nat_invariant :
+  forall (n:nat) (A:Type) (f:A -> A) (Inv:A -> Prop),
+    (forall x:A, Inv x -> Inv (f x)) ->
+    forall x:A, Inv x -> Inv (iter_nat n A f x).
+Proof.
+  induction n; simpl; trivial.
+  intros A f Inv Hf x Hx. apply Hf, IHn; trivial.
 Qed.