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, 17 insertions, 24 deletions

diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v
index 154541164..de41fd3f6 100644
--- a/theories/Reals/RIneq.v
+++ b/theories/Reals/RIneq.v
@@ -1601,7 +1601,7 @@ Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (nat_of_P p).
 Proof.
   intro; apply lt_0_INR.
   simpl in |- *; auto with real.
-  apply lt_O_nat_of_P.
+  apply nat_of_P_pos.
 Qed.
 Hint Resolve pos_INR_nat_of_P: real.
 
@@ -1710,38 +1710,31 @@ Qed.
 Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n).
 Proof.
   simple induction n; auto with real.
-  intros; simpl in |- *; rewrite nat_of_P_o_P_of_succ_nat_eq_succ;
+  intros; simpl in |- *; rewrite nat_of_P_of_succ_nat;
     auto with real.
 Qed.
 
 Lemma plus_IZR_NEG_POS :
   forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q).
 Proof.
-  intros.
-  case (lt_eq_lt_dec (nat_of_P p) (nat_of_P q)).
-  intros [H| H]; simpl in |- *.
-  rewrite nat_of_P_lt_Lt_compare_complement_morphism; simpl in |- *; trivial.
-  rewrite (nat_of_P_minus_morphism q p).
-  rewrite minus_INR; auto with arith; ring.
-  apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial.
-  rewrite (nat_of_P_inj p q); trivial.
-  rewrite Pcompare_refl; simpl in |- *; auto with real.
-  intro H; simpl in |- *.
-  rewrite nat_of_P_gt_Gt_compare_complement_morphism; simpl in |- *;
-    auto with arith.
-  rewrite (nat_of_P_minus_morphism p q).
-  rewrite minus_INR; auto with arith; ring.
-  apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial.
+  intros p q; simpl. case Pcompare_spec; intros H; simpl.
+  subst. ring.
+  rewrite Pminus_minus by trivial.
+  rewrite minus_INR by (now apply lt_le_weak, Plt_lt).
+  ring.
+  rewrite Pminus_minus by trivial.
+  rewrite minus_INR by (now apply lt_le_weak, Plt_lt).
+  ring.
 Qed.
 
 (**********)
 Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m.
 Proof.
   intro z; destruct z; intro t; destruct t; intros; auto with real.
-  simpl in |- *; intros; rewrite nat_of_P_plus_morphism; auto with real.
+  simpl; intros; rewrite Pplus_plus; auto with real.
   apply plus_IZR_NEG_POS.
   rewrite Zplus_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS.
-  simpl in |- *; intros; rewrite nat_of_P_plus_morphism; rewrite plus_INR;
+  simpl; intros; rewrite Pplus_plus; rewrite plus_INR;
     auto with real.
 Qed.
 
@@ -1749,14 +1742,14 @@ Qed.
 Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m.
 Proof.
   intros z t; case z; case t; simpl in |- *; auto with real.
-  intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.
-  intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.
+  intros t1 z1; rewrite Pmult_mult; auto with real.
+  intros t1 z1; rewrite Pmult_mult; auto with real.
   rewrite Rmult_comm.
   rewrite Ropp_mult_distr_l_reverse; auto with real.
   apply Ropp_eq_compat; rewrite mult_comm; auto with real.
-  intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.
+  intros t1 z1; rewrite Pmult_mult; auto with real.
   rewrite Ropp_mult_distr_l_reverse; auto with real.
-  intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.
+  intros t1 z1; rewrite Pmult_mult; auto with real.
   rewrite Rmult_opp_opp; auto with real.
 Qed.
 
@@ -1764,7 +1757,7 @@ Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Zpower z (Z_of_nat n)).
 Proof.
  intros z [|n];simpl;trivial.
  rewrite Zpower_pos_nat.
- rewrite nat_of_P_o_P_of_succ_nat_eq_succ. unfold Zpower_nat;simpl.
+ rewrite nat_of_P_of_succ_nat. unfold Zpower_nat;simpl.
  rewrite mult_IZR.
  induction n;simpl;trivial.
  rewrite mult_IZR;ring[IHn].