Deletion of useless Zdigits_def

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14247 85f007b7-540e-0410-9357-904b9bb8a0f7

5 files changed, 82 insertions, 69 deletions

diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v
index cb16e1291..0142b36be 100644
--- a/theories/Numbers/Integer/BigZ/ZMake.v
+++ b/theories/Numbers/Integer/BigZ/ZMake.v
@@ -678,17 +678,17 @@ Module Make (N:NType) <: ZType.
   destruct (norm_pos x) as [x'|x'];
    specialize (H x' (eq_refl _)) || clear H.
 
- Lemma spec_testbit: forall x p, testbit x p = Ztestbit (to_Z x) (to_Z p).
+ Lemma spec_testbit: forall x p, testbit x p = Z.testbit (to_Z x) (to_Z p).
  Proof.
   intros x p. unfold testbit.
   destr_norm_pos p; simpl. destr_norm_pos x; simpl.
   apply N.spec_testbit.
   rewrite N.spec_testbit, N.spec_pred, Zmax_r by auto with zarith.
   symmetry. apply Z.bits_opp. apply N.spec_pos.
-  symmetry. apply Ztestbit_neg_r; auto with zarith.
+  symmetry. apply Z.testbit_neg_r; auto with zarith.
  Qed.
 
- Lemma spec_shiftl: forall x p, to_Z (shiftl x p) = Zshiftl (to_Z x) (to_Z p).
+ Lemma spec_shiftl: forall x p, to_Z (shiftl x p) = Z.shiftl (to_Z x) (to_Z p).
  Proof.
   intros x p. unfold shiftl.
   destr_norm_pos x; destruct p as [p|p]; simpl;
@@ -703,13 +703,13 @@ Module Make (N:NType) <: ZType.
   now rewrite Zlnot_alt1, Z.lnot_shiftr, Zlnot_alt2.
  Qed.
 
- Lemma spec_shiftr: forall x p, to_Z (shiftr x p) = Zshiftr (to_Z x) (to_Z p).
+ Lemma spec_shiftr: forall x p, to_Z (shiftr x p) = Z.shiftr (to_Z x) (to_Z p).
  Proof.
   intros. unfold shiftr. rewrite spec_shiftl, spec_opp.
   apply Z.shiftl_opp_r.
  Qed.
 
- Lemma spec_land: forall x y, to_Z (land x y) = Zand (to_Z x) (to_Z y).
+ Lemma spec_land: forall x y, to_Z (land x y) = Z.land (to_Z x) (to_Z y).
  Proof.
   intros x y. unfold land.
   destr_norm_pos x; destr_norm_pos y; simpl;
@@ -720,7 +720,7 @@ Module Make (N:NType) <: ZType.
   now rewrite Z.lnot_lor, !Zlnot_alt2.
  Qed.
 
- Lemma spec_lor: forall x y, to_Z (lor x y) = Zor (to_Z x) (to_Z y).
+ Lemma spec_lor: forall x y, to_Z (lor x y) = Z.lor (to_Z x) (to_Z y).
  Proof.
   intros x y. unfold lor.
   destr_norm_pos x; destr_norm_pos y; simpl;
@@ -731,7 +731,7 @@ Module Make (N:NType) <: ZType.
   now rewrite Z.lnot_land, !Zlnot_alt2.
  Qed.
 
- Lemma spec_ldiff: forall x y, to_Z (ldiff x y) = Zdiff (to_Z x) (to_Z y).
+ Lemma spec_ldiff: forall x y, to_Z (ldiff x y) = Z.ldiff (to_Z x) (to_Z y).
  Proof.
   intros x y. unfold ldiff.
   destr_norm_pos x; destr_norm_pos y; simpl;
@@ -742,7 +742,7 @@ Module Make (N:NType) <: ZType.
   now rewrite 2 Z.ldiff_land, Zlnot_alt2, Z.land_comm, Zlnot_alt3.
  Qed.
 
- Lemma spec_lxor: forall x y, to_Z (lxor x y) = Zxor (to_Z x) (to_Z y).
+ Lemma spec_lxor: forall x y, to_Z (lxor x y) = Z.lxor (to_Z x) (to_Z y).
  Proof.
   intros x y. unfold lxor.
   destr_norm_pos x; destr_norm_pos y; simpl;
@@ -753,9 +753,9 @@ Module Make (N:NType) <: ZType.
   now rewrite <- Z.lxor_lnot_lnot, !Zlnot_alt2.
  Qed.
 
- Lemma spec_div2: forall x, to_Z (div2 x) = Zdiv2 (to_Z x).
+ Lemma spec_div2: forall x, to_Z (div2 x) = Z.div2 (to_Z x).
  Proof.
-  intros x. unfold div2. now rewrite spec_shiftr, Zdiv2_spec, spec_1.
+  intros x. unfold div2. now rewrite spec_shiftr, Z.div2_spec, spec_1.
  Qed.
 
 End Make.
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index f40928566..eaab13c2a 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -445,77 +445,77 @@ Program Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit.
 
 Lemma testbit_odd_0 : forall a, testbit (2*a+1) 0 = true.
 Proof.
- intros. zify. apply Ztestbit_odd_0.
+ intros. zify. apply Z.testbit_odd_0.
 Qed.
 
 Lemma testbit_even_0 : forall a, testbit (2*a) 0 = false.
 Proof.
- intros. zify. apply Ztestbit_even_0.
+ intros. zify. apply Z.testbit_even_0.
 Qed.
 
 Lemma testbit_odd_succ : forall a n, 0<=n ->
  testbit (2*a+1) (succ n) = testbit a n.
 Proof.
- intros a n. zify. apply Ztestbit_odd_succ.
+ intros a n. zify. apply Z.testbit_odd_succ.
 Qed.
 
 Lemma testbit_even_succ : forall a n, 0<=n ->
  testbit (2*a) (succ n) = testbit a n.
 Proof.
- intros a n. zify. apply Ztestbit_even_succ.
+ intros a n. zify. apply Z.testbit_even_succ.
 Qed.
 
 Lemma testbit_neg_r : forall a n, n<0 -> testbit a n = false.
 Proof.
- intros a n. zify. apply Ztestbit_neg_r.
+ intros a n. zify. apply Z.testbit_neg_r.
 Qed.
 
 Lemma shiftr_spec : forall a n m, 0<=m ->
  testbit (shiftr a n) m = testbit a (m+n).
 Proof.
- intros a n m. zify. apply Zshiftr_spec.
+ intros a n m. zify. apply Z.shiftr_spec.
 Qed.
 
 Lemma shiftl_spec_high : forall a n m, 0<=m -> n<=m ->
  testbit (shiftl a n) m = testbit a (m-n).
 Proof.
  intros a n m. zify. intros Hn H.
- now apply Zshiftl_spec_high.
+ now apply Z.shiftl_spec_high.
 Qed.
 
 Lemma shiftl_spec_low : forall a n m, m<n ->
  testbit (shiftl a n) m = false.
 Proof.
- intros a n m. zify. intros H. now apply Zshiftl_spec_low.
+ intros a n m. zify. intros H. now apply Z.shiftl_spec_low.
 Qed.
 
 Lemma land_spec : forall a b n,
  testbit (land a b) n = testbit a n && testbit b n.
 Proof.
- intros a n m. zify. now apply Zand_spec.
+ intros a n m. zify. now apply Z.land_spec.
 Qed.
 
 Lemma lor_spec : forall a b n,
  testbit (lor a b) n = testbit a n || testbit b n.
 Proof.
- intros a n m. zify. now apply Zor_spec.
+ intros a n m. zify. now apply Z.lor_spec.
 Qed.
 
 Lemma ldiff_spec : forall a b n,
  testbit (ldiff a b) n = testbit a n && negb (testbit b n).
 Proof.
- intros a n m. zify. now apply Zdiff_spec.
+ intros a n m. zify. now apply Z.ldiff_spec.
 Qed.
 
 Lemma lxor_spec : forall a b n,
  testbit (lxor a b) n = xorb (testbit a n) (testbit b n).
 Proof.
- intros a n m. zify. now apply Zxor_spec.
+ intros a n m. zify. now apply Z.lxor_spec.
 Qed.
 
 Lemma div2_spec : forall a, div2 a == shiftr a 1.
 Proof.
- intros a. zify. now apply Zdiv2_spec.
+ intros a. zify. now apply Z.div2_spec.
 Qed.
 
 End ZTypeIsZAxioms.
diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v
index 64b8ec844..a6eb6ae47 100644
--- a/theories/Numbers/Natural/BigN/NMake.v
+++ b/theories/Numbers/Natural/BigN/NMake.v
@@ -1330,7 +1330,7 @@ Module Make (W0:CyclicType) <: NType.
   generalize (ZnZ.spec_to_Z d); auto with zarith.
  Qed.
 
- Lemma spec_shiftr: forall x p, [shiftr x p] = Zshiftr [x] [p].
+ Lemma spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p].
  Proof.
   intros.
   now rewrite spec_shiftr_pow2, Z.shiftr_div_pow2 by apply spec_pos.
@@ -1603,7 +1603,7 @@ Module Make (W0:CyclicType) <: NType.
  apply Zpower_le_monotone2; auto with zarith.
  Qed.
 
- Lemma spec_shiftl: forall x p, [shiftl x p] = Zshiftl [x] [p].
+ Lemma spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p].
  Proof.
   intros.
   now rewrite spec_shiftl_pow2, Z.shiftl_mul_pow2 by apply spec_pos.
@@ -1613,7 +1613,7 @@ Module Make (W0:CyclicType) <: NType.
 
  Definition testbit x n := odd (shiftr x n).
 
- Lemma spec_testbit: forall x p, testbit x p = Ztestbit [x] [p].
+ Lemma spec_testbit: forall x p, testbit x p = Z.testbit [x] [p].
  Proof.
   intros. unfold testbit. symmetry.
   rewrite spec_odd, spec_shiftr. apply Z.testbit_odd.
@@ -1621,42 +1621,42 @@ Module Make (W0:CyclicType) <: NType.
 
  Definition div2 x := shiftr x one.
 
- Lemma spec_div2: forall x, [div2 x] = Zdiv2 [x].
+ Lemma spec_div2: forall x, [div2 x] = Z.div2 [x].
  Proof.
   intros. unfold div2. symmetry.
-  rewrite spec_shiftr, spec_1. apply Zdiv2_spec.
+  rewrite spec_shiftr, spec_1. apply Z.div2_spec.
  Qed.
 
  (** TODO : provide efficient versions instead of just converting
    from/to N (see with Laurent) *)
 
- Definition lor x y := of_N (Nor (to_N x) (to_N y)).
- Definition land x y := of_N (Nand (to_N x) (to_N y)).
- Definition ldiff x y := of_N (Ndiff (to_N x) (to_N y)).
- Definition lxor x y := of_N (Nxor (to_N x) (to_N y)).
+ Definition lor x y := of_N (N.lor (to_N x) (to_N y)).
+ Definition land x y := of_N (N.land (to_N x) (to_N y)).
+ Definition ldiff x y := of_N (N.ldiff (to_N x) (to_N y)).
+ Definition lxor x y := of_N (N.lxor (to_N x) (to_N y)).
 
- Lemma spec_land: forall x y, [land x y] = Zand [x] [y].
+ Lemma spec_land: forall x y, [land x y] = Z.land [x] [y].
  Proof.
   intros x y. unfold land. rewrite spec_of_N. unfold to_N.
   generalize (spec_pos x), (spec_pos y).
   destruct [x], [y]; trivial; (now destruct 1) || (now destruct 2).
  Qed.
 
- Lemma spec_lor: forall x y, [lor x y] = Zor [x] [y].
+ Lemma spec_lor: forall x y, [lor x y] = Z.lor [x] [y].
  Proof.
   intros x y. unfold lor. rewrite spec_of_N. unfold to_N.
   generalize (spec_pos x), (spec_pos y).
   destruct [x], [y]; trivial; (now destruct 1) || (now destruct 2).
  Qed.
 
- Lemma spec_ldiff: forall x y, [ldiff x y] = Zdiff [x] [y].
+ Lemma spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y].
  Proof.
   intros x y. unfold ldiff. rewrite spec_of_N. unfold to_N.
   generalize (spec_pos x), (spec_pos y).
   destruct [x], [y]; trivial; (now destruct 1) || (now destruct 2).
  Qed.
 
- Lemma spec_lxor: forall x y, [lxor x y] = Zxor [x] [y].
+ Lemma spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y].
  Proof.
   intros x y. unfold lxor. rewrite spec_of_N. unfold to_N.
   generalize (spec_pos x), (spec_pos y).
diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v
index 94f6b32fd..dec8f1fe4 100644
--- a/theories/Numbers/Natural/BigN/Nbasic.v
+++ b/theories/Numbers/Natural/BigN/Nbasic.v
@@ -23,6 +23,19 @@ Implicit Arguments mk_zn2z_specs_karatsuba [t ops].
 Implicit Arguments ZnZ.digits [t].
 Implicit Arguments ZnZ.zdigits [t].
 
+Lemma Pshiftl_nat_Zpower : forall n p,
+  Zpos (Pos.shiftl_nat p n) = Zpos p * 2 ^ Z.of_nat n.
+Proof.
+ intros.
+ rewrite Z.mul_comm.
+ induction n. simpl; auto.
+ transitivity (2 * (2 ^ Z.of_nat n * Zpos p)).
+ rewrite <- IHn. auto.
+ rewrite Z.mul_assoc.
+ rewrite inj_S.
+ rewrite <- Z.pow_succ_r; auto with zarith.
+Qed.
+
 (* To compute the necessary height *)
 
 Fixpoint plength (p: positive) : positive :=
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
index 225c0853e..a1f4ea9a2 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
@@ -11,7 +11,7 @@ Require Import ZArith OrdersFacts Nnat Ndigits NAxioms NDiv NSig.
 
 (** * The interface [NSig.NType] implies the interface [NAxiomsSig] *)
 
-Module NTypeIsNAxioms (Import N : NType').
+Module NTypeIsNAxioms (Import NN : NType').
 
 Hint Rewrite
  spec_0 spec_1 spec_2 spec_succ spec_add spec_mul spec_pred spec_sub
@@ -54,7 +54,7 @@ Definition N_of_Z z := of_N (Zabs_N z).
 
 Section Induction.
 
-Variable A : N.t -> Prop.
+Variable A : NN.t -> Prop.
 Hypothesis A_wd : Proper (eq==>iff) A.
 Hypothesis A0 : A 0.
 Hypothesis AS : forall n, A n <-> A (succ n).
@@ -161,7 +161,7 @@ Proof.
  intros. zify. apply Z.compare_antisym.
 Qed.
 
-Include BoolOrderFacts N N N [no inline].
+Include BoolOrderFacts NN NN NN [no inline].
 
 Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare.
 Proof.
@@ -371,83 +371,83 @@ Program Instance testbit_wd : Proper (eq==>eq==>Logic.eq) testbit.
 
 Lemma testbit_odd_0 : forall a, testbit (2*a+1) 0 = true.
 Proof.
- intros. zify. apply Ztestbit_odd_0.
+ intros. zify. apply Z.testbit_odd_0.
 Qed.
 
 Lemma testbit_even_0 : forall a, testbit (2*a) 0 = false.
 Proof.
- intros. zify. apply Ztestbit_even_0.
+ intros. zify. apply Z.testbit_even_0.
 Qed.
 
 Lemma testbit_odd_succ : forall a n, 0<=n ->
  testbit (2*a+1) (succ n) = testbit a n.
 Proof.
- intros a n. zify. apply Ztestbit_odd_succ.
+ intros a n. zify. apply Z.testbit_odd_succ.
 Qed.
 
 Lemma testbit_even_succ : forall a n, 0<=n ->
  testbit (2*a) (succ n) = testbit a n.
 Proof.
- intros a n. zify. apply Ztestbit_even_succ.
+ intros a n. zify. apply Z.testbit_even_succ.
 Qed.
 
 Lemma testbit_neg_r : forall a n, n<0 -> testbit a n = false.
 Proof.
- intros a n. zify. apply Ztestbit_neg_r.
+ intros a n. zify. apply Z.testbit_neg_r.
 Qed.
 
 Lemma shiftr_spec : forall a n m, 0<=m ->
  testbit (shiftr a n) m = testbit a (m+n).
 Proof.
- intros a n m. zify. apply Zshiftr_spec.
+ intros a n m. zify. apply Z.shiftr_spec.
 Qed.
 
 Lemma shiftl_spec_high : forall a n m, 0<=m -> n<=m ->
  testbit (shiftl a n) m = testbit a (m-n).
 Proof.
- intros a n m. zify. intros Hn H. rewrite Zmax_r by auto with zarith.
- now apply Zshiftl_spec_high.
+ intros a n m. zify. intros Hn H. rewrite Z.max_r by auto with zarith.
+ now apply Z.shiftl_spec_high.
 Qed.
 
 Lemma shiftl_spec_low : forall a n m, m<n ->
  testbit (shiftl a n) m = false.
 Proof.
- intros a n m. zify. intros H. now apply Zshiftl_spec_low.
+ intros a n m. zify. intros H. now apply Z.shiftl_spec_low.
 Qed.
 
 Lemma land_spec : forall a b n,
  testbit (land a b) n = testbit a n && testbit b n.
 Proof.
- intros a n m. zify. now apply Zand_spec.
+ intros a n m. zify. now apply Z.land_spec.
 Qed.
 
 Lemma lor_spec : forall a b n,
  testbit (lor a b) n = testbit a n || testbit b n.
 Proof.
- intros a n m. zify. now apply Zor_spec.
+ intros a n m. zify. now apply Z.lor_spec.
 Qed.
 
 Lemma ldiff_spec : forall a b n,
  testbit (ldiff a b) n = testbit a n && negb (testbit b n).
 Proof.
- intros a n m. zify. now apply Zdiff_spec.
+ intros a n m. zify. now apply Z.ldiff_spec.
 Qed.
 
 Lemma lxor_spec : forall a b n,
  testbit (lxor a b) n = xorb (testbit a n) (testbit b n).
 Proof.
- intros a n m. zify. now apply Zxor_spec.
+ intros a n m. zify. now apply Z.lxor_spec.
 Qed.
 
 Lemma div2_spec : forall a, div2 a == shiftr a 1.
 Proof.
- intros a. zify. now apply Zdiv2_spec.
+ intros a. zify. now apply Z.div2_spec.
 Qed.
 
 (** Recursion *)
 
-Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) :=
-  Nrect (fun _ => A) a (fun n a => f (N.of_N n) a) (N.to_N n).
+Definition recursion (A : Type) (a : A) (f : NN.t -> A -> A) (n : NN.t) :=
+  Nrect (fun _ => A) a (fun n a => f (NN.of_N n) a) (NN.to_N n).
 Implicit Arguments recursion [A].
 
 Instance recursion_wd (A : Type) (Aeq : relation A) :
@@ -456,7 +456,7 @@ Proof.
 unfold eq.
 intros a a' Eaa' f f' Eff' x x' Exx'.
 unfold recursion.
-unfold N.to_N.
+unfold NN.to_N.
 rewrite <- Exx'; clear x' Exx'.
 replace (Zabs_N [x]) with (N_of_nat (Zabs_nat [x])).
 induction (Zabs_nat [x]).
@@ -468,30 +468,30 @@ change (nat_of_P p) with (nat_of_N (Npos p)); apply N_of_nat_of_N.
 Qed.
 
 Theorem recursion_0 :
-  forall (A : Type) (a : A) (f : N.t -> A -> A), recursion a f 0 = a.
+  forall (A : Type) (a : A) (f : NN.t -> A -> A), recursion a f 0 = a.
 Proof.
-intros A a f; unfold recursion, N.to_N; rewrite N.spec_0; simpl; auto.
+intros A a f; unfold recursion, NN.to_N; rewrite NN.spec_0; simpl; auto.
 Qed.
 
 Theorem recursion_succ :
-  forall (A : Type) (Aeq : relation A) (a : A) (f : N.t -> A -> A),
+  forall (A : Type) (Aeq : relation A) (a : A) (f : NN.t -> A -> A),
     Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
       forall n, Aeq (recursion a f (succ n)) (f n (recursion a f n)).
 Proof.
-unfold N.eq, recursion; intros A Aeq a f EAaa f_wd n.
-replace (N.to_N (succ n)) with (Nsucc (N.to_N n)).
+unfold NN.eq, recursion; intros A Aeq a f EAaa f_wd n.
+replace (NN.to_N (succ n)) with (N.succ (NN.to_N n)).
 rewrite Nrect_step.
 apply f_wd; auto.
-unfold N.to_N.
-rewrite N.spec_of_N, Z_of_N_abs, Zabs_eq; auto.
- apply N.spec_pos.
+unfold NN.to_N.
+rewrite NN.spec_of_N, Z_of_N_abs, Zabs_eq; auto.
+ apply NN.spec_pos.
 
 fold (recursion a f n).
 apply recursion_wd; auto.
 red; auto.
-unfold N.to_N.
+unfold NN.to_N.
 
-rewrite N.spec_succ.
+rewrite NN.spec_succ.
 change ([n]+1)%Z with (Zsucc [n]).
 apply Z_of_N_eq_rev.
 rewrite Z_of_N_succ.
@@ -503,6 +503,6 @@ Qed.
 
 End NTypeIsNAxioms.
 
-Module NType_NAxioms (N : NType)
- <: NAxiomsSig <: OrderFunctions N <: HasMinMax N
- := N <+ NTypeIsNAxioms.
+Module NType_NAxioms (NN : NType)
+ <: NAxiomsSig <: OrderFunctions NN <: HasMinMax NN
+ := NN <+ NTypeIsNAxioms.