diff options
Diffstat (limited to 'theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v')
| -rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 66 |
1 files changed, 33 insertions, 33 deletions
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. |