diff options
author | Samuel Mimram <smimram@debian.org> | 2008-07-25 15:12:53 +0200 |
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committer | Samuel Mimram <smimram@debian.org> | 2008-07-25 15:12:53 +0200 |
commit | a0cfa4f118023d35b767a999d5a2ac4b082857b4 (patch) | |
tree | dabcac548e299fee1da464c93b3dba98484f45b1 /theories/Numbers/Natural/Binary | |
parent | 2281410e38ef99d025ea77194585a9bc019fdaa9 (diff) |
Imported Upstream version 8.2~beta3+dfsgupstream/8.2.beta3+dfsg
Diffstat (limited to 'theories/Numbers/Natural/Binary')
-rw-r--r-- | theories/Numbers/Natural/Binary/NBinDefs.v | 267 | ||||
-rw-r--r-- | theories/Numbers/Natural/Binary/NBinary.v | 15 |
2 files changed, 282 insertions, 0 deletions
diff --git a/theories/Numbers/Natural/Binary/NBinDefs.v b/theories/Numbers/Natural/Binary/NBinDefs.v new file mode 100644 index 00000000..fc2bd2df --- /dev/null +++ b/theories/Numbers/Natural/Binary/NBinDefs.v @@ -0,0 +1,267 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Evgeny Makarov, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: NBinDefs.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Import BinPos. +Require Export BinNat. +Require Import NSub. + +Open Local Scope N_scope. + +(** Implementation of [NAxiomsSig] module type via [BinNat.N] *) + +Module NBinaryAxiomsMod <: NAxiomsSig. +Module Export NZOrdAxiomsMod <: NZOrdAxiomsSig. +Module Export NZAxiomsMod <: NZAxiomsSig. + +Definition NZ := N. +Definition NZeq := @eq N. +Definition NZ0 := N0. +Definition NZsucc := Nsucc. +Definition NZpred := Npred. +Definition NZadd := Nplus. +Definition NZsub := Nminus. +Definition NZmul := Nmult. + +Theorem NZeq_equiv : equiv N NZeq. +Proof (eq_equiv N). + +Add Relation N NZeq + reflexivity proved by (proj1 NZeq_equiv) + symmetry proved by (proj2 (proj2 NZeq_equiv)) + transitivity proved by (proj1 (proj2 NZeq_equiv)) +as NZeq_rel. + +Add Morphism NZsucc with signature NZeq ==> NZeq as NZsucc_wd. +Proof. +congruence. +Qed. + +Add Morphism NZpred with signature NZeq ==> NZeq as NZpred_wd. +Proof. +congruence. +Qed. + +Add Morphism NZadd with signature NZeq ==> NZeq ==> NZeq as NZadd_wd. +Proof. +congruence. +Qed. + +Add Morphism NZsub with signature NZeq ==> NZeq ==> NZeq as NZsub_wd. +Proof. +congruence. +Qed. + +Add Morphism NZmul with signature NZeq ==> NZeq ==> NZeq as NZmul_wd. +Proof. +congruence. +Qed. + +Theorem NZinduction : + forall A : NZ -> Prop, predicate_wd NZeq A -> + A N0 -> (forall n, A n <-> A (NZsucc n)) -> forall n : NZ, A n. +Proof. +intros A A_wd A0 AS. apply Nrect. assumption. intros; now apply -> AS. +Qed. + +Theorem NZpred_succ : forall n : NZ, NZpred (NZsucc n) = n. +Proof. +destruct n as [| p]; simpl. reflexivity. +case_eq (Psucc p); try (intros q H; rewrite <- H; now rewrite Ppred_succ). +intro H; false_hyp H Psucc_not_one. +Qed. + +Theorem NZadd_0_l : forall n : NZ, N0 + n = n. +Proof. +reflexivity. +Qed. + +Theorem NZadd_succ_l : forall n m : NZ, (NZsucc n) + m = NZsucc (n + m). +Proof. +destruct n; destruct m. +simpl in |- *; reflexivity. +unfold NZsucc, NZadd, Nsucc, Nplus. rewrite <- Pplus_one_succ_l; reflexivity. +simpl in |- *; reflexivity. +simpl in |- *; rewrite Pplus_succ_permute_l; reflexivity. +Qed. + +Theorem NZsub_0_r : forall n : NZ, n - N0 = n. +Proof. +now destruct n. +Qed. + +Theorem NZsub_succ_r : forall n m : NZ, n - (NZsucc m) = NZpred (n - m). +Proof. +destruct n as [| p]; destruct m as [| q]; try reflexivity. +now destruct p. +simpl. rewrite Pminus_mask_succ_r, Pminus_mask_carry_spec. +now destruct (Pminus_mask p q) as [| r |]; [| destruct r |]. +Qed. + +Theorem NZmul_0_l : forall n : NZ, N0 * n = N0. +Proof. +destruct n; reflexivity. +Qed. + +Theorem NZmul_succ_l : forall n m : NZ, (NZsucc n) * m = n * m + m. +Proof. +destruct n as [| n]; destruct m as [| m]; simpl; try reflexivity. +now rewrite Pmult_Sn_m, Pplus_comm. +Qed. + +End NZAxiomsMod. + +Definition NZlt := Nlt. +Definition NZle := Nle. +Definition NZmin := Nmin. +Definition NZmax := Nmax. + +Add Morphism NZlt with signature NZeq ==> NZeq ==> iff as NZlt_wd. +Proof. +unfold NZeq; intros x1 x2 H1 y1 y2 H2; rewrite H1; now rewrite H2. +Qed. + +Add Morphism NZle with signature NZeq ==> NZeq ==> iff as NZle_wd. +Proof. +unfold NZeq; intros x1 x2 H1 y1 y2 H2; rewrite H1; now rewrite H2. +Qed. + +Add Morphism NZmin with signature NZeq ==> NZeq ==> NZeq as NZmin_wd. +Proof. +congruence. +Qed. + +Add Morphism NZmax with signature NZeq ==> NZeq ==> NZeq as NZmax_wd. +Proof. +congruence. +Qed. + +Theorem NZlt_eq_cases : forall n m : N, n <= m <-> n < m \/ n = m. +Proof. +intros n m. unfold Nle, Nlt. rewrite <- Ncompare_eq_correct. +destruct (n ?= m); split; intro H1; (try discriminate); try (now left); try now right. +now elim H1. destruct H1; discriminate. +Qed. + +Theorem NZlt_irrefl : forall n : NZ, ~ n < n. +Proof. +intro n; unfold Nlt; now rewrite Ncompare_refl. +Qed. + +Theorem NZlt_succ_r : forall n m : NZ, n < (NZsucc m) <-> n <= m. +Proof. +intros n m; unfold Nlt, Nle; destruct n as [| p]; destruct m as [| q]; simpl; +split; intro H; try reflexivity; try discriminate. +destruct p; simpl; intros; discriminate. elimtype False; now apply H. +apply -> Pcompare_p_Sq in H. destruct H as [H | H]. +now rewrite H. now rewrite H, Pcompare_refl. +apply <- Pcompare_p_Sq. case_eq ((p ?= q)%positive Eq); intro H1. +right; now apply Pcompare_Eq_eq. now left. elimtype False; now apply H. +Qed. + +Theorem NZmin_l : forall n m : N, n <= m -> NZmin n m = n. +Proof. +unfold NZmin, Nmin, Nle; intros n m H. +destruct (n ?= m); try reflexivity. now elim H. +Qed. + +Theorem NZmin_r : forall n m : N, m <= n -> NZmin n m = m. +Proof. +unfold NZmin, Nmin, Nle; intros n m H. +case_eq (n ?= m); intro H1; try reflexivity. +now apply -> Ncompare_eq_correct. +rewrite <- Ncompare_antisym, H1 in H; elim H; auto. +Qed. + +Theorem NZmax_l : forall n m : N, m <= n -> NZmax n m = n. +Proof. +unfold NZmax, Nmax, Nle; intros n m H. +case_eq (n ?= m); intro H1; try reflexivity. +symmetry; now apply -> Ncompare_eq_correct. +rewrite <- Ncompare_antisym, H1 in H; elim H; auto. +Qed. + +Theorem NZmax_r : forall n m : N, n <= m -> NZmax n m = m. +Proof. +unfold NZmax, Nmax, Nle; intros n m H. +destruct (n ?= m); try reflexivity. now elim H. +Qed. + +End NZOrdAxiomsMod. + +Definition recursion (A : Type) (a : A) (f : N -> A -> A) (n : N) := + Nrect (fun _ => A) a f n. +Implicit Arguments recursion [A]. + +Theorem pred_0 : Npred N0 = N0. +Proof. +reflexivity. +Qed. + +Theorem recursion_wd : +forall (A : Type) (Aeq : relation A), + forall a a' : A, Aeq a a' -> + forall f f' : N -> A -> A, fun2_eq NZeq Aeq Aeq f f' -> + forall x x' : N, x = x' -> + Aeq (recursion a f x) (recursion a' f' x'). +Proof. +unfold fun2_wd, NZeq, fun2_eq. +intros A Aeq a a' Eaa' f f' Eff'. +intro x; pattern x; apply Nrect. +intros x' H; now rewrite <- H. +clear x. +intros x IH x' H; rewrite <- H. +unfold recursion in *. do 2 rewrite Nrect_step. +now apply Eff'; [| apply IH]. +Qed. + +Theorem recursion_0 : + forall (A : Type) (a : A) (f : N -> A -> A), recursion a f N0 = a. +Proof. +intros A a f; unfold recursion; now rewrite Nrect_base. +Qed. + +Theorem recursion_succ : + forall (A : Type) (Aeq : relation A) (a : A) (f : N -> A -> A), + Aeq a a -> fun2_wd NZeq Aeq Aeq f -> + forall n : N, Aeq (recursion a f (Nsucc n)) (f n (recursion a f n)). +Proof. +unfold NZeq, recursion, fun2_wd; intros A Aeq a f EAaa f_wd n; pattern n; apply Nrect. +rewrite Nrect_step; rewrite Nrect_base; now apply f_wd. +clear n; intro n; do 2 rewrite Nrect_step; intro IH. apply f_wd; [reflexivity|]. +now rewrite Nrect_step. +Qed. + +End NBinaryAxiomsMod. + +Module Export NBinarySubPropMod := NSubPropFunct NBinaryAxiomsMod. + +(* Some fun comparing the efficiency of the generic log defined +by strong (course-of-value) recursion and the log defined by recursion +on notation *) +(* Time Eval compute in (log 100000). *) (* 98 sec *) + +(* +Fixpoint binposlog (p : positive) : N := +match p with +| xH => 0 +| xO p' => Nsucc (binposlog p') +| xI p' => Nsucc (binposlog p') +end. + +Definition binlog (n : N) : N := +match n with +| 0 => 0 +| Npos p => binposlog p +end. +*) +(* Eval compute in (binlog 1000000000000000000). *) (* Works very fast *) + diff --git a/theories/Numbers/Natural/Binary/NBinary.v b/theories/Numbers/Natural/Binary/NBinary.v new file mode 100644 index 00000000..2c99128d --- /dev/null +++ b/theories/Numbers/Natural/Binary/NBinary.v @@ -0,0 +1,15 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Evgeny Makarov, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: NBinary.v 10934 2008-05-15 21:58:20Z letouzey $ i*) + +Require Export NBinDefs. +Require Export NArithRing. + |