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Diffstat (limited to 'theories/NArith/Pnat.v')
-rw-r--r-- | theories/NArith/Pnat.v | 485 |
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diff --git a/theories/NArith/Pnat.v b/theories/NArith/Pnat.v new file mode 100644 index 00000000..f5bbb1c9 --- /dev/null +++ b/theories/NArith/Pnat.v @@ -0,0 +1,485 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(*i $Id: Pnat.v,v 1.3.2.1 2004/07/16 19:31:07 herbelin Exp $ i*) + +Require Import BinPos. + +(**********************************************************************) +(** Properties of the injection from binary positive numbers to Peano + natural numbers *) + +(** Original development by Pierre Crégut, CNET, Lannion, France *) + +Require Import Le. +Require Import Lt. +Require Import Gt. +Require Import Plus. +Require Import Mult. +Require Import Minus. + +(** [nat_of_P] is a morphism for addition *) + +Lemma Pmult_nat_succ_morphism : + forall (p:positive) (n:nat), Pmult_nat (Psucc p) n = n + Pmult_nat p n. +Proof. +intro x; induction x as [p IHp| p IHp| ]; simpl in |- *; auto; intro m; + rewrite IHp; rewrite plus_assoc; trivial. +Qed. + +Lemma nat_of_P_succ_morphism : + forall p:positive, nat_of_P (Psucc p) = S (nat_of_P p). +Proof. + intro; change (S (nat_of_P p)) with (1 + nat_of_P p) in |- *; + unfold nat_of_P in |- *; apply Pmult_nat_succ_morphism. +Qed. + +Theorem Pmult_nat_plus_carry_morphism : + forall (p q:positive) (n:nat), + Pmult_nat (Pplus_carry p q) n = n + Pmult_nat (p + q) n. +Proof. +intro x; induction x as [p IHp| p IHp| ]; intro y; + [ destruct y as [p0| p0| ] + | destruct y as [p0| p0| ] + | destruct y as [p| p| ] ]; simpl in |- *; auto with arith; + intro m; + [ rewrite IHp; rewrite plus_assoc; trivial with arith + | rewrite IHp; rewrite plus_assoc; trivial with arith + | rewrite Pmult_nat_succ_morphism; rewrite plus_assoc; trivial with arith + | rewrite Pmult_nat_succ_morphism; apply plus_assoc_reverse ]. +Qed. + +Theorem nat_of_P_plus_carry_morphism : + forall p q:positive, nat_of_P (Pplus_carry p q) = S (nat_of_P (p + q)). +Proof. +intros; unfold nat_of_P in |- *; rewrite Pmult_nat_plus_carry_morphism; + simpl in |- *; trivial with arith. +Qed. + +Theorem Pmult_nat_l_plus_morphism : + forall (p q:positive) (n:nat), + Pmult_nat (p + q) n = Pmult_nat p n + Pmult_nat q n. +Proof. +intro x; induction x as [p IHp| p IHp| ]; intro y; + [ destruct y as [p0| p0| ] + | destruct y as [p0| p0| ] + | destruct y as [p| p| ] ]; simpl in |- *; auto with arith; + [ intros m; rewrite Pmult_nat_plus_carry_morphism; rewrite IHp; + rewrite plus_assoc_reverse; rewrite plus_assoc_reverse; + rewrite (plus_permute m (Pmult_nat p (m + m))); + trivial with arith + | intros m; rewrite IHp; apply plus_assoc + | intros m; rewrite Pmult_nat_succ_morphism; + rewrite (plus_comm (m + Pmult_nat p (m + m))); + apply plus_assoc_reverse + | intros m; rewrite IHp; apply plus_permute + | intros m; rewrite Pmult_nat_succ_morphism; apply plus_assoc_reverse ]. +Qed. + +Theorem nat_of_P_plus_morphism : + forall p q:positive, nat_of_P (p + q) = nat_of_P p + nat_of_P q. +Proof. +intros x y; exact (Pmult_nat_l_plus_morphism x y 1). +Qed. + +(** [Pmult_nat] is a morphism for addition *) + +Lemma Pmult_nat_r_plus_morphism : + forall (p:positive) (n:nat), + Pmult_nat p (n + n) = Pmult_nat p n + Pmult_nat p n. +Proof. +intro y; induction y as [p H| p H| ]; intro m; + [ simpl in |- *; rewrite H; rewrite plus_assoc_reverse; + rewrite (plus_permute m (Pmult_nat p (m + m))); + rewrite plus_assoc_reverse; auto with arith + | simpl in |- *; rewrite H; auto with arith + | simpl in |- *; trivial with arith ]. +Qed. + +Lemma ZL6 : forall p:positive, Pmult_nat p 2 = nat_of_P p + nat_of_P p. +Proof. +intro p; change 2 with (1 + 1) in |- *; rewrite Pmult_nat_r_plus_morphism; + trivial. +Qed. + +(** [nat_of_P] is a morphism for multiplication *) + +Theorem nat_of_P_mult_morphism : + forall p q:positive, nat_of_P (p * q) = nat_of_P p * nat_of_P q. +Proof. +intros x y; induction x as [x' H| x' H| ]; + [ change (xI x' * y)%positive with (y + xO (x' * y))%positive in |- *; + rewrite nat_of_P_plus_morphism; unfold nat_of_P at 2 3 in |- *; + simpl in |- *; do 2 rewrite ZL6; rewrite H; rewrite mult_plus_distr_r; + reflexivity + | unfold nat_of_P at 1 2 in |- *; simpl in |- *; do 2 rewrite ZL6; rewrite H; + rewrite mult_plus_distr_r; reflexivity + | simpl in |- *; rewrite <- plus_n_O; reflexivity ]. +Qed. + +(** [nat_of_P] maps to the strictly positive subset of [nat] *) + +Lemma ZL4 : forall p:positive, exists h : nat, nat_of_P p = S h. +Proof. +intro y; induction y as [p H| p H| ]; + [ destruct H as [x H1]; exists (S x + S x); unfold nat_of_P in |- *; + simpl in |- *; change 2 with (1 + 1) in |- *; + rewrite Pmult_nat_r_plus_morphism; unfold nat_of_P in H1; + rewrite H1; auto with arith + | destruct H as [x H2]; exists (x + S x); unfold nat_of_P in |- *; + simpl in |- *; change 2 with (1 + 1) in |- *; + rewrite Pmult_nat_r_plus_morphism; unfold nat_of_P in H2; + rewrite H2; auto with arith + | exists 0; auto with arith ]. +Qed. + +(** Extra lemmas on [lt] on Peano natural numbers *) + +Lemma ZL7 : forall n m:nat, n < m -> n + n < m + m. +Proof. +intros m n H; apply lt_trans with (m := m + n); + [ apply plus_lt_compat_l with (1 := H) + | rewrite (plus_comm m n); apply plus_lt_compat_l with (1 := H) ]. +Qed. + +Lemma ZL8 : forall n m:nat, n < m -> S (n + n) < m + m. +Proof. +intros m n H; apply le_lt_trans with (m := m + n); + [ change (m + m < m + n) in |- *; apply plus_lt_compat_l with (1 := H) + | rewrite (plus_comm m n); apply plus_lt_compat_l with (1 := H) ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed + from [compare] on [positive]) + + Part 1: [lt] on [positive] is finer than [lt] on [nat] +*) + +Lemma nat_of_P_lt_Lt_compare_morphism : + forall p q:positive, (p ?= q)%positive Eq = Lt -> nat_of_P p < nat_of_P q. +Proof. +intro x; induction x as [p H| p H| ]; intro y; destruct y as [q| q| ]; + intro H2; + [ unfold nat_of_P in |- *; simpl in |- *; apply lt_n_S; do 2 rewrite ZL6; + apply ZL7; apply H; simpl in H2; assumption + | unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; apply ZL8; + apply H; simpl in H2; apply Pcompare_Gt_Lt; assumption + | simpl in |- *; discriminate H2 + | simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + elim (Pcompare_Lt_Lt p q H2); + [ intros H3; apply lt_S; apply ZL7; apply H; apply H3 + | intros E; rewrite E; apply lt_n_Sn ] + | simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + apply ZL7; apply H; assumption + | simpl in |- *; discriminate H2 + | unfold nat_of_P in |- *; simpl in |- *; apply lt_n_S; rewrite ZL6; + elim (ZL4 q); intros h H3; rewrite H3; simpl in |- *; + apply lt_O_Sn + | unfold nat_of_P in |- *; simpl in |- *; rewrite ZL6; elim (ZL4 q); + intros h H3; rewrite H3; simpl in |- *; rewrite <- plus_n_Sm; + apply lt_n_S; apply lt_O_Sn + | simpl in |- *; discriminate H2 ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed + from [compare] on [positive]) + + Part 1: [gt] on [positive] is finer than [gt] on [nat] +*) + +Lemma nat_of_P_gt_Gt_compare_morphism : + forall p q:positive, (p ?= q)%positive Eq = Gt -> nat_of_P p > nat_of_P q. +Proof. +unfold gt in |- *; intro x; induction x as [p H| p H| ]; intro y; + destruct y as [q| q| ]; intro H2; + [ simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + apply lt_n_S; apply ZL7; apply H; assumption + | simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + elim (Pcompare_Gt_Gt p q H2); + [ intros H3; apply lt_S; apply ZL7; apply H; assumption + | intros E; rewrite E; apply lt_n_Sn ] + | unfold nat_of_P in |- *; simpl in |- *; rewrite ZL6; elim (ZL4 p); + intros h H3; rewrite H3; simpl in |- *; apply lt_n_S; + apply lt_O_Sn + | simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + apply ZL8; apply H; apply Pcompare_Lt_Gt; assumption + | simpl in |- *; unfold nat_of_P in |- *; simpl in |- *; do 2 rewrite ZL6; + apply ZL7; apply H; assumption + | unfold nat_of_P in |- *; simpl in |- *; rewrite ZL6; elim (ZL4 p); + intros h H3; rewrite H3; simpl in |- *; rewrite <- plus_n_Sm; + apply lt_n_S; apply lt_O_Sn + | simpl in |- *; discriminate H2 + | simpl in |- *; discriminate H2 + | simpl in |- *; discriminate H2 ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed + from [compare] on [positive]) + + Part 2: [lt] on [nat] is finer than [lt] on [positive] +*) + +Lemma nat_of_P_lt_Lt_compare_complement_morphism : + forall p q:positive, nat_of_P p < nat_of_P q -> (p ?= q)%positive Eq = Lt. +Proof. +intros x y; unfold gt in |- *; elim (Dcompare ((x ?= y)%positive Eq)); + [ intros E; rewrite (Pcompare_Eq_eq x y E); intros H; + absurd (nat_of_P y < nat_of_P y); [ apply lt_irrefl | assumption ] + | intros H; elim H; + [ auto + | intros H1 H2; absurd (nat_of_P x < nat_of_P y); + [ apply lt_asym; change (nat_of_P x > nat_of_P y) in |- *; + apply nat_of_P_gt_Gt_compare_morphism; assumption + | assumption ] ] ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed + from [compare] on [positive]) + + Part 2: [gt] on [nat] is finer than [gt] on [positive] +*) + +Lemma nat_of_P_gt_Gt_compare_complement_morphism : + forall p q:positive, nat_of_P p > nat_of_P q -> (p ?= q)%positive Eq = Gt. +Proof. +intros x y; unfold gt in |- *; elim (Dcompare ((x ?= y)%positive Eq)); + [ intros E; rewrite (Pcompare_Eq_eq x y E); intros H; + absurd (nat_of_P y < nat_of_P y); [ apply lt_irrefl | assumption ] + | intros H; elim H; + [ intros H1 H2; absurd (nat_of_P y < nat_of_P x); + [ apply lt_asym; apply nat_of_P_lt_Lt_compare_morphism; assumption + | assumption ] + | auto ] ]. +Qed. + +(** [nat_of_P] is strictly positive *) + +Lemma le_Pmult_nat : forall (p:positive) (n:nat), n <= Pmult_nat p n. +induction p; simpl in |- *; auto with arith. +intro m; apply le_trans with (m + m); auto with arith. +Qed. + +Lemma lt_O_nat_of_P : forall p:positive, 0 < nat_of_P p. +intro; unfold nat_of_P in |- *; apply lt_le_trans with 1; auto with arith. +apply le_Pmult_nat. +Qed. + +(** Pmult_nat permutes with multiplication *) + +Lemma Pmult_nat_mult_permute : + forall (p:positive) (n m:nat), Pmult_nat p (m * n) = m * Pmult_nat p n. +Proof. + simple induction p. intros. simpl in |- *. rewrite mult_plus_distr_l. rewrite <- (mult_plus_distr_l m n n). + rewrite (H (n + n) m). reflexivity. + intros. simpl in |- *. rewrite <- (mult_plus_distr_l m n n). apply H. + trivial. +Qed. + +Lemma Pmult_nat_2_mult_2_permute : + forall p:positive, Pmult_nat p 2 = 2 * Pmult_nat p 1. +Proof. + intros. rewrite <- Pmult_nat_mult_permute. reflexivity. +Qed. + +Lemma Pmult_nat_4_mult_2_permute : + forall p:positive, Pmult_nat p 4 = 2 * Pmult_nat p 2. +Proof. + intros. rewrite <- Pmult_nat_mult_permute. reflexivity. +Qed. + +(** Mapping of xH, xO and xI through [nat_of_P] *) + +Lemma nat_of_P_xH : nat_of_P 1 = 1. +Proof. + reflexivity. +Qed. + +Lemma nat_of_P_xO : forall p:positive, nat_of_P (xO p) = 2 * nat_of_P p. +Proof. + simple induction p. unfold nat_of_P in |- *. simpl in |- *. intros. rewrite Pmult_nat_2_mult_2_permute. + rewrite Pmult_nat_4_mult_2_permute. rewrite H. simpl in |- *. rewrite <- plus_Snm_nSm. reflexivity. + unfold nat_of_P in |- *. simpl in |- *. intros. rewrite Pmult_nat_2_mult_2_permute. rewrite Pmult_nat_4_mult_2_permute. + rewrite H. reflexivity. + reflexivity. +Qed. + +Lemma nat_of_P_xI : forall p:positive, nat_of_P (xI p) = S (2 * nat_of_P p). +Proof. + simple induction p. unfold nat_of_P in |- *. simpl in |- *. intro p0. intro. rewrite Pmult_nat_2_mult_2_permute. + rewrite Pmult_nat_4_mult_2_permute; injection H; intro H1; rewrite H1; + rewrite <- plus_Snm_nSm; reflexivity. + unfold nat_of_P in |- *. simpl in |- *. intros. rewrite Pmult_nat_2_mult_2_permute. rewrite Pmult_nat_4_mult_2_permute. + injection H; intro H1; rewrite H1; reflexivity. + reflexivity. +Qed. + +(**********************************************************************) +(** Properties of the shifted injection from Peano natural numbers to + binary positive numbers *) + +(** Composition of [P_of_succ_nat] and [nat_of_P] is successor on [nat] *) + +Theorem nat_of_P_o_P_of_succ_nat_eq_succ : + forall n:nat, nat_of_P (P_of_succ_nat n) = S n. +Proof. +intro m; induction m as [| n H]; + [ reflexivity + | simpl in |- *; rewrite nat_of_P_succ_morphism; rewrite H; auto ]. +Qed. + +(** Miscellaneous lemmas on [P_of_succ_nat] *) + +Lemma ZL3 : + forall n:nat, Psucc (P_of_succ_nat (n + n)) = xO (P_of_succ_nat n). +Proof. +intro x; induction x as [| n H]; + [ simpl in |- *; auto with arith + | simpl in |- *; rewrite plus_comm; simpl in |- *; rewrite H; + rewrite xO_succ_permute; auto with arith ]. +Qed. + +Lemma ZL5 : forall n:nat, P_of_succ_nat (S n + S n) = xI (P_of_succ_nat n). +Proof. +intro x; induction x as [| n H]; simpl in |- *; + [ auto with arith + | rewrite <- plus_n_Sm; simpl in |- *; simpl in H; rewrite H; + auto with arith ]. +Qed. + +(** Composition of [nat_of_P] and [P_of_succ_nat] is successor on [positive] *) + +Theorem P_of_succ_nat_o_nat_of_P_eq_succ : + forall p:positive, P_of_succ_nat (nat_of_P p) = Psucc p. +Proof. +intro x; induction x as [p H| p H| ]; + [ simpl in |- *; rewrite <- H; change 2 with (1 + 1) in |- *; + rewrite Pmult_nat_r_plus_morphism; elim (ZL4 p); + unfold nat_of_P in |- *; intros n H1; rewrite H1; + rewrite ZL3; auto with arith + | unfold nat_of_P in |- *; simpl in |- *; change 2 with (1 + 1) in |- *; + rewrite Pmult_nat_r_plus_morphism; + rewrite <- (Ppred_succ (P_of_succ_nat (Pmult_nat p 1 + Pmult_nat p 1))); + rewrite <- (Ppred_succ (xI p)); simpl in |- *; + rewrite <- H; elim (ZL4 p); unfold nat_of_P in |- *; + intros n H1; rewrite H1; rewrite ZL5; simpl in |- *; + trivial with arith + | unfold nat_of_P in |- *; simpl in |- *; auto with arith ]. +Qed. + +(** Composition of [nat_of_P], [P_of_succ_nat] and [Ppred] is identity + on [positive] *) + +Theorem pred_o_P_of_succ_nat_o_nat_of_P_eq_id : + forall p:positive, Ppred (P_of_succ_nat (nat_of_P p)) = p. +Proof. +intros x; rewrite P_of_succ_nat_o_nat_of_P_eq_succ; rewrite Ppred_succ; + trivial with arith. +Qed. + +(**********************************************************************) +(** Extra properties of the injection from binary positive numbers to Peano + natural numbers *) + +(** [nat_of_P] is a morphism for subtraction on positive numbers *) + +Theorem nat_of_P_minus_morphism : + forall p q:positive, + (p ?= q)%positive Eq = Gt -> nat_of_P (p - q) = nat_of_P p - nat_of_P q. +Proof. +intros x y H; apply plus_reg_l with (nat_of_P y); rewrite le_plus_minus_r; + [ rewrite <- nat_of_P_plus_morphism; rewrite Pplus_minus; auto with arith + | apply lt_le_weak; exact (nat_of_P_gt_Gt_compare_morphism x y H) ]. +Qed. + +(** [nat_of_P] is injective *) + +Lemma nat_of_P_inj : forall p q:positive, nat_of_P p = nat_of_P q -> p = q. +Proof. +intros x y H; rewrite <- (pred_o_P_of_succ_nat_o_nat_of_P_eq_id x); + rewrite <- (pred_o_P_of_succ_nat_o_nat_of_P_eq_id y); + rewrite H; trivial with arith. +Qed. + +Lemma ZL16 : forall p q:positive, nat_of_P p - nat_of_P q < nat_of_P p. +Proof. +intros p q; elim (ZL4 p); elim (ZL4 q); intros h H1 i H2; rewrite H1; + rewrite H2; simpl in |- *; unfold lt in |- *; apply le_n_S; + apply le_minus. +Qed. + +Lemma ZL17 : forall p q:positive, nat_of_P p < nat_of_P (p + q). +Proof. +intros p q; rewrite nat_of_P_plus_morphism; unfold lt in |- *; elim (ZL4 q); + intros k H; rewrite H; rewrite plus_comm; simpl in |- *; + apply le_n_S; apply le_plus_r. +Qed. + +(** Comparison and subtraction *) + +Lemma Pcompare_minus_r : + forall p q r:positive, + (q ?= p)%positive Eq = Lt -> + (r ?= p)%positive Eq = Gt -> + (r ?= q)%positive Eq = Gt -> (r - p ?= r - q)%positive Eq = Lt. +Proof. +intros; apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ apply plus_lt_reg_l with (p := nat_of_P q); rewrite le_plus_minus_r; + [ rewrite plus_comm; apply plus_lt_reg_l with (p := nat_of_P p); + rewrite plus_assoc; rewrite le_plus_minus_r; + [ rewrite (plus_comm (nat_of_P p)); apply plus_lt_compat_l; + apply nat_of_P_lt_Lt_compare_morphism; + assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; + assumption ] + | assumption ] + | assumption ]. +Qed. + +Lemma Pcompare_minus_l : + forall p q r:positive, + (q ?= p)%positive Eq = Lt -> + (p ?= r)%positive Eq = Gt -> + (q ?= r)%positive Eq = Gt -> (q - r ?= p - r)%positive Eq = Lt. +Proof. +intros p q z; intros; apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ unfold gt in |- *; apply plus_lt_reg_l with (p := nat_of_P z); + rewrite le_plus_minus_r; + [ rewrite le_plus_minus_r; + [ apply nat_of_P_lt_Lt_compare_morphism; assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; + assumption ] + | assumption ] + | assumption ]. +Qed. + +(** Distributivity of multiplication over subtraction *) + +Theorem Pmult_minus_distr_l : + forall p q r:positive, + (q ?= r)%positive Eq = Gt -> + (p * (q - r))%positive = (p * q - p * r)%positive. +Proof. +intros x y z H; apply nat_of_P_inj; rewrite nat_of_P_mult_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ do 2 rewrite nat_of_P_mult_morphism; + do 3 rewrite (mult_comm (nat_of_P x)); apply mult_minus_distr_r + | apply nat_of_P_gt_Gt_compare_complement_morphism; + do 2 rewrite nat_of_P_mult_morphism; unfold gt in |- *; + elim (ZL4 x); intros h H1; rewrite H1; apply mult_S_lt_compat_l; + exact (nat_of_P_gt_Gt_compare_morphism y z H) ] + | assumption ]. +Qed. |