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.