diff options
Diffstat (limited to 'theories/PArith')
| -rw-r--r-- | theories/PArith/BinPos.v | 2132 | ||||
| -rw-r--r-- | theories/PArith/BinPosDef.v | 565 | ||||
| -rw-r--r-- | theories/PArith/PArith.v | 11 | ||||
| -rw-r--r-- | theories/PArith/POrderedType.v | 36 | ||||
| -rw-r--r-- | theories/PArith/Pnat.v | 483 | ||||
| -rw-r--r-- | theories/PArith/intro.tex | 4 | ||||
| -rw-r--r-- | theories/PArith/vo.itarget | 5 |
7 files changed, 3236 insertions, 0 deletions
diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v new file mode 100644 index 00000000..2e4d52a2 --- /dev/null +++ b/theories/PArith/BinPos.v @@ -0,0 +1,2132 @@ +(* -*- coding: utf-8 -*- *) +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Export BinNums. +Require Import Eqdep_dec EqdepFacts RelationClasses Morphisms Setoid + Equalities Orders OrdersFacts GenericMinMax Le Plus. + +Require Export BinPosDef. + +(**********************************************************************) +(** * Binary positive numbers, operations and properties *) +(**********************************************************************) + +(** Initial development by Pierre Crégut, CNET, Lannion, France *) + +(** The type [positive] and its constructors [xI] and [xO] and [xH] + are now defined in [BinNums.v] *) + +Local Open Scope positive_scope. +Local Unset Boolean Equality Schemes. +Local Unset Case Analysis Schemes. + +(** Every definitions and early properties about positive numbers + are placed in a module [Pos] for qualification purpose. *) + +Module Pos + <: UsualOrderedTypeFull + <: UsualDecidableTypeFull + <: TotalOrder. + +(** * Definitions of operations, now in a separate file *) + +Include BinPosDef.Pos. + +(** In functor applications that follow, we only inline t and eq *) + +Set Inline Level 30. + +(** * Logical Predicates *) + +Definition eq := @Logic.eq positive. +Definition eq_equiv := @eq_equivalence positive. +Include BackportEq. + +Definition lt x y := (x ?= y) = Lt. +Definition gt x y := (x ?= y) = Gt. +Definition le x y := (x ?= y) <> Gt. +Definition ge x y := (x ?= y) <> Lt. + +Infix "<=" := le : positive_scope. +Infix "<" := lt : positive_scope. +Infix ">=" := ge : positive_scope. +Infix ">" := gt : positive_scope. + +Notation "x <= y <= z" := (x <= y /\ y <= z) : positive_scope. +Notation "x <= y < z" := (x <= y /\ y < z) : positive_scope. +Notation "x < y < z" := (x < y /\ y < z) : positive_scope. +Notation "x < y <= z" := (x < y /\ y <= z) : positive_scope. + +(**********************************************************************) +(** * Properties of operations over positive numbers *) + +(** ** Decidability of equality on binary positive numbers *) + +Lemma eq_dec : forall x y:positive, {x = y} + {x <> y}. +Proof. + decide equality. +Defined. + +(**********************************************************************) +(** * Properties of successor on binary positive numbers *) + +(** ** Specification of [xI] in term of [succ] and [xO] *) + +Lemma xI_succ_xO p : p~1 = succ p~0. +Proof. + reflexivity. +Qed. + +Lemma succ_discr p : p <> succ p. +Proof. + now destruct p. +Qed. + +(** ** Successor and double *) + +Lemma pred_double_spec p : pred_double p = pred (p~0). +Proof. + reflexivity. +Qed. + +Lemma succ_pred_double p : succ (pred_double p) = p~0. +Proof. + induction p; simpl; now f_equal. +Qed. + +Lemma pred_double_succ p : pred_double (succ p) = p~1. +Proof. + induction p; simpl; now f_equal. +Qed. + +Lemma double_succ p : (succ p)~0 = succ (succ p~0). +Proof. + now destruct p. +Qed. + +Lemma pred_double_xO_discr p : pred_double p <> p~0. +Proof. + now destruct p. +Qed. + +(** ** Successor and predecessor *) + +Lemma succ_not_1 p : succ p <> 1. +Proof. + now destruct p. +Qed. + +Lemma pred_succ p : pred (succ p) = p. +Proof. + destruct p; simpl; trivial. apply pred_double_succ. +Qed. + +Lemma succ_pred_or p : p = 1 \/ succ (pred p) = p. +Proof. + destruct p; simpl; auto. + right; apply succ_pred_double. +Qed. + +Lemma succ_pred p : p <> 1 -> succ (pred p) = p. +Proof. + destruct p; intros H; simpl; trivial. + apply succ_pred_double. + now destruct H. +Qed. + +(** ** Injectivity of successor *) + +Lemma succ_inj p q : succ p = succ q -> p = q. +Proof. + revert q. + induction p; intros [q|q| ] H; simpl in H; destr_eq H; f_equal; auto. + elim (succ_not_1 p); auto. + elim (succ_not_1 q); auto. +Qed. + +(** ** Predecessor to [N] *) + +Lemma pred_N_succ p : pred_N (succ p) = Npos p. +Proof. + destruct p; simpl; trivial. f_equal. apply pred_double_succ. +Qed. + + +(**********************************************************************) +(** * Properties of addition on binary positive numbers *) + +(** ** Specification of [succ] in term of [add] *) + +Lemma add_1_r p : p + 1 = succ p. +Proof. + now destruct p. +Qed. + +Lemma add_1_l p : 1 + p = succ p. +Proof. + now destruct p. +Qed. + +(** ** Specification of [add_carry] *) + +Theorem add_carry_spec p q : add_carry p q = succ (p + q). +Proof. + revert q. induction p; destruct q; simpl; now f_equal. +Qed. + +(** ** Commutativity *) + +Theorem add_comm p q : p + q = q + p. +Proof. + revert q. induction p; destruct q; simpl; f_equal; trivial. + rewrite 2 add_carry_spec; now f_equal. +Qed. + +(** ** Permutation of [add] and [succ] *) + +Theorem add_succ_r p q : p + succ q = succ (p + q). +Proof. + revert q. + induction p; destruct q; simpl; f_equal; + auto using add_1_r; rewrite add_carry_spec; auto. +Qed. + +Theorem add_succ_l p q : succ p + q = succ (p + q). +Proof. + rewrite add_comm, (add_comm p). apply add_succ_r. +Qed. + +(** ** No neutral elements for addition *) + +Lemma add_no_neutral p q : q + p <> p. +Proof. + revert q. + induction p as [p IHp|p IHp| ]; intros [q|q| ] H; + destr_eq H; apply (IHp q H). +Qed. + +(** ** Simplification *) + +Lemma add_carry_add p q r s : + add_carry p r = add_carry q s -> p + r = q + s. +Proof. + intros H; apply succ_inj; now rewrite <- 2 add_carry_spec. +Qed. + +Lemma add_reg_r p q r : p + r = q + r -> p = q. +Proof. + revert p q. induction r. + intros [p|p| ] [q|q| ] H; simpl; destr_eq H; f_equal; + auto using add_carry_add; contradict H; + rewrite add_carry_spec, <- add_succ_r; auto using add_no_neutral. + intros [p|p| ] [q|q| ] H; simpl; destr_eq H; f_equal; auto; + contradict H; auto using add_no_neutral. + intros p q H. apply succ_inj. now rewrite <- 2 add_1_r. +Qed. + +Lemma add_reg_l p q r : p + q = p + r -> q = r. +Proof. + rewrite 2 (add_comm p). now apply add_reg_r. +Qed. + +Lemma add_cancel_r p q r : p + r = q + r <-> p = q. +Proof. + split. apply add_reg_r. congruence. +Qed. + +Lemma add_cancel_l p q r : r + p = r + q <-> p = q. +Proof. + split. apply add_reg_l. congruence. +Qed. + +Lemma add_carry_reg_r p q r : + add_carry p r = add_carry q r -> p = q. +Proof. + intros H. apply add_reg_r with (r:=r); now apply add_carry_add. +Qed. + +Lemma add_carry_reg_l p q r : + add_carry p q = add_carry p r -> q = r. +Proof. + intros H; apply add_reg_r with (r:=p); + rewrite (add_comm r), (add_comm q); now apply add_carry_add. +Qed. + +(** ** Addition is associative *) + +Theorem add_assoc p q r : p + (q + r) = p + q + r. +Proof. + revert q r. induction p. + intros [q|q| ] [r|r| ]; simpl; f_equal; trivial; + rewrite ?add_carry_spec, ?add_succ_r, ?add_succ_l, ?add_1_r; + f_equal; trivial. + intros [q|q| ] [r|r| ]; simpl; f_equal; trivial; + rewrite ?add_carry_spec, ?add_succ_r, ?add_succ_l, ?add_1_r; + f_equal; trivial. + intros q r; rewrite 2 add_1_l, add_succ_l; auto. +Qed. + +(** ** Commutation of addition and double *) + +Lemma add_xO p q : (p + q)~0 = p~0 + q~0. +Proof. + now destruct p, q. +Qed. + +Lemma add_xI_pred_double p q : + (p + q)~0 = p~1 + pred_double q. +Proof. + change (p~1) with (p~0 + 1). + now rewrite <- add_assoc, add_1_l, succ_pred_double. +Qed. + +Lemma add_xO_pred_double p q : + pred_double (p + q) = p~0 + pred_double q. +Proof. + revert q. induction p as [p IHp| p IHp| ]; destruct q; simpl; + rewrite ?add_carry_spec, ?pred_double_succ, ?add_xI_pred_double; + try reflexivity. + rewrite IHp; auto. + rewrite <- succ_pred_double, <- add_1_l. reflexivity. +Qed. + +(** ** Miscellaneous *) + +Lemma add_diag p : p + p = p~0. +Proof. + induction p as [p IHp| p IHp| ]; simpl; + now rewrite ?add_carry_spec, ?IHp. +Qed. + +(**********************************************************************) +(** * Peano induction and recursion on binary positive positive numbers *) + +(** The Peano-like recursor function for [positive] (due to Daniel Schepler) *) + +Fixpoint peano_rect (P:positive->Type) (a:P 1) + (f: forall p:positive, P p -> P (succ p)) (p:positive) : P p := +let f2 := peano_rect (fun p:positive => P (p~0)) (f _ a) + (fun (p:positive) (x:P (p~0)) => f _ (f _ x)) +in +match p with + | q~1 => f _ (f2 q) + | q~0 => f2 q + | 1 => a +end. + +Theorem peano_rect_succ (P:positive->Type) (a:P 1) + (f:forall p, P p -> P (succ p)) (p:positive) : + peano_rect P a f (succ p) = f _ (peano_rect P a f p). +Proof. + revert P a f. induction p; trivial. + intros. simpl. now rewrite IHp. +Qed. + +Theorem peano_rect_base (P:positive->Type) (a:P 1) + (f:forall p, P p -> P (succ p)) : + peano_rect P a f 1 = a. +Proof. + trivial. +Qed. + +Definition peano_rec (P:positive->Set) := peano_rect P. + +(** Peano induction *) + +Definition peano_ind (P:positive->Prop) := peano_rect P. + +(** Peano case analysis *) + +Theorem peano_case : + forall P:positive -> Prop, + P 1 -> (forall n:positive, P (succ n)) -> forall p:positive, P p. +Proof. + intros; apply peano_ind; auto. +Qed. + +(** Earlier, the Peano-like recursor was built and proved in a way due to + Conor McBride, see "The view from the left" *) + +Inductive PeanoView : positive -> Type := +| PeanoOne : PeanoView 1 +| PeanoSucc : forall p, PeanoView p -> PeanoView (succ p). + +Fixpoint peanoView_xO p (q:PeanoView p) : PeanoView (p~0) := + match q in PeanoView x return PeanoView (x~0) with + | PeanoOne => PeanoSucc _ PeanoOne + | PeanoSucc _ q => PeanoSucc _ (PeanoSucc _ (peanoView_xO _ q)) + end. + +Fixpoint peanoView_xI p (q:PeanoView p) : PeanoView (p~1) := + match q in PeanoView x return PeanoView (x~1) with + | PeanoOne => PeanoSucc _ (PeanoSucc _ PeanoOne) + | PeanoSucc _ q => PeanoSucc _ (PeanoSucc _ (peanoView_xI _ q)) + end. + +Fixpoint peanoView p : PeanoView p := + match p return PeanoView p with + | 1 => PeanoOne + | p~0 => peanoView_xO p (peanoView p) + | p~1 => peanoView_xI p (peanoView p) + end. + +Definition PeanoView_iter (P:positive->Type) + (a:P 1) (f:forall p, P p -> P (succ p)) := + (fix iter p (q:PeanoView p) : P p := + match q in PeanoView p return P p with + | PeanoOne => a + | PeanoSucc _ q => f _ (iter _ q) + end). + +Theorem eq_dep_eq_positive : + forall (P:positive->Type) (p:positive) (x y:P p), + eq_dep positive P p x p y -> x = y. +Proof. + apply eq_dep_eq_dec. + decide equality. +Qed. + +Theorem PeanoViewUnique : forall p (q q':PeanoView p), q = q'. +Proof. + intros. + induction q as [ | p q IHq ]. + apply eq_dep_eq_positive. + cut (1=1). pattern 1 at 1 2 5, q'. destruct q'. trivial. + destruct p; intros; discriminate. + trivial. + apply eq_dep_eq_positive. + cut (succ p=succ p). pattern (succ p) at 1 2 5, q'. destruct q'. + intro. destruct p; discriminate. + intro. apply succ_inj in H. + generalize q'. rewrite H. intro. + rewrite (IHq q'0). + trivial. + trivial. +Qed. + +Lemma peano_equiv (P:positive->Type) (a:P 1) (f:forall p, P p -> P (succ p)) p : + PeanoView_iter P a f p (peanoView p) = peano_rect P a f p. +Proof. + revert P a f. induction p using peano_rect. + trivial. + intros; simpl. rewrite peano_rect_succ. + rewrite (PeanoViewUnique _ (peanoView (succ p)) (PeanoSucc _ (peanoView p))). + simpl; now f_equal. +Qed. + +(**********************************************************************) +(** * Properties of multiplication on binary positive numbers *) + +(** ** One is neutral for multiplication *) + +Lemma mul_1_l p : 1 * p = p. +Proof. + reflexivity. +Qed. + +Lemma mul_1_r p : p * 1 = p. +Proof. + induction p; simpl; now f_equal. +Qed. + +(** ** Right reduction properties for multiplication *) + +Lemma mul_xO_r p q : p * q~0 = (p * q)~0. +Proof. + induction p; simpl; f_equal; f_equal; trivial. +Qed. + +Lemma mul_xI_r p q : p * q~1 = p + (p * q)~0. +Proof. + induction p as [p IHp|p IHp| ]; simpl; f_equal; trivial. + now rewrite IHp, 2 add_assoc, (add_comm p). +Qed. + +(** ** Commutativity of multiplication *) + +Theorem mul_comm p q : p * q = q * p. +Proof. + induction q as [q IHq|q IHq| ]; simpl; rewrite <- ? IHq; + auto using mul_xI_r, mul_xO_r, mul_1_r. +Qed. + +(** ** Distributivity of multiplication over addition *) + +Theorem mul_add_distr_l p q r : + p * (q + r) = p * q + p * r. +Proof. + induction p as [p IHp|p IHp| ]; simpl. + rewrite IHp. set (m:=(p*q)~0). set (n:=(p*r)~0). + change ((p*q+p*r)~0) with (m+n). + rewrite 2 add_assoc; f_equal. + rewrite <- 2 add_assoc; f_equal. + apply add_comm. + f_equal; auto. + reflexivity. +Qed. + +Theorem mul_add_distr_r p q r : + (p + q) * r = p * r + q * r. +Proof. + rewrite 3 (mul_comm _ r); apply mul_add_distr_l. +Qed. + +(** ** Associativity of multiplication *) + +Theorem mul_assoc p q r : p * (q * r) = p * q * r. +Proof. + induction p as [p IHp| p IHp | ]; simpl; rewrite ?IHp; trivial. + now rewrite mul_add_distr_r. +Qed. + +(** ** Successor and multiplication *) + +Lemma mul_succ_l p q : (succ p) * q = q + p * q. +Proof. + induction p as [p IHp | p IHp | ]; simpl; trivial. + now rewrite IHp, add_assoc, add_diag, <-add_xO. + symmetry; apply add_diag. +Qed. + +Lemma mul_succ_r p q : p * (succ q) = p + p * q. +Proof. + rewrite mul_comm, mul_succ_l. f_equal. apply mul_comm. +Qed. + +(** ** Parity properties of multiplication *) + +Lemma mul_xI_mul_xO_discr p q r : p~1 * r <> q~0 * r. +Proof. + induction r; try discriminate. + rewrite 2 mul_xO_r; intro H; destr_eq H; auto. +Qed. + +Lemma mul_xO_discr p q : p~0 * q <> q. +Proof. + induction q; try discriminate. + rewrite mul_xO_r; injection; assumption. +Qed. + +(** ** Simplification properties of multiplication *) + +Theorem mul_reg_r p q r : p * r = q * r -> p = q. +Proof. + revert q r. + induction p as [p IHp| p IHp| ]; intros [q|q| ] r H; + reflexivity || apply f_equal || exfalso. + apply IHp with (r~0). simpl in *. + rewrite 2 mul_xO_r. apply add_reg_l with (1:=H). + contradict H. apply mul_xI_mul_xO_discr. + contradict H. simpl. rewrite add_comm. apply add_no_neutral. + symmetry in H. contradict H. apply mul_xI_mul_xO_discr. + apply IHp with (r~0). simpl. now rewrite 2 mul_xO_r. + contradict H. apply mul_xO_discr. + symmetry in H. contradict H. simpl. rewrite add_comm. + apply add_no_neutral. + symmetry in H. contradict H. apply mul_xO_discr. +Qed. + +Theorem mul_reg_l p q r : r * p = r * q -> p = q. +Proof. + rewrite 2 (mul_comm r). apply mul_reg_r. +Qed. + +Lemma mul_cancel_r p q r : p * r = q * r <-> p = q. +Proof. + split. apply mul_reg_r. congruence. +Qed. + +Lemma mul_cancel_l p q r : r * p = r * q <-> p = q. +Proof. + split. apply mul_reg_l. congruence. +Qed. + +(** ** Inversion of multiplication *) + +Lemma mul_eq_1_l p q : p * q = 1 -> p = 1. +Proof. + now destruct p, q. +Qed. + +Lemma mul_eq_1_r p q : p * q = 1 -> q = 1. +Proof. + now destruct p, q. +Qed. + +Notation mul_eq_1 := mul_eq_1_l. + +(** ** Square *) + +Lemma square_xO p : p~0 * p~0 = (p*p)~0~0. +Proof. + simpl. now rewrite mul_comm. +Qed. + +Lemma square_xI p : p~1 * p~1 = (p*p+p)~0~1. +Proof. + simpl. rewrite mul_comm. simpl. f_equal. + rewrite add_assoc, add_diag. simpl. now rewrite add_comm. +Qed. + +(** ** Properties of [iter] *) + +Lemma iter_swap_gen : forall A B (f:A->B)(g:A->A)(h:B->B), + (forall a, f (g a) = h (f a)) -> forall p a, + f (iter p g a) = iter p h (f a). +Proof. + induction p; simpl; intros; now rewrite ?H, ?IHp. +Qed. + +Theorem iter_swap : + forall p (A:Type) (f:A -> A) (x:A), + iter p f (f x) = f (iter p f x). +Proof. + intros. symmetry. now apply iter_swap_gen. +Qed. + +Theorem iter_succ : + forall p (A:Type) (f:A -> A) (x:A), + iter (succ p) f x = f (iter p f x). +Proof. + induction p as [p IHp|p IHp|]; intros; simpl; trivial. + now rewrite !IHp, iter_swap. +Qed. + +Theorem iter_add : + forall p q (A:Type) (f:A -> A) (x:A), + iter (p+q) f x = iter p f (iter q f x). +Proof. + induction p using peano_ind; intros. + now rewrite add_1_l, iter_succ. + now rewrite add_succ_l, !iter_succ, IHp. +Qed. + +Theorem iter_invariant : + forall (p:positive) (A:Type) (f:A -> A) (Inv:A -> Prop), + (forall x:A, Inv x -> Inv (f x)) -> + forall x:A, Inv x -> Inv (iter p f x). +Proof. + induction p as [p IHp|p IHp|]; simpl; trivial. + intros A f Inv H x H0. apply H, IHp, IHp; trivial. + intros A f Inv H x H0. apply IHp, IHp; trivial. +Qed. + +(** ** Properties of power *) + +Lemma pow_1_r p : p^1 = p. +Proof. + unfold pow. simpl. now rewrite mul_comm. +Qed. + +Lemma pow_succ_r p q : p^(succ q) = p * p^q. +Proof. + unfold pow. now rewrite iter_succ. +Qed. + +(** ** Properties of square *) + +Lemma square_spec p : square p = p * p. +Proof. + induction p. + - rewrite square_xI. simpl. now rewrite IHp. + - rewrite square_xO. simpl. now rewrite IHp. + - trivial. +Qed. + +(** ** Properties of [sub_mask] *) + +Lemma sub_mask_succ_r p q : + sub_mask p (succ q) = sub_mask_carry p q. +Proof. + revert q. induction p; destruct q; simpl; f_equal; trivial; now destruct p. +Qed. + +Theorem sub_mask_carry_spec p q : + sub_mask_carry p q = pred_mask (sub_mask p q). +Proof. + revert q. induction p as [p IHp|p IHp| ]; destruct q; simpl; + try reflexivity; try rewrite IHp; + destruct (sub_mask p q) as [|[r|r| ]|] || destruct p; auto. +Qed. + +Inductive SubMaskSpec (p q : positive) : mask -> Prop := + | SubIsNul : p = q -> SubMaskSpec p q IsNul + | SubIsPos : forall r, q + r = p -> SubMaskSpec p q (IsPos r) + | SubIsNeg : forall r, p + r = q -> SubMaskSpec p q IsNeg. + +Theorem sub_mask_spec p q : SubMaskSpec p q (sub_mask p q). +Proof. + revert q. induction p; destruct q; simpl; try constructor; trivial. + (* p~1 q~1 *) + destruct (IHp q); subst; try now constructor. + now apply SubIsNeg with r~0. + (* p~1 q~0 *) + destruct (IHp q); subst; try now constructor. + apply SubIsNeg with (pred_double r). symmetry. apply add_xI_pred_double. + (* p~0 q~1 *) + rewrite sub_mask_carry_spec. + destruct (IHp q); subst; try constructor. + now apply SubIsNeg with 1. + destruct r; simpl; try constructor; simpl. + now rewrite add_carry_spec, <- add_succ_r. + now rewrite add_carry_spec, <- add_succ_r, succ_pred_double. + now rewrite add_1_r. + now apply SubIsNeg with r~1. + (* p~0 q~0 *) + destruct (IHp q); subst; try now constructor. + now apply SubIsNeg with r~0. + (* p~0 1 *) + now rewrite add_1_l, succ_pred_double. + (* 1 q~1 *) + now apply SubIsNeg with q~0. + (* 1 q~0 *) + apply SubIsNeg with (pred_double q). now rewrite add_1_l, succ_pred_double. +Qed. + +Theorem sub_mask_nul_iff p q : sub_mask p q = IsNul <-> p = q. +Proof. + split. + now case sub_mask_spec. + intros <-. induction p; simpl; now rewrite ?IHp. +Qed. + +Theorem sub_mask_diag p : sub_mask p p = IsNul. +Proof. + now apply sub_mask_nul_iff. +Qed. + +Lemma sub_mask_add p q r : sub_mask p q = IsPos r -> q + r = p. +Proof. + case sub_mask_spec; congruence. +Qed. + +Lemma sub_mask_add_diag_l p q : sub_mask (p+q) p = IsPos q. +Proof. + case sub_mask_spec. + intros H. rewrite add_comm in H. elim (add_no_neutral _ _ H). + intros r H. apply add_cancel_l in H. now f_equal. + intros r H. rewrite <- add_assoc, add_comm in H. elim (add_no_neutral _ _ H). +Qed. + +Lemma sub_mask_pos_iff p q r : sub_mask p q = IsPos r <-> q + r = p. +Proof. + split. apply sub_mask_add. intros <-; apply sub_mask_add_diag_l. +Qed. + +Lemma sub_mask_add_diag_r p q : sub_mask p (p+q) = IsNeg. +Proof. + case sub_mask_spec; trivial. + intros H. symmetry in H; rewrite add_comm in H. elim (add_no_neutral _ _ H). + intros r H. rewrite <- add_assoc, add_comm in H. elim (add_no_neutral _ _ H). +Qed. + +Lemma sub_mask_neg_iff p q : sub_mask p q = IsNeg <-> exists r, p + r = q. +Proof. + split. + case sub_mask_spec; try discriminate. intros r Hr _; now exists r. + intros (r,<-). apply sub_mask_add_diag_r. +Qed. + +(*********************************************************************) +(** * Properties of boolean comparisons *) + +Theorem eqb_eq p q : (p =? q) = true <-> p=q. +Proof. + revert q. induction p; destruct q; simpl; rewrite ?IHp; split; congruence. +Qed. + +Theorem ltb_lt p q : (p <? q) = true <-> p < q. +Proof. + unfold ltb, lt. destruct compare; easy'. +Qed. + +Theorem leb_le p q : (p <=? q) = true <-> p <= q. +Proof. + unfold leb, le. destruct compare; easy'. +Qed. + +(** More about [eqb] *) + +Include BoolEqualityFacts. + +(**********************************************************************) +(** * Properties of comparison on binary positive numbers *) + +(** First, we express [compare_cont] in term of [compare] *) + +Definition switch_Eq c c' := + match c' with + | Eq => c + | Lt => Lt + | Gt => Gt + end. + +Lemma compare_cont_spec p q c : + compare_cont p q c = switch_Eq c (p ?= q). +Proof. + unfold compare. + revert q c. + induction p; destruct q; simpl; trivial. + intros c. + rewrite 2 IHp. now destruct (compare_cont p q Eq). + intros c. + rewrite 2 IHp. now destruct (compare_cont p q Eq). +Qed. + +(** From this general result, we now describe particular cases + of [compare_cont p q c = c'] : + - When [c=Eq], this is directly [compare] + - When [c<>Eq], we'll show first that [c'<>Eq] + - That leaves only 4 lemmas for [c] and [c'] being [Lt] or [Gt] +*) + +Theorem compare_cont_Eq p q c : + compare_cont p q c = Eq -> c = Eq. +Proof. + rewrite compare_cont_spec. now destruct (p ?= q). +Qed. + +Lemma compare_cont_Lt_Gt p q : + compare_cont p q Lt = Gt <-> p > q. +Proof. + rewrite compare_cont_spec. unfold gt. destruct (p ?= q); now split. +Qed. + +Lemma compare_cont_Lt_Lt p q : + compare_cont p q Lt = Lt <-> p <= q. +Proof. + rewrite compare_cont_spec. unfold le. destruct (p ?= q); easy'. +Qed. + +Lemma compare_cont_Gt_Lt p q : + compare_cont p q Gt = Lt <-> p < q. +Proof. + rewrite compare_cont_spec. unfold lt. destruct (p ?= q); now split. +Qed. + +Lemma compare_cont_Gt_Gt p q : + compare_cont p q Gt = Gt <-> p >= q. +Proof. + rewrite compare_cont_spec. unfold ge. destruct (p ?= q); easy'. +Qed. + +(** We can express recursive equations for [compare] *) + +Lemma compare_xO_xO p q : (p~0 ?= q~0) = (p ?= q). +Proof. reflexivity. Qed. + +Lemma compare_xI_xI p q : (p~1 ?= q~1) = (p ?= q). +Proof. reflexivity. Qed. + +Lemma compare_xI_xO p q : + (p~1 ?= q~0) = switch_Eq Gt (p ?= q). +Proof. exact (compare_cont_spec p q Gt). Qed. + +Lemma compare_xO_xI p q : + (p~0 ?= q~1) = switch_Eq Lt (p ?= q). +Proof. exact (compare_cont_spec p q Lt). Qed. + +Hint Rewrite compare_xO_xO compare_xI_xI compare_xI_xO compare_xO_xI : compare. + +Ltac simpl_compare := autorewrite with compare. +Ltac simpl_compare_in H := autorewrite with compare in H. + +(** Relation between [compare] and [sub_mask] *) + +Definition mask2cmp (p:mask) : comparison := + match p with + | IsNul => Eq + | IsPos _ => Gt + | IsNeg => Lt + end. + +Lemma compare_sub_mask p q : (p ?= q) = mask2cmp (sub_mask p q). +Proof. + revert q. + induction p as [p IHp| p IHp| ]; intros [q|q| ]; simpl; trivial; + specialize (IHp q); rewrite ?sub_mask_carry_spec; + destruct (sub_mask p q) as [|r|]; simpl in *; + try clear r; try destruct r; simpl; trivial; + simpl_compare; now rewrite IHp. +Qed. + +(** Alternative characterisation of strict order in term of addition *) + +Lemma lt_iff_add p q : p < q <-> exists r, p + r = q. +Proof. + unfold "<". rewrite <- sub_mask_neg_iff, compare_sub_mask. + destruct sub_mask; now split. +Qed. + +Lemma gt_iff_add p q : p > q <-> exists r, q + r = p. +Proof. + unfold ">". rewrite compare_sub_mask. + split. + case_eq (sub_mask p q); try discriminate; intros r Hr _. + exists r. now apply sub_mask_pos_iff. + intros (r,Hr). apply sub_mask_pos_iff in Hr. now rewrite Hr. +Qed. + +(** Basic facts about [compare_cont] *) + +Theorem compare_cont_refl p c : + compare_cont p p c = c. +Proof. + now induction p. +Qed. + +Lemma compare_cont_antisym p q c : + CompOpp (compare_cont p q c) = compare_cont q p (CompOpp c). +Proof. + revert q c. + induction p as [p IHp|p IHp| ]; intros [q|q| ] c; simpl; + trivial; apply IHp. +Qed. + +(** Basic facts about [compare] *) + +Lemma compare_eq_iff p q : (p ?= q) = Eq <-> p = q. +Proof. + rewrite compare_sub_mask, <- sub_mask_nul_iff. + destruct sub_mask; now split. +Qed. + +Lemma compare_antisym p q : (q ?= p) = CompOpp (p ?= q). +Proof. + unfold compare. now rewrite compare_cont_antisym. +Qed. + +Lemma compare_lt_iff p q : (p ?= q) = Lt <-> p < q. +Proof. reflexivity. Qed. + +Lemma compare_le_iff p q : (p ?= q) <> Gt <-> p <= q. +Proof. reflexivity. Qed. + +(** More properties about [compare] and boolean comparisons, + including [compare_spec] and [lt_irrefl] and [lt_eq_cases]. *) + +Include BoolOrderFacts. + +Definition le_lteq := lt_eq_cases. + +(** ** Facts about [gt] and [ge] *) + +(** The predicates [lt] and [le] are now favored in the statements + of theorems, the use of [gt] and [ge] is hence not recommended. + We provide here the bare minimal results to related them with + [lt] and [le]. *) + +Lemma gt_lt_iff p q : p > q <-> q < p. +Proof. + unfold lt, gt. now rewrite compare_antisym, CompOpp_iff. +Qed. + +Lemma gt_lt p q : p > q -> q < p. +Proof. + apply gt_lt_iff. +Qed. + +Lemma lt_gt p q : p < q -> q > p. +Proof. + apply gt_lt_iff. +Qed. + +Lemma ge_le_iff p q : p >= q <-> q <= p. +Proof. + unfold le, ge. now rewrite compare_antisym, CompOpp_iff. +Qed. + +Lemma ge_le p q : p >= q -> q <= p. +Proof. + apply ge_le_iff. +Qed. + +Lemma le_ge p q : p <= q -> q >= p. +Proof. + apply ge_le_iff. +Qed. + +(** ** Comparison and the successor *) + +Lemma compare_succ_r p q : + switch_Eq Gt (p ?= succ q) = switch_Eq Lt (p ?= q). +Proof. + revert q. + induction p as [p IH|p IH| ]; intros [q|q| ]; simpl; + simpl_compare; rewrite ?IH; trivial; + (now destruct compare) || (now destruct p). +Qed. + +Lemma compare_succ_l p q : + switch_Eq Lt (succ p ?= q) = switch_Eq Gt (p ?= q). +Proof. + rewrite 2 (compare_antisym q). generalize (compare_succ_r q p). + now do 2 destruct compare. +Qed. + +Theorem lt_succ_r p q : p < succ q <-> p <= q. +Proof. + unfold lt, le. generalize (compare_succ_r p q). + do 2 destruct compare; try discriminate; now split. +Qed. + +Lemma lt_succ_diag_r p : p < succ p. +Proof. + rewrite lt_iff_add. exists 1. apply add_1_r. +Qed. + +Lemma compare_succ_succ p q : (succ p ?= succ q) = (p ?= q). +Proof. + revert q. + induction p; destruct q; simpl; simpl_compare; trivial; + apply compare_succ_l || apply compare_succ_r || + (now destruct p) || (now destruct q). +Qed. + +(** ** 1 is the least positive number *) + +Lemma le_1_l p : 1 <= p. +Proof. + now destruct p. +Qed. + +Lemma nlt_1_r p : ~ p < 1. +Proof. + now destruct p. +Qed. + +Lemma lt_1_succ p : 1 < succ p. +Proof. + apply lt_succ_r, le_1_l. +Qed. + +(** ** Properties of the order *) + +Lemma le_nlt p q : p <= q <-> ~ q < p. +Proof. + now rewrite <- ge_le_iff. +Qed. + +Lemma lt_nle p q : p < q <-> ~ q <= p. +Proof. + intros. unfold lt, le. rewrite compare_antisym. + destruct compare; split; auto; easy'. +Qed. + +Lemma lt_le_incl p q : p<q -> p<=q. +Proof. + intros. apply le_lteq. now left. +Qed. + +Lemma lt_lt_succ n m : n < m -> n < succ m. +Proof. + intros. now apply lt_succ_r, lt_le_incl. +Qed. + +Lemma succ_lt_mono n m : n < m <-> succ n < succ m. +Proof. + unfold lt. now rewrite compare_succ_succ. +Qed. + +Lemma succ_le_mono n m : n <= m <-> succ n <= succ m. +Proof. + unfold le. now rewrite compare_succ_succ. +Qed. + +Lemma lt_trans n m p : n < m -> m < p -> n < p. +Proof. + rewrite 3 lt_iff_add. intros (r,Hr) (s,Hs). + exists (r+s). now rewrite add_assoc, Hr, Hs. +Qed. + +Theorem lt_ind : forall (A : positive -> Prop) (n : positive), + A (succ n) -> + (forall m : positive, n < m -> A m -> A (succ m)) -> + forall m : positive, n < m -> A m. +Proof. + intros A n AB AS m. induction m using peano_ind; intros H. + elim (nlt_1_r _ H). + apply lt_succ_r, le_lteq in H. destruct H as [H|H]; subst; auto. +Qed. + +Instance lt_strorder : StrictOrder lt. +Proof. split. exact lt_irrefl. exact lt_trans. Qed. + +Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt. +Proof. repeat red. intros. subst; auto. Qed. + +Lemma lt_total p q : p < q \/ p = q \/ q < p. +Proof. + case (compare_spec p q); intuition. +Qed. + +Lemma le_refl p : p <= p. +Proof. + intros. unfold le. now rewrite compare_refl. +Qed. + +Lemma le_lt_trans n m p : n <= m -> m < p -> n < p. +Proof. + intros H H'. apply le_lteq in H. destruct H. + now apply lt_trans with m. now subst. +Qed. + +Lemma lt_le_trans n m p : n < m -> m <= p -> n < p. +Proof. + intros H H'. apply le_lteq in H'. destruct H'. + now apply lt_trans with m. now subst. +Qed. + +Lemma le_trans n m p : n <= m -> m <= p -> n <= p. +Proof. + intros H H'. + apply le_lteq in H. destruct H. + apply le_lteq; left. now apply lt_le_trans with m. + now subst. +Qed. + +Lemma le_succ_l n m : succ n <= m <-> n < m. +Proof. + rewrite <- lt_succ_r. symmetry. apply succ_lt_mono. +Qed. + +Lemma le_antisym p q : p <= q -> q <= p -> p = q. +Proof. + rewrite le_lteq; destruct 1; auto. + rewrite le_lteq; destruct 1; auto. + elim (lt_irrefl p). now transitivity q. +Qed. + +Instance le_preorder : PreOrder le. +Proof. split. exact le_refl. exact le_trans. Qed. + +Instance le_partorder : PartialOrder Logic.eq le. +Proof. + intros x y. change (x=y <-> x <= y <= x). + split. intros; now subst. + destruct 1; now apply le_antisym. +Qed. + +(** ** Comparison and addition *) + +Lemma add_compare_mono_l p q r : (p+q ?= p+r) = (q ?= r). +Proof. + revert p q r. induction p using peano_ind; intros q r. + rewrite 2 add_1_l. apply compare_succ_succ. + now rewrite 2 add_succ_l, compare_succ_succ. +Qed. + +Lemma add_compare_mono_r p q r : (q+p ?= r+p) = (q ?= r). +Proof. + rewrite 2 (add_comm _ p). apply add_compare_mono_l. +Qed. + +(** ** Order and addition *) + +Lemma lt_add_diag_r p q : p < p + q. +Proof. + rewrite lt_iff_add. now exists q. +Qed. + +Lemma add_lt_mono_l p q r : q<r <-> p+q < p+r. +Proof. + unfold lt. rewrite add_compare_mono_l. apply iff_refl. +Qed. + +Lemma add_lt_mono_r p q r : q<r <-> q+p < r+p. +Proof. + unfold lt. rewrite add_compare_mono_r. apply iff_refl. +Qed. + +Lemma add_lt_mono p q r s : p<q -> r<s -> p+r<q+s. +Proof. + intros. apply lt_trans with (p+s). + now apply add_lt_mono_l. + now apply add_lt_mono_r. +Qed. + +Lemma add_le_mono_l p q r : q<=r <-> p+q<=p+r. +Proof. + unfold le. rewrite add_compare_mono_l. apply iff_refl. +Qed. + +Lemma add_le_mono_r p q r : q<=r <-> q+p<=r+p. +Proof. + unfold le. rewrite add_compare_mono_r. apply iff_refl. +Qed. + +Lemma add_le_mono p q r s : p<=q -> r<=s -> p+r <= q+s. +Proof. + intros. apply le_trans with (p+s). + now apply add_le_mono_l. + now apply add_le_mono_r. +Qed. + +(** ** Comparison and multiplication *) + +Lemma mul_compare_mono_l p q r : (p*q ?= p*r) = (q ?= r). +Proof. + revert q r. induction p; simpl; trivial. + intros q r. specialize (IHp q r). + destruct (compare_spec q r). + subst. apply compare_refl. + now apply add_lt_mono. + now apply lt_gt, add_lt_mono, gt_lt. +Qed. + +Lemma mul_compare_mono_r p q r : (q*p ?= r*p) = (q ?= r). +Proof. + rewrite 2 (mul_comm _ p). apply mul_compare_mono_l. +Qed. + +(** ** Order and multiplication *) + +Lemma mul_lt_mono_l p q r : q<r <-> p*q < p*r. +Proof. + unfold lt. rewrite mul_compare_mono_l. apply iff_refl. +Qed. + +Lemma mul_lt_mono_r p q r : q<r <-> q*p < r*p. +Proof. + unfold lt. rewrite mul_compare_mono_r. apply iff_refl. +Qed. + +Lemma mul_lt_mono p q r s : p<q -> r<s -> p*r < q*s. +Proof. + intros. apply lt_trans with (p*s). + now apply mul_lt_mono_l. + now apply mul_lt_mono_r. +Qed. + +Lemma mul_le_mono_l p q r : q<=r <-> p*q<=p*r. +Proof. + unfold le. rewrite mul_compare_mono_l. apply iff_refl. +Qed. + +Lemma mul_le_mono_r p q r : q<=r <-> q*p<=r*p. +Proof. + unfold le. rewrite mul_compare_mono_r. apply iff_refl. +Qed. + +Lemma mul_le_mono p q r s : p<=q -> r<=s -> p*r <= q*s. +Proof. + intros. apply le_trans with (p*s). + now apply mul_le_mono_l. + now apply mul_le_mono_r. +Qed. + +Lemma lt_add_r p q : p < p+q. +Proof. + induction q using peano_ind. + rewrite add_1_r. apply lt_succ_diag_r. + apply lt_trans with (p+q); auto. + apply add_lt_mono_l, lt_succ_diag_r. +Qed. + +Lemma lt_not_add_l p q : ~ p+q < p. +Proof. + intro H. elim (lt_irrefl p). + apply lt_trans with (p+q); auto using lt_add_r. +Qed. + +Lemma pow_gt_1 n p : 1<n -> 1<n^p. +Proof. + intros H. induction p using peano_ind. + now rewrite pow_1_r. + rewrite pow_succ_r. + apply lt_trans with (n*1). + now rewrite mul_1_r. + now apply mul_lt_mono_l. +Qed. + +(**********************************************************************) +(** * Properties of subtraction on binary positive numbers *) + +Lemma sub_1_r p : sub p 1 = pred p. +Proof. + now destruct p. +Qed. + +Lemma pred_sub p : pred p = sub p 1. +Proof. + symmetry. apply sub_1_r. +Qed. + +Theorem sub_succ_r p q : p - (succ q) = pred (p - q). +Proof. + unfold sub; rewrite sub_mask_succ_r, sub_mask_carry_spec. + destruct (sub_mask p q) as [|[r|r| ]|]; auto. +Qed. + +(** ** Properties of subtraction without underflow *) + +Lemma sub_mask_pos' p q : + q < p -> exists r, sub_mask p q = IsPos r /\ q + r = p. +Proof. + rewrite lt_iff_add. intros (r,Hr). exists r. split; trivial. + now apply sub_mask_pos_iff. +Qed. + +Lemma sub_mask_pos p q : + q < p -> exists r, sub_mask p q = IsPos r. +Proof. + intros H. destruct (sub_mask_pos' p q H) as (r & Hr & _). now exists r. +Qed. + +Theorem sub_add p q : q < p -> (p-q)+q = p. +Proof. + intros H. destruct (sub_mask_pos p q H) as (r,U). + unfold sub. rewrite U. rewrite add_comm. now apply sub_mask_add. +Qed. + +Lemma add_sub p q : (p+q)-q = p. +Proof. + intros. apply add_reg_r with q. + rewrite sub_add; trivial. + rewrite add_comm. apply lt_add_r. +Qed. + +Lemma mul_sub_distr_l p q r : r<q -> p*(q-r) = p*q-p*r. +Proof. + intros H. + apply add_reg_r with (p*r). + rewrite <- mul_add_distr_l. + rewrite sub_add; trivial. + symmetry. apply sub_add; trivial. + now apply mul_lt_mono_l. +Qed. + +Lemma mul_sub_distr_r p q r : q<p -> (p-q)*r = p*r-q*r. +Proof. + intros H. rewrite 3 (mul_comm _ r). now apply mul_sub_distr_l. +Qed. + +Lemma sub_lt_mono_l p q r: q<p -> p<r -> r-p < r-q. +Proof. + intros Hqp Hpr. + apply (add_lt_mono_r p). + rewrite sub_add by trivial. + apply le_lt_trans with ((r-q)+q). + rewrite sub_add by (now apply lt_trans with p). + apply le_refl. + now apply add_lt_mono_l. +Qed. + +Lemma sub_compare_mono_l p q r : + q<p -> r<p -> (p-q ?= p-r) = (r ?= q). +Proof. + intros Hqp Hrp. + case (compare_spec r q); intros H. subst. apply compare_refl. + apply sub_lt_mono_l; trivial. + apply lt_gt, sub_lt_mono_l; trivial. +Qed. + +Lemma sub_compare_mono_r p q r : + p<q -> p<r -> (q-p ?= r-p) = (q ?= r). +Proof. + intros. rewrite <- (add_compare_mono_r p), 2 sub_add; trivial. +Qed. + +Lemma sub_lt_mono_r p q r : q<p -> r<q -> q-r < p-r. +Proof. + intros. unfold lt. rewrite sub_compare_mono_r; trivial. + now apply lt_trans with q. +Qed. + +Lemma sub_decr n m : m<n -> n-m < n. +Proof. + intros. + apply add_lt_mono_r with m. + rewrite sub_add; trivial. + apply lt_add_r. +Qed. + +Lemma add_sub_assoc p q r : r<q -> p+(q-r) = p+q-r. +Proof. + intros. + apply add_reg_r with r. + rewrite <- add_assoc, !sub_add; trivial. + rewrite add_comm. apply lt_trans with q; trivial using lt_add_r. +Qed. + +Lemma sub_add_distr p q r : q+r < p -> p-(q+r) = p-q-r. +Proof. + intros. + assert (q < p) + by (apply lt_trans with (q+r); trivial using lt_add_r). + rewrite (add_comm q r) in *. + apply add_reg_r with (r+q). + rewrite sub_add by trivial. + rewrite add_assoc, !sub_add; trivial. + apply (add_lt_mono_r q). rewrite sub_add; trivial. +Qed. + +Lemma sub_sub_distr p q r : r<q -> q-r < p -> p-(q-r) = p+r-q. +Proof. + intros. + apply add_reg_r with ((q-r)+r). + rewrite add_assoc, !sub_add; trivial. + rewrite <- (sub_add q r); trivial. + now apply add_lt_mono_r. +Qed. + +(** Recursive equations for [sub] *) + +Lemma sub_xO_xO n m : m<n -> n~0 - m~0 = (n-m)~0. +Proof. + intros H. unfold sub. simpl. + now destruct (sub_mask_pos n m H) as (p, ->). +Qed. + +Lemma sub_xI_xI n m : m<n -> n~1 - m~1 = (n-m)~0. +Proof. + intros H. unfold sub. simpl. + now destruct (sub_mask_pos n m H) as (p, ->). +Qed. + +Lemma sub_xI_xO n m : m<n -> n~1 - m~0 = (n-m)~1. +Proof. + intros H. unfold sub. simpl. + now destruct (sub_mask_pos n m) as (p, ->). +Qed. + +Lemma sub_xO_xI n m : n~0 - m~1 = pred_double (n-m). +Proof. + unfold sub. simpl. rewrite sub_mask_carry_spec. + now destruct (sub_mask n m) as [|[r|r|]|]. +Qed. + +(** Properties of subtraction with underflow *) + +Lemma sub_mask_neg_iff' p q : sub_mask p q = IsNeg <-> p < q. +Proof. + rewrite lt_iff_add. apply sub_mask_neg_iff. +Qed. + +Lemma sub_mask_neg p q : p<q -> sub_mask p q = IsNeg. +Proof. + apply sub_mask_neg_iff'. +Qed. + +Lemma sub_le p q : p<=q -> p-q = 1. +Proof. + unfold le, sub. rewrite compare_sub_mask. + destruct sub_mask; easy'. +Qed. + +Lemma sub_lt p q : p<q -> p-q = 1. +Proof. + intros. now apply sub_le, lt_le_incl. +Qed. + +Lemma sub_diag p : p-p = 1. +Proof. + unfold sub. now rewrite sub_mask_diag. +Qed. + +(** ** Results concerning [size] and [size_nat] *) + +Lemma size_nat_monotone p q : p<q -> (size_nat p <= size_nat q)%nat. +Proof. + assert (le0 : forall n, (0<=n)%nat) by (induction n; auto). + assert (leS : forall n m, (n<=m -> S n <= S m)%nat) by (induction 1; auto). + revert q. + induction p; destruct q; simpl; intros; auto; easy || apply leS; + red in H; simpl_compare_in H. + apply IHp. red. now destruct (p?=q). + destruct (compare_spec p q); subst; now auto. +Qed. + +Lemma size_gt p : p < 2^(size p). +Proof. + induction p; simpl; try rewrite pow_succ_r; try easy. + apply le_succ_l in IHp. now apply le_succ_l. +Qed. + +Lemma size_le p : 2^(size p) <= p~0. +Proof. + induction p; simpl; try rewrite pow_succ_r; try easy. + apply mul_le_mono_l. + apply le_lteq; left. rewrite xI_succ_xO. apply lt_succ_r, IHp. +Qed. + +(** ** Properties of [min] and [max] *) + +(** First, the specification *) + +Lemma max_l : forall x y, y<=x -> max x y = x. +Proof. + intros x y H. unfold max. case compare_spec; auto. + intros H'. apply le_nlt in H. now elim H. +Qed. + +Lemma max_r : forall x y, x<=y -> max x y = y. +Proof. + unfold le, max. intros x y. destruct compare; easy'. +Qed. + +Lemma min_l : forall x y, x<=y -> min x y = x. +Proof. + unfold le, min. intros x y. destruct compare; easy'. +Qed. + +Lemma min_r : forall x y, y<=x -> min x y = y. +Proof. + intros x y H. unfold min. case compare_spec; auto. + intros H'. apply le_nlt in H. now elim H'. +Qed. + +(** We hence obtain all the generic properties of [min] and [max]. *) + +Include !UsualMinMaxLogicalProperties. +Include !UsualMinMaxDecProperties. + +(** Minimum, maximum and constant one *) + +Lemma max_1_l n : max 1 n = n. +Proof. + unfold max. case compare_spec; auto. + intros H. apply lt_nle in H. elim H. apply le_1_l. +Qed. + +Lemma max_1_r n : max n 1 = n. +Proof. rewrite max_comm. apply max_1_l. Qed. + +Lemma min_1_l n : min 1 n = 1. +Proof. + unfold min. case compare_spec; auto. + intros H. apply lt_nle in H. elim H. apply le_1_l. +Qed. + +Lemma min_1_r n : min n 1 = 1. +Proof. rewrite min_comm. apply min_1_l. Qed. + +(** Minimum, maximum and operations (consequences of monotonicity) *) + +Lemma succ_max_distr n m : succ (max n m) = max (succ n) (succ m). +Proof. + symmetry. apply max_monotone. + intros x x'. apply succ_le_mono. +Qed. + +Lemma succ_min_distr n m : succ (min n m) = min (succ n) (succ m). +Proof. + symmetry. apply min_monotone. + intros x x'. apply succ_le_mono. +Qed. + +Lemma add_max_distr_l n m p : max (p + n) (p + m) = p + max n m. +Proof. + apply max_monotone. intros x x'. apply add_le_mono_l. +Qed. + +Lemma add_max_distr_r n m p : max (n + p) (m + p) = max n m + p. +Proof. + rewrite 3 (add_comm _ p). apply add_max_distr_l. +Qed. + +Lemma add_min_distr_l n m p : min (p + n) (p + m) = p + min n m. +Proof. + apply min_monotone. intros x x'. apply add_le_mono_l. +Qed. + +Lemma add_min_distr_r n m p : min (n + p) (m + p) = min n m + p. +Proof. + rewrite 3 (add_comm _ p). apply add_min_distr_l. +Qed. + +Lemma mul_max_distr_l n m p : max (p * n) (p * m) = p * max n m. +Proof. + apply max_monotone. intros x x'. apply mul_le_mono_l. +Qed. + +Lemma mul_max_distr_r n m p : max (n * p) (m * p) = max n m * p. +Proof. + rewrite 3 (mul_comm _ p). apply mul_max_distr_l. +Qed. + +Lemma mul_min_distr_l n m p : min (p * n) (p * m) = p * min n m. +Proof. + apply min_monotone. intros x x'. apply mul_le_mono_l. +Qed. + +Lemma mul_min_distr_r n m p : min (n * p) (m * p) = min n m * p. +Proof. + rewrite 3 (mul_comm _ p). apply mul_min_distr_l. +Qed. + + +(** ** Results concerning [iter_op] *) + +Lemma iter_op_succ : forall A (op:A->A->A), + (forall x y z, op x (op y z) = op (op x y) z) -> + forall p a, + iter_op op (succ p) a = op a (iter_op op p a). +Proof. + induction p; simpl; intros; trivial. + rewrite H. apply IHp. +Qed. + +(** ** Results about [of_nat] and [of_succ_nat] *) + +Lemma of_nat_succ (n:nat) : of_succ_nat n = of_nat (S n). +Proof. + induction n. trivial. simpl. f_equal. now rewrite IHn. +Qed. + +Lemma pred_of_succ_nat (n:nat) : pred (of_succ_nat n) = of_nat n. +Proof. + destruct n. trivial. simpl pred. rewrite pred_succ. apply of_nat_succ. +Qed. + +Lemma succ_of_nat (n:nat) : n<>O -> succ (of_nat n) = of_succ_nat n. +Proof. + rewrite of_nat_succ. destruct n; trivial. now destruct 1. +Qed. + +(** ** Correctness proofs for the square root function *) + +Inductive SqrtSpec : positive*mask -> positive -> Prop := + | SqrtExact s x : x=s*s -> SqrtSpec (s,IsNul) x + | SqrtApprox s r x : x=s*s+r -> r <= s~0 -> SqrtSpec (s,IsPos r) x. + +Lemma sqrtrem_step_spec f g p x : + (f=xO \/ f=xI) -> (g=xO \/ g=xI) -> + SqrtSpec p x -> SqrtSpec (sqrtrem_step f g p) (g (f x)). +Proof. +intros Hf Hg [ s _ -> | s r _ -> Hr ]. +(* exact *) +unfold sqrtrem_step. +destruct Hf,Hg; subst; simpl; constructor; now rewrite ?square_xO. +(* approx *) +assert (Hfg : forall p q, g (f (p+q)) = p~0~0 + g (f q)) + by (intros; destruct Hf, Hg; now subst). +unfold sqrtrem_step, leb. +case compare_spec; [intros EQ | intros LT | intros GT]. +(* - EQ *) +rewrite <- EQ, sub_mask_diag. constructor. +destruct Hg; subst g; destr_eq EQ. +destruct Hf; subst f; destr_eq EQ. +subst. now rewrite square_xI. +(* - LT *) +destruct (sub_mask_pos' _ _ LT) as (y & -> & H). constructor. +rewrite Hfg, <- H. now rewrite square_xI, add_assoc. clear Hfg. +rewrite <- lt_succ_r in Hr. change (r < s~1) in Hr. +rewrite <- lt_succ_r, (add_lt_mono_l (s~0~1)), H. simpl. +rewrite add_carry_spec, add_diag. simpl. +destruct Hf,Hg; subst; red; simpl_compare; now rewrite Hr. +(* - GT *) +constructor. now rewrite Hfg, square_xO. apply lt_succ_r, GT. +Qed. + +Lemma sqrtrem_spec p : SqrtSpec (sqrtrem p) p. +Proof. +revert p. fix 1. + destruct p; try destruct p; try (constructor; easy); + apply sqrtrem_step_spec; auto. +Qed. + +Lemma sqrt_spec p : + let s := sqrt p in s*s <= p < (succ s)*(succ s). +Proof. + simpl. + assert (H:=sqrtrem_spec p). + unfold sqrt in *. destruct sqrtrem as (s,rm); simpl. + inversion_clear H; subst. + (* exact *) + split. reflexivity. apply mul_lt_mono; apply lt_succ_diag_r. + (* approx *) + split. + apply lt_le_incl, lt_add_r. + rewrite <- add_1_l, mul_add_distr_r, !mul_add_distr_l, !mul_1_r, !mul_1_l. + rewrite add_assoc, (add_comm _ r). apply add_lt_mono_r. + now rewrite <- add_assoc, add_diag, add_1_l, lt_succ_r. +Qed. + +(** ** Correctness proofs for the gcd function *) + +Lemma divide_add_cancel_l p q r : (p | r) -> (p | q + r) -> (p | q). +Proof. + intros (s,Hs) (t,Ht). + exists (t-s). + rewrite mul_sub_distr_r. + rewrite <- Hs, <- Ht. + symmetry. apply add_sub. + apply mul_lt_mono_r with p. + rewrite <- Hs, <- Ht, add_comm. + apply lt_add_r. +Qed. + +Lemma divide_xO_xI p q r : (p | q~0) -> (p | r~1) -> (p | q). +Proof. + intros (s,Hs) (t,Ht). + destruct p. + destruct s; try easy. simpl in Hs. destr_eq Hs. now exists s. + rewrite mul_xO_r in Ht; discriminate. + exists q; now rewrite mul_1_r. +Qed. + +Lemma divide_xO_xO p q : (p~0|q~0) <-> (p|q). +Proof. + split; intros (r,H); simpl in *. + rewrite mul_xO_r in H. destr_eq H. now exists r. + exists r; simpl. rewrite mul_xO_r. f_equal; auto. +Qed. + +Lemma divide_mul_l p q r : (p|q) -> (p|q*r). +Proof. + intros (s,H). exists (s*r). + rewrite <- mul_assoc, (mul_comm r p), mul_assoc. now f_equal. +Qed. + +Lemma divide_mul_r p q r : (p|r) -> (p|q*r). +Proof. + rewrite mul_comm. apply divide_mul_l. +Qed. + +(** The first component of ggcd is gcd *) + +Lemma ggcdn_gcdn : forall n a b, + fst (ggcdn n a b) = gcdn n a b. +Proof. + induction n. + simpl; auto. + destruct a, b; simpl; auto; try case compare_spec; simpl; trivial; + rewrite <- IHn; destruct ggcdn as (g,(u,v)); simpl; auto. +Qed. + +Lemma ggcd_gcd : forall a b, fst (ggcd a b) = gcd a b. +Proof. + unfold ggcd, gcd. intros. apply ggcdn_gcdn. +Qed. + +(** The other components of ggcd are indeed the correct factors. *) + +Ltac destr_pggcdn IHn := + match goal with |- context [ ggcdn _ ?x ?y ] => + generalize (IHn x y); destruct ggcdn as (g,(u,v)); simpl + end. + +Lemma ggcdn_correct_divisors : forall n a b, + let '(g,(aa,bb)) := ggcdn n a b in + a = g*aa /\ b = g*bb. +Proof. + induction n. + simpl; auto. + destruct a, b; simpl; auto; try case compare_spec; try destr_pggcdn IHn. + (* Eq *) + intros ->. now rewrite mul_comm. + (* Lt *) + intros (H',H) LT; split; auto. + rewrite mul_add_distr_l, mul_xO_r, <- H, <- H'. + simpl. f_equal. symmetry. + rewrite add_comm. now apply sub_add. + (* Gt *) + intros (H',H) LT; split; auto. + rewrite mul_add_distr_l, mul_xO_r, <- H, <- H'. + simpl. f_equal. symmetry. + rewrite add_comm. now apply sub_add. + (* Then... *) + intros (H,H'); split; auto. rewrite mul_xO_r, H'; auto. + intros (H,H'); split; auto. rewrite mul_xO_r, H; auto. + intros (H,H'); split; subst; auto. +Qed. + +Lemma ggcd_correct_divisors : forall a b, + let '(g,(aa,bb)) := ggcd a b in + a=g*aa /\ b=g*bb. +Proof. + unfold ggcd. intros. apply ggcdn_correct_divisors. +Qed. + +(** We can use this fact to prove a part of the gcd correctness *) + +Lemma gcd_divide_l : forall a b, (gcd a b | a). +Proof. + intros a b. rewrite <- ggcd_gcd. generalize (ggcd_correct_divisors a b). + destruct ggcd as (g,(aa,bb)); simpl. intros (H,_). exists aa. + now rewrite mul_comm. +Qed. + +Lemma gcd_divide_r : forall a b, (gcd a b | b). +Proof. + intros a b. rewrite <- ggcd_gcd. generalize (ggcd_correct_divisors a b). + destruct ggcd as (g,(aa,bb)); simpl. intros (_,H). exists bb. + now rewrite mul_comm. +Qed. + +(** We now prove directly that gcd is the greatest amongst common divisors *) + +Lemma gcdn_greatest : forall n a b, (size_nat a + size_nat b <= n)%nat -> + forall p, (p|a) -> (p|b) -> (p|gcdn n a b). +Proof. + induction n. + destruct a, b; simpl; inversion 1. + destruct a, b; simpl; try case compare_spec; simpl; auto. + (* Lt *) + intros LT LE p Hp1 Hp2. apply IHn; clear IHn; trivial. + apply le_S_n in LE. eapply Le.le_trans; [|eapply LE]. + rewrite plus_comm, <- plus_n_Sm, <- plus_Sn_m. + apply plus_le_compat; trivial. + apply size_nat_monotone, sub_decr, LT. + apply divide_xO_xI with a; trivial. + apply (divide_add_cancel_l p _ a~1); trivial. + now rewrite <- sub_xI_xI, sub_add. + (* Gt *) + intros LT LE p Hp1 Hp2. apply IHn; clear IHn; trivial. + apply le_S_n in LE. eapply Le.le_trans; [|eapply LE]. + apply plus_le_compat; trivial. + apply size_nat_monotone, sub_decr, LT. + apply divide_xO_xI with b; trivial. + apply (divide_add_cancel_l p _ b~1); trivial. + now rewrite <- sub_xI_xI, sub_add. + (* a~1 b~0 *) + intros LE p Hp1 Hp2. apply IHn; clear IHn; trivial. + apply le_S_n in LE. simpl. now rewrite plus_n_Sm. + apply divide_xO_xI with a; trivial. + (* a~0 b~1 *) + intros LE p Hp1 Hp2. apply IHn; clear IHn; trivial. + simpl. now apply le_S_n. + apply divide_xO_xI with b; trivial. + (* a~0 b~0 *) + intros LE p Hp1 Hp2. + destruct p. + change (gcdn n a b)~0 with (2*(gcdn n a b)). + apply divide_mul_r. + apply IHn; clear IHn. + apply le_S_n in LE. apply le_Sn_le. now rewrite plus_n_Sm. + apply divide_xO_xI with p; trivial. now exists 1. + apply divide_xO_xI with p; trivial. now exists 1. + apply divide_xO_xO. + apply IHn; clear IHn. + apply le_S_n in LE. apply le_Sn_le. now rewrite plus_n_Sm. + now apply divide_xO_xO. + now apply divide_xO_xO. + exists (gcdn n a b)~0. now rewrite mul_1_r. +Qed. + +Lemma gcd_greatest : forall a b p, (p|a) -> (p|b) -> (p|gcd a b). +Proof. + intros. apply gcdn_greatest; auto. +Qed. + +(** As a consequence, the rests after division by gcd are relatively prime *) + +Lemma ggcd_greatest : forall a b, + let (aa,bb) := snd (ggcd a b) in + forall p, (p|aa) -> (p|bb) -> p=1. +Proof. + intros. generalize (gcd_greatest a b) (ggcd_correct_divisors a b). + rewrite <- ggcd_gcd. destruct ggcd as (g,(aa,bb)); simpl. + intros H (EQa,EQb) p Hp1 Hp2; subst. + assert (H' : (g*p | g)). + apply H. + destruct Hp1 as (r,Hr). exists r. + now rewrite mul_assoc, (mul_comm r g), <- mul_assoc, <- Hr. + destruct Hp2 as (r,Hr). exists r. + now rewrite mul_assoc, (mul_comm r g), <- mul_assoc, <- Hr. + destruct H' as (q,H'). + rewrite (mul_comm g p), mul_assoc in H'. + apply mul_eq_1 with q; rewrite mul_comm. + now apply mul_reg_r with g. +Qed. + +End Pos. + +(** Exportation of notations *) + +Infix "+" := Pos.add : positive_scope. +Infix "-" := Pos.sub : positive_scope. +Infix "*" := Pos.mul : positive_scope. +Infix "^" := Pos.pow : positive_scope. +Infix "?=" := Pos.compare (at level 70, no associativity) : positive_scope. +Infix "=?" := Pos.eqb (at level 70, no associativity) : positive_scope. +Infix "<=?" := Pos.leb (at level 70, no associativity) : positive_scope. +Infix "<?" := Pos.ltb (at level 70, no associativity) : positive_scope. +Infix "<=" := Pos.le : positive_scope. +Infix "<" := Pos.lt : positive_scope. +Infix ">=" := Pos.ge : positive_scope. +Infix ">" := Pos.gt : positive_scope. + +Notation "x <= y <= z" := (x <= y /\ y <= z) : positive_scope. +Notation "x <= y < z" := (x <= y /\ y < z) : positive_scope. +Notation "x < y < z" := (x < y /\ y < z) : positive_scope. +Notation "x < y <= z" := (x < y /\ y <= z) : positive_scope. + +Notation "( p | q )" := (Pos.divide p q) (at level 0) : positive_scope. + +(** Compatibility notations *) + +Notation positive := positive (only parsing). +Notation positive_rect := positive_rect (only parsing). +Notation positive_rec := positive_rec (only parsing). +Notation positive_ind := positive_ind (only parsing). +Notation xI := xI (only parsing). +Notation xO := xO (only parsing). +Notation xH := xH (only parsing). + +Notation Psucc := Pos.succ (only parsing). +Notation Pplus := Pos.add (only parsing). +Notation Pplus_carry := Pos.add_carry (only parsing). +Notation Ppred := Pos.pred (only parsing). +Notation Piter_op := Pos.iter_op (only parsing). +Notation Piter_op_succ := Pos.iter_op_succ (only parsing). +Notation Pmult_nat := (Pos.iter_op plus) (only parsing). +Notation nat_of_P := Pos.to_nat (only parsing). +Notation P_of_succ_nat := Pos.of_succ_nat (only parsing). +Notation Pdouble_minus_one := Pos.pred_double (only parsing). +Notation positive_mask := Pos.mask (only parsing). +Notation IsNul := Pos.IsNul (only parsing). +Notation IsPos := Pos.IsPos (only parsing). +Notation IsNeg := Pos.IsNeg (only parsing). +Notation positive_mask_rect := Pos.mask_rect (only parsing). +Notation positive_mask_ind := Pos.mask_ind (only parsing). +Notation positive_mask_rec := Pos.mask_rec (only parsing). +Notation Pdouble_plus_one_mask := Pos.succ_double_mask (only parsing). +Notation Pdouble_mask := Pos.double_mask (only parsing). +Notation Pdouble_minus_two := Pos.double_pred_mask (only parsing). +Notation Pminus_mask := Pos.sub_mask (only parsing). +Notation Pminus_mask_carry := Pos.sub_mask_carry (only parsing). +Notation Pminus := Pos.sub (only parsing). +Notation Pmult := Pos.mul (only parsing). +Notation iter_pos := @Pos.iter (only parsing). +Notation Ppow := Pos.pow (only parsing). +Notation Pdiv2 := Pos.div2 (only parsing). +Notation Pdiv2_up := Pos.div2_up (only parsing). +Notation Psize := Pos.size_nat (only parsing). +Notation Psize_pos := Pos.size (only parsing). +Notation Pcompare := Pos.compare_cont (only parsing). +Notation Plt := Pos.lt (only parsing). +Notation Pgt := Pos.gt (only parsing). +Notation Ple := Pos.le (only parsing). +Notation Pge := Pos.ge (only parsing). +Notation Pmin := Pos.min (only parsing). +Notation Pmax := Pos.max (only parsing). +Notation Peqb := Pos.eqb (only parsing). +Notation positive_eq_dec := Pos.eq_dec (only parsing). +Notation xI_succ_xO := Pos.xI_succ_xO (only parsing). +Notation Psucc_discr := Pos.succ_discr (only parsing). +Notation Psucc_o_double_minus_one_eq_xO := + Pos.succ_pred_double (only parsing). +Notation Pdouble_minus_one_o_succ_eq_xI := + Pos.pred_double_succ (only parsing). +Notation xO_succ_permute := Pos.double_succ (only parsing). +Notation double_moins_un_xO_discr := + Pos.pred_double_xO_discr (only parsing). +Notation Psucc_not_one := Pos.succ_not_1 (only parsing). +Notation Ppred_succ := Pos.pred_succ (only parsing). +Notation Psucc_pred := Pos.succ_pred_or (only parsing). +Notation Psucc_inj := Pos.succ_inj (only parsing). +Notation Pplus_carry_spec := Pos.add_carry_spec (only parsing). +Notation Pplus_comm := Pos.add_comm (only parsing). +Notation Pplus_succ_permute_r := Pos.add_succ_r (only parsing). +Notation Pplus_succ_permute_l := Pos.add_succ_l (only parsing). +Notation Pplus_no_neutral := Pos.add_no_neutral (only parsing). +Notation Pplus_carry_plus := Pos.add_carry_add (only parsing). +Notation Pplus_reg_r := Pos.add_reg_r (only parsing). +Notation Pplus_reg_l := Pos.add_reg_l (only parsing). +Notation Pplus_carry_reg_r := Pos.add_carry_reg_r (only parsing). +Notation Pplus_carry_reg_l := Pos.add_carry_reg_l (only parsing). +Notation Pplus_assoc := Pos.add_assoc (only parsing). +Notation Pplus_xO := Pos.add_xO (only parsing). +Notation Pplus_xI_double_minus_one := Pos.add_xI_pred_double (only parsing). +Notation Pplus_xO_double_minus_one := Pos.add_xO_pred_double (only parsing). +Notation Pplus_diag := Pos.add_diag (only parsing). +Notation PeanoView := Pos.PeanoView (only parsing). +Notation PeanoOne := Pos.PeanoOne (only parsing). +Notation PeanoSucc := Pos.PeanoSucc (only parsing). +Notation PeanoView_rect := Pos.PeanoView_rect (only parsing). +Notation PeanoView_ind := Pos.PeanoView_ind (only parsing). +Notation PeanoView_rec := Pos.PeanoView_rec (only parsing). +Notation peanoView_xO := Pos.peanoView_xO (only parsing). +Notation peanoView_xI := Pos.peanoView_xI (only parsing). +Notation peanoView := Pos.peanoView (only parsing). +Notation PeanoView_iter := Pos.PeanoView_iter (only parsing). +Notation eq_dep_eq_positive := Pos.eq_dep_eq_positive (only parsing). +Notation PeanoViewUnique := Pos.PeanoViewUnique (only parsing). +Notation Prect := Pos.peano_rect (only parsing). +Notation Prect_succ := Pos.peano_rect_succ (only parsing). +Notation Prect_base := Pos.peano_rect_base (only parsing). +Notation Prec := Pos.peano_rec (only parsing). +Notation Pind := Pos.peano_ind (only parsing). +Notation Pcase := Pos.peano_case (only parsing). +Notation Pmult_1_r := Pos.mul_1_r (only parsing). +Notation Pmult_Sn_m := Pos.mul_succ_l (only parsing). +Notation Pmult_xO_permute_r := Pos.mul_xO_r (only parsing). +Notation Pmult_xI_permute_r := Pos.mul_xI_r (only parsing). +Notation Pmult_comm := Pos.mul_comm (only parsing). +Notation Pmult_plus_distr_l := Pos.mul_add_distr_l (only parsing). +Notation Pmult_plus_distr_r := Pos.mul_add_distr_r (only parsing). +Notation Pmult_assoc := Pos.mul_assoc (only parsing). +Notation Pmult_xI_mult_xO_discr := Pos.mul_xI_mul_xO_discr (only parsing). +Notation Pmult_xO_discr := Pos.mul_xO_discr (only parsing). +Notation Pmult_reg_r := Pos.mul_reg_r (only parsing). +Notation Pmult_reg_l := Pos.mul_reg_l (only parsing). +Notation Pmult_1_inversion_l := Pos.mul_eq_1_l (only parsing). +Notation Psquare_xO := Pos.square_xO (only parsing). +Notation Psquare_xI := Pos.square_xI (only parsing). +Notation iter_pos_swap_gen := Pos.iter_swap_gen (only parsing). +Notation iter_pos_swap := Pos.iter_swap (only parsing). +Notation iter_pos_succ := Pos.iter_succ (only parsing). +Notation iter_pos_plus := Pos.iter_add (only parsing). +Notation iter_pos_invariant := Pos.iter_invariant (only parsing). +Notation Ppow_1_r := Pos.pow_1_r (only parsing). +Notation Ppow_succ_r := Pos.pow_succ_r (only parsing). +Notation Peqb_refl := Pos.eqb_refl (only parsing). +Notation Peqb_eq := Pos.eqb_eq (only parsing). +Notation Pcompare_refl_id := Pos.compare_cont_refl (only parsing). +Notation Pcompare_eq_iff := Pos.compare_eq_iff (only parsing). +Notation Pcompare_Gt_Lt := Pos.compare_cont_Gt_Lt (only parsing). +Notation Pcompare_eq_Lt := Pos.compare_lt_iff (only parsing). +Notation Pcompare_Lt_Gt := Pos.compare_cont_Lt_Gt (only parsing). + +Notation Pcompare_antisym := Pos.compare_cont_antisym (only parsing). +Notation ZC1 := Pos.gt_lt (only parsing). +Notation ZC2 := Pos.lt_gt (only parsing). +Notation Pcompare_spec := Pos.compare_spec (only parsing). +Notation Pcompare_p_Sp := Pos.lt_succ_diag_r (only parsing). +Notation Pcompare_succ_succ := Pos.compare_succ_succ (only parsing). +Notation Pcompare_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1_succ := Pos.lt_1_succ (only parsing). +Notation Plt_lt_succ := Pos.lt_lt_succ (only parsing). +Notation Plt_irrefl := Pos.lt_irrefl (only parsing). +Notation Plt_trans := Pos.lt_trans (only parsing). +Notation Plt_ind := Pos.lt_ind (only parsing). +Notation Ple_lteq := Pos.le_lteq (only parsing). +Notation Ple_refl := Pos.le_refl (only parsing). +Notation Ple_lt_trans := Pos.le_lt_trans (only parsing). +Notation Plt_le_trans := Pos.lt_le_trans (only parsing). +Notation Ple_trans := Pos.le_trans (only parsing). +Notation Plt_succ_r := Pos.lt_succ_r (only parsing). +Notation Ple_succ_l := Pos.le_succ_l (only parsing). +Notation Pplus_compare_mono_l := Pos.add_compare_mono_l (only parsing). +Notation Pplus_compare_mono_r := Pos.add_compare_mono_r (only parsing). +Notation Pplus_lt_mono_l := Pos.add_lt_mono_l (only parsing). +Notation Pplus_lt_mono_r := Pos.add_lt_mono_r (only parsing). +Notation Pplus_lt_mono := Pos.add_lt_mono (only parsing). +Notation Pplus_le_mono_l := Pos.add_le_mono_l (only parsing). +Notation Pplus_le_mono_r := Pos.add_le_mono_r (only parsing). +Notation Pplus_le_mono := Pos.add_le_mono (only parsing). +Notation Pmult_compare_mono_l := Pos.mul_compare_mono_l (only parsing). +Notation Pmult_compare_mono_r := Pos.mul_compare_mono_r (only parsing). +Notation Pmult_lt_mono_l := Pos.mul_lt_mono_l (only parsing). +Notation Pmult_lt_mono_r := Pos.mul_lt_mono_r (only parsing). +Notation Pmult_lt_mono := Pos.mul_lt_mono (only parsing). +Notation Pmult_le_mono_l := Pos.mul_le_mono_l (only parsing). +Notation Pmult_le_mono_r := Pos.mul_le_mono_r (only parsing). +Notation Pmult_le_mono := Pos.mul_le_mono (only parsing). +Notation Plt_plus_r := Pos.lt_add_r (only parsing). +Notation Plt_not_plus_l := Pos.lt_not_add_l (only parsing). +Notation Ppow_gt_1 := Pos.pow_gt_1 (only parsing). +Notation Ppred_mask := Pos.pred_mask (only parsing). +Notation Pminus_mask_succ_r := Pos.sub_mask_succ_r (only parsing). +Notation Pminus_mask_carry_spec := Pos.sub_mask_carry_spec (only parsing). +Notation Pminus_succ_r := Pos.sub_succ_r (only parsing). +Notation Pminus_mask_diag := Pos.sub_mask_diag (only parsing). + +Notation Pplus_minus_eq := Pos.add_sub (only parsing). +Notation Pmult_minus_distr_l := Pos.mul_sub_distr_l (only parsing). +Notation Pminus_lt_mono_l := Pos.sub_lt_mono_l (only parsing). +Notation Pminus_compare_mono_l := Pos.sub_compare_mono_l (only parsing). +Notation Pminus_compare_mono_r := Pos.sub_compare_mono_r (only parsing). +Notation Pminus_lt_mono_r := Pos.sub_lt_mono_r (only parsing). +Notation Pminus_decr := Pos.sub_decr (only parsing). +Notation Pminus_xI_xI := Pos.sub_xI_xI (only parsing). +Notation Pplus_minus_assoc := Pos.add_sub_assoc (only parsing). +Notation Pminus_plus_distr := Pos.sub_add_distr (only parsing). +Notation Pminus_minus_distr := Pos.sub_sub_distr (only parsing). +Notation Pminus_mask_Lt := Pos.sub_mask_neg (only parsing). +Notation Pminus_Lt := Pos.sub_lt (only parsing). +Notation Pminus_Eq := Pos.sub_diag (only parsing). +Notation Psize_monotone := Pos.size_nat_monotone (only parsing). +Notation Psize_pos_gt := Pos.size_gt (only parsing). +Notation Psize_pos_le := Pos.size_le (only parsing). + +(** More complex compatibility facts, expressed as lemmas + (to preserve scopes for instance) *) + +Lemma Peqb_true_eq x y : Pos.eqb x y = true -> x=y. +Proof. apply Pos.eqb_eq. Qed. +Lemma Pcompare_eq_Gt p q : (p ?= q) = Gt <-> p > q. +Proof. reflexivity. Qed. +Lemma Pplus_one_succ_r p : Psucc p = p + 1. +Proof (eq_sym (Pos.add_1_r p)). +Lemma Pplus_one_succ_l p : Psucc p = 1 + p. +Proof (eq_sym (Pos.add_1_l p)). +Lemma Pcompare_refl p : Pcompare p p Eq = Eq. +Proof (Pos.compare_cont_refl p Eq). +Lemma Pcompare_Eq_eq : forall p q, Pcompare p q Eq = Eq -> p = q. +Proof Pos.compare_eq. +Lemma ZC4 p q : Pcompare p q Eq = CompOpp (Pcompare q p Eq). +Proof (Pos.compare_antisym q p). +Lemma Ppred_minus p : Ppred p = p - 1. +Proof (eq_sym (Pos.sub_1_r p)). + +Lemma Pminus_mask_Gt p q : + p > q -> + exists h : positive, + Pminus_mask p q = IsPos h /\ + q + h = p /\ (h = 1 \/ Pminus_mask_carry p q = IsPos (Ppred h)). +Proof. + intros H. apply Pos.gt_lt in H. + destruct (Pos.sub_mask_pos p q H) as (r & U). + exists r. repeat split; trivial. + now apply Pos.sub_mask_pos_iff. + destruct (Pos.eq_dec r 1) as [EQ|NE]; [now left|right]. + rewrite Pos.sub_mask_carry_spec, U. destruct r; trivial. now elim NE. +Qed. + +Lemma Pplus_minus : forall p q, p > q -> q+(p-q) = p. +Proof. + intros. rewrite Pos.add_comm. now apply Pos.sub_add, Pos.gt_lt. +Qed. + +(** Discontinued results of little interest and little/zero use + in user contributions: + + Pplus_carry_no_neutral + Pplus_carry_pred_eq_plus + Pcompare_not_Eq + Pcompare_Lt_Lt + Pcompare_Lt_eq_Lt + Pcompare_Gt_Gt + Pcompare_Gt_eq_Gt + Psucc_lt_compat + Psucc_le_compat + ZC3 + Pcompare_p_Sq + Pminus_mask_carry_diag + Pminus_mask_IsNeg + ZL10 + ZL11 + double_eq_zero_inversion + double_plus_one_zero_discr + double_plus_one_eq_one_inversion + double_eq_one_discr + + Infix "/" := Pdiv2 : positive_scope. +*) + +(** Old stuff, to remove someday *) + +Lemma Dcompare : forall r:comparison, r = Eq \/ r = Lt \/ r = Gt. +Proof. + destruct r; auto. +Qed. + +(** Incompatibilities : + + - [(_ ?= _)%positive] expects no arg now, and designates [Pos.compare] + which is convertible but syntactically distinct to + [Pos.compare_cont .. .. Eq]. + + - [Pmult_nat] cannot be unfolded (unfold [Pos.iter_op] instead). + +*) diff --git a/theories/PArith/BinPosDef.v b/theories/PArith/BinPosDef.v new file mode 100644 index 00000000..7916511a --- /dev/null +++ b/theories/PArith/BinPosDef.v @@ -0,0 +1,565 @@ +(* -*- coding: utf-8 -*- *) +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(**********************************************************************) +(** * Binary positive numbers, operations *) +(**********************************************************************) + +(** Initial development by Pierre Crégut, CNET, Lannion, France *) + +(** The type [positive] and its constructors [xI] and [xO] and [xH] + are now defined in [BinNums.v] *) + +Require Export BinNums. + +(** Postfix notation for positive numbers, allowing to mimic + the position of bits in a big-endian representation. + For instance, we can write [1~1~0] instead of [(xO (xI xH))] + for the number 6 (which is 110 in binary notation). +*) + +Notation "p ~ 1" := (xI p) + (at level 7, left associativity, format "p '~' '1'") : positive_scope. +Notation "p ~ 0" := (xO p) + (at level 7, left associativity, format "p '~' '0'") : positive_scope. + +Local Open Scope positive_scope. +Local Unset Boolean Equality Schemes. +Local Unset Case Analysis Schemes. + +Module Pos. + +Definition t := positive. + +(** * Operations over positive numbers *) + +(** ** Successor *) + +Fixpoint succ x := + match x with + | p~1 => (succ p)~0 + | p~0 => p~1 + | 1 => 1~0 + end. + +(** ** Addition *) + +Fixpoint add x y := + match x, y with + | p~1, q~1 => (add_carry p q)~0 + | p~1, q~0 => (add p q)~1 + | p~1, 1 => (succ p)~0 + | p~0, q~1 => (add p q)~1 + | p~0, q~0 => (add p q)~0 + | p~0, 1 => p~1 + | 1, q~1 => (succ q)~0 + | 1, q~0 => q~1 + | 1, 1 => 1~0 + end + +with add_carry x y := + match x, y with + | p~1, q~1 => (add_carry p q)~1 + | p~1, q~0 => (add_carry p q)~0 + | p~1, 1 => (succ p)~1 + | p~0, q~1 => (add_carry p q)~0 + | p~0, q~0 => (add p q)~1 + | p~0, 1 => (succ p)~0 + | 1, q~1 => (succ q)~1 + | 1, q~0 => (succ q)~0 + | 1, 1 => 1~1 + end. + +Infix "+" := add : positive_scope. + +(** ** Operation [x -> 2*x-1] *) + +Fixpoint pred_double x := + match x with + | p~1 => p~0~1 + | p~0 => (pred_double p)~1 + | 1 => 1 + end. + +(** ** Predecessor *) + +Definition pred x := + match x with + | p~1 => p~0 + | p~0 => pred_double p + | 1 => 1 + end. + +(** ** The predecessor of a positive number can be seen as a [N] *) + +Definition pred_N x := + match x with + | p~1 => Npos (p~0) + | p~0 => Npos (pred_double p) + | 1 => N0 + end. + +(** ** An auxiliary type for subtraction *) + +Inductive mask : Set := +| IsNul : mask +| IsPos : positive -> mask +| IsNeg : mask. + +(** ** Operation [x -> 2*x+1] *) + +Definition succ_double_mask (x:mask) : mask := + match x with + | IsNul => IsPos 1 + | IsNeg => IsNeg + | IsPos p => IsPos p~1 + end. + +(** ** Operation [x -> 2*x] *) + +Definition double_mask (x:mask) : mask := + match x with + | IsNul => IsNul + | IsNeg => IsNeg + | IsPos p => IsPos p~0 + end. + +(** ** Operation [x -> 2*x-2] *) + +Definition double_pred_mask x : mask := + match x with + | p~1 => IsPos p~0~0 + | p~0 => IsPos (pred_double p)~0 + | 1 => IsNul + end. + +(** ** Predecessor with mask *) + +Definition pred_mask (p : mask) : mask := + match p with + | IsPos 1 => IsNul + | IsPos q => IsPos (pred q) + | IsNul => IsNeg + | IsNeg => IsNeg + end. + +(** ** Subtraction, result as a mask *) + +Fixpoint sub_mask (x y:positive) {struct y} : mask := + match x, y with + | p~1, q~1 => double_mask (sub_mask p q) + | p~1, q~0 => succ_double_mask (sub_mask p q) + | p~1, 1 => IsPos p~0 + | p~0, q~1 => succ_double_mask (sub_mask_carry p q) + | p~0, q~0 => double_mask (sub_mask p q) + | p~0, 1 => IsPos (pred_double p) + | 1, 1 => IsNul + | 1, _ => IsNeg + end + +with sub_mask_carry (x y:positive) {struct y} : mask := + match x, y with + | p~1, q~1 => succ_double_mask (sub_mask_carry p q) + | p~1, q~0 => double_mask (sub_mask p q) + | p~1, 1 => IsPos (pred_double p) + | p~0, q~1 => double_mask (sub_mask_carry p q) + | p~0, q~0 => succ_double_mask (sub_mask_carry p q) + | p~0, 1 => double_pred_mask p + | 1, _ => IsNeg + end. + +(** ** Subtraction, result as a positive, returning 1 if [x<=y] *) + +Definition sub x y := + match sub_mask x y with + | IsPos z => z + | _ => 1 + end. + +Infix "-" := sub : positive_scope. + +(** ** Multiplication *) + +Fixpoint mul x y := + match x with + | p~1 => y + (mul p y)~0 + | p~0 => (mul p y)~0 + | 1 => y + end. + +Infix "*" := mul : positive_scope. + +(** ** Iteration over a positive number *) + +Fixpoint iter (n:positive) {A} (f:A -> A) (x:A) : A := + match n with + | xH => f x + | xO n' => iter n' f (iter n' f x) + | xI n' => f (iter n' f (iter n' f x)) + end. + +(** ** Power *) + +Definition pow (x y:positive) := iter y (mul x) 1. + +Infix "^" := pow : positive_scope. + +(** ** Square *) + +Fixpoint square p := + match p with + | p~1 => (square p + p)~0~1 + | p~0 => (square p)~0~0 + | 1 => 1 + end. + +(** ** Division by 2 rounded below but for 1 *) + +Definition div2 p := + match p with + | 1 => 1 + | p~0 => p + | p~1 => p + end. + +(** Division by 2 rounded up *) + +Definition div2_up p := + match p with + | 1 => 1 + | p~0 => p + | p~1 => succ p + end. + +(** ** Number of digits in a positive number *) + +Fixpoint size_nat p : nat := + match p with + | 1 => S O + | p~1 => S (size_nat p) + | p~0 => S (size_nat p) + end. + +(** Same, with positive output *) + +Fixpoint size p := + match p with + | 1 => 1 + | p~1 => succ (size p) + | p~0 => succ (size p) + end. + +(** ** Comparison on binary positive numbers *) + +Fixpoint compare_cont (x y:positive) (r:comparison) {struct y} : comparison := + match x, y with + | p~1, q~1 => compare_cont p q r + | p~1, q~0 => compare_cont p q Gt + | p~1, 1 => Gt + | p~0, q~1 => compare_cont p q Lt + | p~0, q~0 => compare_cont p q r + | p~0, 1 => Gt + | 1, q~1 => Lt + | 1, q~0 => Lt + | 1, 1 => r + end. + +Definition compare x y := compare_cont x y Eq. + +Infix "?=" := compare (at level 70, no associativity) : positive_scope. + +Definition min p p' := + match p ?= p' with + | Lt | Eq => p + | Gt => p' + end. + +Definition max p p' := + match p ?= p' with + | Lt | Eq => p' + | Gt => p + end. + +(** ** Boolean equality and comparisons *) + +(** Nota: this [eqb] is not convertible with the generated [positive_beq], due + to a different guard argument. We keep this version for compatibility. *) + +Fixpoint eqb p q {struct q} := + match p, q with + | p~1, q~1 => eqb p q + | p~0, q~0 => eqb p q + | 1, 1 => true + | _, _ => false + end. + +Definition leb x y := + match x ?= y with Gt => false | _ => true end. + +Definition ltb x y := + match x ?= y with Lt => true | _ => false end. + +Infix "=?" := eqb (at level 70, no associativity) : positive_scope. +Infix "<=?" := leb (at level 70, no associativity) : positive_scope. +Infix "<?" := ltb (at level 70, no associativity) : positive_scope. + +(** ** A Square Root function for positive numbers *) + +(** We procede by blocks of two digits : if p is written qbb' + then sqrt(p) will be sqrt(q)~0 or sqrt(q)~1. + For deciding easily in which case we are, we store the remainder + (as a mask, since it can be null). + Instead of copy-pasting the following code four times, we + factorize as an auxiliary function, with f and g being either + xO or xI depending of the initial digits. + NB: (sub_mask (g (f 1)) 4) is a hack, morally it's g (f 0). +*) + +Definition sqrtrem_step (f g:positive->positive) p := + match p with + | (s, IsPos r) => + let s' := s~0~1 in + let r' := g (f r) in + if s' <=? r' then (s~1, sub_mask r' s') + else (s~0, IsPos r') + | (s,_) => (s~0, sub_mask (g (f 1)) 4) + end. + +Fixpoint sqrtrem p : positive * mask := + match p with + | 1 => (1,IsNul) + | 2 => (1,IsPos 1) + | 3 => (1,IsPos 2) + | p~0~0 => sqrtrem_step xO xO (sqrtrem p) + | p~0~1 => sqrtrem_step xO xI (sqrtrem p) + | p~1~0 => sqrtrem_step xI xO (sqrtrem p) + | p~1~1 => sqrtrem_step xI xI (sqrtrem p) + end. + +Definition sqrt p := fst (sqrtrem p). + + +(** ** Greatest Common Divisor *) + +Definition divide p q := exists r, q = r*p. +Notation "( p | q )" := (divide p q) (at level 0) : positive_scope. + +(** Instead of the Euclid algorithm, we use here the Stein binary + algorithm, which is faster for this representation. This algorithm + is almost structural, but in the last cases we do some recursive + calls on subtraction, hence the need for a counter. +*) + +Fixpoint gcdn (n : nat) (a b : positive) : positive := + match n with + | O => 1 + | S n => + match a,b with + | 1, _ => 1 + | _, 1 => 1 + | a~0, b~0 => (gcdn n a b)~0 + | _ , b~0 => gcdn n a b + | a~0, _ => gcdn n a b + | a'~1, b'~1 => + match a' ?= b' with + | Eq => a + | Lt => gcdn n (b'-a') a + | Gt => gcdn n (a'-b') b + end + end + end. + +(** We'll show later that we need at most (log2(a.b)) loops *) + +Definition gcd (a b : positive) := gcdn (size_nat a + size_nat b)%nat a b. + +(** Generalized Gcd, also computing the division of a and b by the gcd *) + +Fixpoint ggcdn (n : nat) (a b : positive) : (positive*(positive*positive)) := + match n with + | O => (1,(a,b)) + | S n => + match a,b with + | 1, _ => (1,(1,b)) + | _, 1 => (1,(a,1)) + | a~0, b~0 => + let (g,p) := ggcdn n a b in + (g~0,p) + | _, b~0 => + let '(g,(aa,bb)) := ggcdn n a b in + (g,(aa, bb~0)) + | a~0, _ => + let '(g,(aa,bb)) := ggcdn n a b in + (g,(aa~0, bb)) + | a'~1, b'~1 => + match a' ?= b' with + | Eq => (a,(1,1)) + | Lt => + let '(g,(ba,aa)) := ggcdn n (b'-a') a in + (g,(aa, aa + ba~0)) + | Gt => + let '(g,(ab,bb)) := ggcdn n (a'-b') b in + (g,(bb + ab~0, bb)) + end + end + end. + +Definition ggcd (a b: positive) := ggcdn (size_nat a + size_nat b)%nat a b. + +(** Local copies of the not-yet-available [N.double] and [N.succ_double] *) + +Definition Nsucc_double x := + match x with + | N0 => Npos 1 + | Npos p => Npos p~1 + end. + +Definition Ndouble n := + match n with + | N0 => N0 + | Npos p => Npos p~0 + end. + +(** Operation over bits. *) + +(** Logical [or] *) + +Fixpoint lor (p q : positive) : positive := + match p, q with + | 1, q~0 => q~1 + | 1, _ => q + | p~0, 1 => p~1 + | _, 1 => p + | p~0, q~0 => (lor p q)~0 + | p~0, q~1 => (lor p q)~1 + | p~1, q~0 => (lor p q)~1 + | p~1, q~1 => (lor p q)~1 + end. + +(** Logical [and] *) + +Fixpoint land (p q : positive) : N := + match p, q with + | 1, q~0 => N0 + | 1, _ => Npos 1 + | p~0, 1 => N0 + | _, 1 => Npos 1 + | p~0, q~0 => Ndouble (land p q) + | p~0, q~1 => Ndouble (land p q) + | p~1, q~0 => Ndouble (land p q) + | p~1, q~1 => Nsucc_double (land p q) + end. + +(** Logical [diff] *) + +Fixpoint ldiff (p q:positive) : N := + match p, q with + | 1, q~0 => Npos 1 + | 1, _ => N0 + | _~0, 1 => Npos p + | p~1, 1 => Npos (p~0) + | p~0, q~0 => Ndouble (ldiff p q) + | p~0, q~1 => Ndouble (ldiff p q) + | p~1, q~1 => Ndouble (ldiff p q) + | p~1, q~0 => Nsucc_double (ldiff p q) + end. + +(** [xor] *) + +Fixpoint lxor (p q:positive) : N := + match p, q with + | 1, 1 => N0 + | 1, q~0 => Npos (q~1) + | 1, q~1 => Npos (q~0) + | p~0, 1 => Npos (p~1) + | p~0, q~0 => Ndouble (lxor p q) + | p~0, q~1 => Nsucc_double (lxor p q) + | p~1, 1 => Npos (p~0) + | p~1, q~0 => Nsucc_double (lxor p q) + | p~1, q~1 => Ndouble (lxor p q) + end. + +(** Shifts. NB: right shift of 1 stays at 1. *) + +Definition shiftl_nat (p:positive)(n:nat) := nat_iter n xO p. +Definition shiftr_nat (p:positive)(n:nat) := nat_iter n div2 p. + +Definition shiftl (p:positive)(n:N) := + match n with + | N0 => p + | Npos n => iter n xO p + end. + +Definition shiftr (p:positive)(n:N) := + match n with + | N0 => p + | Npos n => iter n div2 p + end. + +(** Checking whether a particular bit is set or not *) + +Fixpoint testbit_nat (p:positive) : nat -> bool := + match p with + | 1 => fun n => match n with + | O => true + | S _ => false + end + | p~0 => fun n => match n with + | O => false + | S n' => testbit_nat p n' + end + | p~1 => fun n => match n with + | O => true + | S n' => testbit_nat p n' + end + end. + +(** Same, but with index in N *) + +Fixpoint testbit (p:positive)(n:N) := + match p, n with + | p~0, N0 => false + | _, N0 => true + | 1, _ => false + | p~0, Npos n => testbit p (pred_N n) + | p~1, Npos n => testbit p (pred_N n) + end. + +(** ** From binary positive numbers to Peano natural numbers *) + +Definition iter_op {A}(op:A->A->A) := + fix iter (p:positive)(a:A) : A := + match p with + | 1 => a + | p~0 => iter p (op a a) + | p~1 => op a (iter p (op a a)) + end. + +Definition to_nat (x:positive) : nat := iter_op plus x (S O). + +(** ** From Peano natural numbers to binary positive numbers *) + +(** A version preserving positive numbers, and sending 0 to 1. *) + +Fixpoint of_nat (n:nat) : positive := + match n with + | O => 1 + | S O => 1 + | S x => succ (of_nat x) + end. + +(* Another version that converts [n] into [n+1] *) + +Fixpoint of_succ_nat (n:nat) : positive := + match n with + | O => 1 + | S x => succ (of_succ_nat x) + end. + +End Pos.
\ No newline at end of file diff --git a/theories/PArith/PArith.v b/theories/PArith/PArith.v new file mode 100644 index 00000000..26b8265b --- /dev/null +++ b/theories/PArith/PArith.v @@ -0,0 +1,11 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(** Library for positive natural numbers *) + +Require Export BinNums BinPos Pnat POrderedType. diff --git a/theories/PArith/POrderedType.v b/theories/PArith/POrderedType.v new file mode 100644 index 00000000..de7b2b82 --- /dev/null +++ b/theories/PArith/POrderedType.v @@ -0,0 +1,36 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import BinPos Equalities Orders OrdersTac. + +Local Open Scope positive_scope. + +(** * DecidableType structure for [positive] numbers *) + +Module Positive_as_DT <: UsualDecidableTypeFull := Pos. + +(** Note that the last module fulfills by subtyping many other + interfaces, such as [DecidableType] or [EqualityType]. *) + + +(** * OrderedType structure for [positive] numbers *) + +Module Positive_as_OT <: OrderedTypeFull := Pos. + +(** Note that [Positive_as_OT] can also be seen as a [UsualOrderedType] + and a [OrderedType] (and also as a [DecidableType]). *) + + + +(** * An [order] tactic for positive numbers *) + +Module PositiveOrder := OTF_to_OrderTac Positive_as_OT. +Ltac p_order := PositiveOrder.order. + +(** Note that [p_order] is domain-agnostic: it will not prove + [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *) diff --git a/theories/PArith/Pnat.v b/theories/PArith/Pnat.v new file mode 100644 index 00000000..f9df70bd --- /dev/null +++ b/theories/PArith/Pnat.v @@ -0,0 +1,483 @@ +(* -*- coding: utf-8 -*- *) +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Require Import BinPos Le Lt Gt Plus Mult Minus Compare_dec. + +(** Properties of the injection from binary positive numbers + to Peano natural numbers *) + +(** Original development by Pierre Crégut, CNET, Lannion, France *) + +Local Open Scope positive_scope. +Local Open Scope nat_scope. + +Module Pos2Nat. + Import Pos. + +(** [Pos.to_nat] is a morphism for successor, addition, multiplication *) + +Lemma inj_succ p : to_nat (succ p) = S (to_nat p). +Proof. + unfold to_nat. rewrite iter_op_succ. trivial. + apply plus_assoc. +Qed. + +Theorem inj_add p q : to_nat (p + q) = to_nat p + to_nat q. +Proof. + revert q. induction p using Pind; intros q. + now rewrite add_1_l, inj_succ. + now rewrite add_succ_l, !inj_succ, IHp. +Qed. + +Theorem inj_mul p q : to_nat (p * q) = to_nat p * to_nat q. +Proof. + revert q. induction p using peano_ind; simpl; intros; trivial. + now rewrite mul_succ_l, inj_add, IHp, inj_succ. +Qed. + +(** Mapping of xH, xO and xI through [Pos.to_nat] *) + +Lemma inj_1 : to_nat 1 = 1. +Proof. + reflexivity. +Qed. + +Lemma inj_xO p : to_nat (xO p) = 2 * to_nat p. +Proof. + exact (inj_mul 2 p). +Qed. + +Lemma inj_xI p : to_nat (xI p) = S (2 * to_nat p). +Proof. + now rewrite xI_succ_xO, inj_succ, inj_xO. +Qed. + +(** [Pos.to_nat] maps to the strictly positive subset of [nat] *) + +Lemma is_succ : forall p, exists n, to_nat p = S n. +Proof. + induction p using peano_ind. + now exists 0. + destruct IHp as (n,Hn). exists (S n). now rewrite inj_succ, Hn. +Qed. + +(** [Pos.to_nat] is strictly positive *) + +Lemma is_pos p : 0 < to_nat p. +Proof. + destruct (is_succ p) as (n,->). auto with arith. +Qed. + +(** [Pos.to_nat] is a bijection between [positive] and + non-zero [nat], with [Pos.of_nat] as reciprocal. + See [Nat2Pos.id] below for the dual equation. *) + +Theorem id p : of_nat (to_nat p) = p. +Proof. + induction p using peano_ind. trivial. + rewrite inj_succ. rewrite <- IHp at 2. + now destruct (is_succ p) as (n,->). +Qed. + +(** [Pos.to_nat] is hence injective *) + +Lemma inj p q : to_nat p = to_nat q -> p = q. +Proof. + intros H. now rewrite <- (id p), <- (id q), H. +Qed. + +Lemma inj_iff p q : to_nat p = to_nat q <-> p = q. +Proof. + split. apply inj. intros; now subst. +Qed. + +(** [Pos.to_nat] is a morphism for comparison *) + +Lemma inj_compare p q : (p ?= q) = nat_compare (to_nat p) (to_nat q). +Proof. + revert q. induction p as [ |p IH] using peano_ind; intros q. + destruct (succ_pred_or q) as [Hq|Hq]; [now subst|]. + rewrite <- Hq, lt_1_succ, inj_succ, inj_1, nat_compare_S. + symmetry. apply nat_compare_lt, is_pos. + destruct (succ_pred_or q) as [Hq|Hq]; [subst|]. + rewrite compare_antisym, lt_1_succ, inj_succ. simpl. + symmetry. apply nat_compare_gt, is_pos. + now rewrite <- Hq, 2 inj_succ, compare_succ_succ, IH. +Qed. + +(** [Pos.to_nat] is a morphism for [lt], [le], etc *) + +Lemma inj_lt p q : (p < q)%positive <-> to_nat p < to_nat q. +Proof. + unfold lt. now rewrite inj_compare, nat_compare_lt. +Qed. + +Lemma inj_le p q : (p <= q)%positive <-> to_nat p <= to_nat q. +Proof. + unfold le. now rewrite inj_compare, nat_compare_le. +Qed. + +Lemma inj_gt p q : (p > q)%positive <-> to_nat p > to_nat q. +Proof. + unfold gt. now rewrite inj_compare, nat_compare_gt. +Qed. + +Lemma inj_ge p q : (p >= q)%positive <-> to_nat p >= to_nat q. +Proof. + unfold ge. now rewrite inj_compare, nat_compare_ge. +Qed. + +(** [Pos.to_nat] is a morphism for subtraction *) + +Theorem inj_sub p q : (q < p)%positive -> + to_nat (p - q) = to_nat p - to_nat q. +Proof. + intro H; apply plus_reg_l with (to_nat q); rewrite le_plus_minus_r. + now rewrite <- inj_add, add_comm, sub_add. + now apply lt_le_weak, inj_lt. +Qed. + +Theorem inj_sub_max p q : + to_nat (p - q) = Peano.max 1 (to_nat p - to_nat q). +Proof. + destruct (ltb_spec q p). + rewrite <- inj_sub by trivial. + now destruct (is_succ (p - q)) as (m,->). + rewrite sub_le by trivial. + replace (to_nat p - to_nat q) with 0; trivial. + apply le_n_0_eq. + rewrite <- (minus_diag (to_nat p)). + now apply minus_le_compat_l, inj_le. +Qed. + +Theorem inj_pred p : (1 < p)%positive -> + to_nat (pred p) = Peano.pred (to_nat p). +Proof. + intros H. now rewrite <- Pos.sub_1_r, inj_sub, pred_of_minus. +Qed. + +Theorem inj_pred_max p : + to_nat (pred p) = Peano.max 1 (Peano.pred (to_nat p)). +Proof. + rewrite <- Pos.sub_1_r, pred_of_minus. apply inj_sub_max. +Qed. + +(** [Pos.to_nat] and other operations *) + +Lemma inj_min p q : + to_nat (min p q) = Peano.min (to_nat p) (to_nat q). +Proof. + unfold min. rewrite inj_compare. + case nat_compare_spec; intros H; symmetry. + apply Peano.min_l. now rewrite H. + now apply Peano.min_l, lt_le_weak. + now apply Peano.min_r, lt_le_weak. +Qed. + +Lemma inj_max p q : + to_nat (max p q) = Peano.max (to_nat p) (to_nat q). +Proof. + unfold max. rewrite inj_compare. + case nat_compare_spec; intros H; symmetry. + apply Peano.max_r. now rewrite H. + now apply Peano.max_r, lt_le_weak. + now apply Peano.max_l, lt_le_weak. +Qed. + +Theorem inj_iter : + forall p {A} (f:A->A) (x:A), + Pos.iter p f x = nat_iter (to_nat p) f x. +Proof. + induction p using peano_ind. trivial. + intros. rewrite inj_succ, iter_succ. simpl. now f_equal. +Qed. + +End Pos2Nat. + +Module Nat2Pos. + +(** [Pos.of_nat] is a bijection between non-zero [nat] and + [positive], with [Pos.to_nat] as reciprocal. + See [Pos2Nat.id] above for the dual equation. *) + +Theorem id (n:nat) : n<>0 -> Pos.to_nat (Pos.of_nat n) = n. +Proof. + induction n as [|n H]; trivial. now destruct 1. + intros _. simpl. destruct n. trivial. + rewrite Pos2Nat.inj_succ. f_equal. now apply H. +Qed. + +Theorem id_max (n:nat) : Pos.to_nat (Pos.of_nat n) = max 1 n. +Proof. + destruct n. trivial. now rewrite id. +Qed. + +(** [Pos.of_nat] is hence injective for non-zero numbers *) + +Lemma inj (n m : nat) : n<>0 -> m<>0 -> Pos.of_nat n = Pos.of_nat m -> n = m. +Proof. + intros Hn Hm H. now rewrite <- (id n), <- (id m), H. +Qed. + +Lemma inj_iff (n m : nat) : n<>0 -> m<>0 -> + (Pos.of_nat n = Pos.of_nat m <-> n = m). +Proof. + split. now apply inj. intros; now subst. +Qed. + +(** Usual operations are morphisms with respect to [Pos.of_nat] + for non-zero numbers. *) + +Lemma inj_succ (n:nat) : n<>0 -> Pos.of_nat (S n) = Pos.succ (Pos.of_nat n). +Proof. +intro H. apply Pos2Nat.inj. now rewrite Pos2Nat.inj_succ, !id. +Qed. + +Lemma inj_pred (n:nat) : Pos.of_nat (pred n) = Pos.pred (Pos.of_nat n). +Proof. + destruct n as [|[|n]]; trivial. simpl. now rewrite Pos.pred_succ. +Qed. + +Lemma inj_add (n m : nat) : n<>0 -> m<>0 -> + Pos.of_nat (n+m) = (Pos.of_nat n + Pos.of_nat m)%positive. +Proof. +intros Hn Hm. apply Pos2Nat.inj. +rewrite Pos2Nat.inj_add, !id; trivial. +intros H. destruct n. now destruct Hn. now simpl in H. +Qed. + +Lemma inj_mul (n m : nat) : n<>0 -> m<>0 -> + Pos.of_nat (n*m) = (Pos.of_nat n * Pos.of_nat m)%positive. +Proof. +intros Hn Hm. apply Pos2Nat.inj. +rewrite Pos2Nat.inj_mul, !id; trivial. +intros H. apply mult_is_O in H. destruct H. now elim Hn. now elim Hm. +Qed. + +Lemma inj_compare (n m : nat) : n<>0 -> m<>0 -> + nat_compare n m = (Pos.of_nat n ?= Pos.of_nat m). +Proof. +intros Hn Hm. rewrite Pos2Nat.inj_compare, !id; trivial. +Qed. + +Lemma inj_sub (n m : nat) : m<>0 -> + Pos.of_nat (n-m) = (Pos.of_nat n - Pos.of_nat m)%positive. +Proof. + intros Hm. + apply Pos2Nat.inj. + rewrite Pos2Nat.inj_sub_max. + rewrite (id m) by trivial. rewrite !id_max. + destruct n, m; trivial. +Qed. + +Lemma inj_min (n m : nat) : + Pos.of_nat (min n m) = Pos.min (Pos.of_nat n) (Pos.of_nat m). +Proof. + destruct n as [|n]. simpl. symmetry. apply Pos.min_l, Pos.le_1_l. + destruct m as [|m]. simpl. symmetry. apply Pos.min_r, Pos.le_1_l. + unfold Pos.min. rewrite <- inj_compare by easy. + case nat_compare_spec; intros H; f_equal; apply min_l || apply min_r. + rewrite H; auto. now apply lt_le_weak. now apply lt_le_weak. +Qed. + +Lemma inj_max (n m : nat) : + Pos.of_nat (max n m) = Pos.max (Pos.of_nat n) (Pos.of_nat m). +Proof. + destruct n as [|n]. simpl. symmetry. apply Pos.max_r, Pos.le_1_l. + destruct m as [|m]. simpl. symmetry. apply Pos.max_l, Pos.le_1_l. + unfold Pos.max. rewrite <- inj_compare by easy. + case nat_compare_spec; intros H; f_equal; apply max_l || apply max_r. + rewrite H; auto. now apply lt_le_weak. now apply lt_le_weak. +Qed. + +End Nat2Pos. + +(**********************************************************************) +(** Properties of the shifted injection from Peano natural numbers + to binary positive numbers *) + +Module Pos2SuccNat. + +(** Composition of [Pos.to_nat] and [Pos.of_succ_nat] is successor + on [positive] *) + +Theorem id_succ p : Pos.of_succ_nat (Pos.to_nat p) = Pos.succ p. +Proof. +rewrite Pos.of_nat_succ, <- Pos2Nat.inj_succ. apply Pos2Nat.id. +Qed. + +(** Composition of [Pos.to_nat], [Pos.of_succ_nat] and [Pos.pred] + is identity on [positive] *) + +Theorem pred_id p : Pos.pred (Pos.of_succ_nat (Pos.to_nat p)) = p. +Proof. +now rewrite id_succ, Pos.pred_succ. +Qed. + +End Pos2SuccNat. + +Module SuccNat2Pos. + +(** Composition of [Pos.of_succ_nat] and [Pos.to_nat] is successor on [nat] *) + +Theorem id_succ (n:nat) : Pos.to_nat (Pos.of_succ_nat n) = S n. +Proof. +rewrite Pos.of_nat_succ. now apply Nat2Pos.id. +Qed. + +Theorem pred_id (n:nat) : pred (Pos.to_nat (Pos.of_succ_nat n)) = n. +Proof. +now rewrite id_succ. +Qed. + +(** [Pos.of_succ_nat] is hence injective *) + +Lemma inj (n m : nat) : Pos.of_succ_nat n = Pos.of_succ_nat m -> n = m. +Proof. + intro H. apply (f_equal Pos.to_nat) in H. rewrite !id_succ in H. + now injection H. +Qed. + +Lemma inj_iff (n m : nat) : Pos.of_succ_nat n = Pos.of_succ_nat m <-> n = m. +Proof. + split. apply inj. intros; now subst. +Qed. + +(** Another formulation *) + +Theorem inv n p : Pos.to_nat p = S n -> Pos.of_succ_nat n = p. +Proof. + intros H. apply Pos2Nat.inj. now rewrite id_succ. +Qed. + +(** Successor and comparison are morphisms with respect to + [Pos.of_succ_nat] *) + +Lemma inj_succ n : Pos.of_succ_nat (S n) = Pos.succ (Pos.of_succ_nat n). +Proof. +apply Pos2Nat.inj. now rewrite Pos2Nat.inj_succ, !id_succ. +Qed. + +Lemma inj_compare n m : + nat_compare n m = (Pos.of_succ_nat n ?= Pos.of_succ_nat m). +Proof. +rewrite Pos2Nat.inj_compare, !id_succ; trivial. +Qed. + +(** Other operations, for instance [Pos.add] and [plus] aren't + directly related this way (we would need to compensate for + the successor hidden in [Pos.of_succ_nat] *) + +End SuccNat2Pos. + +(** For compatibility, old names and old-style lemmas *) + +Notation Psucc_S := Pos2Nat.inj_succ (only parsing). +Notation Pplus_plus := Pos2Nat.inj_add (only parsing). +Notation Pmult_mult := Pos2Nat.inj_mul (only parsing). +Notation Pcompare_nat_compare := Pos2Nat.inj_compare (only parsing). +Notation nat_of_P_xH := Pos2Nat.inj_1 (only parsing). +Notation nat_of_P_xO := Pos2Nat.inj_xO (only parsing). +Notation nat_of_P_xI := Pos2Nat.inj_xI (only parsing). +Notation nat_of_P_is_S := Pos2Nat.is_succ (only parsing). +Notation nat_of_P_pos := Pos2Nat.is_pos (only parsing). +Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (only parsing). +Notation nat_of_P_inj := Pos2Nat.inj (only parsing). +Notation Plt_lt := Pos2Nat.inj_lt (only parsing). +Notation Pgt_gt := Pos2Nat.inj_gt (only parsing). +Notation Ple_le := Pos2Nat.inj_le (only parsing). +Notation Pge_ge := Pos2Nat.inj_ge (only parsing). +Notation Pminus_minus := Pos2Nat.inj_sub (only parsing). +Notation iter_nat_of_P := @Pos2Nat.inj_iter (only parsing). + +Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (only parsing). +Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (only parsing). + +Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (only parsing). +Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (only parsing). +Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (only parsing). +Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (only parsing). +Notation lt_O_nat_of_P := Pos2Nat.is_pos (only parsing). +Notation ZL4 := Pos2Nat.is_succ (only parsing). +Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (only parsing). +Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (only parsing). +Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (only parsing). + +Lemma nat_of_P_minus_morphism p q : + Pcompare p q Eq = Gt -> Pos.to_nat (p - q) = Pos.to_nat p - Pos.to_nat q. +Proof (fun H => Pos2Nat.inj_sub p q (ZC1 _ _ H)). + +Lemma nat_of_P_lt_Lt_compare_morphism p q : + Pcompare p q Eq = Lt -> Pos.to_nat p < Pos.to_nat q. +Proof (proj1 (Pos2Nat.inj_lt p q)). + +Lemma nat_of_P_gt_Gt_compare_morphism p q : + Pcompare p q Eq = Gt -> Pos.to_nat p > Pos.to_nat q. +Proof (proj1 (Pos2Nat.inj_gt p q)). + +Lemma nat_of_P_lt_Lt_compare_complement_morphism p q : + Pos.to_nat p < Pos.to_nat q -> Pcompare p q Eq = Lt. +Proof (proj2 (Pos2Nat.inj_lt p q)). + +Definition nat_of_P_gt_Gt_compare_complement_morphism p q : + Pos.to_nat p > Pos.to_nat q -> Pcompare p q Eq = Gt. +Proof (proj2 (Pos2Nat.inj_gt p q)). + +(** Old intermediate results about [Pmult_nat] *) + +Section ObsoletePmultNat. + +Lemma Pmult_nat_mult : forall p n, + Pmult_nat p n = Pos.to_nat p * n. +Proof. + induction p; intros n; unfold Pos.to_nat; simpl. + f_equal. rewrite 2 IHp. rewrite <- mult_assoc. + f_equal. simpl. now rewrite <- plus_n_O. + rewrite 2 IHp. rewrite <- mult_assoc. + f_equal. simpl. now rewrite <- plus_n_O. + simpl. now rewrite <- plus_n_O. +Qed. + +Lemma Pmult_nat_succ_morphism : + forall p n, Pmult_nat (Psucc p) n = n + Pmult_nat p n. +Proof. + intros. now rewrite !Pmult_nat_mult, Pos2Nat.inj_succ. +Qed. + +Theorem Pmult_nat_l_plus_morphism : + forall p q n, Pmult_nat (p + q) n = Pmult_nat p n + Pmult_nat q n. +Proof. + intros. rewrite !Pmult_nat_mult, Pos2Nat.inj_add. apply mult_plus_distr_r. +Qed. + +Theorem Pmult_nat_plus_carry_morphism : + forall p q n, Pmult_nat (Pplus_carry p q) n = n + Pmult_nat (p + q) n. +Proof. + intros. now rewrite Pos.add_carry_spec, Pmult_nat_succ_morphism. +Qed. + +Lemma Pmult_nat_r_plus_morphism : + forall p n, Pmult_nat p (n + n) = Pmult_nat p n + Pmult_nat p n. +Proof. + intros. rewrite !Pmult_nat_mult. apply mult_plus_distr_l. +Qed. + +Lemma ZL6 : forall p, Pmult_nat p 2 = Pos.to_nat p + Pos.to_nat p. +Proof. + intros. rewrite Pmult_nat_mult, mult_comm. simpl. now rewrite <- plus_n_O. +Qed. + +Lemma le_Pmult_nat : forall p n, n <= Pmult_nat p n. +Proof. + intros. rewrite Pmult_nat_mult. + apply le_trans with (1*n). now rewrite mult_1_l. + apply mult_le_compat_r. apply Pos2Nat.is_pos. +Qed. + +End ObsoletePmultNat. diff --git a/theories/PArith/intro.tex b/theories/PArith/intro.tex new file mode 100644 index 00000000..ffce881e --- /dev/null +++ b/theories/PArith/intro.tex @@ -0,0 +1,4 @@ +\section{Binary positive integers : PArith}\label{PArith} + +Here are defined various arithmetical notions and their properties, +similar to those of {\tt Arith}. diff --git a/theories/PArith/vo.itarget b/theories/PArith/vo.itarget new file mode 100644 index 00000000..73044e2c --- /dev/null +++ b/theories/PArith/vo.itarget @@ -0,0 +1,5 @@ +BinPosDef.vo +BinPos.vo +Pnat.vo +POrderedType.vo +PArith.vo
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