path: root/theories/NArith/BinNat.v

(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

Require Export BinNums.
Require Import BinPos RelationClasses Morphisms Setoid
 Equalities OrdersFacts GenericMinMax Bool NAxioms NProperties.
Require BinNatDef.

(**********************************************************************)
(** * Binary natural numbers, operations and properties *)
(**********************************************************************)

(** The type [N] and its constructors [N0] and [Npos] are now
    defined in [BinNums.v] *)

(** Every definitions and properties about binary natural numbers
    are placed in a module [N] for qualification purpose. *)

Local Open Scope N_scope.

(** Every definitions and early properties about positive numbers
    are placed in a module [N] for qualification purpose. *)

Module N
 <: NAxiomsSig
 <: UsualOrderedTypeFull
 <: UsualDecidableTypeFull
 <: TotalOrder.

(** Definitions of operations, now in a separate file *)

Include BinNatDef.N.

(** When including property functors, only inline t eq zero one two *)

Set Inline Level 30.

(** Logical predicates *)

Definition eq := @Logic.eq N.
Definition eq_equiv := @eq_equivalence N.

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 : N_scope.
Infix "<" := lt : N_scope.
Infix ">=" := ge : N_scope.
Infix ">" := gt : N_scope.

Notation "x <= y <= z" := (x <= y /\ y <= z) : N_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : N_scope.
Notation "x < y < z" := (x < y /\ y < z) : N_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : N_scope.

Definition divide p q := exists r, q = r*p.
Notation "( p | q )" := (divide p q) (at level 0) : N_scope.

Definition Even n := exists m, n = 2*m.
Definition Odd n := exists m, n = 2*m+1.

(** Decidability of equality. *)

Definition eq_dec : forall n m : N, { n = m } + { n <> m }.
Proof.
 decide equality.
 apply Pos.eq_dec.
Defined.

(** Discrimination principle *)

Definition discr n : { p:positive | n = pos p } + { n = 0 }.
Proof.
 destruct n; auto.
 left; exists p; auto.
Defined.

(** Convenient induction principles *)

Definition binary_rect (P:N -> Type) (f0 : P 0)
  (f2 : forall n, P n -> P (double n))
  (fS2 : forall n, P n -> P (succ_double n)) (n : N) : P n :=
 let P' p := P (pos p) in
 let f2' p := f2 (pos p) in
 let fS2' p := fS2 (pos p) in
 match n with
 | 0 => f0
 | pos p => positive_rect P' fS2' f2' (fS2 0 f0) p
 end.

Definition binary_rec (P:N -> Set) := binary_rect P.
Definition binary_ind (P:N -> Prop) := binary_rect P.

(** Peano induction on binary natural numbers *)

Definition peano_rect
  (P : N -> Type) (f0 : P 0)
    (f : forall n : N, P n -> P (succ n)) (n : N) : P n :=
let P' p := P (pos p) in
let f' p := f (pos p) in
match n with
| 0 => f0
| pos p => Pos.peano_rect P' (f 0 f0) f' p
end.

Theorem peano_rect_base P a f : peano_rect P a f 0 = a.
Proof.
reflexivity.
Qed.

Theorem peano_rect_succ P a f n :
 peano_rect P a f (succ n) = f n (peano_rect P a f n).
Proof.
destruct n; simpl.
trivial.
now rewrite Pos.peano_rect_succ.
Qed.

Definition peano_ind (P : N -> Prop) := peano_rect P.

Definition peano_rec (P : N -> Set) := peano_rect P.

Theorem peano_rec_base P a f : peano_rec P a f 0 = a.
Proof.
apply peano_rect_base.
Qed.

Theorem peano_rec_succ P a f n :
 peano_rec P a f (succ n) = f n (peano_rec P a f n).
Proof.
apply peano_rect_succ.
Qed.

(** Properties of mixed successor and predecessor. *)

Lemma pos_pred_spec p : Pos.pred_N p = pred (pos p).
Proof.
 now destruct p.
Qed.

Lemma succ_pos_spec n : pos (succ_pos n) = succ n.
Proof.
 now destruct n.
Qed.

Lemma pos_pred_succ n : Pos.pred_N (succ_pos n) = n.
Proof.
 destruct n. trivial. apply Pos.pred_N_succ.
Qed.

Lemma succ_pos_pred p : succ (Pos.pred_N p) = pos p.
Proof.
 destruct p; simpl; trivial. f_equal. apply Pos.succ_pred_double.
Qed.

(** Properties of successor and predecessor *)

Theorem pred_succ n : pred (succ n) = n.
Proof.
destruct n; trivial. simpl. apply Pos.pred_N_succ.
Qed.

Theorem pred_sub n : pred n = sub n 1.
Proof.
 now destruct n as [|[p|p|]].
Qed.

Theorem succ_0_discr n : succ n <> 0.
Proof.
now destruct n.
Qed.

(** Specification of addition *)

Theorem add_0_l n : 0 + n = n.
Proof.
reflexivity.
Qed.

Theorem add_succ_l n m : succ n + m = succ (n + m).
Proof.
destruct n, m; unfold succ, add; now rewrite ?Pos.add_1_l, ?Pos.add_succ_l.
Qed.

(** Specification of subtraction. *)

Theorem sub_0_r n : n - 0 = n.
Proof.
now destruct n.
Qed.

Theorem sub_succ_r n m : n - succ m = pred (n - m).
Proof.
destruct n as [|p], m as [|q]; trivial.
now destruct p.
simpl. rewrite Pos.sub_mask_succ_r, Pos.sub_mask_carry_spec.
now destruct (Pos.sub_mask p q) as [|[r|r|]|].
Qed.

(** Specification of multiplication *)

Theorem mul_0_l n : 0 * n = 0.
Proof.
reflexivity.
Qed.

Theorem mul_succ_l n m : (succ n) * m = n * m + m.
Proof.
destruct n, m; simpl; trivial. f_equal. rewrite Pos.add_comm.
apply Pos.mul_succ_l.
Qed.

(** Specification of boolean comparisons. *)

Lemma eqb_eq n m : eqb n m = true <-> n=m.
Proof.
destruct n as [|n], m as [|m]; simpl; try easy'.
rewrite Pos.eqb_eq. split; intro H. now subst. now destr_eq H.
Qed.

Lemma ltb_lt n m : (n <? m) = true <-> n < m.
Proof.
  unfold ltb, lt. destruct compare; easy'.
Qed.

Lemma leb_le n m : (n <=? m) = true <-> n <= m.
Proof.
  unfold leb, le. destruct compare; easy'.
Qed.

(** Basic properties of comparison *)

Theorem compare_eq_iff n m : (n ?= m) = Eq <-> n = m.
Proof.
destruct n, m; simpl; rewrite ?Pos.compare_eq_iff; split; congruence.
Qed.

Theorem compare_lt_iff n m : (n ?= m) = Lt <-> n < m.
Proof.
reflexivity.
Qed.

Theorem compare_le_iff n m : (n ?= m) <> Gt <-> n <= m.
Proof.
reflexivity.
Qed.

Theorem compare_antisym n m : (m ?= n) = CompOpp (n ?= m).
Proof.
destruct n, m; simpl; trivial. apply Pos.compare_antisym.
Qed.

(** Some more advanced properties of comparison and orders,
    including [compare_spec] and [lt_irrefl] and [lt_eq_cases]. *)

Include BoolOrderFacts.

(** We regroup here some results used for proving the correctness
    of more advanced functions. These results will also be provided
    by the generic functor of properties about natural numbers
    instantiated at the end of the file. *)

Module Import Private_BootStrap.

Theorem add_0_r n : n + 0 = n.
Proof.
now destruct n.
Qed.

Theorem add_comm n m : n + m = m + n.
Proof.
destruct n, m; simpl; try reflexivity. simpl. f_equal. apply Pos.add_comm.
Qed.

Theorem add_assoc n m p : n + (m + p) = n + m + p.
Proof.
destruct n; try reflexivity.
destruct m; try reflexivity.
destruct p; try reflexivity.
simpl. f_equal. apply Pos.add_assoc.
Qed.

Lemma sub_add n m : n <= m -> m - n + n = m.
Proof.
 destruct n as [|p], m as [|q]; simpl; try easy'. intros H.
 case Pos.sub_mask_spec; intros; simpl; subst; trivial.
 now rewrite Pos.add_comm.
 apply Pos.le_nlt in H. elim H. apply Pos.lt_add_r.
Qed.

Theorem mul_comm n m : n * m = m * n.
Proof.
destruct n, m; simpl; trivial. f_equal. apply Pos.mul_comm.
Qed.

Lemma le_0_l n : 0<=n.
Proof.
now destruct n.
Qed.

Lemma leb_spec n m : BoolSpec (n<=m) (m<n) (n <=? m).
Proof.
 unfold le, lt, leb. rewrite (compare_antisym n m).
 case compare; now constructor.
Qed.

Lemma add_lt_cancel_l n m p : p+n < p+m -> n<m.
Proof.
 intro H. destruct p. simpl; auto.
 destruct n; destruct m.
 elim (Pos.lt_irrefl _ H).
 red; auto.
 rewrite add_0_r in H. simpl in H.
 red in H. simpl in H.
 elim (Pos.lt_not_add_l _ _ H).
 now apply (Pos.add_lt_mono_l p).
Qed.

End Private_BootStrap.

(** Specification of lt and le. *)

Lemma lt_succ_r n m : n < succ m <-> n<=m.
Proof.
destruct n as [|p], m as [|q]; simpl; try easy'.
split. now destruct p. now destruct 1.
apply Pos.lt_succ_r.
Qed.

(** Properties of [double] and [succ_double] *)

Lemma double_spec n : double n = 2 * n.
Proof.
 reflexivity.
Qed.

Lemma succ_double_spec n : succ_double n = 2 * n + 1.
Proof.
 now destruct n.
Qed.

Lemma double_add n m : double (n+m) = double n + double m.
Proof.
 now destruct n, m.
Qed.

Lemma succ_double_add n m : succ_double (n+m) = double n + succ_double m.
Proof.
 now destruct n, m.
Qed.

Lemma double_mul n m : double (n*m) = double n * m.
Proof.
 now destruct n, m.
Qed.

Lemma succ_double_mul n m :
 succ_double n * m = double n * m + m.
Proof.
 destruct n; simpl; destruct m; trivial.
 now rewrite Pos.add_comm.
Qed.

Lemma div2_double n : div2 (double n) = n.
Proof.
now destruct n.
Qed.

Lemma div2_succ_double n : div2 (succ_double n) = n.
Proof.
now destruct n.
Qed.

Lemma double_inj n m : double n = double m -> n = m.
Proof.
intro H. rewrite <- (div2_double n), H. apply div2_double.
Qed.

Lemma succ_double_inj n m : succ_double n = succ_double m -> n = m.
Proof.
intro H. rewrite <- (div2_succ_double n), H. apply div2_succ_double.
Qed.

Lemma succ_double_lt n m : n<m -> succ_double n < double m.
Proof.
 destruct n as [|n], m as [|m]; intros H; try easy.
 unfold lt in *; simpl in *. now rewrite Pos.compare_xI_xO, H.
Qed.


(** Specification of minimum and maximum *)

Theorem min_l n m : n <= m -> min n m = n.
Proof.
unfold min, le. case compare; trivial. now destruct 1.
Qed.

Theorem min_r n m : m <= n -> min n m = m.
Proof.
unfold min, le. rewrite compare_antisym.
case compare_spec; trivial. now destruct 2.
Qed.

Theorem max_l n m : m <= n -> max n m = n.
Proof.
unfold max, le. rewrite compare_antisym.
case compare_spec; auto. now destruct 2.
Qed.

Theorem max_r n m : n <= m -> max n m = m.
Proof.
unfold max, le. case compare; trivial. now destruct 1.
Qed.

(** 0 is the least natural number *)

Theorem compare_0_r n : (n ?= 0) <> Lt.
Proof.
now destruct n.
Qed.

(** Specifications of power *)

Lemma pow_0_r n : n ^ 0 = 1.
Proof. reflexivity. Qed.

Lemma pow_succ_r n p : 0<=p -> n^(succ p) = n * n^p.
Proof.
 intros _.
 destruct n, p; simpl; trivial; f_equal. apply Pos.pow_succ_r.
Qed.

Lemma pow_neg_r n p : p<0 -> n^p = 0.
Proof.
 now destruct p.
Qed.

(** Specification of square *)

Lemma square_spec n : square n = n * n.
Proof.
 destruct n; trivial. simpl. f_equal. apply Pos.square_spec.
Qed.

(** Specification of Base-2 logarithm *)

Lemma size_log2 n : n<>0 -> size n = succ (log2 n).
Proof.
 destruct n as [|[n|n| ]]; trivial. now destruct 1.
Qed.

Lemma size_gt n : n < 2^(size n).
Proof.
 destruct n. reflexivity. simpl. apply Pos.size_gt.
Qed.

Lemma size_le n : 2^(size n) <= succ_double n.
Proof.
 destruct n. discriminate. simpl.
 change (2^Pos.size p <= Pos.succ (p~0))%positive.
 apply Pos.lt_le_incl, Pos.lt_succ_r, Pos.size_le.
Qed.

Lemma log2_spec n : 0 < n ->
 2^(log2 n) <= n < 2^(succ (log2 n)).
Proof.
 destruct n as [|[p|p|]]; discriminate || intros _; simpl; split.
 apply (size_le (pos p)).
 apply Pos.size_gt.
 apply Pos.size_le.
 apply Pos.size_gt.
 discriminate.
 reflexivity.
Qed.

Lemma log2_nonpos n : n<=0 -> log2 n = 0.
Proof.
 destruct n; intros Hn. reflexivity. now destruct Hn.
Qed.

(** Specification of parity functions *)

Lemma even_spec n : even n = true <-> Even n.
Proof.
 destruct n.
 split. now exists 0.
 trivial.
 destruct p; simpl; split; try easy.
 intros (m,H). now destruct m.
 now exists (pos p).
 intros (m,H). now destruct m.
Qed.

Lemma odd_spec n : odd n = true <-> Odd n.
Proof.
 destruct n.
 split. discriminate.
 intros (m,H). now destruct m.
 destruct p; simpl; split; try easy.
 now exists (pos p).
 intros (m,H). now destruct m.
 now exists 0.
Qed.

(** Specification of the euclidean division *)

Theorem pos_div_eucl_spec (a:positive)(b:N) :
  let (q,r) := pos_div_eucl a b in pos a = q * b + r.
Proof.
  induction a; cbv beta iota delta [pos_div_eucl]; fold pos_div_eucl; cbv zeta.
  (* a~1 *)
  destruct pos_div_eucl as (q,r).
  change (pos a~1) with (succ_double (pos a)).
  rewrite IHa, succ_double_add, double_mul.
  case leb_spec; intros H; trivial.
  rewrite succ_double_mul, <- add_assoc. f_equal.
  now rewrite (add_comm b), sub_add.
  (* a~0 *)
  destruct pos_div_eucl as (q,r).
  change (pos a~0) with (double (pos a)).
  rewrite IHa, double_add, double_mul.
  case leb_spec; intros H; trivial.
  rewrite succ_double_mul, <- add_assoc. f_equal.
  now rewrite (add_comm b), sub_add.
  (* 1 *)
  now destruct b as [|[ | | ]].
Qed.

Theorem div_eucl_spec a b :
 let (q,r) := div_eucl a b in a = b * q + r.
Proof.
  destruct a as [|a], b as [|b]; unfold div_eucl; trivial.
  generalize (pos_div_eucl_spec a (pos b)).
  destruct pos_div_eucl. now rewrite mul_comm.
Qed.

Theorem div_mod' a b : a = b * (a/b) + (a mod b).
Proof.
  generalize (div_eucl_spec a b).
  unfold div, modulo. now destruct div_eucl.
Qed.

Definition div_mod a b : b<>0 -> a = b * (a/b) + (a mod b).
Proof.
 intros _. apply div_mod'.
Qed.

Theorem pos_div_eucl_remainder (a:positive) (b:N) :
  b<>0 -> snd (pos_div_eucl a b) < b.
Proof.
  intros Hb.
  induction a; cbv beta iota delta [pos_div_eucl]; fold pos_div_eucl; cbv zeta.
  (* a~1 *)
  destruct pos_div_eucl as (q,r); simpl in *.
  case leb_spec; intros H; simpl; trivial.
  apply add_lt_cancel_l with b. rewrite add_comm, sub_add by trivial.
  destruct b as [|b]; [now destruct Hb| simpl; rewrite Pos.add_diag ].
  apply (succ_double_lt _ _ IHa).
  (* a~0 *)
  destruct pos_div_eucl as (q,r); simpl in *.
  case leb_spec; intros H; simpl; trivial.
  apply add_lt_cancel_l with b. rewrite add_comm, sub_add by trivial.
  destruct b as [|b]; [now destruct Hb| simpl; rewrite Pos.add_diag ].
  now destruct r.
  (* 1 *)
  destruct b as [|[ | | ]]; easy || (now destruct Hb).
Qed.

Theorem mod_lt a b : b<>0 -> a mod b < b.
Proof.
  destruct b as [ |b]. now destruct 1.
  destruct a as [ |a]. reflexivity.
  unfold modulo. simpl. apply pos_div_eucl_remainder.
Qed.

Theorem mod_bound_pos a b : 0<=a -> 0<b -> 0 <= a mod b < b.
Proof.
 intros _ H. split. apply le_0_l. apply mod_lt. now destruct b.
Qed.

(** Specification of square root *)

Lemma sqrtrem_sqrt n : fst (sqrtrem n) = sqrt n.
Proof.
 destruct n. reflexivity.
 unfold sqrtrem, sqrt, Pos.sqrt.
 destruct (Pos.sqrtrem p) as (s,r). now destruct r.
Qed.

Lemma sqrtrem_spec n :
 let (s,r) := sqrtrem n in n = s*s + r /\ r <= 2*s.
Proof.
 destruct n. now split.
 generalize (Pos.sqrtrem_spec p). simpl.
 destruct 1; simpl; subst; now split.
Qed.

Lemma sqrt_spec n : 0<=n ->
 let s := sqrt n in s*s <= n < (succ s)*(succ s).
Proof.
 intros _. destruct n. now split. apply (Pos.sqrt_spec p).
Qed.

Lemma sqrt_neg n : n<0 -> sqrt n = 0.
Proof.
 now destruct n.
Qed.

(** Specification of gcd *)

(** The first component of ggcd is gcd *)

Lemma ggcd_gcd a b : fst (ggcd a b) = gcd a b.
Proof.
 destruct a as [|p], b as [|q]; simpl; auto.
 assert (H := Pos.ggcd_gcd p q).
 destruct Pos.ggcd as (g,(aa,bb)); simpl; now f_equal.
Qed.

(** The other components of ggcd are indeed the correct factors. *)

Lemma ggcd_correct_divisors a b :
  let '(g,(aa,bb)) := ggcd a b in
  a=g*aa /\ b=g*bb.
Proof.
 destruct a as [|p], b as [|q]; simpl; auto.
 now rewrite Pos.mul_1_r.
 now rewrite Pos.mul_1_r.
 generalize (Pos.ggcd_correct_divisors p q).
 destruct Pos.ggcd as (g,(aa,bb)); simpl.
 destruct 1; split; now f_equal.
Qed.

(** We can use this fact to prove a part of the gcd correctness *)

Lemma gcd_divide_l a b : (gcd a b | a).
Proof.
 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 a b : (gcd a b | b).
Proof.
 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 gcd_greatest a b c : (c|a) -> (c|b) -> (c|gcd a b).
Proof.
 destruct a as [ |p], b as [ |q]; simpl; trivial.
 destruct c as [ |r]. intros (s,H). destruct s; discriminate.
 intros ([ |s],Hs) ([ |t],Ht); try discriminate; simpl in *.
 destruct (Pos.gcd_greatest p q r) as (u,H).
 exists s. now inversion Hs.
 exists t. now inversion Ht.
 exists (pos u). simpl; now f_equal.
Qed.

Lemma gcd_nonneg a b : 0 <= gcd a b.
Proof. apply le_0_l. Qed.

(** Specification of bitwise functions *)

(** Correctness proofs for [testbit]. *)

Lemma testbit_even_0 a : testbit (2*a) 0 = false.
Proof.
 now destruct a.
Qed.

Lemma testbit_odd_0 a : testbit (2*a+1) 0 = true.
Proof.
 now destruct a.
Qed.

Lemma testbit_succ_r_div2 a n : 0<=n ->
 testbit a (succ n) = testbit (div2 a) n.
Proof.
 intros _. destruct a as [|[a|a| ]], n as [|n]; simpl; trivial;
  f_equal; apply Pos.pred_N_succ.
Qed.

Lemma testbit_odd_succ a n : 0<=n ->
 testbit (2*a+1) (succ n) = testbit a n.
Proof.
 intros H. rewrite testbit_succ_r_div2 by trivial. f_equal. now destruct a.
Qed.

Lemma testbit_even_succ a n : 0<=n ->
 testbit (2*a) (succ n) = testbit a n.
Proof.
 intros H. rewrite testbit_succ_r_div2 by trivial. f_equal. now destruct a.
Qed.

Lemma testbit_neg_r a n : n<0 -> testbit a n = false.
Proof.
 now destruct n.
Qed.

(** Correctness proofs for shifts *)

Lemma shiftr_succ_r a n :
 shiftr a (succ n) = div2 (shiftr a n).
Proof.
 destruct n; simpl; trivial. apply Pos.iter_succ.
Qed.

Lemma shiftl_succ_r a n :
 shiftl a (succ n) = double (shiftl a n).
Proof.
 destruct n, a; simpl; trivial. f_equal. apply Pos.iter_succ.
Qed.

Lemma shiftr_spec a n m : 0<=m ->
 testbit (shiftr a n) m = testbit a (m+n).
Proof.
 intros _. revert a m.
 induction n using peano_ind; intros a m. now rewrite add_0_r.
 rewrite add_comm, add_succ_l, add_comm, <- add_succ_l.
 now rewrite <- IHn, testbit_succ_r_div2, shiftr_succ_r by apply le_0_l.
Qed.

Lemma shiftl_spec_high a n m : 0<=m -> n<=m ->
 testbit (shiftl a n) m = testbit a (m-n).
Proof.
 intros _ H.
 rewrite <- (sub_add n m H) at 1.
 set (m' := m-n). clearbody m'. clear H m. revert a m'.
 induction n using peano_ind; intros a m.
 rewrite add_0_r; now destruct a.
 rewrite shiftl_succ_r.
 rewrite add_comm, add_succ_l, add_comm.
 now rewrite testbit_succ_r_div2, div2_double by apply le_0_l.
Qed.

Lemma shiftl_spec_low a n m : m<n ->
 testbit (shiftl a n) m = false.
Proof.
 revert a m.
 induction n using peano_ind; intros a m H.
 elim (le_0_l m). now rewrite compare_antisym, H.
 rewrite shiftl_succ_r.
 destruct m. now destruct (shiftl a n).
 rewrite <- (succ_pos_pred p), testbit_succ_r_div2, div2_double by apply le_0_l.
 apply IHn.
 apply add_lt_cancel_l with 1. rewrite 2 (add_succ_l 0). simpl.
 now rewrite succ_pos_pred.
Qed.

Definition div2_spec a : div2 a = shiftr a 1.
Proof.
 reflexivity.
Qed.

(** Semantics of bitwise operations *)

Lemma pos_lxor_spec p p' n :
 testbit (Pos.lxor p p') n = xorb (Pos.testbit p n) (Pos.testbit p' n).
Proof.
 revert p' n.
 induction p as [p IH|p IH|]; intros [p'|p'|] [|n]; trivial; simpl;
 (specialize (IH p'); destruct Pos.lxor; trivial; now rewrite <-IH) ||
 (now destruct Pos.testbit).
Qed.

Lemma lxor_spec a a' n :
 testbit (lxor a a') n = xorb (testbit a n) (testbit a' n).
Proof.
 destruct a, a'; simpl; trivial.
 now destruct Pos.testbit.
 now destruct Pos.testbit.
 apply pos_lxor_spec.
Qed.

Lemma pos_lor_spec p p' n :
 Pos.testbit (Pos.lor p p') n = (Pos.testbit p n) || (Pos.testbit p' n).
Proof.
 revert p' n.
 induction p as [p IH|p IH|]; intros [p'|p'|] [|n]; trivial; simpl;
  apply IH || now rewrite orb_false_r.
Qed.

Lemma lor_spec a a' n :
 testbit (lor a a') n = (testbit a n) || (testbit a' n).
Proof.
 destruct a, a'; simpl; trivial.
 now rewrite orb_false_r.
 apply pos_lor_spec.
Qed.

Lemma pos_land_spec p p' n :
 testbit (Pos.land p p') n = (Pos.testbit p n) && (Pos.testbit p' n).
Proof.
 revert p' n.
 induction p as [p IH|p IH|]; intros [p'|p'|] [|n]; trivial; simpl;
 (specialize (IH p'); destruct Pos.land; trivial; now rewrite <-IH) ||
 (now rewrite andb_false_r).
Qed.

Lemma land_spec a a' n :
 testbit (land a a') n = (testbit a n) && (testbit a' n).
Proof.
 destruct a, a'; simpl; trivial.
 now rewrite andb_false_r.
 apply pos_land_spec.
Qed.

Lemma pos_ldiff_spec p p' n :
 testbit (Pos.ldiff p p') n = (Pos.testbit p n) && negb (Pos.testbit p' n).
Proof.
 revert p' n.
 induction p as [p IH|p IH|]; intros [p'|p'|] [|n]; trivial; simpl;
 (specialize (IH p'); destruct Pos.ldiff; trivial; now rewrite <-IH) ||
 (now rewrite andb_true_r).
Qed.

Lemma ldiff_spec a a' n :
 testbit (ldiff a a') n = (testbit a n) && negb (testbit a' n).
Proof.
 destruct a, a'; simpl; trivial.
 now rewrite andb_true_r.
 apply pos_ldiff_spec.
Qed.

(** Specification of constants *)

Lemma one_succ : 1 = succ 0.
Proof. reflexivity. Qed.

Lemma two_succ : 2 = succ 1.
Proof. reflexivity. Qed.

Definition pred_0 : pred 0 = 0.
Proof. reflexivity. Qed.

(** Proofs of morphisms, obvious since eq is Leibniz *)

Local Obligation Tactic := simpl_relation.
Program Definition succ_wd : Proper (eq==>eq) succ := _.
Program Definition pred_wd : Proper (eq==>eq) pred := _.
Program Definition add_wd : Proper (eq==>eq==>eq) add := _.
Program Definition sub_wd : Proper (eq==>eq==>eq) sub := _.
Program Definition mul_wd : Proper (eq==>eq==>eq) mul := _.
Program Definition lt_wd : Proper (eq==>eq==>iff) lt := _.
Program Definition div_wd : Proper (eq==>eq==>eq) div := _.
Program Definition mod_wd : Proper (eq==>eq==>eq) modulo := _.
Program Definition pow_wd : Proper (eq==>eq==>eq) pow := _.
Program Definition testbit_wd : Proper (eq==>eq==>Logic.eq) testbit := _.

(** Generic induction / recursion *)

Theorem bi_induction :
  forall A : N -> Prop, Proper (Logic.eq==>iff) A ->
    A 0 -> (forall n, A n <-> A (succ n)) -> forall n : N, A n.
Proof.
intros A A_wd A0 AS. apply peano_rect. assumption. intros; now apply -> AS.
Qed.

Definition recursion {A} : A -> (N -> A -> A) -> N -> A :=
  peano_rect (fun _ => A).

Instance recursion_wd {A} (Aeq : relation A) :
 Proper (Aeq==>(Logic.eq==>Aeq==>Aeq)==>Logic.eq==>Aeq) recursion.
Proof.
intros a a' Ea f f' Ef x x' Ex. subst x'.
induction x using peano_ind.
trivial.
unfold recursion in *. rewrite 2 peano_rect_succ. now apply Ef.
Qed.

Theorem recursion_0 {A} (a:A) (f:N->A->A) : recursion a f 0 = a.
Proof. reflexivity. Qed.

Theorem recursion_succ {A} (Aeq : relation A) (a : A) (f : N -> A -> A):
 Aeq a a -> Proper (Logic.eq==>Aeq==>Aeq) f ->
  forall n : N, Aeq (recursion a f (succ n)) (f n (recursion a f n)).
Proof.
unfold recursion; intros a_wd f_wd n. induction n using peano_ind.
rewrite peano_rect_succ. now apply f_wd.
rewrite !peano_rect_succ in *. now apply f_wd.
Qed.

(** Instantiation of generic properties of natural numbers *)

(** The Bind Scope prevents N to stay associated with abstract_scope.
    (TODO FIX) *)

Include NProp. Bind Scope N_scope with N.
Include UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.

(** In generic statements, the predicates [lt] and [le] have been
  favored, whereas [gt] and [ge] don't even exist in the abstract
  layers. 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 n m : n > m <-> m < n.
Proof.
 unfold lt, gt. now rewrite compare_antisym, CompOpp_iff.
Qed.

Lemma gt_lt n m : n > m -> m < n.
Proof.
 apply gt_lt_iff.
Qed.

Lemma lt_gt n m : n < m -> m > n.
Proof.
 apply gt_lt_iff.
Qed.

Lemma ge_le_iff n m : n >= m <-> m <= n.
Proof.
 unfold le, ge. now rewrite compare_antisym, CompOpp_iff.
Qed.

Lemma ge_le n m : n >= m -> m <= n.
Proof.
 apply ge_le_iff.
Qed.

Lemma le_ge n m : n <= m -> m >= n.
Proof.
 apply ge_le_iff.
Qed.

(** Auxiliary results about right shift on positive numbers,
    used in BinInt *)

Lemma pos_pred_shiftl_low : forall p n m, m<n ->
 testbit (Pos.pred_N (Pos.shiftl p n)) m = true.
Proof.
 induction n using peano_ind.
 now destruct m.
 intros m H. unfold Pos.shiftl.
 destruct n as [|n]; simpl in *.
 destruct m. now destruct p. elim (Pos.nlt_1_r _ H).
 rewrite Pos.iter_succ. simpl.
 set (u:=Pos.iter n xO p) in *; clearbody u.
 destruct m as [|m]. now destruct u.
 rewrite <- (IHn (Pos.pred_N m)).
 rewrite <- (testbit_odd_succ _ (Pos.pred_N m)).
 rewrite succ_pos_pred. now destruct u.
 apply le_0_l.
 apply succ_lt_mono. now rewrite succ_pos_pred.
Qed.

Lemma pos_pred_shiftl_high : forall p n m, n<=m ->
 testbit (Pos.pred_N (Pos.shiftl p n)) m =
 testbit (shiftl (Pos.pred_N p) n) m.
Proof.
 induction n using peano_ind; intros m H.
 unfold shiftl. simpl. now destruct (Pos.pred_N p).
 rewrite shiftl_succ_r.
 destruct n as [|n].
 destruct m as [|m]. now destruct H. now destruct p.
 destruct m as [|m]. now destruct H.
 rewrite <- (succ_pos_pred m).
 rewrite double_spec, testbit_even_succ by apply le_0_l.
 rewrite <- IHn.
 rewrite testbit_succ_r_div2 by apply le_0_l.
 f_equal. simpl. rewrite Pos.iter_succ.
 now destruct (Pos.iter n xO p).
 apply succ_le_mono. now rewrite succ_pos_pred.
Qed.

Lemma pred_div2_up p : Pos.pred_N (Pos.div2_up p) = div2 (Pos.pred_N p).
Proof.
 destruct p as [p|p| ]; trivial.
 simpl. apply Pos.pred_N_succ.
 destruct p; simpl; trivial.
Qed.

End N.

(** Exportation of notations *)

Infix "+" := N.add : N_scope.
Infix "-" := N.sub : N_scope.
Infix "*" := N.mul : N_scope.
Infix "^" := N.pow : N_scope.

Infix "?=" := N.compare (at level 70, no associativity) : N_scope.

Infix "<=" := N.le : N_scope.
Infix "<" := N.lt : N_scope.
Infix ">=" := N.ge : N_scope.
Infix ">" := N.gt : N_scope.

Notation "x <= y <= z" := (x <= y /\ y <= z) : N_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : N_scope.
Notation "x < y < z" := (x < y /\ y < z) : N_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : N_scope.

Infix "=?" := N.eqb (at level 70, no associativity) : N_scope.
Infix "<=?" := N.leb (at level 70, no associativity) : N_scope.
Infix "<?" := N.ltb (at level 70, no associativity) : N_scope.

Infix "/" := N.div : N_scope.
Infix "mod" := N.modulo (at level 40, no associativity) : N_scope.

Notation "( p | q )" := (N.divide p q) (at level 0) : N_scope.

(** Compatibility notations *)

(*Notation N := N (compat "8.3").*) (*hidden by module N above *)
Notation N_rect := N_rect (only parsing).
Notation N_rec := N_rec (only parsing).
Notation N_ind := N_ind (only parsing).
Notation N0 := N0 (only parsing).
Notation Npos := N.pos (only parsing).

Notation Ndiscr := N.discr (compat "8.3").
Notation Ndouble_plus_one := N.succ_double (compat "8.3").
Notation Ndouble := N.double (compat "8.3").
Notation Nsucc := N.succ (compat "8.3").
Notation Npred := N.pred (compat "8.3").
Notation Nsucc_pos := N.succ_pos (compat "8.3").
Notation Ppred_N := Pos.pred_N (compat "8.3").
Notation Nplus := N.add (compat "8.3").
Notation Nminus := N.sub (compat "8.3").
Notation Nmult := N.mul (compat "8.3").
Notation Neqb := N.eqb (compat "8.3").
Notation Ncompare := N.compare (compat "8.3").
Notation Nlt := N.lt (compat "8.3").
Notation Ngt := N.gt (compat "8.3").
Notation Nle := N.le (compat "8.3").
Notation Nge := N.ge (compat "8.3").
Notation Nmin := N.min (compat "8.3").
Notation Nmax := N.max (compat "8.3").
Notation Ndiv2 := N.div2 (compat "8.3").
Notation Neven := N.even (compat "8.3").
Notation Nodd := N.odd (compat "8.3").
Notation Npow := N.pow (compat "8.3").
Notation Nlog2 := N.log2 (compat "8.3").

Notation nat_of_N := N.to_nat (compat "8.3").
Notation N_of_nat := N.of_nat (compat "8.3").
Notation N_eq_dec := N.eq_dec (compat "8.3").
Notation Nrect := N.peano_rect (compat "8.3").
Notation Nrect_base := N.peano_rect_base (compat "8.3").
Notation Nrect_step := N.peano_rect_succ (compat "8.3").
Notation Nind := N.peano_ind (compat "8.3").
Notation Nrec := N.peano_rec (compat "8.3").
Notation Nrec_base := N.peano_rec_base (compat "8.3").
Notation Nrec_succ := N.peano_rec_succ (compat "8.3").

Notation Npred_succ := N.pred_succ (compat "8.3").
Notation Npred_minus := N.pred_sub (compat "8.3").
Notation Nsucc_pred := N.succ_pred (compat "8.3").
Notation Ppred_N_spec := N.pos_pred_spec (compat "8.3").
Notation Nsucc_pos_spec := N.succ_pos_spec (compat "8.3").
Notation Ppred_Nsucc := N.pos_pred_succ (compat "8.3").
Notation Nplus_0_l := N.add_0_l (compat "8.3").
Notation Nplus_0_r := N.add_0_r (compat "8.3").
Notation Nplus_comm := N.add_comm (compat "8.3").
Notation Nplus_assoc := N.add_assoc (compat "8.3").
Notation Nplus_succ := N.add_succ_l (compat "8.3").
Notation Nsucc_0 := N.succ_0_discr (compat "8.3").
Notation Nsucc_inj := N.succ_inj (compat "8.3").
Notation Nminus_N0_Nle := N.sub_0_le (compat "8.3").
Notation Nminus_0_r := N.sub_0_r (compat "8.3").
Notation Nminus_succ_r:= N.sub_succ_r (compat "8.3").
Notation Nmult_0_l := N.mul_0_l (compat "8.3").
Notation Nmult_1_l := N.mul_1_l (compat "8.3").
Notation Nmult_1_r := N.mul_1_r (compat "8.3").
Notation Nmult_comm := N.mul_comm (compat "8.3").
Notation Nmult_assoc := N.mul_assoc (compat "8.3").
Notation Nmult_plus_distr_r := N.mul_add_distr_r (compat "8.3").
Notation Neqb_eq := N.eqb_eq (compat "8.3").
Notation Nle_0 := N.le_0_l (compat "8.3").
Notation Ncompare_refl := N.compare_refl (compat "8.3").
Notation Ncompare_Eq_eq := N.compare_eq (compat "8.3").
Notation Ncompare_eq_correct := N.compare_eq_iff (compat "8.3").
Notation Nlt_irrefl := N.lt_irrefl (compat "8.3").
Notation Nlt_trans := N.lt_trans (compat "8.3").
Notation Nle_lteq := N.lt_eq_cases (compat "8.3").
Notation Nlt_succ_r := N.lt_succ_r (compat "8.3").
Notation Nle_trans := N.le_trans (compat "8.3").
Notation Nle_succ_l := N.le_succ_l (compat "8.3").
Notation Ncompare_spec := N.compare_spec (compat "8.3").
Notation Ncompare_0 := N.compare_0_r (compat "8.3").
Notation Ndouble_div2 := N.div2_double (compat "8.3").
Notation Ndouble_plus_one_div2 := N.div2_succ_double (compat "8.3").
Notation Ndouble_inj := N.double_inj (compat "8.3").
Notation Ndouble_plus_one_inj := N.succ_double_inj (compat "8.3").
Notation Npow_0_r := N.pow_0_r (compat "8.3").
Notation Npow_succ_r := N.pow_succ_r (compat "8.3").
Notation Nlog2_spec := N.log2_spec (compat "8.3").
Notation Nlog2_nonpos := N.log2_nonpos (compat "8.3").
Notation Neven_spec := N.even_spec (compat "8.3").
Notation Nodd_spec := N.odd_spec (compat "8.3").
Notation Nlt_not_eq := N.lt_neq (compat "8.3").
Notation Ngt_Nlt := N.gt_lt (compat "8.3").

(** More complex compatibility facts, expressed as lemmas
    (to preserve scopes for instance) *)

Lemma Nplus_reg_l n m p : n + m = n + p -> m = p.
Proof (proj1 (N.add_cancel_l m p n)).
Lemma Nmult_Sn_m n m : N.succ n * m = m + n * m.
Proof (eq_trans (N.mul_succ_l n m) (N.add_comm _ _)).
Lemma Nmult_plus_distr_l n m p : p * (n + m) = p * n + p * m.
Proof (N.mul_add_distr_l p n m).
Lemma Nmult_reg_r n m p : p <> 0 -> n * p = m * p -> n = m.
Proof (fun H => proj1 (N.mul_cancel_r n m p H)).
Lemma Ncompare_antisym n m : CompOpp (n ?= m) = (m ?= n).
Proof (eq_sym (N.compare_antisym n m)).

Definition N_ind_double a P f0 f2 fS2 := N.binary_ind P f0 f2 fS2 a.
Definition N_rec_double a P f0 f2 fS2 := N.binary_rec P f0 f2 fS2 a.

(** Not kept : Ncompare_n_Sm Nplus_lt_cancel_l *)