Library Coq.ZArith.BinInt


Require Export BinNums BinPos Pnat.
Require Import BinNat Bool Equalities GenericMinMax
 OrdersFacts ZAxioms ZProperties.
Require BinIntDef.

Binary Integers

Initial author: Pierre Crégut, CNET, Lannion, France
The type Z and its constructors Z0 and Zpos and Zneg are now defined in BinNums.v

Local Open Scope Z_scope.

Every definitions and early properties about binary integers are placed in a module Z for qualification purpose.

Module Z
 <: ZAxiomsSig
 <: UsualOrderedTypeFull
 <: UsualDecidableTypeFull
 <: TotalOrder.

Definitions of operations, now in a separate file


Include BinIntDef.Z.

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

Set Inline Level 30.

Logic Predicates


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

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

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

Definition divide x y := exists z, y = z*x.
Notation "( x | y )" := (divide x y) (at level 0).

Definition Even a := exists b, a = 2*b.
Definition Odd a := exists b, a = 2*b+1.

Decidability of equality.


Definition eq_dec (x y : Z) : {x = y} + {x <> y}.
Proof.
 decide equality; apply Pos.eq_dec.
Defined.

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 opp_wd : Proper (eq==>eq) opp := _.
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 quot_wd : Proper (eq==>eq==>eq) quot := _.
Program Definition rem_wd : Proper (eq==>eq==>eq) rem := _.
Program Definition pow_wd : Proper (eq==>eq==>eq) pow := _.
Program Definition testbit_wd : Proper (eq==>eq==>Logic.eq) testbit := _.

Properties of pos_sub

pos_sub can be written in term of positive comparison and subtraction (cf. earlier definition of addition of Z)

Lemma pos_sub_spec p q :
 pos_sub p q =
 match (p ?= q)%positive with
   | Eq => 0
   | Lt => neg (q - p)
   | Gt => pos (p - q)
 end.
Proof.
 revert q. induction p; destruct q; simpl; trivial;
 rewrite ?Pos.compare_xI_xI, ?Pos.compare_xO_xI,
  ?Pos.compare_xI_xO, ?Pos.compare_xO_xO, IHp; simpl;
 case Pos.compare_spec; intros; simpl; trivial;
  (now rewrite Pos.sub_xI_xI) || (now rewrite Pos.sub_xO_xO) ||
  (now rewrite Pos.sub_xO_xI) || (now rewrite Pos.sub_xI_xO) ||
  subst; unfold Pos.sub; simpl; now rewrite Pos.sub_mask_diag.
Qed.

Lemma pos_sub_discr p q :
  match pos_sub p q with
  | Z0 => p = q
  | pos k => p = q + k
  | neg k => q = p + k
  end%positive.
Proof.
 rewrite pos_sub_spec.
 case Pos.compare_spec; auto; intros;
  now rewrite Pos.add_comm, Pos.sub_add.
Qed.

Particular cases of the previous result

Lemma pos_sub_diag p : pos_sub p p = 0.
Proof.
 now rewrite pos_sub_spec, Pos.compare_refl.
Qed.

Lemma pos_sub_lt p q : (p < q)%positive -> pos_sub p q = neg (q - p).
Proof.
 intros H. now rewrite pos_sub_spec, H.
Qed.

Lemma pos_sub_gt p q : (q < p)%positive -> pos_sub p q = pos (p - q).
Proof.
 intros H. now rewrite pos_sub_spec, Pos.compare_antisym, H.
Qed.

The opposite of pos_sub is pos_sub with reversed arguments

Lemma pos_sub_opp p q : - pos_sub p q = pos_sub q p.
Proof.
 revert q; induction p; destruct q; simpl; trivial;
 rewrite <- IHp; now destruct pos_sub.
Qed.

In the following module, we group results that are needed now to prove specifications of operations, but will also be provided later by the generic functor of properties.

Module Import Private_BootStrap.

Operations and constants


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

Lemma mul_0_r n : n * 0 = 0.
Proof.
 now destruct n.
Qed.

Lemma mul_1_l n : 1 * n = n.
Proof.
 now destruct n.
Qed.

Addition is commutative


Lemma add_comm n m : n + m = m + n.
Proof.
 destruct n, m; simpl; trivial; now rewrite Pos.add_comm.
Qed.

Opposite distributes over addition


Lemma opp_add_distr n m : - (n + m) = - n + - m.
Proof.
 destruct n, m; simpl; trivial using pos_sub_opp.
Qed.

Opposite is injective


Lemma opp_inj n m : -n = -m -> n = m.
Proof.
 destruct n, m; simpl; intros H; destr_eq H; now f_equal.
Qed.

Addition is associative


Lemma pos_sub_add p q r :
  pos_sub (p + q) r = pos p + pos_sub q r.
Proof.
 simpl. rewrite !pos_sub_spec.
 case (Pos.compare_spec q r); intros E0.
 -
   subst.
   assert (H := Pos.lt_add_r r p).
   rewrite Pos.add_comm in H. apply Pos.lt_gt in H.
   now rewrite H, Pos.add_sub.
 -
   rewrite pos_sub_spec.
   assert (Hr : (r = (r-q)+q)%positive) by (now rewrite Pos.sub_add).
   rewrite Hr at 1. rewrite Pos.add_compare_mono_r.
   case Pos.compare_spec; intros E1; trivial; f_equal.
   rewrite Pos.add_comm. apply Pos.sub_add_distr.
   rewrite Hr, Pos.add_comm. now apply Pos.add_lt_mono_r.
   symmetry. apply Pos.sub_sub_distr; trivial.
 -
   assert (LT : (r < p + q)%positive).
   { apply Pos.lt_trans with q; trivial.
     rewrite Pos.add_comm. apply Pos.lt_add_r. }
   apply Pos.lt_gt in LT. rewrite LT. f_equal.
   symmetry. now apply Pos.add_sub_assoc.
Qed.

Local Arguments add !x !y.

Lemma add_assoc_pos p n m : pos p + (n + m) = pos p + n + m.
Proof.
 destruct n as [|n|n], m as [|m|m]; simpl; trivial.
 - now rewrite Pos.add_assoc.
 - symmetry. apply pos_sub_add.
 - symmetry. apply add_0_r.
 - now rewrite <- pos_sub_add, add_comm, <- pos_sub_add, Pos.add_comm.
 - apply opp_inj. rewrite !opp_add_distr, !pos_sub_opp.
   rewrite add_comm, Pos.add_comm. apply pos_sub_add.
Qed.

Lemma add_assoc n m p : n + (m + p) = n + m + p.
Proof.
 destruct n.
 - trivial.
 - apply add_assoc_pos.
 - apply opp_inj. rewrite !opp_add_distr. simpl. apply add_assoc_pos.
Qed.

Opposite is inverse for addition


Lemma add_opp_diag_r n : n + - n = 0.
Proof.
 destruct n; simpl; trivial; now rewrite pos_sub_diag.
Qed.

Multiplication and Opposite


Lemma mul_opp_r n m : n * - m = - (n * m).
Proof.
 now destruct n, m.
Qed.

Distributivity of multiplication over addition


Lemma mul_add_distr_pos (p:positive) n m :
 (n + m) * pos p = n * pos p + m * pos p.
Proof.
 destruct n as [|n|n], m as [|m|m]; simpl; trivial.
 - now rewrite Pos.mul_add_distr_r.
 - rewrite ?pos_sub_spec, ?Pos.mul_compare_mono_r; case Pos.compare_spec;
   simpl; trivial; intros; now rewrite Pos.mul_sub_distr_r.
 - rewrite ?pos_sub_spec, ?Pos.mul_compare_mono_r; case Pos.compare_spec;
   simpl; trivial; intros; now rewrite Pos.mul_sub_distr_r.
 - now rewrite Pos.mul_add_distr_r.
Qed.

Lemma mul_add_distr_r n m p : (n + m) * p = n * p + m * p.
Proof.
 destruct p as [|p|p].
 - now rewrite !mul_0_r.
 - apply mul_add_distr_pos.
 - apply opp_inj. rewrite opp_add_distr, <- !mul_opp_r.
   apply mul_add_distr_pos.
Qed.

End Private_BootStrap.

Proofs of specifications

Specification of constants


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

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

Specification of addition


Lemma add_0_l n : 0 + n = n.
Proof.
 now destruct n.
Qed.

Lemma add_succ_l n m : succ n + m = succ (n + m).
Proof.
 unfold succ. now rewrite 2 (add_comm _ 1), add_assoc.
Qed.

Specification of opposite


Lemma opp_0 : -0 = 0.
Proof.
 reflexivity.
Qed.

Lemma opp_succ n : -(succ n) = pred (-n).
Proof.
 unfold succ, pred. apply opp_add_distr.
Qed.

Specification of successor and predecessor


Local Arguments pos_sub : simpl nomatch.

Lemma succ_pred n : succ (pred n) = n.
Proof.
 unfold succ, pred. now rewrite <- add_assoc, add_opp_diag_r, add_0_r.
Qed.

Lemma pred_succ n : pred (succ n) = n.
Proof.
 unfold succ, pred. now rewrite <- add_assoc, add_opp_diag_r, add_0_r.
Qed.

Specification of subtraction


Lemma sub_0_r n : n - 0 = n.
Proof.
 apply add_0_r.
Qed.

Lemma sub_succ_r n m : n - succ m = pred (n - m).
Proof.
 unfold sub, succ, pred. now rewrite opp_add_distr, add_assoc.
Qed.

Specification of multiplication


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

Lemma mul_succ_l n m : succ n * m = n * m + m.
Proof.
 unfold succ. now rewrite mul_add_distr_r, mul_1_l.
Qed.

Specification of comparisons and order


Lemma eqb_eq n m : (n =? m) = true <-> n = m.
Proof.
 destruct n, m; simpl; try (now split); rewrite Pos.eqb_eq;
 split; (now injection 1) || (intros; now f_equal).
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.

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

Lemma compare_sub n m : (n ?= m) = (n - m ?= 0).
Proof.
 destruct n as [|n|n], m as [|m|m]; simpl; trivial;
 rewrite <- ? Pos.compare_antisym, ?pos_sub_spec;
 case Pos.compare_spec; trivial.
Qed.

Lemma compare_antisym n m : (m ?= n) = CompOpp (n ?= m).
Proof.
destruct n, m; simpl; trivial; now rewrite <- ?Pos.compare_antisym.
Qed.

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

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

Some more advanced properties of comparison and orders, including compare_spec and lt_irrefl and lt_eq_cases.

Include BoolOrderFacts.

Remaining specification of lt and le

Lemma lt_succ_r n m : n < succ m <-> n<=m.
Proof.
 unfold lt, le. rewrite compare_sub, sub_succ_r.
 rewrite (compare_sub n m).
 destruct (n-m) as [|[ | | ]|]; easy'.
Qed.

Specification of minimum and maximum


Lemma max_l n m : m<=n -> max n m = n.
Proof.
 unfold le, max. rewrite (compare_antisym n m).
 case compare; intuition.
Qed.

Lemma max_r n m : n<=m -> max n m = m.
Proof.
 unfold le, max. case compare_spec; intuition.
Qed.

Lemma min_l n m : n<=m -> min n m = n.
Proof.
 unfold le, min. case compare_spec; intuition.
Qed.

Lemma min_r n m : m<=n -> min n m = m.
Proof.
 unfold le, min.
 rewrite (compare_antisym n m). case compare_spec; intuition.
Qed.

Induction principles based on successor / predecessor


Lemma peano_ind (P : Z -> Prop) :
  P 0 ->
  (forall x, P x -> P (succ x)) ->
  (forall x, P x -> P (pred x)) ->
  forall z, P z.
Proof.
 intros H0 Hs Hp z; destruct z.
 assumption.
 induction p using Pos.peano_ind.
  now apply (Hs 0).
  rewrite <- Pos.add_1_r.
  now apply (Hs (pos p)).
 induction p using Pos.peano_ind.
  now apply (Hp 0).
  rewrite <- Pos.add_1_r.
  now apply (Hp (neg p)).
Qed.

Lemma bi_induction (P : Z -> Prop) :
  Proper (eq ==> iff) P ->
  P 0 ->
  (forall x, P x <-> P (succ x)) ->
  forall z, P z.
Proof.
 intros _ H0 Hs. induction z using peano_ind.
 assumption.
 now apply -> Hs.
 apply Hs. now rewrite succ_pred.
Qed.

We can now derive all properties of basic functions and orders, and use these properties for proving the specs of more advanced functions.

Include ZBasicProp <+ UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.

Specification of absolute value


Lemma abs_eq n : 0 <= n -> abs n = n.
Proof.
 destruct n; trivial. now destruct 1.
Qed.

Lemma abs_neq n : n <= 0 -> abs n = - n.
Proof.
 destruct n; trivial. now destruct 1.
Qed.

Specification of sign


Lemma sgn_null n : n = 0 -> sgn n = 0.
Proof.
 intros. now subst.
Qed.

Lemma sgn_pos n : 0 < n -> sgn n = 1.
Proof.
 now destruct n.
Qed.

Lemma sgn_neg n : n < 0 -> sgn n = -1.
Proof.
 now destruct n.
Qed.

Specification of power


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

Lemma pow_succ_r n m : 0<=m -> n^(succ m) = n * n^m.
Proof.
 destruct m as [|m|m]; (now destruct 1) || (intros _); simpl; trivial.
 unfold pow_pos. now rewrite Pos.add_comm, Pos.iter_add.
Qed.

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

For folding back a pow_pos into a pow

Lemma pow_pos_fold n p : pow_pos n p = n ^ (pos p).
Proof.
 reflexivity.
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 square root


Lemma sqrtrem_spec n : 0<=n ->
 let (s,r) := sqrtrem n in n = s*s + r /\ 0 <= r <= 2*s.
Proof.
 destruct n. now repeat split.
 generalize (Pos.sqrtrem_spec p). simpl.
 destruct 1; simpl; subst; now repeat split.
 now destruct 1.
Qed.

Lemma sqrt_spec n : 0<=n ->
 let s := sqrt n in s*s <= n < (succ s)*(succ s).
Proof.
 destruct n. now repeat split. unfold sqrt.
 intros _. simpl succ. rewrite Pos.add_1_r. apply (Pos.sqrt_spec p).
 now destruct 1.
Qed.

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

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

Specification of logarithm


Lemma log2_spec n : 0 < n -> 2^(log2 n) <= n < 2^(succ (log2 n)).
Proof.
 assert (Pow : forall p q, pos (p^q) = (pos p)^(pos q)).
 { intros. now apply Pos.iter_swap_gen. }
 destruct n as [|[p|p|]|]; intros Hn; split; try easy; unfold log2;
  simpl succ; rewrite ?Pos.add_1_r, <- Pow.
 change (2^Pos.size p <= Pos.succ (p~0))%positive.
 apply Pos.lt_le_incl, Pos.lt_succ_r, Pos.size_le.
 apply Pos.size_gt.
 apply Pos.size_le.
 apply Pos.size_gt.
Qed.

Lemma log2_nonpos n : n<=0 -> log2 n = 0.
Proof.
 destruct n as [|p|p]; trivial; now destruct 1.
Qed.

Specification of parity functions

Lemma even_spec n : even n = true <-> Even n.
Proof.
 split.
 exists (div2 n). now destruct n as [|[ | | ]|[ | | ]].
 intros (m,->). now destruct m.
Qed.

Lemma odd_spec n : odd n = true <-> Odd n.
Proof.
 split.
 exists (div2 n). destruct n as [|[ | | ]|[ | | ]]; simpl; try easy.
 now rewrite Pos.pred_double_succ.
 intros (m,->). now destruct m as [|[ | | ]|[ | | ]].
Qed.

Multiplication and Doubling


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 pred_double_spec n : pred_double n = 2*n - 1.
Proof.
 now destruct n.
Qed.

Correctness proofs for Trunc division


Lemma pos_div_eucl_eq a b : 0 < b ->
  let (q, r) := pos_div_eucl a b in pos a = q * b + r.
Proof.
 intros Hb.
 induction a; unfold pos_div_eucl; fold pos_div_eucl.
 -
   destruct pos_div_eucl as (q,r).
   change (pos a~1) with (2*(pos a)+1).
   rewrite IHa, mul_add_distr_l, mul_assoc.
   destruct ltb.
   now rewrite add_assoc.
   rewrite mul_add_distr_r, mul_1_l, <- !add_assoc. f_equal.
   unfold sub. now rewrite (add_comm _ (-b)), add_assoc, add_opp_diag_r.
 -
   destruct pos_div_eucl as (q,r).
   change (pos a~0) with (2*pos a).
   rewrite IHa, mul_add_distr_l, mul_assoc.
   destruct ltb.
   trivial.
   rewrite mul_add_distr_r, mul_1_l, <- !add_assoc. f_equal.
   unfold sub. now rewrite (add_comm _ (-b)), add_assoc, add_opp_diag_r.
 -
   case leb_spec; trivial.
   intros Hb'.
   destruct b as [|b|b]; try easy; clear Hb.
   replace b with 1%positive; trivial.
   apply Pos.le_antisym. apply Pos.le_1_l. now apply Pos.lt_succ_r.
Qed.

Lemma div_eucl_eq a b : b<>0 ->
 let (q, r) := div_eucl a b in a = b * q + r.
Proof.
 destruct a as [ |a|a], b as [ |b|b]; unfold div_eucl; trivial;
  (now destruct 1) || intros _;
  generalize (pos_div_eucl_eq a (pos b) Logic.eq_refl);
  destruct pos_div_eucl as (q,r); rewrite mul_comm.
 -
   trivial.
 -
   intros ->.
   destruct r as [ |r|r]; rewrite <- !mul_opp_comm; trivial;
    rewrite mul_add_distr_l, mul_1_r, <- add_assoc; f_equal;
   now rewrite add_assoc, add_opp_diag_r.
 -
   change (neg a) with (- pos a). intros ->.
   rewrite (opp_add_distr _ r), <- mul_opp_r.
   destruct r as [ |r|r]; trivial;
    rewrite opp_add_distr, mul_add_distr_l, <- add_assoc; f_equal;
    unfold sub; now rewrite add_assoc, mul_opp_r, mul_1_r, add_opp_diag_l.
 -
   change (neg a) with (- pos a). intros ->.
   now rewrite opp_add_distr, <- mul_opp_l.
Qed.

Lemma div_mod a b : b<>0 -> a = b*(a/b) + (a mod b).
Proof.
 intros Hb. generalize (div_eucl_eq a b Hb).
 unfold div, modulo. now destruct div_eucl.
Qed.

Lemma pos_div_eucl_bound a b : 0<b -> 0 <= snd (pos_div_eucl a b) < b.
Proof.
 assert (AUX : forall m p, m < pos (p~0) -> m - pos p < pos p).
  intros m p. unfold lt.
  rewrite (compare_sub m), (compare_sub _ (pos _)). unfold sub.
  rewrite <- add_assoc. simpl opp; simpl (neg _ + _).
  now rewrite Pos.add_diag.
 intros Hb.
 destruct b as [|b|b]; discriminate Hb || clear Hb.
 induction a; unfold pos_div_eucl; fold pos_div_eucl.
 destruct pos_div_eucl as (q,r).
 simpl in IHa; destruct IHa as (Hr,Hr').
 case ltb_spec; intros H; unfold snd. split; trivial. now destruct r.
 split. unfold le.
  now rewrite compare_antisym, <- compare_sub, <- compare_antisym.
 apply AUX. rewrite <- succ_double_spec.
 destruct r; try easy. unfold lt in *; simpl in *.
  now rewrite Pos.compare_xI_xO, Hr'.
 destruct pos_div_eucl as (q,r).
 simpl in IHa; destruct IHa as (Hr,Hr').
 case ltb_spec; intros H; unfold snd. split; trivial. now destruct r.
 split. unfold le.
  now rewrite compare_antisym, <- compare_sub, <- compare_antisym.
 apply AUX. destruct r; try easy.
 case leb_spec; intros H; simpl; split; try easy.
 red; simpl. now apply Pos.le_succ_l.
Qed.

Lemma mod_pos_bound a b : 0 < b -> 0 <= a mod b < b.
Proof.
 destruct b as [|b|b]; try easy; intros _.
 destruct a as [|a|a]; unfold modulo, div_eucl.
 now split.
 now apply pos_div_eucl_bound.
 generalize (pos_div_eucl_bound a (pos b) Logic.eq_refl).
 destruct pos_div_eucl as (q,r); unfold snd; intros (Hr,Hr').
 destruct r as [|r|r]; (now destruct Hr) || clear Hr.
 now split.
 split. unfold le.
  now rewrite compare_antisym, <- compare_sub, <- compare_antisym, Hr'.
 unfold lt in *; simpl in *. rewrite pos_sub_gt by trivial.
 simpl. now apply Pos.sub_decr.
Qed.

Definition mod_bound_pos a b (_:0<=a) := mod_pos_bound a b.

Lemma mod_neg_bound a b : b < 0 -> b < a mod b <= 0.
Proof.
 destruct b as [|b|b]; try easy; intros _.
 destruct a as [|a|a]; unfold modulo, div_eucl.
 now split.
 generalize (pos_div_eucl_bound a (pos b) Logic.eq_refl).
 destruct pos_div_eucl as (q,r); unfold snd; intros (Hr,Hr').
 destruct r as [|r|r]; (now destruct Hr) || clear Hr.
 now split.
 split.
 unfold lt in *; simpl in *. rewrite pos_sub_lt by trivial.
 rewrite <- Pos.compare_antisym. now apply Pos.sub_decr.
 change (neg b - neg r <= 0). unfold le, lt in *.
  rewrite <- compare_sub. simpl in *.
  now rewrite <- Pos.compare_antisym, Hr'.
 generalize (pos_div_eucl_bound a (pos b) Logic.eq_refl).
 destruct pos_div_eucl as (q,r); unfold snd; intros (Hr,Hr').
 split; destruct r; try easy.
  red; simpl; now rewrite <- Pos.compare_antisym.
Qed.

Correctness proofs for Floor division


Theorem quotrem_eq a b : let (q,r) := quotrem a b in a = q * b + r.
Proof.
 destruct a as [|a|a], b as [|b|b]; simpl; trivial;
 generalize (N.pos_div_eucl_spec a (N.pos b)); case N.pos_div_eucl; trivial;
  intros q r;
  try change (neg a) with (-pos a);
  change (pos a) with (of_N (N.pos a)); intros ->; now destruct q, r.
Qed.

Lemma quot_rem' a b : a = b*(a÷b) + rem a b.
Proof.
 rewrite mul_comm. generalize (quotrem_eq a b).
 unfold quot, rem. now destruct quotrem.
Qed.

Lemma quot_rem a b : b<>0 -> a = b*(a÷b) + rem a b.
Proof. intros _. apply quot_rem'. Qed.

Lemma rem_bound_pos a b : 0<=a -> 0<b -> 0 <= rem a b < b.
Proof.
 intros Ha Hb.
 destruct b as [|b|b]; (now discriminate Hb) || clear Hb;
 destruct a as [|a|a]; (now destruct Ha) || clear Ha.
 compute. now split.
 unfold rem, quotrem.
 assert (H := N.pos_div_eucl_remainder a (N.pos b)).
 destruct N.pos_div_eucl as (q,[|r]); simpl; split; try easy.
 now apply H.
Qed.

Lemma rem_opp_l' a b : rem (-a) b = - (rem a b).
Proof.
 destruct a, b; trivial; unfold rem; simpl;
  now destruct N.pos_div_eucl as (q,[|r]).
Qed.

Lemma rem_opp_r' a b : rem a (-b) = rem a b.
Proof.
 destruct a, b; trivial; unfold rem; simpl;
  now destruct N.pos_div_eucl as (q,[|r]).
Qed.

Lemma rem_opp_l a b : b<>0 -> rem (-a) b = - (rem a b).
Proof. intros _. apply rem_opp_l'. Qed.

Lemma rem_opp_r a b : b<>0 -> rem a (-b) = rem a b.
Proof. intros _. apply rem_opp_r'. Qed.

Extra properties about divide


Lemma divide_Zpos p q : (pos p|pos q) <-> (p|q)%positive.
Proof.
 split.
 intros ([ |r|r],H); simpl in *; destr_eq H. exists r; auto.
 intros (r,H). exists (pos r); simpl; now f_equal.
Qed.

Lemma divide_Zpos_Zneg_r n p : (n|pos p) <-> (n|neg p).
Proof.
 split; intros (m,H); exists (-m); now rewrite mul_opp_l, <- H.
Qed.

Lemma divide_Zpos_Zneg_l n p : (pos p|n) <-> (neg p|n).
Proof.
 split; intros (m,H); exists (-m); now rewrite mul_opp_l, <- mul_opp_r.
Qed.

Correctness proofs for gcd


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

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|p], b as [ |q|q]; simpl; rewrite ?Pos.mul_1_r; auto;
  generalize (Pos.ggcd_correct_divisors p q);
  destruct Pos.ggcd as (g,(aa,bb)); simpl; destruct 1; now subst.
Qed.

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.

Lemma gcd_greatest a b c : (c|a) -> (c|b) -> (c | gcd a b).
Proof.
 assert (H : forall p q r, (r|pos p) -> (r|pos q) -> (r|pos (Pos.gcd p q))).
 { intros p q [|r|r] H H'.
   destruct H; now rewrite mul_comm in *.
   apply divide_Zpos, Pos.gcd_greatest; now apply divide_Zpos.
   apply divide_Zpos_Zneg_l, divide_Zpos, Pos.gcd_greatest;
    now apply divide_Zpos, divide_Zpos_Zneg_l.
 }
 destruct a, b; simpl; auto; intros; try apply H; trivial;
  now apply divide_Zpos_Zneg_r.
Qed.

Lemma gcd_nonneg a b : 0 <= gcd a b.
Proof.
 now destruct a, b.
Qed.

ggcd and opp : an auxiliary result used in QArith

Theorem ggcd_opp a b :
  ggcd (-a) b = (let '(g,(aa,bb)) := ggcd a b in (g,(-aa,bb))).
Proof.
 destruct a as [|a|a], b as [|b|b]; unfold ggcd, opp; auto;
  destruct (Pos.ggcd a b) as (g,(aa,bb)); auto.
Qed.

Extra properties about testbit


Lemma testbit_of_N a n :
 testbit (of_N a) (of_N n) = N.testbit a n.
Proof.
 destruct a as [|a], n; simpl; trivial. now destruct a.
Qed.

Lemma testbit_of_N' a n : 0<=n ->
 testbit (of_N a) n = N.testbit a (to_N n).
Proof.
 intro Hn. rewrite <- testbit_of_N. f_equal.
 destruct n; trivial; now destruct Hn.
Qed.

Lemma testbit_Zpos a n : 0<=n ->
 testbit (pos a) n = N.testbit (N.pos a) (to_N n).
Proof.
 intro Hn. now rewrite <- testbit_of_N'.
Qed.

Lemma testbit_Zneg a n : 0<=n ->
 testbit (neg a) n = negb (N.testbit (Pos.pred_N a) (to_N n)).
Proof.
 intro Hn.
 rewrite <- testbit_of_N' by trivial.
 destruct n as [ |n|n];
  [ | simpl; now destruct (Pos.pred_N a) | now destruct Hn].
 unfold testbit.
 now destruct a as [|[ | | ]| ].
Qed.

Proofs of specifications for bitwise operations


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

Lemma testbit_0_l n : testbit 0 n = false.
Proof.
 now destruct n.
Qed.

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

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

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

Lemma testbit_odd_succ a n : 0<=n ->
 testbit (2*a+1) (succ n) = testbit a n.
Proof.
 destruct n as [|n|n]; (now destruct 1) || intros _.
 destruct a as [|[a|a|]|[a|a|]]; simpl; trivial. now destruct a.
 unfold testbit; simpl.
 destruct a as [|a|[a|a|]]; simpl; trivial;
  rewrite ?Pos.add_1_r, ?Pos.pred_N_succ; now destruct n.
Qed.

Lemma testbit_even_succ a n : 0<=n ->
 testbit (2*a) (succ n) = testbit a n.
Proof.
 destruct n as [|n|n]; (now destruct 1) || intros _.
 destruct a as [|[a|a|]|[a|a|]]; simpl; trivial. now destruct a.
 unfold testbit; simpl.
 destruct a as [|a|[a|a|]]; simpl; trivial;
  rewrite ?Pos.add_1_r, ?Pos.pred_N_succ; now destruct n.
Qed.

Correctness proofs about Z.shiftr and Z.shiftl

Lemma shiftr_spec_aux a n m : 0<=n -> 0<=m ->
 testbit (shiftr a n) m = testbit a (m+n).
Proof.
 intros Hn Hm. unfold shiftr.
 destruct n as [ |n|n]; (now destruct Hn) || clear Hn; simpl.
 now rewrite add_0_r.
 assert (forall p, to_N (m + pos p) = (to_N m + N.pos p)%N).
  destruct m; trivial; now destruct Hm.
 assert (forall p, 0 <= m + pos p).
  destruct m; easy || now destruct Hm.
 destruct a as [ |a|a].
 replace (Pos.iter div2 0 n) with 0
  by (apply Pos.iter_invariant; intros; subst; trivial).
 now rewrite 2 testbit_0_l.
 change (pos a) with (of_N (N.pos a)) at 1.
 rewrite <- (Pos.iter_swap_gen _ _ _ N.div2) by now intros [|[ | | ]].
 rewrite testbit_Zpos, testbit_of_N', H; trivial.
 exact (N.shiftr_spec' (N.pos a) (N.pos n) (to_N m)).
 rewrite <- (Pos.iter_swap_gen _ _ _ Pos.div2_up) by trivial.
 rewrite 2 testbit_Zneg, H; trivial. f_equal.
 rewrite (Pos.iter_swap_gen _ _ _ _ N.div2) by exact N.pred_div2_up.
 exact (N.shiftr_spec' (Pos.pred_N a) (N.pos n) (to_N m)).
Qed.

Lemma shiftl_spec_low a n m : m<n ->
 testbit (shiftl a n) m = false.
Proof.
 intros H. destruct n as [|n|n], m as [|m|m]; try easy; simpl shiftl.
 destruct (Pos.succ_pred_or n) as [-> | <-];
  rewrite ?Pos.iter_succ; apply testbit_even_0.
 destruct a as [ |a|a].
 replace (Pos.iter (mul 2) 0 n) with 0
  by (apply Pos.iter_invariant; intros; subst; trivial).
 apply testbit_0_l.
 rewrite <- (Pos.iter_swap_gen _ _ _ xO) by trivial.
 rewrite testbit_Zpos by easy.
 exact (N.shiftl_spec_low (N.pos a) (N.pos n) (N.pos m) H).
 rewrite <- (Pos.iter_swap_gen _ _ _ xO) by trivial.
 rewrite testbit_Zneg by easy.
 now rewrite (N.pos_pred_shiftl_low a (N.pos n)).
Qed.

Lemma shiftl_spec_high a n m : 0<=m -> n<=m ->
 testbit (shiftl a n) m = testbit a (m-n).
Proof.
 intros Hm H.
 destruct n as [ |n|n]. simpl. now rewrite sub_0_r.
 destruct m as [ |m|m]; try (now destruct H).
 assert (0 <= pos m - pos n).
  red. now rewrite compare_antisym, <- compare_sub, <- compare_antisym.
 assert (EQ : to_N (pos m - pos n) = (N.pos m - N.pos n)%N).
  red in H. simpl in H. simpl to_N.
  rewrite pos_sub_spec, Pos.compare_antisym.
  destruct (Pos.compare_spec n m) as [H'|H'|H']; try (now destruct H).
  subst. now rewrite N.sub_diag.
  simpl. destruct (Pos.sub_mask_pos' m n H') as (p & -> & <-).
  f_equal. now rewrite Pos.add_comm, Pos.add_sub.
 destruct a; unfold shiftl.
 replace (Pos.iter (mul 2) 0 n) with 0
  by (apply Pos.iter_invariant; intros; subst; trivial).
 now rewrite 2 testbit_0_l.
 rewrite <- (Pos.iter_swap_gen _ _ _ xO) by trivial.
 rewrite 2 testbit_Zpos, EQ by easy.
 exact (N.shiftl_spec_high' (N.pos p) (N.pos n) (N.pos m) H).
 rewrite <- (Pos.iter_swap_gen _ _ _ xO) by trivial.
 rewrite 2 testbit_Zneg, EQ by easy. f_equal.
 simpl to_N.
 rewrite <- N.shiftl_spec_high by easy.
 now apply (N.pos_pred_shiftl_high p (N.pos n)).
 unfold sub. simpl.
 now apply (shiftr_spec_aux a (pos n) m).
Qed.

Lemma shiftr_spec a n m : 0<=m ->
 testbit (shiftr a n) m = testbit a (m+n).
Proof.
 intros Hm.
 destruct (leb_spec 0 n).
 now apply shiftr_spec_aux.
 destruct (leb_spec (-n) m) as [LE|GT].
 unfold shiftr.
 rewrite (shiftl_spec_high a (-n) m); trivial. now destruct n.
 unfold shiftr.
 rewrite (shiftl_spec_low a (-n) m); trivial.
 rewrite testbit_neg_r; trivial.
 red in GT. rewrite compare_sub in GT. now destruct n.
Qed.

Correctness proofs for bitwise operations

Lemma lor_spec a b n :
 testbit (lor a b) n = testbit a n || testbit b n.
Proof.
 destruct (leb_spec 0 n) as [Hn|Hn]; [|now rewrite !testbit_neg_r].
 destruct a as [ |a|a], b as [ |b|b];
  rewrite ?testbit_0_l, ?orb_false_r; trivial; unfold lor;
  rewrite ?testbit_Zpos, ?testbit_Zneg, ?N.pos_pred_succ by trivial.
 now rewrite <- N.lor_spec.
 now rewrite N.ldiff_spec, negb_andb, negb_involutive, orb_comm.
 now rewrite N.ldiff_spec, negb_andb, negb_involutive.
 now rewrite N.land_spec, negb_andb.
Qed.

Lemma land_spec a b n :
 testbit (land a b) n = testbit a n && testbit b n.
Proof.
 destruct (leb_spec 0 n) as [Hn|Hn]; [|now rewrite !testbit_neg_r].
 destruct a as [ |a|a], b as [ |b|b];
  rewrite ?testbit_0_l, ?andb_false_r; trivial; unfold land;
  rewrite ?testbit_Zpos, ?testbit_Zneg, ?testbit_of_N', ?N.pos_pred_succ
   by trivial.
 now rewrite <- N.land_spec.
 now rewrite N.ldiff_spec.
 now rewrite N.ldiff_spec, andb_comm.
 now rewrite N.lor_spec, negb_orb.
Qed.

Lemma ldiff_spec a b n :
 testbit (ldiff a b) n = testbit a n && negb (testbit b n).
Proof.
 destruct (leb_spec 0 n) as [Hn|Hn]; [|now rewrite !testbit_neg_r].
 destruct a as [ |a|a], b as [ |b|b];
  rewrite ?testbit_0_l, ?andb_true_r; trivial; unfold ldiff;
  rewrite ?testbit_Zpos, ?testbit_Zneg, ?testbit_of_N', ?N.pos_pred_succ
   by trivial.
 now rewrite <- N.ldiff_spec.
 now rewrite N.land_spec, negb_involutive.
 now rewrite N.lor_spec, negb_orb.
 now rewrite N.ldiff_spec, negb_involutive, andb_comm.
Qed.

Lemma lxor_spec a b n :
 testbit (lxor a b) n = xorb (testbit a n) (testbit b n).
Proof.
 destruct (leb_spec 0 n) as [Hn|Hn]; [|now rewrite !testbit_neg_r].
 destruct a as [ |a|a], b as [ |b|b];
  rewrite ?testbit_0_l, ?xorb_false_l, ?xorb_false_r; trivial; unfold lxor;
  rewrite ?testbit_Zpos, ?testbit_Zneg, ?testbit_of_N', ?N.pos_pred_succ
   by trivial.
 now rewrite <- N.lxor_spec.
 now rewrite N.lxor_spec, negb_xorb_r.
 now rewrite N.lxor_spec, negb_xorb_l.
 now rewrite N.lxor_spec, xorb_negb_negb.
Qed.

Generic properties of advanced functions.

Include ZExtraProp.

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.

We provide a tactic converting from one style to the other.

Ltac swap_greater := rewrite ?gt_lt_iff in *; rewrite ?ge_le_iff in *.

Similarly, the boolean comparisons ltb and leb are favored over their dual gtb and geb. We prove here the equivalence and a few minimal results.

Lemma gtb_ltb n m : (n >? m) = (m <? n).
Proof.
 unfold gtb, ltb. rewrite compare_antisym. now case compare.
Qed.

Lemma geb_leb n m : (n >=? m) = (m <=? n).
Proof.
 unfold geb, leb. rewrite compare_antisym. now case compare.
Qed.

Lemma gtb_lt n m : (n >? m) = true <-> m < n.
Proof.
 rewrite gtb_ltb. apply ltb_lt.
Qed.

Lemma geb_le n m : (n >=? m) = true <-> m <= n.
Proof.
 rewrite geb_leb. apply leb_le.
Qed.

Lemma gtb_spec n m : BoolSpec (m<n) (n<=m) (n >? m).
Proof.
 rewrite gtb_ltb. apply ltb_spec.
Qed.

Lemma geb_spec n m : BoolSpec (m<=n) (n<m) (n >=? m).
Proof.
 rewrite geb_leb. apply leb_spec.
Qed.

TODO : to add in Numbers ?

Lemma add_reg_l n m p : n + m = n + p -> m = p.
Proof.
 exact (proj1 (add_cancel_l m p n)).
Qed.

Lemma mul_reg_l n m p : p <> 0 -> p * n = p * m -> n = m.
Proof.
 exact (fun Hp => proj1 (mul_cancel_l n m p Hp)).
Qed.

Lemma mul_reg_r n m p : p <> 0 -> n * p = m * p -> n = m.
Proof.
 exact (fun Hp => proj1 (mul_cancel_r n m p Hp)).
Qed.

Lemma opp_eq_mul_m1 n : - n = n * -1.
Proof.
 rewrite mul_comm. now destruct n.
Qed.

Lemma add_diag n : n + n = 2 * n.
Proof.
 change 2 with (1+1). now rewrite mul_add_distr_r, !mul_1_l.
Qed.

Comparison and opposite


Lemma compare_opp n m : (- n ?= - m) = (m ?= n).
Proof.
 destruct n, m; simpl; trivial; intros; now rewrite <- Pos.compare_antisym.
Qed.

Comparison and addition


Lemma add_compare_mono_l n m p : (n + m ?= n + p) = (m ?= p).
Proof.
 rewrite (compare_sub m p), compare_sub. f_equal.
 unfold sub. rewrite opp_add_distr, (add_comm n m), add_assoc.
 f_equal. now rewrite <- add_assoc, add_opp_diag_r, add_0_r.
Qed.

End Z.

Bind Scope Z_scope with Z.t Z.

Re-export Notations

Numeral Notation Z Z.of_int Z.to_int : Z_scope.

Infix "+" := Z.add : Z_scope.
Notation "- x" := (Z.opp x) : Z_scope.
Infix "-" := Z.sub : Z_scope.
Infix "*" := Z.mul : Z_scope.
Infix "^" := Z.pow : Z_scope.
Infix "/" := Z.div : Z_scope.
Infix "mod" := Z.modulo (at level 40, no associativity) : Z_scope.
Infix "÷" := Z.quot (at level 40, left associativity) : Z_scope.
Infix "?=" := Z.compare (at level 70, no associativity) : Z_scope.
Infix "=?" := Z.eqb (at level 70, no associativity) : Z_scope.
Infix "<=?" := Z.leb (at level 70, no associativity) : Z_scope.
Infix "<?" := Z.ltb (at level 70, no associativity) : Z_scope.
Infix ">=?" := Z.geb (at level 70, no associativity) : Z_scope.
Infix ">?" := Z.gtb (at level 70, no associativity) : Z_scope.
Notation "( x | y )" := (Z.divide x y) (at level 0) : Z_scope.
Infix "<=" := Z.le : Z_scope.
Infix "<" := Z.lt : Z_scope.
Infix ">=" := Z.ge : Z_scope.
Infix ">" := Z.gt : Z_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : Z_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : Z_scope.
Notation "x < y < z" := (x < y /\ y < z) : Z_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : Z_scope.

Conversions from / to positive numbers

Module Pos2Z.

Lemma id p : Z.to_pos (Z.pos p) = p.
Proof. reflexivity. Qed.

Lemma inj p q : Z.pos p = Z.pos q -> p = q.
Proof. now injection 1. Qed.

Lemma inj_iff p q : Z.pos p = Z.pos q <-> p = q.
Proof. split. apply inj. intros; now f_equal. Qed.

Lemma is_pos p : 0 < Z.pos p.
Proof. reflexivity. Qed.

Lemma is_nonneg p : 0 <= Z.pos p.
Proof. easy. Qed.

Lemma inj_1 : Z.pos 1 = 1.
Proof. reflexivity. Qed.

Lemma inj_xO p : Z.pos p~0 = 2 * Z.pos p.
Proof. reflexivity. Qed.

Lemma inj_xI p : Z.pos p~1 = 2 * Z.pos p + 1.
Proof. reflexivity. Qed.

Lemma inj_succ p : Z.pos (Pos.succ p) = Z.succ (Z.pos p).
Proof. simpl. now rewrite Pos.add_1_r. Qed.

Lemma inj_add p q : Z.pos (p+q) = Z.pos p + Z.pos q.
Proof. reflexivity. Qed.

Lemma inj_sub p q : (p < q)%positive ->
 Z.pos (q-p) = Z.pos q - Z.pos p.
Proof. intros. simpl. now rewrite Z.pos_sub_gt. Qed.

Lemma inj_sub_max p q : Z.pos (p - q) = Z.max 1 (Z.pos p - Z.pos q).
Proof.
  simpl. rewrite Z.pos_sub_spec. case Pos.compare_spec; intros.
  - subst; now rewrite Pos.sub_diag.
  - now rewrite Pos.sub_lt.
  - now destruct (p-q)%positive.
Qed.

Lemma inj_pred p : p <> 1%positive ->
 Z.pos (Pos.pred p) = Z.pred (Z.pos p).
Proof. destruct p; easy || now destruct 1. Qed.

Lemma inj_mul p q : Z.pos (p*q) = Z.pos p * Z.pos q.
Proof. reflexivity. Qed.

Lemma inj_pow_pos p q : Z.pos (p^q) = Z.pow_pos (Z.pos p) q.
Proof. now apply Pos.iter_swap_gen. Qed.

Lemma inj_pow p q : Z.pos (p^q) = (Z.pos p)^(Z.pos q).
Proof. apply inj_pow_pos. Qed.

Lemma inj_square p : Z.pos (Pos.square p) = Z.square (Z.pos p).
Proof. reflexivity. Qed.

Lemma inj_compare p q : (p ?= q)%positive = (Z.pos p ?= Z.pos q).
Proof. reflexivity. Qed.

Lemma inj_leb p q : (p <=? q)%positive = (Z.pos p <=? Z.pos q).
Proof. reflexivity. Qed.

Lemma inj_ltb p q : (p <? q)%positive = (Z.pos p <? Z.pos q).
Proof. reflexivity. Qed.

Lemma inj_eqb p q : (p =? q)%positive = (Z.pos p =? Z.pos q).
Proof. reflexivity. Qed.

Lemma inj_max p q : Z.pos (Pos.max p q) = Z.max (Z.pos p) (Z.pos q).
Proof.
 unfold Z.max, Pos.max. rewrite inj_compare. now case Z.compare_spec.
Qed.

Lemma inj_min p q : Z.pos (Pos.min p q) = Z.min (Z.pos p) (Z.pos q).
Proof.
 unfold Z.min, Pos.min. rewrite inj_compare. now case Z.compare_spec.
Qed.

Lemma inj_sqrt p : Z.pos (Pos.sqrt p) = Z.sqrt (Z.pos p).
Proof. reflexivity. Qed.

Lemma inj_gcd p q : Z.pos (Pos.gcd p q) = Z.gcd (Z.pos p) (Z.pos q).
Proof. reflexivity. Qed.

Definition inj_divide p q : (Z.pos p|Z.pos q) <-> (p|q)%positive.
Proof. apply Z.divide_Zpos. Qed.

Lemma inj_testbit a n : 0<=n ->
 Z.testbit (Z.pos a) n = N.testbit (N.pos a) (Z.to_N n).
Proof. apply Z.testbit_Zpos. Qed.

Some results concerning Z.neg and Z.pos

Lemma inj_neg p q : Z.neg p = Z.neg q -> p = q.
Proof. now injection 1. Qed.

Lemma inj_neg_iff p q : Z.neg p = Z.neg q <-> p = q.
Proof. split. apply inj_neg. intros; now f_equal. Qed.

Lemma inj_pos p q : Z.pos p = Z.pos q -> p = q.
Proof. now injection 1. Qed.

Lemma inj_pos_iff p q : Z.pos p = Z.pos q <-> p = q.
Proof. split. apply inj_pos. intros; now f_equal. Qed.

Lemma neg_is_neg p : Z.neg p < 0.
Proof. reflexivity. Qed.

Lemma neg_is_nonpos p : Z.neg p <= 0.
Proof. easy. Qed.

Lemma pos_is_pos p : 0 < Z.pos p.
Proof. reflexivity. Qed.

Lemma pos_is_nonneg p : 0 <= Z.pos p.
Proof. easy. Qed.

Lemma neg_le_pos p q : Zneg p <= Zpos q.
Proof. easy. Qed.

Lemma neg_lt_pos p q : Zneg p < Zpos q.
Proof. easy. Qed.

Lemma neg_le_neg p q : (q <= p)%positive -> Zneg p <= Zneg q.
Proof. intros; unfold Z.le; simpl. now rewrite <- Pos.compare_antisym. Qed.

Lemma neg_lt_neg p q : (q < p)%positive -> Zneg p < Zneg q.
Proof. intros; unfold Z.lt; simpl. now rewrite <- Pos.compare_antisym. Qed.

Lemma pos_le_pos p q : (p <= q)%positive -> Zpos p <= Zpos q.
Proof. easy. Qed.

Lemma pos_lt_pos p q : (p < q)%positive -> Zpos p < Zpos q.
Proof. easy. Qed.

Lemma neg_xO p : Z.neg p~0 = 2 * Z.neg p.
Proof. reflexivity. Qed.

Lemma neg_xI p : Z.neg p~1 = 2 * Z.neg p - 1.
Proof. reflexivity. Qed.

Lemma pos_xO p : Z.pos p~0 = 2 * Z.pos p.
Proof. reflexivity. Qed.

Lemma pos_xI p : Z.pos p~1 = 2 * Z.pos p + 1.
Proof. reflexivity. Qed.

Lemma opp_neg p : - Z.neg p = Z.pos p.
Proof. reflexivity. Qed.

Lemma opp_pos p : - Z.pos p = Z.neg p.
Proof. reflexivity. Qed.

Lemma add_neg_neg p q : Z.neg p + Z.neg q = Z.neg (p+q).
Proof. reflexivity. Qed.

Lemma add_pos_neg p q : Z.pos p + Z.neg q = Z.pos_sub p q.
Proof. reflexivity. Qed.

Lemma add_neg_pos p q : Z.neg p + Z.pos q = Z.pos_sub q p.
Proof. reflexivity. Qed.

Lemma add_pos_pos p q : Z.pos p + Z.pos q = Z.pos (p+q).
Proof. reflexivity. Qed.

Lemma divide_pos_neg_r n p : (n|Z.pos p) <-> (n|Z.neg p).
Proof. apply Z.divide_Zpos_Zneg_r. Qed.

Lemma divide_pos_neg_l n p : (Z.pos p|n) <-> (Z.neg p|n).
Proof. apply Z.divide_Zpos_Zneg_l. Qed.

Lemma testbit_neg a n : 0<=n ->
 Z.testbit (Z.neg a) n = negb (N.testbit (Pos.pred_N a) (Z.to_N n)).
Proof. apply Z.testbit_Zneg. Qed.

Lemma testbit_pos a n : 0<=n ->
 Z.testbit (Z.pos a) n = N.testbit (N.pos a) (Z.to_N n).
Proof. apply Z.testbit_Zpos. Qed.

End Pos2Z.

Module Z2Pos.

Lemma id x : 0 < x -> Z.pos (Z.to_pos x) = x.
Proof. now destruct x. Qed.

Lemma inj x y : 0 < x -> 0 < y -> Z.to_pos x = Z.to_pos y -> x = y.
Proof.
 destruct x; simpl; try easy. intros _ H ->. now apply id.
Qed.

Lemma inj_iff x y : 0 < x -> 0 < y -> (Z.to_pos x = Z.to_pos y <-> x = y).
Proof. split. now apply inj. intros; now f_equal. Qed.

Lemma to_pos_nonpos x : x <= 0 -> Z.to_pos x = 1%positive.
Proof. destruct x; trivial. now destruct 1. Qed.

Lemma inj_1 : Z.to_pos 1 = 1%positive.
Proof. reflexivity. Qed.

Lemma inj_double x : 0 < x ->
 Z.to_pos (Z.double x) = (Z.to_pos x)~0%positive.
Proof. now destruct x. Qed.

Lemma inj_succ_double x : 0 < x ->
 Z.to_pos (Z.succ_double x) = (Z.to_pos x)~1%positive.
Proof. now destruct x. Qed.

Lemma inj_succ x : 0 < x -> Z.to_pos (Z.succ x) = Pos.succ (Z.to_pos x).
Proof.
 destruct x; try easy. simpl. now rewrite Pos.add_1_r.
Qed.

Lemma inj_add x y : 0 < x -> 0 < y ->
 Z.to_pos (x+y) = (Z.to_pos x + Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_sub x y : 0 < x < y ->
 Z.to_pos (y-x) = (Z.to_pos y - Z.to_pos x)%positive.
Proof.
 destruct x; try easy. destruct y; try easy. simpl.
 intros. now rewrite Z.pos_sub_gt.
Qed.

Lemma inj_pred x : 1 < x -> Z.to_pos (Z.pred x) = Pos.pred (Z.to_pos x).
Proof. now destruct x as [|[x|x|]|]. Qed.

Lemma inj_mul x y : 0 < x -> 0 < y ->
 Z.to_pos (x*y) = (Z.to_pos x * Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_pow x y : 0 < x -> 0 < y ->
 Z.to_pos (x^y) = (Z.to_pos x ^ Z.to_pos y)%positive.
Proof.
 intros. apply Pos2Z.inj. rewrite Pos2Z.inj_pow, !id; trivial.
 apply Z.pow_pos_nonneg. trivial. now apply Z.lt_le_incl.
Qed.

Lemma inj_pow_pos x p : 0 < x ->
 Z.to_pos (Z.pow_pos x p) = ((Z.to_pos x)^p)%positive.
Proof. intros. now apply (inj_pow x (Z.pos p)). Qed.

Lemma inj_compare x y : 0 < x -> 0 < y ->
 (x ?= y) = (Z.to_pos x ?= Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_leb x y : 0 < x -> 0 < y ->
 (x <=? y) = (Z.to_pos x <=? Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_ltb x y : 0 < x -> 0 < y ->
 (x <? y) = (Z.to_pos x <? Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_eqb x y : 0 < x -> 0 < y ->
 (x =? y) = (Z.to_pos x =? Z.to_pos y)%positive.
Proof. destruct x; easy || now destruct y. Qed.

Lemma inj_max x y :
 Z.to_pos (Z.max x y) = Pos.max (Z.to_pos x) (Z.to_pos y).
Proof.
 destruct x; simpl; try rewrite Pos.max_1_l.
 - now destruct y.
 - destruct y; simpl; now rewrite ?Pos.max_1_r, <- ?Pos2Z.inj_max.
 - destruct y; simpl; rewrite ?Pos.max_1_r; trivial.
   apply to_pos_nonpos. now apply Z.max_lub.
Qed.

Lemma inj_min x y :
 Z.to_pos (Z.min x y) = Pos.min (Z.to_pos x) (Z.to_pos y).
Proof.
 destruct x; simpl; try rewrite Pos.min_1_l.
 - now destruct y.
 - destruct y; simpl; now rewrite ?Pos.min_1_r, <- ?Pos2Z.inj_min.
 - destruct y; simpl; rewrite ?Pos.min_1_r; trivial.
   apply to_pos_nonpos. apply Z.min_le_iff. now left.
Qed.

Lemma inj_sqrt x : Z.to_pos (Z.sqrt x) = Pos.sqrt (Z.to_pos x).
Proof. now destruct x. Qed.

Lemma inj_gcd x y : 0 < x -> 0 < y ->
 Z.to_pos (Z.gcd x y) = Pos.gcd (Z.to_pos x) (Z.to_pos y).
Proof. destruct x; easy || now destruct y. Qed.

End Z2Pos.

Compatibility Notations

Notation Zdouble_plus_one := Z.succ_double (only parsing).
Notation Zdouble_minus_one := Z.pred_double (only parsing).
Notation Zdouble := Z.double (compat "8.7").
Notation ZPminus := Z.pos_sub (only parsing).
Notation Zsucc' := Z.succ (compat "8.7").
Notation Zpred' := Z.pred (compat "8.7").
Notation Zplus' := Z.add (compat "8.7").
Notation Zplus := Z.add (only parsing). Notation Zopp := Z.opp (compat "8.7").
Notation Zsucc := Z.succ (compat "8.7").
Notation Zpred := Z.pred (compat "8.7").
Notation Zminus := Z.sub (only parsing).
Notation Zmult := Z.mul (only parsing).
Notation Zcompare := Z.compare (compat "8.7").
Notation Zsgn := Z.sgn (compat "8.7").
Notation Zle := Z.le (compat "8.7").
Notation Zge := Z.ge (compat "8.7").
Notation Zlt := Z.lt (compat "8.7").
Notation Zgt := Z.gt (compat "8.7").
Notation Zmax := Z.max (compat "8.7").
Notation Zmin := Z.min (compat "8.7").
Notation Zabs := Z.abs (compat "8.7").
Notation Zabs_nat := Z.abs_nat (compat "8.7").
Notation Zabs_N := Z.abs_N (compat "8.7").
Notation Z_of_nat := Z.of_nat (only parsing).
Notation Z_of_N := Z.of_N (only parsing).

Notation Zind := Z.peano_ind (only parsing).
Notation Zopp_0 := Z.opp_0 (compat "8.7").
Notation Zopp_involutive := Z.opp_involutive (compat "8.7").
Notation Zopp_inj := Z.opp_inj (compat "8.7").
Notation Zplus_0_l := Z.add_0_l (only parsing).
Notation Zplus_0_r := Z.add_0_r (only parsing).
Notation Zplus_comm := Z.add_comm (only parsing).
Notation Zopp_plus_distr := Z.opp_add_distr (only parsing).
Notation Zopp_succ := Z.opp_succ (compat "8.7").
Notation Zplus_opp_r := Z.add_opp_diag_r (only parsing).
Notation Zplus_opp_l := Z.add_opp_diag_l (only parsing).
Notation Zplus_assoc := Z.add_assoc (only parsing).
Notation Zplus_permute := Z.add_shuffle3 (only parsing).
Notation Zplus_reg_l := Z.add_reg_l (only parsing).
Notation Zplus_succ_l := Z.add_succ_l (only parsing).
Notation Zplus_succ_comm := Z.add_succ_comm (only parsing).
Notation Zsucc_discr := Z.neq_succ_diag_r (only parsing).
Notation Zsucc_inj := Z.succ_inj (compat "8.7").
Notation Zsucc'_inj := Z.succ_inj (compat "8.7").
Notation Zsucc'_pred' := Z.succ_pred (compat "8.7").
Notation Zpred'_succ' := Z.pred_succ (compat "8.7").
Notation Zpred'_inj := Z.pred_inj (compat "8.7").
Notation Zsucc'_discr := Z.neq_succ_diag_r (only parsing).
Notation Zminus_0_r := Z.sub_0_r (only parsing).
Notation Zminus_diag := Z.sub_diag (only parsing).
Notation Zminus_plus_distr := Z.sub_add_distr (only parsing).
Notation Zminus_succ_r := Z.sub_succ_r (only parsing).
Notation Zminus_plus := Z.add_simpl_l (only parsing).
Notation Zmult_0_l := Z.mul_0_l (only parsing).
Notation Zmult_0_r := Z.mul_0_r (only parsing).
Notation Zmult_1_l := Z.mul_1_l (only parsing).
Notation Zmult_1_r := Z.mul_1_r (only parsing).
Notation Zmult_comm := Z.mul_comm (only parsing).
Notation Zmult_assoc := Z.mul_assoc (only parsing).
Notation Zmult_permute := Z.mul_shuffle3 (only parsing).
Notation Zmult_1_inversion_l := Z.mul_eq_1 (only parsing).
Notation Zdouble_mult := Z.double_spec (only parsing).
Notation Zdouble_plus_one_mult := Z.succ_double_spec (only parsing).
Notation Zopp_mult_distr_l_reverse := Z.mul_opp_l (only parsing).
Notation Zmult_opp_opp := Z.mul_opp_opp (only parsing).
Notation Zmult_opp_comm := Z.mul_opp_comm (only parsing).
Notation Zopp_eq_mult_neg_1 := Z.opp_eq_mul_m1 (only parsing).
Notation Zmult_plus_distr_r := Z.mul_add_distr_l (only parsing).
Notation Zmult_plus_distr_l := Z.mul_add_distr_r (only parsing).
Notation Zmult_minus_distr_r := Z.mul_sub_distr_r (only parsing).
Notation Zmult_reg_l := Z.mul_reg_l (only parsing).
Notation Zmult_reg_r := Z.mul_reg_r (only parsing).
Notation Zmult_succ_l := Z.mul_succ_l (only parsing).
Notation Zmult_succ_r := Z.mul_succ_r (only parsing).

Notation Zpos_xI := Pos2Z.inj_xI (only parsing).
Notation Zpos_xO := Pos2Z.inj_xO (only parsing).
Notation Zneg_xI := Pos2Z.neg_xI (only parsing).
Notation Zneg_xO := Pos2Z.neg_xO (only parsing).
Notation Zopp_neg := Pos2Z.opp_neg (only parsing).
Notation Zpos_succ_morphism := Pos2Z.inj_succ (only parsing).
Notation Zpos_mult_morphism := Pos2Z.inj_mul (only parsing).
Notation Zpos_minus_morphism := Pos2Z.inj_sub (only parsing).
Notation Zpos_eq_rev := Pos2Z.inj (only parsing).
Notation Zpos_plus_distr := Pos2Z.inj_add (only parsing).
Notation Zneg_plus_distr := Pos2Z.add_neg_neg (only parsing).

Notation Z := Z (only parsing).
Notation Z_rect := Z_rect (only parsing).
Notation Z_rec := Z_rec (only parsing).
Notation Z_ind := Z_ind (only parsing).
Notation Z0 := Z0 (only parsing).
Notation Zpos := Zpos (only parsing).
Notation Zneg := Zneg (only parsing).

Compatibility lemmas. These could be notations, but scope information would be lost.

Notation SYM1 lem := (fun n => eq_sym (lem n)).
Notation SYM2 lem := (fun n m => eq_sym (lem n m)).
Notation SYM3 lem := (fun n m p => eq_sym (lem n m p)).

Lemma Zplus_assoc_reverse : forall n m p, n+m+p = n+(m+p).
Proof (SYM3 Z.add_assoc).
Lemma Zplus_succ_r_reverse : forall n m, Z.succ (n+m) = n+Z.succ m.
Proof (SYM2 Z.add_succ_r).
Notation Zplus_succ_r := Zplus_succ_r_reverse (only parsing).
Lemma Zplus_0_r_reverse : forall n, n = n + 0.
Proof (SYM1 Z.add_0_r).
Lemma Zplus_eq_compat : forall n m p q, n=m -> p=q -> n+p=m+q.
Proof (f_equal2 Z.add).
Lemma Zsucc_pred : forall n, n = Z.succ (Z.pred n).
Proof (SYM1 Z.succ_pred).
Lemma Zpred_succ : forall n, n = Z.pred (Z.succ n).
Proof (SYM1 Z.pred_succ).
Lemma Zsucc_eq_compat : forall n m, n = m -> Z.succ n = Z.succ m.
Proof (f_equal Z.succ).
Lemma Zminus_0_l_reverse : forall n, n = n - 0.
Proof (SYM1 Z.sub_0_r).
Lemma Zminus_diag_reverse : forall n, 0 = n-n.
Proof (SYM1 Z.sub_diag).
Lemma Zminus_succ_l : forall n m, Z.succ (n - m) = Z.succ n - m.
Proof (SYM2 Z.sub_succ_l).
Lemma Zplus_minus_eq : forall n m p, n = m + p -> p = n - m.
Proof. intros. now apply Z.add_move_l. Qed.
Lemma Zplus_minus : forall n m, n + (m - n) = m.
Proof (fun n m => eq_trans (Z.add_comm n (m-n)) (Z.sub_add n m)).
Lemma Zminus_plus_simpl_l : forall n m p, p + n - (p + m) = n - m.
Proof (fun n m p => Z.add_add_simpl_l_l p n m).
Lemma Zminus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m).
Proof (SYM3 Zminus_plus_simpl_l).
Lemma Zminus_plus_simpl_r : forall n m p, n + p - (m + p) = n - m.
Proof (fun n m p => Z.add_add_simpl_r_r n p m).
Lemma Zeq_minus : forall n m, n = m -> n - m = 0.
Proof (fun n m => proj2 (Z.sub_move_0_r n m)).
Lemma Zminus_eq : forall n m, n - m = 0 -> n = m.
Proof (fun n m => proj1 (Z.sub_move_0_r n m)).
Lemma Zmult_0_r_reverse : forall n, 0 = n * 0.
Proof (SYM1 Z.mul_0_r).
Lemma Zmult_assoc_reverse : forall n m p, n * m * p = n * (m * p).
Proof (SYM3 Z.mul_assoc).
Lemma Zmult_integral : forall n m, n * m = 0 -> n = 0 \/ m = 0.
Proof (fun n m => proj1 (Z.mul_eq_0 n m)).
Lemma Zmult_integral_l : forall n m, n <> 0 -> m * n = 0 -> m = 0.
Proof (fun n m H H' => Z.mul_eq_0_l m n H' H).
Lemma Zopp_mult_distr_l : forall n m, - (n * m) = - n * m.
Proof (SYM2 Z.mul_opp_l).
Lemma Zopp_mult_distr_r : forall n m, - (n * m) = n * - m.
Proof (SYM2 Z.mul_opp_r).
Lemma Zmult_minus_distr_l : forall n m p, p * (n - m) = p * n - p * m.
Proof (fun n m p => Z.mul_sub_distr_l p n m).
Lemma Zmult_succ_r_reverse : forall n m, n * m + n = n * Z.succ m.
Proof (SYM2 Z.mul_succ_r).
Lemma Zmult_succ_l_reverse : forall n m, n * m + m = Z.succ n * m.
Proof (SYM2 Z.mul_succ_l).
Lemma Zpos_eq : forall p q, p = q -> Z.pos p = Z.pos q.
Proof. congruence. Qed.
Lemma Zpos_eq_iff : forall p q, p = q <-> Z.pos p = Z.pos q.
Proof (fun p q => iff_sym (Pos2Z.inj_iff p q)).

Hint Immediate Zsucc_pred: zarith.



Obsolete stuff

Definition Zne (x y:Z) := x <> y.
Ltac elim_compare com1 com2 :=
  case (Dcompare (com1 ?= com2)%Z);
    [ idtac | let x := fresh "H" in
      (intro x; case x; clear x) ].

Lemma ZL0 : 2%nat = (1 + 1)%nat.
Proof.
  reflexivity.
Qed.

Lemma Zplus_diag_eq_mult_2 n : n + n = n * 2.
Proof.
 rewrite Z.mul_comm. apply Z.add_diag.
Qed.

Lemma Z_eq_mult n m : m = 0 -> m * n = 0.
Proof.
 intros; now subst.
Qed.