Library Coq.Numbers.Integer.NatPairs.ZNatPairs

Require Import NSub ZAxioms.
Require Export Ring.

Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope.
Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope.
Local Open Scope pair_scope.

Module ZPairsAxiomsMod (Import N : NAxiomsMiniSig) <: ZAxiomsMiniSig.
Module Import NProp.
Include NSubProp N.
End NProp.

Delimit Scope NScope with N.
Bind Scope NScope with N.t.
Infix "==" := N.eq (at level 70) : NScope.
Notation "x ~= y" := (~ N.eq x y) (at level 70) : NScope.
Notation "0" := N.zero : NScope.
Notation "1" := N.one : NScope.
Notation "2" := N.two : NScope.
Infix "+" := N.add : NScope.
Infix "-" := N.sub : NScope.
Infix "*" := N.mul : NScope.
Infix "<" := N.lt : NScope.
Infix "<=" := N.le : NScope.
Local Open Scope NScope.

The definitions of functions (add, mul, etc.) will be unfolded by the properties functor. Since we don't want add_comm to refer to unfolded definitions of equality: fun p1 p2 => (fst p1 + snd p2) = (fst p2 + snd p1), we will provide an extra layer of definitions.

Module Z.

Definition t := (N.t * N.t)%type.
Definition zero : t := (0, 0).
Definition one : t := (1,0).
Definition two : t := (2,0).
Definition eq (p q : t) := (p #1 + q #2 == q #1 + p #2).
Definition succ (n : t) : t := (N.succ n #1 , n #2 ).
Definition pred (n : t) : t := (n #1 , N.succ n #2 ).
Definition opp (n : t) : t := (n #2 , n #1 ).
Definition add (n m : t) : t := (n #1 + m #1 , n #2 + m #2 ).
Definition sub (n m : t) : t := (n #1 + m #2 , n #2 + m #1 ).
Definition mul (n m : t) : t :=
(n #1 * m #1 + n #2 * m #2 , n #1 * m #2 + n #2 * m #1 ).
Definition lt (n m : t) := n #1 + m #2 < m #1 + n #2.
Definition le (n m : t) := n #1 + m #2 <= m #1 + n #2.
Definition min (n m : t) : t := (min (n #1 + m #2) (m #1 + n #2), n #2 + m #2 ).
Definition max (n m : t) : t := (max (n #1 + m #2) (m #1 + n #2), n #2 + m #2 ).

NB : We do not have Z.pred (Z.succ n) = n but only Z.pred (Z.succ n) == n. It could be possible to consider as canonical only pairs where one of the elements is 0, and make all operations convert canonical values into other canonical values. In that case, we could get rid of setoids and arrive at integers as signed natural numbers.

NB : Unfortunately, the elements of the pair keep increasing during many operations, even during subtraction.

End Z.

Delimit Scope ZScope with Z.
Bind Scope ZScope with Z.t.
Infix "==" := Z.eq (at level 70) : ZScope.
Notation "x ~= y" := (~ Z.eq x y) (at level 70) : ZScope.
Notation "0" := Z.zero : ZScope.
Notation "1" := Z.one : ZScope.
Notation "2" := Z.two : ZScope.
Infix "+" := Z.add : ZScope.
Infix "-" := Z.sub : ZScope.
Infix "*" := Z.mul : ZScope.
Notation "- x" := (Z.opp x) : ZScope.
Infix "<" := Z.lt : ZScope.
Infix "<=" := Z.le : ZScope.
Local Open Scope ZScope.

Lemma sub_add_opp : forall n m, Z.sub n m = Z.add n (Z.opp m).
Proof. reflexivity. Qed.

Instance eq_equiv : Equivalence Z.eq.
Proof.
split.
unfold Reflexive, Z.eq. reflexivity.
unfold Symmetric, Z.eq; now symmetry.
unfold Transitive, Z.eq. intros (n1,n2) (m1,m2) (p1,p2) H1 H2; simpl in *.
apply (add_cancel_r _ _ (m1+m2)%N).
rewrite add_shuffle2, H1, add_shuffle1, H2.
now rewrite add_shuffle1, (add_comm m1).
Qed.

Instance pair_wd : Proper (N.eq ==>N.eq ==>Z.eq) (@pair N.t N.t).
Proof.
intros n1 n2 H1 m1 m2 H2; unfold Z.eq; simpl; now rewrite H1, H2.
Qed.

Instance succ_wd : Proper (Z.eq ==> Z.eq) Z.succ.
Proof.
unfold Z.succ, Z.eq; intros n m H; simpl.
do 2 rewrite add_succ_l; now rewrite H.
Qed.

Instance pred_wd : Proper (Z.eq ==> Z.eq) Z.pred.
Proof.
unfold Z.pred, Z.eq; intros n m H; simpl.
do 2 rewrite add_succ_r; now rewrite H.
Qed.

Instance add_wd : Proper (Z.eq ==> Z.eq ==> Z.eq) Z.add.
Proof.
unfold Z.eq, Z.add; intros n1 m1 H1 n2 m2 H2; simpl.
now rewrite add_shuffle1, H1, H2, add_shuffle1.
Qed.

Instance opp_wd : Proper (Z.eq ==> Z.eq) Z.opp.
Proof.
unfold Z.eq, Z.opp; intros (n1,n2) (m1,m2) H; simpl in *.
now rewrite (add_comm n2), (add_comm m2).
Qed.

Instance sub_wd : Proper (Z.eq ==> Z.eq ==> Z.eq) Z.sub.
Proof.
intros n1 m1 H1 n2 m2 H2. rewrite 2 sub_add_opp. now do 2 f_equiv.
Qed.

Lemma mul_comm : forall n m, n *m == m *n.
Proof.
intros (n1,n2) (m1,m2); compute.
rewrite (add_comm (m1*n2)%N).
do 2 f_equiv; apply mul_comm.
Qed.

Instance mul_wd : Proper (Z.eq ==> Z.eq ==> Z.eq) Z.mul.
Proof.
assert (forall n, Proper (Z.eq ==> Z.eq) (Z.mul n)).
unfold Z.mul, Z.eq. intros (n1,n2) (p1,p2) (q1,q2) H; simpl in *.
rewrite add_shuffle1, (add_comm (n1*p1)%N).
symmetry. rewrite add_shuffle1.
rewrite <- ! mul_add_distr_l.
rewrite (add_comm p2), (add_comm q2), H.
reflexivity.
intros n n' Hn m m' Hm.
rewrite Hm, (mul_comm n), (mul_comm n'), Hn.
reflexivity.
Qed.

Section Induction.
Variable A : Z.t -> Prop.
Hypothesis A_wd : Proper (Z.eq ==>iff) A.

Theorem bi_induction :
A 0 -> (forall n, A n <-> A (Z.succ n)) -> forall n, A n.
Proof.
Open Scope NScope.
intros A0 AS n; unfold Z.zero, Z.succ, Z.eq in *.
destruct n as [n m].
cut (forall p, A (p , 0)); [intro H1 |].
cut (forall p, A (0, p )); [intro H2 |].
destruct (add_dichotomy n m) as [[p H] | [p H]].
rewrite (A_wd (n, m) (0, p)) by (rewrite add_0_l; now rewrite add_comm).
apply H2.
rewrite (A_wd (n, m) (p, 0)) by now rewrite add_0_r. apply H1.
induct p. assumption. intros p IH.
apply (A_wd (0, p) (1, N.succ p)) in IH; [| now rewrite add_0_l, add_1_l].
rewrite one_succ in IH. now apply AS.
induct p. assumption. intros p IH.
replace 0 with (snd (p, 0)); [| reflexivity].
replace (N.succ p) with (N.succ (fst (p, 0))); [| reflexivity]. now apply -> AS.
Close Scope NScope.
Qed.

End Induction.

Theorem pred_succ : forall n, Z.pred (Z.succ n) == n.
Proof.
unfold Z.pred, Z.succ, Z.eq; intro n; simpl; now nzsimpl.
Qed.

Theorem succ_pred : forall n, Z.succ (Z.pred n) == n.
Proof.
intro n; unfold Z.succ, Z.pred, Z.eq; simpl; now nzsimpl.
Qed.

Theorem one_succ : 1 == Z.succ 0.
Proof.
unfold Z.eq; simpl. now nzsimpl'.
Qed.

Theorem two_succ : 2 == Z.succ 1.
Proof.
unfold Z.eq; simpl. now nzsimpl'.
Qed.

Theorem opp_0 : - 0 == 0.
Proof.
unfold Z.opp, Z.eq; simpl. now nzsimpl.
Qed.

Theorem opp_succ : forall n, - (Z.succ n ) == Z.pred (- n).
Proof.
reflexivity.
Qed.

Theorem add_0_l : forall n, 0 + n == n.
Proof.
intro n; unfold Z.add, Z.eq; simpl. now nzsimpl.
Qed.

Theorem add_succ_l : forall n m, (Z.succ n ) + m == Z.succ (n + m).
Proof.
intros n m; unfold Z.add, Z.eq; simpl. now nzsimpl.
Qed.

Theorem sub_0_r : forall n, n - 0 == n.
Proof.
intro n; unfold Z.sub, Z.eq; simpl. now nzsimpl.
Qed.

Theorem sub_succ_r : forall n m, n - (Z.succ m ) == Z.pred (n - m).
Proof.
intros n m; unfold Z.sub, Z.eq; simpl. symmetry; now rewrite add_succ_r.
Qed.

Theorem mul_0_l : forall n, 0 * n == 0.
Proof.
intros (n1,n2); unfold Z.mul, Z.eq; simpl; now nzsimpl.
Qed.

Theorem mul_succ_l : forall n m, (Z.succ n ) * m == n * m + m.
Proof.
intros (n1,n2) (m1,m2); unfold Z.mul, Z.succ, Z.eq; simpl; nzsimpl.
rewrite <- (add_assoc _ m1), (add_comm m1), (add_assoc _ _ m1).
now rewrite <- (add_assoc _ m2), (add_comm m2), (add_assoc _ (n2*m1)%N m2).
Qed.

Order

Lemma lt_eq_cases : forall n m, n <=m <-> n <m \/ n ==m.
Proof.
intros; apply N.lt_eq_cases.
Qed.

Theorem lt_irrefl : forall n, ~ (n < n ).
Proof.
intros; apply N.lt_irrefl.
Qed.

Theorem lt_succ_r : forall n m, n < (Z.succ m ) <-> n <= m.
Proof.
intros n m; unfold Z.lt, Z.le, Z.eq; simpl; nzsimpl. apply lt_succ_r.
Qed.

Theorem min_l : forall n m, n <= m -> Z.min n m == n.
Proof.
unfold Z.min, Z.le, Z.eq; simpl; intros (n1,n2) (m1,m2) H; simpl in *.
rewrite min_l by assumption.
now rewrite <- add_assoc, (add_comm m2).
Qed.

Theorem min_r : forall n m, m <= n -> Z.min n m == m.
Proof.
unfold Z.min, Z.le, Z.eq; simpl; intros (n1,n2) (m1,m2) H; simpl in *.
rewrite min_r by assumption.
now rewrite add_assoc.
Qed.

Theorem max_l : forall n m, m <= n -> Z.max n m == n.
Proof.
unfold Z.max, Z.le, Z.eq; simpl; intros (n1,n2) (m1,m2) H; simpl in *.
rewrite max_l by assumption.
now rewrite <- add_assoc, (add_comm m2).
Qed.

Theorem max_r : forall n m, n <= m -> Z.max n m == m.
Proof.
unfold Z.max, Z.le, Z.eq; simpl; intros n m H.
rewrite max_r by assumption.
now rewrite add_assoc.
Qed.

Theorem lt_nge : forall n m, n < m <-> ~(m <=n ).
Proof.
intros. apply lt_nge.
Qed.

Instance lt_wd : Proper (Z.eq ==> Z.eq ==> iff) Z.lt.
Proof.
assert (forall n, Proper (Z.eq ==>iff) (Z.lt n)).
intros (n1,n2). apply proper_sym_impl_iff; auto with *.
unfold Z.lt, Z.eq; intros (r1,r2) (s1,s2) Eq H; simpl in *.
apply le_lt_add_lt with (r1+r2)%N (r1+r2)%N; [apply le_refl; auto with *|].
rewrite add_shuffle2, (add_comm s2), Eq.
rewrite (add_comm s1 n2), (add_shuffle1 n2), (add_comm n2 r1).
now rewrite <- add_lt_mono_r.
intros n n' Hn m m' Hm.
rewrite Hm. rewrite 2 lt_nge, 2 lt_eq_cases, Hn; auto with *.
Qed.

Definition t := Z.t.
Definition eq := Z.eq.
Definition zero := Z.zero.
Definition one := Z.one.
Definition two := Z.two.
Definition succ := Z.succ.
Definition pred := Z.pred.
Definition add := Z.add.
Definition sub := Z.sub.
Definition mul := Z.mul.
Definition opp := Z.opp.
Definition lt := Z.lt.
Definition le := Z.le.
Definition min := Z.min.
Definition max := Z.max.

End ZPairsAxiomsMod.