(****************************************************************************  IEEE754  :  Zenum  Laurent Thery  ***************************************************************************** Simple functions to enumerate relative numbers *)
Require Export Faux.
Require Export Omega.
Require Export List.
(* Returns the list of relative numbers from z to z+n *)
Fixpoint mZlist_aux (p : Z) (n : nat) {struct n} :
list Z :=
match n with
| O => p :: nil
| S n1 => p :: mZlist_aux (Z.succ p) n1
end.
intros n; elim n; clear n; auto.
intros p q; try rewrite <- Zplus_0_r_reverse.
intros H' H'0; simpl in |- *; left.
apply Zle_antisym; auto.
intros n H' p q H'0 H'1; case (Zle_lt_or_eq _ _ H'0); intros H'2.
simpl in |- *; right.
apply H'; auto with zarith.
rewrite Zplus_succ_comm.
rewrite <- inj_S; auto.
simpl in |- *; auto.
Qed.
intros n; elim n; clear n; simpl in |- *; auto.
intros p q H'; elim H'; auto with zarith.
intros n H' p q H'0; elim H'0; auto with zarith.
intros H'1; apply Zle_succ_le; auto with zarith.
Qed.
intros n; elim n; clear n; auto.
intros p q H'; elim H'; auto with zarith.
intros H'0; elim H'0.
intros n H' p q H'0; elim H'0; auto with zarith.
intros H'1; rewrite inj_S; rewrite <- Zplus_succ_comm; auto.
Qed.
(* Return the list of of relative numbres from p to p+q if p=<q, otherwise the empty list *)
Definition mZlist (p q : Z) : list Z :=
match (q - p)%Z with
| Z0 => p :: nil
| Zpos d => mZlist_aux p (nat_of_P d)
| Zneg _ => nil (A:=Z)
end.
intros p q r H' H'0; unfold mZlist in |- *; CaseEq (q - p)%Z;
auto with zarith.
intros H'1; rewrite (Zle_antisym r p); auto with datatypes.
auto with zarith.
intros p0 H'1; apply mZlist_aux_correct; auto.
rewrite inject_nat_convert with (1 := H'1); auto with zarith.
intros p0 H'1; absurd (p <= q)%Z; auto.
apply Zlt_not_le; auto.
apply Zlt_O_minus_lt; auto.
replace (p - q)%Z with (- (q - p))%Z; auto with zarith.
rewrite H'1; simpl in |- *; auto with zarith.
unfold Z.lt in |- *; simpl in |- *; auto.
apply Z.le_trans with (m := r); auto.
Qed.
intros p q r; unfold mZlist in |- *; CaseEq (q - p)%Z.
intros H' H'0; elim H'0; auto with zarith.
intros H'1; elim H'1.
intros p0 H' H'0.
apply mZlist_aux_correct_rev1 with (n := nat_of_P p0); auto.
intros p0 H' H'0; elim H'0.
Qed.
intros p q r; unfold mZlist in |- *; CaseEq (q - p)%Z.
intros H' H'0; elim H'0; auto with zarith.
intros H'1; elim H'1.
intros p0 H' H'0.
rewrite <- (Zplus_minus p q).
rewrite <- inject_nat_convert with (1 := H').
apply mZlist_aux_correct_rev2; auto.
intros p0 H' H'0; elim H'0.
Qed.
(* Given two list returns the list of possible product of an element of the first list with an element of the second list *)
Fixpoint mProd (A B C : Set) (l1 : list A) (l2 : list B) {struct l2} :
list (A * B) :=
match l2 with
| nil => nil
| b :: l2' => map (fun a : A => (a, b)) l1 ++ mProd A B C l1 l2'
end.

Theorem mProd_correct :
forall (A B C : Set) (l1 : list A) (l2 : list B) (a : A) (b : B),
In a l1 -> In b l2 -> In (a, b) (mProd A B C l1 l2).