Zenum

# Library Zenum

`Require Export Faux.
Require Export Omega.
Require Export List.
Fixpoint mZlist_aux (p : Z) (n : nat) {struct n} :
list Z :=
match n with
| O => p :: nil
| S n1 => p :: mZlist_aux (Zsucc p) n1
end.
Theorem mZlist_aux_correct :
forall (n : nat) (p q : Z),
(p <= q)%Z -> (q <= p + Z_of_nat n)%Z -> In q (mZlist_aux p n).
Theorem mZlist_aux_correct_rev1 :
forall (n : nat) (p q : Z), In q (mZlist_aux p n) -> (p <= q)%Z.
Theorem mZlist_aux_correct_rev2 :
forall (n : nat) (p q : Z),
In q (mZlist_aux p n) -> (q <= p + Z_of_nat n)%Z.
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.
Theorem mZlist_correct :
forall p q r : Z, (p <= r)%Z -> (r <= q)%Z -> In r (mZlist p q).
Theorem mZlist_correct_rev1 :
forall p q r : Z, In r (mZlist p q) -> (p <= r)%Z.
Theorem mZlist_correct_rev2 :
forall p q r : Z, In r (mZlist p q) -> (r <= q)%Z.
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).
Theorem mProd_correct_rev1 :
forall (A B C : Set) (l1 : list A) (l2 : list B) (a : A) (b : B),
In (a, b) (mProd A B C l1 l2) -> In a l1.
Theorem mProd_correct_rev2 :
forall (A B C : Set) (l1 : list A) (l2 : list B) (a : A) (b : B),
In (a, b) (mProd A B C l1 l2) -> In b l2.
Theorem in_map_inv :
forall (A B : Set) (f : A -> B) (l : list A) (x : A),
(forall a b : A, f a = f b -> a = b) -> In (f x) (map f l) -> In x l.
`