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(* as published by the Free Software Foundation; either version 2.1   *)
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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** list. Polymorphic lists with some operations.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Polymorphic lists *)
Inductive list (A : Set) : Set :=
| Nil : list A
| Cons : A -> list A -> list A.
(** Length of a list *)
Fixpoint length (A : Set) (l : list A) {struct l} : nat :=
match l with
| Nil => 0
| Cons a r => S (length A r)
end.
(** Map a function over a list *)
Fixpoint map (A B : Set) (f : A -> B) (l : list A) {struct l} :
list B :=
match l with
| Nil => Nil B
| Cons a r => Cons B (f a) (map A B f r)
end.
Notation Map := (map _ _) (only parsing).
(** Checks that all members of a list are P *)
Fixpoint alllist (A : Set) (P : A -> Prop) (qlist : list A) {struct qlist} :
Prop :=
match qlist with
| Nil => True
| Cons m l => P m /\ alllist A P l
end.
(** Checks that some member of a list is P *)
Fixpoint exlist (A : Set) (P : A -> Prop) (qlist : list A) {struct qlist} :
Prop :=
match qlist with
| Nil => False
| Cons m l => P m \/ exlist A P l
end.
(** Membership *)
Definition inlist (A : Set) (a : A) := exlist A (fun b : A => a = b).

Lemma inlist_head_neq :
forall (A : Set) (x y : A) (l : list A),
x <> y -> (inlist A x (Cons A y l) <-> inlist A x l).