(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)

(* $Id$ *)

Inductive list [A:Type] : Type :=
  nil : (list A) | cons : A->(list A)->(list A).

Fixpoint app [A:Type][l:(list A)] : (list A) -> (list A) :=
  [m:(list A)]
  Cases l of
  | nil => m 
  | (cons a l1) => (cons A a (app A l1 m))
  end.

Inductive Sprod [A:Type;B:Set] : Type :=
  Spair : A -> B -> (Sprod A B).

Definition assoc_2nd :=
Fix assoc_2nd_rec {assoc_2nd_rec/4:
  (A:Type)(B:Set)((e1,e2:B){e1=e2}+{~e1=e2})->(list (Sprod A B))->B->A->A:=
    [A:Type;B:Set;eq_dec:(e1,e2:B){e1=e2}+{~e1=e2};lst:(list (Sprod A B));
      key:B;default:A]
  Cases lst of
  | nil => default
  | (cons (Spair v e) l) =>
    (Cases (eq_dec e key) of
    | (left _) => v
    | (right _) => (assoc_2nd_rec A B eq_dec l key default)
    end)
  end}.

Inductive prodT [A,B:Type] : Type :=
  pairT: A->B->(prodT A B).

Definition fstT [A,B:Type;c:(prodT A B)] :=
  Cases c of
  | (pairT a _) => a
  end.

Definition sndT [A,B:Type;c:(prodT A B)] :=
  Cases c of
  | (pairT _ a) => a
  end.

Definition mem :=
Fix mem {mem/4:(A:Set)((e1,e2:A){e1=e2}+{~e1=e2})->(a:A)(list A)->bool :=
  [A:Set;eq_dec:(e1,e2:A){e1=e2}+{~e1=e2};a:A;l:(list A)]
  Cases l of
  | nil => false
  | (cons a1 l1) =>
    Cases (eq_dec a a1) of
    | (left _) => true
    | (right _) => (mem A eq_dec a l1)
    end
  end}.

Inductive option [A:Type] : Type :=
  | None : (option A)
  | Some : (A -> A -> A) -> (option A).