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(***********************************************************************)
(* 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 listT [A:Type] : Type :=
nilT : (listT A) | consT : A->(listT A)->(listT A).
Fixpoint appT [A:Type][l:(listT A)] : (listT A) -> (listT A) :=
[m:(listT A)]
Cases l of
| nilT => m
| (consT a l1) => (consT A a (appT 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})->(listT (Sprod A B))->B->A->A:=
[A:Type;B:Set;eq_dec:(e1,e2:B){e1=e2}+{~e1=e2};lst:(listT (Sprod A B));
key:B;default:A]
Cases lst of
| nilT => default
| (consT (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)(listT A)->bool :=
[A:Set;eq_dec:(e1,e2:A){e1=e2}+{~e1=e2};a:A;l:(listT A)]
Cases l of
| nilT => false
| (consT 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).
|
