Section RndListElem.
Variable A : Set.
Hypothesis eqd : EqDec A.
Local Open Scope list_scope.
Definition rndListElem(ls : list A) : Comp (option A) :=
match (length ls) with
| O => ret None
| S _ =>
i <-$ [0 .. (length ls));
ret (nth_option ls i)
end.
intuition.
unfold rndListElem.
case_eq (length ls); intuition; wftac.
Qed.
End RndListElem.
Local Open Scope list_scope.
induction ls; intuition; simpl in *.
discriminate.
destruct i.
inversion H; clear H; subst.
intuition.
right.
eapply IHls.
eauto.
Qed.
Fixpoint firstIndexOf(A : Set)(eqd : eq_dec A)(ls : list A)(a : A)(def : nat) :=
match ls with
| nil => def
| a' :: ls' =>
if (eqd a a') then O else (S (firstIndexOf eqd ls' a def))
end.
induction ls; intuition; simpl in *;
intuition.
subst.
destruct (eqd a0 a0); subst.
omega.
intuition.
destruct (eqd a0 a); subst.
omega.
eapply lt_n_S.
eauto.
Qed.

Theorem nth_firstIndexOf :
forall (A : Set)(eqd : eq_dec A)(ls : list A)(a : A)(def : nat),
In a ls ->
nth_option ls (firstIndexOf eqd ls a def) = Some a.