Section ListArray.
Variable D R : Set.
Hypothesis D_EqDec : EqDec D.
Definition Array := list (D * R).
Fixpoint arrayLookup(ls : list (D * R)) d :=
match ls with
| nil => None
| (d', r') :: ls' =>
if (eqb d d')
then Some r'
else (arrayLookup ls' d)
end.
Definition arrayEquiv(arr1 arr2 : Array) :=
forall d,
(arrayLookup arr1 d) = (arrayLookup arr2 d).
Definition arrayLookupDef(ls : list (D * R)) d def :=
match arrayLookup ls d with
| None => def
| Some r => r
end.
Definition arrayUpdate (arr_opt : option Array) d r_opt :=
let arr := match arr_opt with
| None => nil
| Some arr => arr
end in
match r_opt with
| None => arr
| Some r =>
(d, r) :: arr
end.
induction ls; intuition; simpl in *.
rewrite eqb_refl.
trivial.
case_eq (eqb d a0); intuition;
rewrite H0 in H.
discriminate.
eauto.
Qed.

Lemma arrayLookup_app_none : forall ls d d' v',
arrayLookup ls d = None ->
arrayLookup (ls ++ (d, v') :: nil) d' = arrayLookup ((d, v') :: ls) d'.