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.
induction ls; intuition; simpl in *.
case_eq (eqb d a0); intuition;
rewrite H0 in H.
discriminate.
case_eq (eqb d' d); intuition.
rewrite eqb_leibniz in H1.
subst.
case_eq (eqb d a0); intuition.
rewrite eqb_leibniz in H1.
subst.
rewrite eqb_refl in H0.
discriminate.
eapply arrayLookup_app_some_eq.
trivial.
specialize (IHls d d').
case_eq (eqb d' a0); intuition.
rewrite H1 in IHls.
eapply IHls.
trivial.
Qed.
induction ls; intuition; simpl in *.
case_eq (eqb d' d); intuition.
rewrite eqb_leibniz in H0.
subst.
intuition.
case_eq (eqb d' a0); intuition.
Qed.
induction ls; intuition; simpl in *.
discriminate.
case_eq (eqb d a0); intuition.
rewrite H0 in H.
inversion H; clear H; subst.
rewrite eqb_leibniz in H0. subst.
case_eq (eqb d' a0); intuition.
eapply arrayLookup_app_ne.
intuition.
subst.
rewrite eqb_refl in H.
discriminate.
rewrite H0 in H.
case_eq (eqb d' a0); intuition.
eapply IHls; eauto.
Qed.
intuition.
unfold arrayUpdate.
destruct arr.
simpl.
rewrite eqb_refl.
trivial.
simpl.
rewrite eqb_refl.
trivial.
Qed.
intuition.
unfold arrayUpdate.
simpl.
case_eq (eqb p2 p1); intuition.
rewrite eqb_leibniz in H0.
subst.
intuition.
Qed.
intuition.
unfold arrayUpdate.
simpl.
case_eq (eqb p2 p1); intuition.
rewrite eqb_leibniz in H0.
subst.
intuition.
Qed.
End ListArray.
(* Some notation for array lookups *)
Notation "f '#' i " := (arrayLookup _ f i) (at level 70) : array_scope.
Local Open Scope array_scope.
(* A special case for array lookup when the range is lists -- returns nil instead of None when not found.  This is useful for multidimensional arrays. *)
Definition arrayLookupList(A B : Set)(eqd : EqDec A)(ls : list (A * (list B)))(a : A) :=
match (arrayLookup _ ls a) with
| None => nil
| Some ls => ls
end.
induction arr; intuition; simpl in *.
discriminate.
case_eq (eqb a a0); intuition;
rewrite H0 in *.
inversion H; clear H; subst.
rewrite eqb_leibniz in H0.
subst.
intuition.
right.
eapply IHarr.
eauto.
Qed.
induction arr; intuition; simpl in *.
case_eq (eqb a a0); intuition.
rewrite eqb_leibniz in H0.
subst.
intuition.
Qed.
Fixpoint arrayLookupOpt(A B : Set)(eqd : EqDec A)(ls : list (option (A * B)))(a : A) :=
match ls with
| nil => None
| o :: ls' =>
match o with
| None => arrayLookupOpt _ ls' a
| Some (a', b') =>
if (eqb a a') then (Some b') else (arrayLookupOpt _ ls' a)
end
end.
induction 1; intuition; simpl in *.
trivial.
destruct a1.
destruct a2.
destruct p.
simpl in *.
intuition.
subst.
destruct p0. simpl in *.
case_eq (eqb a a0); intuition.
econstructor.
intuition.
eapply IHlist_pred.
destruct p.
case_eq (eqb a a0); intuition.
subst.
eapply IHlist_pred.
Qed.
induction ls; intuition; simpl in *.
destruct a.
destruct p.
simpl.
case_eq (eqb a0 a); intuition.
eapply IHls.
Qed.
induction ls1; intuition; simpl in *.
discriminate.
case_eq (eqb a a0); intuition.
rewrite H0 in H.
trivial.
rewrite H0 in H.
eauto.
Qed.
induction ls1; intuition; simpl in *.
case_eq (eqb a a0); intuition;
rewrite H0 in H.
discriminate.
eauto.
Qed.
induction ls; intuition; simpl in *.
unfold arrayLookupList.
simpl.
case_eq (eqb a a0); intuition.
specialize (X (a0, b)).
intuition.
eapply IHls; intuition.
Qed.
induction ls; intuition; simpl in *.
discriminate.
case_eq (eqb a a0); intuition;
rewrite H0 in H.
inversion H; clear H; subst.
rewrite eqb_leibniz in H0.
subst.
intuition.
right.
eauto.
Qed.

Theorem list_pred_impl_arrayLookup :
forall (A B C : Set) P (eqda : EqDec A)(ls1 : list (A * B))(ls2 : list (A * C)),
list_pred (fun x y => fst x = fst y /\ P (snd x) (snd y)) ls1 ls2 ->
(forall v z, arrayLookup _ ls1 v = Some z -> exists z', arrayLookup _ ls2 v = Some z' /\ P z z') /\
(forall v, arrayLookup _ ls1 v = None -> arrayLookup _ ls2 v = None).