intuition.
unfold arrayUpdate.
simpl.
case_eq (eqb p2 p1); intuition.
rewrite eqb_leibniz in H0.
subst.
intuition.
Qed.

Theorem arrayLookup_arrayUpdate_None_ne :
forall p1 p2 v,
p1 <> p2 ->
arrayLookup (arrayUpdate None p1 (Some v)) p2 = None.