Module PartialMap.
Export NatList.
Inductive partial_map : Type :=
| empty  : partial_map
| record : id -> nat -> partial_map -> partial_map.
Instance dec_partial_map (p q : partial_map) : Dec (p = q).

Theorem update_neq :
forall (d : partial_map) (x y : id) (o: nat),
beq_id x y = false -> find x (update d y o) = find x d.