(** Similarly: *)

(* Specifications: *)
Theorem beq_id_false_iff : forall x y : id,
beq_id x y = false
<-> x <> y.
Lemma update_eq : forall A (m: partial_map A) x v,
(update m x v) x = Some v.
Theorem update_neq : forall (X:Type) v x1 x2
(m : partial_map X),
x2 <> x1 ->
(update m x2 v) x1 = m x1.
Theorem update_same : forall X v x (m : partial_map X),
m x = Some v ->
update m x v = m.
Theorem update_permute : forall (X:Type) v1 v2 x1 x2
(m : partial_map X),
x2 <> x1 ->
(update (update m x2 v2) x1 v1)
= (update (update m x1 v1) x2 v2).