Section update.
Variables A B : Type.
Hypothesis A_eq_dec : forall x y : A, {x = y} + {x <> y}.
Proof using.
unfold update.
intros.
break_if; congruence.
Qed.
Proof using.
unfold update.
intros. break_if; congruence.
Qed.
Proof using.
intros. subst.
unfold update.
break_if; congruence.
Qed.
Proof using.
intros.
rewrite update_eq; auto.
Qed.
Proof using.
intros.
apply functional_extensionality.
intros.
apply update_nop.
Qed.

Lemma update_nop_ext' :
forall (sigma : A -> B) y v,
sigma y = v ->
update A_eq_dec sigma y v = sigma.