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.

Lemma update_same :
forall (sigma : A -> B) x v,
update A_eq_dec sigma x v x = v.