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.

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