Section Update2.
Variables A B : Type.
Hypothesis A_eq_dec : forall x y : A, {x = y} + {x <> y}.
Proof using.
unfold update2.
intros.
break_if; intuition congruence.
Qed.
Proof using.
unfold update2.
intros.
break_if; intuition congruence.
Qed.
Proof using.
unfold update2.
intros.
break_if; intuition congruence.
Qed.

Lemma update2_nop :
forall (sigma : A -> A -> B) x y x' y',
update2 A_eq_dec sigma x y (sigma x y) x' y' = sigma x' y'.