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.
Proof using.
unfold update2.
intros. break_if; intuition congruence.
Qed.
Proof using.
intros. subst.
unfold update2.
break_if; intuition congruence.
Qed.

Lemma update2_eq_prod :
forall (sigma : A -> A -> B) x y x' y' v,
(x, y) = (x', y') ->
update2 A_eq_dec sigma x y v x' y' = v.