Section DECIDABLE_EQUALITY.
Variable A: Type.
Variable dec_eq: forall (x y: A), {x=y} + {x<>y}.
Variable B: Type.

Lemma dec_eq_sym:
forall (x y: A) (ifso ifnot: B),
(if dec_eq x y then ifso else ifnot) =
(if dec_eq y x then ifso else ifnot).