End DECIDABLE_EQUALITY.
Section DECIDABLE_PREDICATE.
Variable P: Prop.
Variable dec: {P} + {~P}.
Variable A: Type.

(* Specifications: *)
Lemma pred_dec_true:
forall (a b: A), P -> (if dec then a else b) = a.
Lemma pred_dec_false:
forall (a b: A), ~P -> (if dec then a else b) = b.
Lemma pred_dec_true:
forall (a b: A), P -> (if dec then a else b) = a.
Lemma pred_dec_false:
forall (a b: A), ~P -> (if dec then a else b) = b.