Section DecBool.
Definition decb P {D: Dec P} := if dec P then true else false.
Arguments decb _ {_}.
Definition decb1 {X: Type} (Q: X -> Prop) {D: Dec1 Q} x := decb (Q x).
Arguments decb1 {_} _ {_} _.
Definition decb2 {X Y: Type} (R: X -> Y -> Prop) {D: Dec2 R} x := decb1 (R x).
Arguments decb2 {_} {_} _ {_} _.

Lemma decb_dec (P: Prop) {D: Dec P}: decb P = true -> P.