Library Coq.Bool.DecBool

Set Implicit Arguments.

Definition ifdec (A B:Prop) (C:Type) (H:{A} + {B}) (x y:C) : C :=
  if H then x else y.

Theorem ifdec_left :
  forall (A B:Prop) (C:Set) (H:{A} + {B}),
    ~ B -> forall x y:C, ifdec H x y = x.
Proof.
  intros; case H; auto.
  intro; absurd B; trivial.
Qed.

Theorem ifdec_right :
  forall (A B:Prop) (C:Set) (H:{A} + {B}),
    ~ A -> forall x y:C, ifdec H x y = y.
Proof.
  intros; case H; auto.
  intro; absurd A; trivial.
Qed.

Unset Implicit Arguments.