(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

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.