DecBool.v

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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.forall (A B : Prop) (C : Set) (H : {A} + {B}),
~ B -> forall x y : C, ifdec H x y = x
Proof.forall (A B : Prop) (C : Set) (H : {A} + {B}),
~ B -> forall x y : C, ifdec H x y = x
  intros; case H; auto.A, B:Prop
C:Set
H:{A} + {B}
H0:~ B
x, y:C
forall b : B, ifdec (right b) x y = x
  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.forall (A B : Prop) (C : Set) (H : {A} + {B}),
~ A -> forall x y : C, ifdec H x y = y
Proof.forall (A B : Prop) (C : Set) (H : {A} + {B}),
~ A -> forall x y : C, ifdec H x y = y
  intros; case H; auto.A, B:Prop
C:Set
H:{A} + {B}
H0:~ A
x, y:C
forall a : A, ifdec (left a) x y = y
  intro; absurd A; trivial.
Qed.

Unset Implicit Arguments.