Library Coq.Bool.IfProp


Require Import Bool.




Inductive IfProp (A B:Prop) : bool -> Prop :=

  | Iftrue : A -> IfProp A B true

  | Iffalse : B -> IfProp A B false.




Hint Resolve Iftrue Iffalse: bool v62.




Lemma Iftrue_inv : forall (A B:Prop) (b:bool), IfProp A B b -> b = true -> A.

destruct 1; intros; auto with bool.

case diff_true_false; auto with bool.

Qed.




Lemma Iffalse_inv :

 forall (A B:Prop) (b:bool), IfProp A B b -> b = false -> B.

destruct 1; intros; auto with bool.

case diff_true_false; trivial with bool.

Qed.




Lemma IfProp_true : forall A B:Prop, IfProp A B true -> A.

intros.

inversion H.

assumption.

Qed.




Lemma IfProp_false : forall A B:Prop, IfProp A B false -> B.

intros.

inversion H.

assumption.

Qed.




Lemma IfProp_or : forall (A B:Prop) (b:bool), IfProp A B b -> A \/ B.

destruct 1; auto with bool.

Qed.




Lemma IfProp_sum : forall (A B:Prop) (b:bool), IfProp A B b -> {A} + {B}.

destruct b; intro H.

left; inversion H; auto with bool.

right; inversion H; auto with bool.

Qed.