(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
(************************************************************************)

Require Import Bool.

Inductive IfProp (A B:Prop) : bool -> Prop :=
  | Iftrue : A -> IfProp A B true
  | Iffalse : B -> IfProp A B false.

#[global]
Hint Resolve Iftrue Iffalse: bool.

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 A B H.
inversion H.
assumption.
Qed.

Lemma IfProp_false : forall A B:Prop, IfProp A B false -> B.
intros A B H.
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}.
intros A B b; destruct b; intro H.
- left; inversion H; auto with bool.
- right; inversion H; auto with bool.
Qed.
