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(************************************************************************)
(* * 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.
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