IfProp.v

Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.

(************************************************************************)
(*         *   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)         *)
(************************************************************************)

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.

Lemma Iftrue_inv : forall (A B:Prop) (b:bool), IfProp A B b -> b = true -> A.forall (A B : Prop) (b : bool),
IfProp A B b -> b = true -> A
destruct 1; intros; auto with bool.A, B:Prop
H:B
H0:false = true
A
case diff_true_false; auto with bool.
Qed.

Lemma Iffalse_inv :
 forall (A B:Prop) (b:bool), IfProp A B b -> b = false -> B.forall (A B : Prop) (b : bool),
IfProp A B b -> b = false -> B
destruct 1; intros; auto with bool.A, B:Prop
H:A
H0:true = false
B
case diff_true_false; trivial with bool.
Qed.

Lemma IfProp_true : forall A B:Prop, IfProp A B true -> A.forall A B : Prop, IfProp A B true -> A
intros.A, B:Prop
H:IfProp A B true
A
inversion H.A, B:Prop
H:IfProp A B true
H0:A
A
assumption.
Qed.

Lemma IfProp_false : forall A B:Prop, IfProp A B false -> B.forall A B : Prop, IfProp A B false -> B
intros.A, B:Prop
H:IfProp A B false
B
inversion H.A, B:Prop
H:IfProp A B false
H0:B
B
assumption.
Qed.

Lemma IfProp_or : forall (A B:Prop) (b:bool), IfProp A B b -> A \/ B.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}.forall (A B : Prop) (b : bool),
IfProp A B b -> {A} + {B}
destruct b; intro H.A, B:Prop
H:IfProp A B true
{A} + {B}
A, B:Prop
H:IfProp A B false
{A} + {B}
-A, B:Prop
H:IfProp A B true
{A} + {B} left; inversion H; auto with bool.
-A, B:Prop
H:IfProp A B false
{A} + {B} right; inversion H; auto with bool.
Qed.