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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Sumbool.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*)
(** Here are collected some results about the type sumbool (see INIT/Specif.v)
[sumbool A B], which is written [{A}+{B}], is the informative
disjunction "A or B", where A and B are logical propositions.
Its extraction is isomorphic to the type of booleans. *)
(** A boolean is either [true] or [false], and this is decidable *)
Definition sumbool_of_bool : (b:bool) {b=true}+{b=false}.
Proof.
NewDestruct b; Auto.
Defined.
Hints Resolve sumbool_of_bool : bool.
Definition bool_eq_rec : (b:bool)(P:bool->Set)
((b=true)->(P true))->((b=false)->(P false))->(P b).
NewDestruct b; Auto.
Defined.
Definition bool_eq_ind : (b:bool)(P:bool->Prop)
((b=true)->(P true))->((b=false)->(P false))->(P b).
NewDestruct b; Auto.
Defined.
(*i pourquoi ce machin-la est dans BOOL et pas dans LOGIC ? Papageno i*)
(** Logic connectives on type [sumbool] *)
Section connectives.
Variables A,B,C,D : Prop.
Hypothesis H1 : {A}+{B}.
Hypothesis H2 : {C}+{D}.
Definition sumbool_and : {A/\C}+{B\/D}.
Proof.
Case H1; Case H2; Auto.
Defined.
Definition sumbool_or : {A\/C}+{B/\D}.
Proof.
Case H1; Case H2; Auto.
Defined.
Definition sumbool_not : {B}+{A}.
Proof.
Case H1; Auto.
Defined.
End connectives.
Hints Resolve sumbool_and sumbool_or sumbool_not : core.
(** Any decidability function in type [sumbool] can be turned into a function
returning a boolean with the corresponding specification: *)
Definition bool_of_sumbool :
(A,B:Prop) {A}+{B} -> { b:bool | if b then A else B }.
Proof.
Intros A B H.
Elim H; [ Intro; Exists true; Assumption
| Intro; Exists false; Assumption ].
Defined.
Implicits bool_of_sumbool.
|
