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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* $Id$ *)
(* 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 *)
Lemma sumbool_of_bool : (b:bool) {b=true}+{b=false}.
Proof.
Induction b; Auto.
Save.
Hints Resolve sumbool_of_bool : bool.
(* pourquoi ce machin-la est dans BOOL et pas dans LOGIC ? Papageno *)
(* Logic connectives on type sumbool *)
Section connectives.
Variables A,B,C,D : Prop.
Hypothesis H1 : {A}+{B}.
Hypothesis H2 : {C}+{D}.
Lemma sumbool_and : {A/\C}+{B\/D}.
Proof.
Case H1; Case H2; Auto.
Save.
Lemma sumbool_or : {A\/C}+{B/\D}.
Proof.
Case H1; Case H2; Auto.
Save.
Lemma sumbool_not : {B}+{A}.
Proof.
Case H1; Auto.
Save.
End connectives.
Hints Resolve sumbool_and sumbool_or sumbool_not : core.
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