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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 *)
(***********************************************************************)
(*i $Id$ 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 *)
Lemma sumbool_of_bool : (b:bool) {b=true}+{b=false}.
Proof.
Induction b; Auto.
Qed.
Hints Resolve sumbool_of_bool : bool.
Lemma bool_eq_rec : (b:bool)(P:bool->Set)
((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Qed.
Lemma bool_eq_ind : (b:bool)(P:bool->Prop)
((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Qed.
(*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}.
Lemma sumbool_and : {A/\C}+{B\/D}.
Proof.
Case H1; Case H2; Auto.
Qed.
Lemma sumbool_or : {A\/C}+{B/\D}.
Proof.
Case H1; Case H2; Auto.
Qed.
Lemma sumbool_not : {B}+{A}.
Proof.
Case H1; Auto.
Qed.
End connectives.
Hints Resolve sumbool_and sumbool_or sumbool_not : core.
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