(* $Id$ *)

(* 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.