# Library Coq.Bool.Sumbool

 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 : forall b:bool, {b = true} + {b = false}.
Proof.
destruct b; auto.
Defined.

Hint Resolve sumbool_of_bool: bool.

Definition bool_eq_rec :
forall (b:bool) (P:bool -> Set),
(b = true -> P true) -> (b = false -> P false) -> P b.
destruct b; auto.
Defined.

Definition bool_eq_ind :
forall (b:bool) (P:bool -> Prop),
(b = true -> P true) -> (b = false -> P false) -> P b.
destruct b; auto.
Defined.

 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.

Hint 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 :
forall 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.
Implicit Arguments bool_of_sumbool.

Index