# 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 }.
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.
Defined.
Definition bool_eq_ind :
forall (b:bool) (P:bool -> Prop),
(b = true -> P true) -> (b = false -> P false) -> P b.
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 }.
Defined.
Definition sumbool_or : { A \/ C } + { B /\ D }.
Defined.
Definition sumbool_not : { B } + { A }.
Defined.
End connectives.
Hint Resolve sumbool_and sumbool_or: core.
Hint Immediate 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 }.
Defined.
Implicit Arguments bool_of_sumbool.