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.