Library Corelib.Init.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
Require Import Logic Datatypes Specif.
Definition sumbool_of_bool (b : bool) : {b = true} + {b = false} :=
if b return {b = true} + {b = false} then left eq_refl else right eq_refl.
#[global]
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 :=
fun b =>
if b return forall P, (b = true -> P true) -> (b = false -> P false) -> P b
then fun _ H _ => H eq_refl else fun _ _ H => H eq_refl.
Definition bool_eq_ind :
forall (b:bool) (P:bool -> Prop),
(b = true -> P true) -> (b = false -> P false) -> P b :=
fun b =>
if b return forall P, (b = true -> P true) -> (b = false -> P false) -> P b
then fun _ H _ => H eq_refl else fun _ _ H => H eq_refl.
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} :=
match H1, H2 with
| left a, left c => left (conj a c)
| left a, right d => right (or_intror d)
| right b, left c => right (or_introl b)
| right b, right d => right (or_intror d)
end.
Definition sumbool_or : {A \/ C} + {B /\ D} :=
match H1, H2 with
| left a, left c => left (or_intror c)
| left a, right d => left (or_introl a)
| right b, left c => left (or_intror c)
| right b, right d => right (conj b d)
end.
Definition sumbool_not : {B} + {A} :=
match H1 with
| left a => right a
| right b => left b
end.
End connectives.
#[global]
Hint Resolve sumbool_and sumbool_or: core.
#[global]
Hint Immediate sumbool_not : core.
Any decidability function in type sumbool can be turned into a function
returning a boolean with the corresponding specification: