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.
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A boolean is either true or false, and this is decidable
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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
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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:
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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.