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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** 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}.
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}.
case H1; case H2; auto.
Defined.
Definition sumbool_or : {A \/ C} + {B /\ D}.
case H1; case H2; auto.
Defined.
Definition sumbool_not : {B} + {A}.
case H1; auto.
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}.
intros A B H.
elim H; intro; [exists true | exists false]; assumption.
Defined.
Arguments bool_of_sumbool : default implicits.
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