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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Properties of decidable propositions *)
Definition decidable (P:Prop) := P \/ ~ P.
Theorem dec_not_not : forall P:Prop, decidable P -> (~ P -> False) -> P.
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_True : decidable True.
Proof.
unfold decidable; auto.
Qed.
Theorem dec_False : decidable False.
Proof.
unfold decidable, not; auto.
Qed.
Theorem dec_or :
forall A B:Prop, decidable A -> decidable B -> decidable (A \/ B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_and :
forall A B:Prop, decidable A -> decidable B -> decidable (A /\ B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_not : forall A:Prop, decidable A -> decidable (~ A).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_imp :
forall A B:Prop, decidable A -> decidable B -> decidable (A -> B).
Proof.
unfold decidable; tauto.
Qed.
Theorem dec_iff :
forall A B:Prop, decidable A -> decidable B -> decidable (A<->B).
Proof.
unfold decidable. tauto.
Qed.
Theorem not_not : forall P:Prop, decidable P -> ~ ~ P -> P.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_or : forall A B:Prop, ~ (A \/ B) -> ~ A /\ ~ B.
Proof.
tauto.
Qed.
Theorem not_and : forall A B:Prop, decidable A -> ~ (A /\ B) -> ~ A \/ ~ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_imp : forall A B:Prop, decidable A -> ~ (A -> B) -> A /\ ~ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem imp_simp : forall A B:Prop, decidable A -> (A -> B) -> ~ A \/ B.
Proof.
unfold decidable; tauto.
Qed.
Theorem not_iff :
forall A B:Prop, decidable A -> decidable B ->
~ (A <-> B) -> (A /\ ~ B) \/ (~ A /\ B).
Proof.
unfold decidable; tauto.
Qed.
(** Results formulated with iff, used in FSetDecide.
Negation are expanded since it is unclear whether setoid rewrite
will always perform conversion. *)
(** We begin with lemmas that, when read from left to right,
can be understood as ways to eliminate uses of [not]. *)
Theorem not_true_iff : (True -> False) <-> False.
Proof.
tauto.
Qed.
Theorem not_false_iff : (False -> False) <-> True.
Proof.
tauto.
Qed.
Theorem not_not_iff : forall A:Prop, decidable A ->
(((A -> False) -> False) <-> A).
Proof.
unfold decidable; tauto.
Qed.
Theorem contrapositive : forall A B:Prop, decidable A ->
(((A -> False) -> (B -> False)) <-> (B -> A)).
Proof.
unfold decidable; tauto.
Qed.
Lemma or_not_l_iff_1 : forall A B: Prop, decidable A ->
((A -> False) \/ B <-> (A -> B)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_l_iff_2 : forall A B: Prop, decidable B ->
((A -> False) \/ B <-> (A -> B)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_r_iff_1 : forall A B: Prop, decidable A ->
(A \/ (B -> False) <-> (B -> A)).
Proof.
unfold decidable. tauto.
Qed.
Lemma or_not_r_iff_2 : forall A B: Prop, decidable B ->
(A \/ (B -> False) <-> (B -> A)).
Proof.
unfold decidable. tauto.
Qed.
Lemma imp_not_l : forall A B: Prop, decidable A ->
(((A -> False) -> B) <-> (A \/ B)).
Proof.
unfold decidable. tauto.
Qed.
(** Moving Negations Around:
We have four lemmas that, when read from left to right,
describe how to push negations toward the leaves of a
proposition and, when read from right to left, describe
how to pull negations toward the top of a proposition. *)
Theorem not_or_iff : forall A B:Prop,
(A \/ B -> False) <-> (A -> False) /\ (B -> False).
Proof.
tauto.
Qed.
Lemma not_and_iff : forall A B:Prop,
(A /\ B -> False) <-> (A -> B -> False).
Proof.
tauto.
Qed.
Lemma not_imp_iff : forall A B:Prop, decidable A ->
(((A -> B) -> False) <-> A /\ (B -> False)).
Proof.
unfold decidable. tauto.
Qed.
Lemma not_imp_rev_iff : forall A B : Prop, decidable A ->
(((A -> B) -> False) <-> (B -> False) /\ A).
Proof.
unfold decidable. tauto.
Qed.
(* Functional relations on decidable co-domains are decidable *)
Theorem dec_functional_relation :
forall (X Y : Type) (A:X->Y->Prop), (forall y y' : Y, decidable (y=y')) ->
(forall x, exists! y, A x y) -> forall x y, decidable (A x y).
Proof.
intros X Y A Hdec H x y.
destruct (H x) as (y',(Hex,Huniq)).
destruct (Hdec y y') as [->|Hnot]; firstorder.
Qed.
(** With the following hint database, we can leverage [auto] to check
decidability of propositions. *)
Hint Resolve dec_True dec_False dec_or dec_and dec_imp dec_not dec_iff
: decidable_prop.
(** [solve_decidable using lib] will solve goals about the
decidability of a proposition, assisted by an auxiliary
database of lemmas. The database is intended to contain
lemmas stating the decidability of base propositions,
(e.g., the decidability of equality on a particular
inductive type). *)
Tactic Notation "solve_decidable" "using" ident(db) :=
match goal with
| |- decidable _ =>
solve [ auto 100 with decidable_prop db ]
end.
Tactic Notation "solve_decidable" :=
solve_decidable using core.
|
