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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 *)
(************************************************************************)
(**** Tactics Tauto and Intuition ****)
(**** Tauto:
Tactic for automating proof in Intuionnistic Propositional Calculus, based on
the contraction-free LJT* of Dickhoff ****)
(**** Intuition:
Simplifications of goals, based on LJT* calcul ****)
(**** Examples of intuitionistic tautologies ****)
Parameter A B C D E F : Prop.
Parameter even : nat -> Prop.
Parameter P : nat -> Prop.
Lemma Ex_Wallen : (A -> B /\ C) -> (A -> B) \/ (A -> C).
Proof.
tauto.
Qed.
Lemma Ex_Klenne : ~ ~ (A \/ ~ A).
Proof.
tauto.
Qed.
Lemma Ex_Klenne' : forall n : nat, ~ ~ (even n \/ ~ even n).
Proof.
tauto.
Qed.
Lemma Ex_Klenne'' :
~ ~ ((forall n : nat, even n) \/ ~ (forall m : nat, even m)).
Proof.
tauto.
Qed.
Lemma tauto : (forall x : nat, P x) -> forall y : nat, P y.
Proof.
tauto.
Qed.
Lemma tauto1 : A -> A.
Proof.
tauto.
Qed.
Lemma tauto2 : (A -> B -> C) -> (A -> B) -> A -> C.
Proof.
tauto.
Qed.
Lemma a : forall (x0 : A \/ B) (x1 : B /\ C), A -> B.
Proof.
tauto.
Qed.
Lemma a2 : (A -> B /\ C) -> (A -> B) \/ (A -> C).
Proof.
tauto.
Qed.
Lemma a4 : ~ A -> ~ A.
Proof.
tauto.
Qed.
Lemma e2 : ~ ~ (A \/ ~ A).
Proof.
tauto.
Qed.
Lemma e4 : ~ ~ (A \/ B -> A \/ B).
Proof.
tauto.
Qed.
Lemma y0 :
forall (x0 : A) (x1 : ~ A) (x2 : A -> B) (x3 : A \/ B) (x4 : A /\ B),
A -> False.
Proof.
tauto.
Qed.
Lemma y1 : forall x0 : (A /\ B) /\ C, B.
Proof.
tauto.
Qed.
Lemma y2 : forall (x0 : A) (x1 : B), C \/ B.
Proof.
tauto.
Qed.
Lemma y3 : forall x0 : A /\ B, B /\ A.
Proof.
tauto.
Qed.
Lemma y5 : forall x0 : A \/ B, B \/ A.
Proof.
tauto.
Qed.
Lemma y6 : forall (x0 : A -> B) (x1 : A), B.
Proof.
tauto.
Qed.
Lemma y7 : forall (x0 : A /\ B -> C) (x1 : B) (x2 : A), C.
Proof.
tauto.
Qed.
Lemma y8 : forall (x0 : A \/ B -> C) (x1 : A), C.
Proof.
tauto.
Qed.
Lemma y9 : forall (x0 : A \/ B -> C) (x1 : B), C.
Proof.
tauto.
Qed.
Lemma y10 : forall (x0 : (A -> B) -> C) (x1 : B), C.
Proof.
tauto.
Qed.
(* This example took much time with the old version of Tauto *)
Lemma critical_example0 : (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.
Proof.
tauto.
Qed.
(* Same remark as previously *)
Lemma critical_example1 : (~ ~ B -> B) -> (~ B -> ~ A) -> ~ ~ A -> B.
Proof.
tauto.
Qed.
(* This example took very much time (about 3mn on a PIII 450MHz in bytecode)
with the old Tauto. Now, it's immediate (less than 1s). *)
Lemma critical_example2 : (~ A <-> B) -> (~ B <-> A) -> (~ ~ A <-> A).
Proof.
tauto.
Qed.
(* This example was a bug *)
Lemma old_bug0 :
(~ A <-> B) -> (~ (C \/ E) <-> D /\ F) -> (~ (C \/ A \/ E) <-> D /\ B /\ F).
Proof.
tauto.
Qed.
(* Another bug *)
Lemma old_bug1 : ((A -> B -> False) -> False) -> (B -> False) -> False.
Proof.
tauto.
Qed.
(* A bug again *)
Lemma old_bug2 :
((((C -> False) -> A) -> ((B -> False) -> A) -> False) -> False) ->
(((C -> B -> False) -> False) -> False) -> ~ A -> A.
Proof.
tauto.
Qed.
(* A bug from CNF form *)
Lemma old_bug3 :
((~ A \/ B) /\ (~ B \/ B) /\ (~ A \/ ~ B) /\ (~ B \/ ~ B) -> False) ->
~ ((A -> B) -> B) -> False.
Proof.
tauto.
Qed.
(* sometimes, the behaviour of Tauto depends on the order of the hyps *)
Lemma old_bug3bis :
~ ((A -> B) -> B) ->
((~ B \/ ~ B) /\ (~ B \/ ~ A) /\ (B \/ ~ B) /\ (B \/ ~ A) -> False) -> False.
Proof.
tauto.
Qed.
(* A bug found by Freek Wiedijk <freek@cs.kun.nl> *)
Lemma new_bug :
((A <-> B) -> (B <-> C)) ->
((B <-> C) -> (C <-> A)) -> ((C <-> A) -> (A <-> B)) -> (A <-> B).
Proof.
tauto.
Qed.
(* A private club has the following rules :
*
* . rule 1 : Every non-scottish member wears red socks
* . rule 2 : Every member wears a kilt or doesn't wear red socks
* . rule 3 : The married members don't go out on sunday
* . rule 4 : A member goes out on sunday if and only if he is scottish
* . rule 5 : Every member who wears a kilt is scottish and married
* . rule 6 : Every scottish member wears a kilt
*
* Actually, no one can be accepted !
*)
Section club.
Variable Scottish RedSocks WearKilt Married GoOutSunday : Prop.
Hypothesis rule1 : ~ Scottish -> RedSocks.
Hypothesis rule2 : WearKilt \/ ~ RedSocks.
Hypothesis rule3 : Married -> ~ GoOutSunday.
Hypothesis rule4 : GoOutSunday <-> Scottish.
Hypothesis rule5 : WearKilt -> Scottish /\ Married.
Hypothesis rule6 : Scottish -> WearKilt.
Lemma NoMember : False.
tauto.
Qed.
End club.
(**** Use of Intuition ****)
Lemma intu0 :
(forall x : nat, P x) /\ B -> (forall y : nat, P y) /\ P 0 \/ B /\ P 0.
Proof.
intuition.
Qed.
Lemma intu1 :
(forall A : Prop, A \/ ~ A) -> forall x y : nat, x = y \/ x <> y.
Proof.
intuition.
Qed.
|
