TSTP Solution File: SET023-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET023-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:04:55 EDT 2022
% Result : Unsatisfiable 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 34
% Syntax : Number of formulae : 72 ( 18 unt; 7 typ; 0 def)
% Number of atoms : 243 ( 55 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 321 ( 149 ~; 145 |; 0 &)
% ( 27 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of FOOLs : 6 ( 6 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 7 ( 6 >; 1 *; 0 +; 0 <<)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 1 con; 0-2 aty)
% Number of variables : 85 ( 77 !; 0 ?; 85 :)
% Comments :
%------------------------------------------------------------------------------
tff(little_set_type,type,
little_set: $i > $o ).
tff(second_type,type,
second: $i > $i ).
tff(a_type,type,
a: $i ).
tff(f3_type,type,
f3: $i > $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(f2_type,type,
f2: $i > $i ).
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(1,plain,
( ordered_pair_predicate(a)
<=> ordered_pair_predicate(a) ),
inference(rewrite,[status(thm)],]) ).
tff(2,axiom,
ordered_pair_predicate(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',an_ordered_pair_predicate) ).
tff(3,plain,
ordered_pair_predicate(a),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [X: $i] :
refl(
( ( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) )),
inference(bind,[status(th)],]) ).
tff(5,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(7,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate2) ).
tff(8,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(a)
| little_set(f3(a)) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(a)
| little_set(f3(a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(12,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(a)
| little_set(f3(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(13,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(a)
| little_set(f3(a)) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
little_set(f3(a)),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
^ [X: $i] :
refl(
( ( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) )),
inference(bind,[status(th)],]) ).
tff(16,plain,
( ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(18,plain,
^ [X: $i] :
rewrite(
( ( ~ ordered_pair_predicate(X)
| little_set(f2(X)) )
<=> ( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) )),
inference(bind,[status(th)],]) ).
tff(19,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f2(X)) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f2(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate1) ).
tff(21,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[23,16]) ).
tff(25,plain,
( ( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(a))
| ~ ordered_pair_predicate(a) )
<=> ( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(a))
| ~ ordered_pair_predicate(a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(26,plain,
( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(a))
| ~ ordered_pair_predicate(a) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(a))
| ~ ordered_pair_predicate(a) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
little_set(f2(a)),
inference(unit_resolution,[status(thm)],[27,24,3]) ).
tff(29,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) )),
inference(bind,[status(th)],]) ).
tff(30,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(quant_intro,[status(thm)],[29]) ).
tff(31,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(32,plain,
^ [Y: $i,X: $i] :
rewrite(
( ( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) )),
inference(bind,[status(th)],]) ).
tff(33,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_second) ).
tff(35,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(skolemize,[status(sab)],[36]) ).
tff(38,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[37,30]) ).
tff(39,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,plain,
( ( ~ little_set(f3(a))
| ~ little_set(f2(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) )
<=> ( ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(41,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(a))
| ~ little_set(f2(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ) ),
inference(monotonicity,[status(thm)],[40]) ).
tff(42,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(a))
| ~ little_set(f2(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ) ),
inference(transitivity,[status(thm)],[41,39]) ).
tff(43,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(a))
| ~ little_set(f2(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(44,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(a))
| ~ little_set(f3(a))
| ( second(ordered_pair(f2(a),f3(a))) = f3(a) ) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
second(ordered_pair(f2(a),f3(a))) = f3(a),
inference(unit_resolution,[status(thm)],[44,38,28,14]) ).
tff(46,plain,
^ [X: $i] :
refl(
( ( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) )),
inference(bind,[status(th)],]) ).
tff(47,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) ),
inference(quant_intro,[status(thm)],[46]) ).
tff(48,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(49,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate3) ).
tff(50,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(skolemize,[status(sab)],[50]) ).
tff(52,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[51,47]) ).
tff(53,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(a)
| ( a = ordered_pair(f2(a),f3(a)) ) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(a)
| ( a = ordered_pair(f2(a),f3(a)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(54,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(a)
| ( a = ordered_pair(f2(a),f3(a)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(55,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(a)
| ( a = ordered_pair(f2(a),f3(a)) ) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
a = ordered_pair(f2(a),f3(a)),
inference(unit_resolution,[status(thm)],[55,52,3]) ).
tff(57,plain,
second(a) = second(ordered_pair(f2(a),f3(a))),
inference(monotonicity,[status(thm)],[56]) ).
tff(58,plain,
second(a) = f3(a),
inference(transitivity,[status(thm)],[57,45]) ).
tff(59,plain,
( little_set(second(a))
<=> little_set(f3(a)) ),
inference(monotonicity,[status(thm)],[58]) ).
tff(60,plain,
( little_set(f3(a))
<=> little_set(second(a)) ),
inference(symmetry,[status(thm)],[59]) ).
tff(61,plain,
little_set(second(a)),
inference(modus_ponens,[status(thm)],[14,60]) ).
tff(62,plain,
( ~ little_set(second(a))
<=> ~ little_set(second(a)) ),
inference(rewrite,[status(thm)],]) ).
tff(63,axiom,
~ little_set(second(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_second_component_is_small) ).
tff(64,plain,
~ little_set(second(a)),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
$false,
inference(unit_resolution,[status(thm)],[64,61]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SET023-3 : TPTP v8.1.0. Released v1.0.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Sep 3 01:20:03 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.43 % SZS status Unsatisfiable
% 0.19/0.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------