TSTP Solution File: NUM009-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : NUM009-1 : TPTP v8.1.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n010.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 : Sun Sep 18 13:02:37 EDT 2022
% Result : Unsatisfiable 0.20s 0.46s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 29
% Syntax : Number of formulae : 57 ( 13 unt; 6 typ; 0 def)
% Number of atoms : 227 ( 0 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 270 ( 106 ~; 138 |; 0 &)
% ( 26 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 12 ( 12 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 4 ( 3 >; 1 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 3 con; 0-1 aty)
% Number of variables : 72 ( 64 !; 0 ?; 72 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(f44_type,type,
f44: $i > $i ).
tff(empty_set_type,type,
empty_set: $i ).
tff(natural_numbers_type,type,
natural_numbers: $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(infinity_type,type,
infinity: $i ).
tff(1,plain,
( ~ member(empty_set,natural_numbers)
<=> ~ member(empty_set,natural_numbers) ),
inference(rewrite,[status(thm)],]) ).
tff(2,axiom,
~ member(empty_set,natural_numbers),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_zero_is_a_natural) ).
tff(3,plain,
~ member(empty_set,natural_numbers),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Z: $i] :
refl(
( ( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
<=> ( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ) )),
inference(bind,[status(th)],]) ).
tff(5,plain,
( ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
<=> ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
<=> ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ) ),
inference(rewrite,[status(thm)],]) ).
tff(7,plain,
^ [Z: $i] :
rewrite(
( ( member(Z,natural_numbers)
| ~ member(Z,f44(Z)) )
<=> ( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ) )),
inference(bind,[status(th)],]) ).
tff(8,plain,
( ! [Z: $i] :
( member(Z,natural_numbers)
| ~ member(Z,f44(Z)) )
<=> ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [Z: $i] :
( member(Z,natural_numbers)
| ~ member(Z,f44(Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/NUM003-0.ax',natural_numbers6) ).
tff(10,plain,
! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
| ~ member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) )
<=> ( ~ ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
| ~ member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) ) ),
inference(rewrite,[status(thm)],]) ).
tff(15,plain,
( ~ ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
| ~ member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) ),
inference(quant_inst,[status(thm)],]) ).
tff(16,plain,
( ~ ! [Z: $i] :
( ~ member(Z,f44(Z))
| member(Z,natural_numbers) )
| ~ member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
~ member(empty_set,f44(empty_set)),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
( member(empty_set,infinity)
<=> member(empty_set,infinity) ),
inference(rewrite,[status(thm)],]) ).
tff(19,axiom,
member(empty_set,infinity),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',infinity2) ).
tff(20,plain,
member(empty_set,infinity),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
^ [Y: $i,X: $i] :
refl(
( ( little_set(X)
| ~ member(X,Y) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(22,plain,
( ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(24,plain,
^ [Y: $i,X: $i] :
rewrite(
( ( ~ member(X,Y)
| little_set(X) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(25,plain,
( ! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,axiom,
! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',a2) ).
tff(27,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[27,23]) ).
tff(29,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[29,22]) ).
tff(31,plain,
( ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(empty_set)
| ~ member(empty_set,infinity) )
<=> ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(empty_set)
| ~ member(empty_set,infinity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(32,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(empty_set)
| ~ member(empty_set,infinity) ),
inference(quant_inst,[status(thm)],]) ).
tff(33,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(empty_set)
| ~ member(empty_set,infinity) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
little_set(empty_set),
inference(unit_resolution,[status(thm)],[33,30,20]) ).
tff(35,plain,
^ [Z: $i] :
refl(
( ( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
<=> ( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) )),
inference(bind,[status(th)],]) ).
tff(36,plain,
( ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
<=> ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) ),
inference(quant_intro,[status(thm)],[35]) ).
tff(37,plain,
( ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
<=> ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) ),
inference(rewrite,[status(thm)],]) ).
tff(38,plain,
^ [Z: $i] :
trans(
monotonicity(
rewrite(
( ( member(Z,natural_numbers)
| ~ little_set(Z) )
<=> ( ~ little_set(Z)
| member(Z,natural_numbers) ) )),
( ( member(Z,natural_numbers)
| ~ little_set(Z)
| member(empty_set,f44(Z)) )
<=> ( ~ little_set(Z)
| member(Z,natural_numbers)
| member(empty_set,f44(Z)) ) )),
rewrite(
( ( ~ little_set(Z)
| member(Z,natural_numbers)
| member(empty_set,f44(Z)) )
<=> ( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) )),
( ( member(Z,natural_numbers)
| ~ little_set(Z)
| member(empty_set,f44(Z)) )
<=> ( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) )),
inference(bind,[status(th)],]) ).
tff(39,plain,
( ! [Z: $i] :
( member(Z,natural_numbers)
| ~ little_set(Z)
| member(empty_set,f44(Z)) )
<=> ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,axiom,
! [Z: $i] :
( member(Z,natural_numbers)
| ~ little_set(Z)
| member(empty_set,f44(Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/NUM003-0.ax',natural_numbers4) ).
tff(41,plain,
! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[41,37]) ).
tff(43,plain,
! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) ),
inference(modus_ponens,[status(thm)],[43,36]) ).
tff(45,plain,
( ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) )
<=> ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) ) ),
inference(rewrite,[status(thm)],]) ).
tff(46,plain,
( ( ~ little_set(empty_set)
| member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) )
<=> ( member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) ) ),
inference(rewrite,[status(thm)],]) ).
tff(47,plain,
( ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| ~ little_set(empty_set)
| member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) )
<=> ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) ) ),
inference(monotonicity,[status(thm)],[46]) ).
tff(48,plain,
( ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| ~ little_set(empty_set)
| member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) )
<=> ( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) ) ),
inference(transitivity,[status(thm)],[47,45]) ).
tff(49,plain,
( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| ~ little_set(empty_set)
| member(empty_set,f44(empty_set))
| member(empty_set,natural_numbers) ),
inference(quant_inst,[status(thm)],]) ).
tff(50,plain,
( ~ ! [Z: $i] :
( ~ little_set(Z)
| member(empty_set,f44(Z))
| member(Z,natural_numbers) )
| member(empty_set,natural_numbers)
| member(empty_set,f44(empty_set))
| ~ little_set(empty_set) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
$false,
inference(unit_resolution,[status(thm)],[50,44,3,34,17]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : NUM009-1 : TPTP v8.1.0. Bugfixed v1.2.1.
% 0.11/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n010.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 : Fri Sep 2 06:16:28 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.
% 0.20/0.46 % SZS status Unsatisfiable
% 0.20/0.46 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------