TSTP Solution File: SET018-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET018-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n023.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 05:04:50 EDT 2022
% Result : Unsatisfiable 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 46
% Syntax : Number of formulae : 101 ( 45 unt; 8 typ; 0 def)
% Number of atoms : 397 ( 195 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 548 ( 262 ~; 247 |; 0 &)
% ( 39 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 18 ( 18 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 6 ( 4 >; 2 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 4 con; 0-2 aty)
% Number of variables : 187 ( 169 !; 0 ?; 187 :)
% Comments :
%------------------------------------------------------------------------------
tff(non_ordered_pair_type,type,
non_ordered_pair: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(singleton_set_type,type,
singleton_set: $i > $i ).
tff(c_type,type,
c: $i ).
tff(d_type,type,
d: $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(1,plain,
( ( ordered_pair(a,b) = ordered_pair(c,d) )
<=> ( ordered_pair(a,b) = ordered_pair(c,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(2,axiom,
ordered_pair(a,b) = ordered_pair(c,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_ordered_pair) ).
tff(3,plain,
ordered_pair(a,b) = ordered_pair(c,d),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( little_set(c)
<=> little_set(c) ),
inference(rewrite,[status(thm)],]) ).
tff(5,axiom,
little_set(c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',c_little_set) ).
tff(6,plain,
little_set(c),
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,plain,
( little_set(a)
<=> little_set(a) ),
inference(rewrite,[status(thm)],]) ).
tff(8,axiom,
little_set(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_little_set) ).
tff(9,plain,
little_set(a),
inference(modus_ponens,[status(thm)],[8,7]) ).
tff(10,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
<=> ( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(13,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ little_set(X)
| ~ little_set(U)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
<=> ( ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) )),
( ( ~ little_set(X)
| ~ little_set(U)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) )
<=> ( ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) ) )),
rewrite(
( ( ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) )
<=> ( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) )),
( ( ~ little_set(X)
| ~ little_set(U)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) )
<=> ( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) )),
inference(bind,[status(th)],]) ).
tff(14,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(U)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(U)
| ( ordered_pair(X,Y) != ordered_pair(U,V) )
| ( X = U ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',first_components_are_equal) ).
tff(16,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ),
inference(modus_ponens,[status(thm)],[16,12]) ).
tff(18,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) ),
inference(modus_ponens,[status(thm)],[18,11]) ).
tff(20,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
| ( a = c )
| ~ little_set(c)
| ~ little_set(a)
| ( ordered_pair(a,b) != ordered_pair(c,d) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
| ( a = c )
| ~ little_set(c)
| ~ little_set(a)
| ( ordered_pair(a,b) != ordered_pair(c,d) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(21,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
| ( a = c )
| ~ little_set(c)
| ~ little_set(a)
| ( ordered_pair(a,b) != ordered_pair(c,d) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(22,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ( X = U )
| ~ little_set(U)
| ~ little_set(X)
| ( ordered_pair(X,Y) != ordered_pair(U,V) ) )
| ( a = c )
| ~ little_set(c)
| ~ little_set(a)
| ( ordered_pair(a,b) != ordered_pair(c,d) ) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
a = c,
inference(unit_resolution,[status(thm)],[22,19,9,6,3]) ).
tff(24,plain,
singleton_set(a) = singleton_set(c),
inference(monotonicity,[status(thm)],[23]) ).
tff(25,plain,
singleton_set(c) = singleton_set(a),
inference(symmetry,[status(thm)],[24]) ).
tff(26,plain,
non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) = non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)),
inference(monotonicity,[status(thm)],[25]) ).
tff(27,plain,
non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) = non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)),
inference(symmetry,[status(thm)],[26]) ).
tff(28,plain,
^ [Y: $i,X: $i] :
refl(
( ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) )),
inference(bind,[status(th)],]) ).
tff(29,plain,
( ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[28]) ).
tff(30,plain,
( ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(31,axiom,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair) ).
tff(32,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
( ~ ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
| ( ordered_pair(a,b) = non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(36,plain,
ordered_pair(a,b) = non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
ordered_pair(c,d) = ordered_pair(a,b),
inference(symmetry,[status(thm)],[3]) ).
tff(38,plain,
( ~ ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
| ( ordered_pair(c,d) = non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(39,plain,
ordered_pair(c,d) = non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)),
inference(unit_resolution,[status(thm)],[38,34]) ).
tff(40,plain,
non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) = ordered_pair(c,d),
inference(symmetry,[status(thm)],[39]) ).
tff(41,plain,
non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) = non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)),
inference(transitivity,[status(thm)],[40,37,36,27]) ).
tff(42,plain,
^ [Y: $i,X: $i] :
refl(
( little_set(non_ordered_pair(X,Y))
<=> little_set(non_ordered_pair(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(43,plain,
( ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y))
<=> ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,plain,
( ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y))
<=> ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)) ),
inference(rewrite,[status(thm)],]) ).
tff(45,axiom,
! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',non_ordered_pair4) ).
tff(46,plain,
! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[47,43]) ).
tff(49,plain,
( ~ ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y))
| little_set(non_ordered_pair(c,d)) ),
inference(quant_inst,[status(thm)],]) ).
tff(50,plain,
little_set(non_ordered_pair(c,d)),
inference(unit_resolution,[status(thm)],[49,48]) ).
tff(51,plain,
c = a,
inference(symmetry,[status(thm)],[23]) ).
tff(52,plain,
non_ordered_pair(c,b) = non_ordered_pair(a,b),
inference(monotonicity,[status(thm)],[51]) ).
tff(53,plain,
( ( non_ordered_pair(c,d) = non_ordered_pair(c,b) )
<=> ( non_ordered_pair(c,d) = non_ordered_pair(a,b) ) ),
inference(monotonicity,[status(thm)],[52]) ).
tff(54,plain,
( ( non_ordered_pair(c,d) != non_ordered_pair(c,b) )
<=> ( non_ordered_pair(c,d) != non_ordered_pair(a,b) ) ),
inference(monotonicity,[status(thm)],[53]) ).
tff(55,plain,
( ( non_ordered_pair(c,b) = non_ordered_pair(c,d) )
<=> ( non_ordered_pair(c,d) = non_ordered_pair(c,b) ) ),
inference(commutativity,[status(thm)],]) ).
tff(56,plain,
( ( non_ordered_pair(c,b) != non_ordered_pair(c,d) )
<=> ( non_ordered_pair(c,d) != non_ordered_pair(c,b) ) ),
inference(monotonicity,[status(thm)],[55]) ).
tff(57,plain,
( ( b != d )
<=> ( b != d ) ),
inference(rewrite,[status(thm)],]) ).
tff(58,axiom,
b != d,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_second_components_equal) ).
tff(59,plain,
b != d,
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
( little_set(d)
<=> little_set(d) ),
inference(rewrite,[status(thm)],]) ).
tff(61,axiom,
little_set(d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d_little_set) ).
tff(62,plain,
little_set(d),
inference(modus_ponens,[status(thm)],[61,60]) ).
tff(63,plain,
( little_set(b)
<=> little_set(b) ),
inference(rewrite,[status(thm)],]) ).
tff(64,axiom,
little_set(b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_little_set) ).
tff(65,plain,
little_set(b),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
<=> ( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) )),
inference(bind,[status(th)],]) ).
tff(67,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(69,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ little_set(X)
| ~ little_set(Y)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) ) )),
rewrite(
( ( ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) )
<=> ( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) )
<=> ( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) )),
inference(bind,[status(th)],]) ).
tff(70,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) )
| ( X = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',left_cancellation) ).
tff(72,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) ),
inference(modus_ponens,[status(thm)],[74,67]) ).
tff(76,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(77,plain,
( ( ( b = d )
| ~ little_set(d)
| ~ little_set(b)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) )
<=> ( ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(78,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(d)
| ~ little_set(b)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ) ),
inference(monotonicity,[status(thm)],[77]) ).
tff(79,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(d)
| ~ little_set(b)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ) ),
inference(transitivity,[status(thm)],[78,76]) ).
tff(80,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(d)
| ~ little_set(b)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(81,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( b = d )
| ~ little_set(b)
| ~ little_set(d)
| ( non_ordered_pair(c,b) != non_ordered_pair(c,d) ) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
non_ordered_pair(c,b) != non_ordered_pair(c,d),
inference(unit_resolution,[status(thm)],[81,75,65,62,59]) ).
tff(83,plain,
non_ordered_pair(c,d) != non_ordered_pair(c,b),
inference(modus_ponens,[status(thm)],[82,56]) ).
tff(84,plain,
non_ordered_pair(c,d) != non_ordered_pair(a,b),
inference(modus_ponens,[status(thm)],[83,54]) ).
tff(85,plain,
( little_set(non_ordered_pair(c,b))
<=> little_set(non_ordered_pair(a,b)) ),
inference(monotonicity,[status(thm)],[52]) ).
tff(86,plain,
( ~ ! [Y: $i,X: $i] : little_set(non_ordered_pair(X,Y))
| little_set(non_ordered_pair(c,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(87,plain,
little_set(non_ordered_pair(c,b)),
inference(unit_resolution,[status(thm)],[86,48]) ).
tff(88,plain,
little_set(non_ordered_pair(a,b)),
inference(modus_ponens,[status(thm)],[87,85]) ).
tff(89,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( non_ordered_pair(c,d) = non_ordered_pair(a,b) )
| ~ little_set(non_ordered_pair(a,b))
| ~ little_set(non_ordered_pair(c,d))
| ( non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) != non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( non_ordered_pair(c,d) = non_ordered_pair(a,b) )
| ~ little_set(non_ordered_pair(a,b))
| ~ little_set(non_ordered_pair(c,d))
| ( non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) != non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(90,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( non_ordered_pair(c,d) = non_ordered_pair(a,b) )
| ~ little_set(non_ordered_pair(a,b))
| ~ little_set(non_ordered_pair(c,d))
| ( non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) != non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(91,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ little_set(Y)
| ~ little_set(X)
| ( non_ordered_pair(Z,X) != non_ordered_pair(Z,Y) ) )
| ( non_ordered_pair(c,d) = non_ordered_pair(a,b) )
| ~ little_set(non_ordered_pair(a,b))
| ~ little_set(non_ordered_pair(c,d))
| ( non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) != non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) ) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
( ( non_ordered_pair(c,d) = non_ordered_pair(a,b) )
| ~ little_set(non_ordered_pair(a,b))
| ~ little_set(non_ordered_pair(c,d))
| ( non_ordered_pair(singleton_set(c),non_ordered_pair(c,d)) != non_ordered_pair(singleton_set(c),non_ordered_pair(a,b)) ) ),
inference(unit_resolution,[status(thm)],[91,75]) ).
tff(93,plain,
$false,
inference(unit_resolution,[status(thm)],[92,88,84,50,41]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET018-3 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n023.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 01:33:18 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
%------------------------------------------------------------------------------