TSTP Solution File: SET964+1 by CSE_E---1.5
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%------------------------------------------------------------------------------
% File : CSE_E---1.5
% Problem : SET964+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -jar /export/starexec/sandbox2/solver/bin/mcs_scs.jar %d %s
% Computer : n001.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 : Thu Aug 31 14:36:28 EDT 2023
% Result : Theorem 0.20s 0.72s
% Output : CNFRefutation 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 27
% Syntax : Number of formulae : 84 ( 14 unt; 20 typ; 0 def)
% Number of atoms : 186 ( 40 equ)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 207 ( 85 ~; 94 |; 21 &)
% ( 6 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 32 ( 14 >; 18 *; 0 +; 0 <<)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 17 ( 17 usr; 6 con; 0-4 aty)
% Number of variables : 166 ( 19 sgn; 53 !; 2 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
tff(decl_22,type,
in: ( $i * $i ) > $o ).
tff(decl_23,type,
unordered_pair: ( $i * $i ) > $i ).
tff(decl_24,type,
empty_set: $i ).
tff(decl_25,type,
cartesian_product2: ( $i * $i ) > $i ).
tff(decl_26,type,
ordered_pair: ( $i * $i ) > $i ).
tff(decl_27,type,
subset: ( $i * $i ) > $o ).
tff(decl_28,type,
singleton: $i > $i ).
tff(decl_29,type,
empty: $i > $o ).
tff(decl_30,type,
esk1_1: $i > $i ).
tff(decl_31,type,
esk2_4: ( $i * $i * $i * $i ) > $i ).
tff(decl_32,type,
esk3_4: ( $i * $i * $i * $i ) > $i ).
tff(decl_33,type,
esk4_3: ( $i * $i * $i ) > $i ).
tff(decl_34,type,
esk5_3: ( $i * $i * $i ) > $i ).
tff(decl_35,type,
esk6_3: ( $i * $i * $i ) > $i ).
tff(decl_36,type,
esk7_2: ( $i * $i ) > $i ).
tff(decl_37,type,
esk8_0: $i ).
tff(decl_38,type,
esk9_0: $i ).
tff(decl_39,type,
esk10_0: $i ).
tff(decl_40,type,
esk11_0: $i ).
tff(decl_41,type,
esk12_0: $i ).
fof(d1_xboole_0,axiom,
! [X1] :
( X1 = empty_set
<=> ! [X2] : ~ in(X2,X1) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_xboole_0) ).
fof(t117_zfmisc_1,conjecture,
! [X1,X2,X3] :
~ ( X1 != empty_set
& ( subset(cartesian_product2(X2,X1),cartesian_product2(X3,X1))
| subset(cartesian_product2(X1,X2),cartesian_product2(X1,X3)) )
& ~ subset(X2,X3) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t117_zfmisc_1) ).
fof(d2_zfmisc_1,axiom,
! [X1,X2,X3] :
( X3 = cartesian_product2(X1,X2)
<=> ! [X4] :
( in(X4,X3)
<=> ? [X5,X6] :
( in(X5,X1)
& in(X6,X2)
& X4 = ordered_pair(X5,X6) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_zfmisc_1) ).
fof(d3_tarski,axiom,
! [X1,X2] :
( subset(X1,X2)
<=> ! [X3] :
( in(X3,X1)
=> in(X3,X2) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_tarski) ).
fof(l55_zfmisc_1,axiom,
! [X1,X2,X3,X4] :
( in(ordered_pair(X1,X2),cartesian_product2(X3,X4))
<=> ( in(X1,X3)
& in(X2,X4) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l55_zfmisc_1) ).
fof(d5_tarski,axiom,
! [X1,X2] : ordered_pair(X1,X2) = unordered_pair(unordered_pair(X1,X2),singleton(X1)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_tarski) ).
fof(commutativity_k2_tarski,axiom,
! [X1,X2] : unordered_pair(X1,X2) = unordered_pair(X2,X1),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
fof(c_0_7,plain,
! [X1] :
( X1 = empty_set
<=> ! [X2] : ~ in(X2,X1) ),
inference(fof_simplification,[status(thm)],[d1_xboole_0]) ).
fof(c_0_8,negated_conjecture,
~ ! [X1,X2,X3] :
~ ( X1 != empty_set
& ( subset(cartesian_product2(X2,X1),cartesian_product2(X3,X1))
| subset(cartesian_product2(X1,X2),cartesian_product2(X1,X3)) )
& ~ subset(X2,X3) ),
inference(fof_simplification,[status(thm)],[inference(assume_negation,[status(cth)],[t117_zfmisc_1])]) ).
fof(c_0_9,plain,
! [X11,X12,X13] :
( ( X11 != empty_set
| ~ in(X12,X11) )
& ( in(esk1_1(X13),X13)
| X13 = empty_set ) ),
inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_7])])])])]) ).
fof(c_0_10,plain,
! [X15,X16,X17,X18,X21,X22,X23,X24,X25,X26,X28,X29] :
( ( in(esk2_4(X15,X16,X17,X18),X15)
| ~ in(X18,X17)
| X17 != cartesian_product2(X15,X16) )
& ( in(esk3_4(X15,X16,X17,X18),X16)
| ~ in(X18,X17)
| X17 != cartesian_product2(X15,X16) )
& ( X18 = ordered_pair(esk2_4(X15,X16,X17,X18),esk3_4(X15,X16,X17,X18))
| ~ in(X18,X17)
| X17 != cartesian_product2(X15,X16) )
& ( ~ in(X22,X15)
| ~ in(X23,X16)
| X21 != ordered_pair(X22,X23)
| in(X21,X17)
| X17 != cartesian_product2(X15,X16) )
& ( ~ in(esk4_3(X24,X25,X26),X26)
| ~ in(X28,X24)
| ~ in(X29,X25)
| esk4_3(X24,X25,X26) != ordered_pair(X28,X29)
| X26 = cartesian_product2(X24,X25) )
& ( in(esk5_3(X24,X25,X26),X24)
| in(esk4_3(X24,X25,X26),X26)
| X26 = cartesian_product2(X24,X25) )
& ( in(esk6_3(X24,X25,X26),X25)
| in(esk4_3(X24,X25,X26),X26)
| X26 = cartesian_product2(X24,X25) )
& ( esk4_3(X24,X25,X26) = ordered_pair(esk5_3(X24,X25,X26),esk6_3(X24,X25,X26))
| in(esk4_3(X24,X25,X26),X26)
| X26 = cartesian_product2(X24,X25) ) ),
inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[d2_zfmisc_1])])])])])]) ).
fof(c_0_11,plain,
! [X32,X33,X34,X35,X36] :
( ( ~ subset(X32,X33)
| ~ in(X34,X32)
| in(X34,X33) )
& ( in(esk7_2(X35,X36),X35)
| subset(X35,X36) )
& ( ~ in(esk7_2(X35,X36),X36)
| subset(X35,X36) ) ),
inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(fof_nnf,[status(thm)],[d3_tarski])])])])])]) ).
fof(c_0_12,negated_conjecture,
( esk10_0 != empty_set
& ( subset(cartesian_product2(esk11_0,esk10_0),cartesian_product2(esk12_0,esk10_0))
| subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0)) )
& ~ subset(esk11_0,esk12_0) ),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_8])])]) ).
fof(c_0_13,plain,
! [X42,X43,X44,X45] :
( ( in(X42,X44)
| ~ in(ordered_pair(X42,X43),cartesian_product2(X44,X45)) )
& ( in(X43,X45)
| ~ in(ordered_pair(X42,X43),cartesian_product2(X44,X45)) )
& ( ~ in(X42,X44)
| ~ in(X43,X45)
| in(ordered_pair(X42,X43),cartesian_product2(X44,X45)) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[l55_zfmisc_1])])]) ).
fof(c_0_14,plain,
! [X38,X39] : ordered_pair(X38,X39) = unordered_pair(unordered_pair(X38,X39),singleton(X38)),
inference(variable_rename,[status(thm)],[d5_tarski]) ).
cnf(c_0_15,plain,
( X1 != empty_set
| ~ in(X2,X1) ),
inference(split_conjunct,[status(thm)],[c_0_9]) ).
cnf(c_0_16,plain,
( in(esk2_4(X1,X2,X3,X4),X1)
| ~ in(X4,X3)
| X3 != cartesian_product2(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_10]) ).
cnf(c_0_17,plain,
( in(X3,X2)
| ~ subset(X1,X2)
| ~ in(X3,X1) ),
inference(split_conjunct,[status(thm)],[c_0_11]) ).
cnf(c_0_18,negated_conjecture,
( subset(cartesian_product2(esk11_0,esk10_0),cartesian_product2(esk12_0,esk10_0))
| subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0)) ),
inference(split_conjunct,[status(thm)],[c_0_12]) ).
cnf(c_0_19,plain,
( in(ordered_pair(X1,X3),cartesian_product2(X2,X4))
| ~ in(X1,X2)
| ~ in(X3,X4) ),
inference(split_conjunct,[status(thm)],[c_0_13]) ).
cnf(c_0_20,plain,
ordered_pair(X1,X2) = unordered_pair(unordered_pair(X1,X2),singleton(X1)),
inference(split_conjunct,[status(thm)],[c_0_14]) ).
cnf(c_0_21,plain,
~ in(X1,empty_set),
inference(er,[status(thm)],[c_0_15]) ).
cnf(c_0_22,plain,
( in(esk2_4(X1,X2,cartesian_product2(X1,X2),X3),X1)
| ~ in(X3,cartesian_product2(X1,X2)) ),
inference(er,[status(thm)],[c_0_16]) ).
cnf(c_0_23,plain,
( in(X1,X2)
| ~ in(ordered_pair(X1,X3),cartesian_product2(X2,X4)) ),
inference(split_conjunct,[status(thm)],[c_0_13]) ).
cnf(c_0_24,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(X1,cartesian_product2(esk12_0,esk10_0))
| ~ in(X1,cartesian_product2(esk11_0,esk10_0)) ),
inference(spm,[status(thm)],[c_0_17,c_0_18]) ).
cnf(c_0_25,plain,
( in(unordered_pair(unordered_pair(X1,X3),singleton(X1)),cartesian_product2(X2,X4))
| ~ in(X3,X4)
| ~ in(X1,X2) ),
inference(rw,[status(thm)],[c_0_19,c_0_20]) ).
cnf(c_0_26,plain,
~ in(X1,cartesian_product2(empty_set,X2)),
inference(spm,[status(thm)],[c_0_21,c_0_22]) ).
cnf(c_0_27,plain,
( in(esk1_1(X1),X1)
| X1 = empty_set ),
inference(split_conjunct,[status(thm)],[c_0_9]) ).
fof(c_0_28,plain,
! [X9,X10] : unordered_pair(X9,X10) = unordered_pair(X10,X9),
inference(variable_rename,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_0_29,plain,
( X1 = ordered_pair(esk2_4(X2,X3,X4,X1),esk3_4(X2,X3,X4,X1))
| ~ in(X1,X4)
| X4 != cartesian_product2(X2,X3) ),
inference(split_conjunct,[status(thm)],[c_0_10]) ).
cnf(c_0_30,plain,
( in(X1,X2)
| ~ in(unordered_pair(unordered_pair(X1,X3),singleton(X1)),cartesian_product2(X2,X4)) ),
inference(rw,[status(thm)],[c_0_23,c_0_20]) ).
cnf(c_0_31,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(unordered_pair(unordered_pair(X1,X2),singleton(X1)),cartesian_product2(esk12_0,esk10_0))
| ~ in(X2,esk10_0)
| ~ in(X1,esk11_0) ),
inference(spm,[status(thm)],[c_0_24,c_0_25]) ).
cnf(c_0_32,plain,
~ in(X1,cartesian_product2(cartesian_product2(empty_set,X2),X3)),
inference(spm,[status(thm)],[c_0_26,c_0_22]) ).
cnf(c_0_33,plain,
( in(esk5_3(X1,X2,X3),X1)
| in(esk4_3(X1,X2,X3),X3)
| X3 = cartesian_product2(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_10]) ).
cnf(c_0_34,plain,
cartesian_product2(empty_set,X1) = empty_set,
inference(spm,[status(thm)],[c_0_26,c_0_27]) ).
cnf(c_0_35,plain,
unordered_pair(X1,X2) = unordered_pair(X2,X1),
inference(split_conjunct,[status(thm)],[c_0_28]) ).
cnf(c_0_36,plain,
( X1 = unordered_pair(unordered_pair(esk2_4(X2,X3,X4,X1),esk3_4(X2,X3,X4,X1)),singleton(esk2_4(X2,X3,X4,X1)))
| X4 != cartesian_product2(X2,X3)
| ~ in(X1,X4) ),
inference(rw,[status(thm)],[c_0_29,c_0_20]) ).
cnf(c_0_37,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(X1,esk12_0)
| ~ in(X2,esk10_0)
| ~ in(X1,esk11_0) ),
inference(spm,[status(thm)],[c_0_30,c_0_31]) ).
cnf(c_0_38,plain,
( X1 = empty_set
| in(esk4_3(empty_set,X2,X1),X1) ),
inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_32,c_0_33]),c_0_34]),c_0_34]),c_0_34]),c_0_34]),c_0_34]) ).
cnf(c_0_39,negated_conjecture,
esk10_0 != empty_set,
inference(split_conjunct,[status(thm)],[c_0_12]) ).
cnf(c_0_40,plain,
( in(esk3_4(X1,X2,X3,X4),X2)
| ~ in(X4,X3)
| X3 != cartesian_product2(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_10]) ).
cnf(c_0_41,plain,
( in(unordered_pair(singleton(X1),unordered_pair(X1,X2)),cartesian_product2(X3,X4))
| ~ in(X2,X4)
| ~ in(X1,X3) ),
inference(spm,[status(thm)],[c_0_25,c_0_35]) ).
cnf(c_0_42,plain,
( unordered_pair(singleton(esk2_4(X1,X2,cartesian_product2(X1,X2),X3)),unordered_pair(esk2_4(X1,X2,cartesian_product2(X1,X2),X3),esk3_4(X1,X2,cartesian_product2(X1,X2),X3))) = X3
| ~ in(X3,cartesian_product2(X1,X2)) ),
inference(er,[status(thm)],[inference(rw,[status(thm)],[c_0_36,c_0_35])]) ).
cnf(c_0_43,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(X1,esk12_0)
| ~ in(X1,esk11_0) ),
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_37,c_0_38]),c_0_39]) ).
cnf(c_0_44,plain,
( in(esk3_4(X1,X2,cartesian_product2(X1,X2),X3),X2)
| ~ in(X3,cartesian_product2(X1,X2)) ),
inference(er,[status(thm)],[c_0_40]) ).
cnf(c_0_45,plain,
( in(X1,cartesian_product2(X2,X3))
| ~ in(esk3_4(X4,X5,cartesian_product2(X4,X5),X1),X3)
| ~ in(esk2_4(X4,X5,cartesian_product2(X4,X5),X1),X2)
| ~ in(X1,cartesian_product2(X4,X5)) ),
inference(spm,[status(thm)],[c_0_41,c_0_42]) ).
cnf(c_0_46,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(esk3_4(X1,esk11_0,cartesian_product2(X1,esk11_0),X2),esk12_0)
| ~ in(X2,cartesian_product2(X1,esk11_0)) ),
inference(spm,[status(thm)],[c_0_43,c_0_44]) ).
cnf(c_0_47,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(X1,cartesian_product2(X2,esk12_0))
| ~ in(esk2_4(X3,esk11_0,cartesian_product2(X3,esk11_0),X1),X2)
| ~ in(X1,cartesian_product2(X3,esk11_0)) ),
inference(spm,[status(thm)],[c_0_45,c_0_46]) ).
cnf(c_0_48,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| in(X1,cartesian_product2(X2,esk12_0))
| ~ in(X1,cartesian_product2(X2,esk11_0)) ),
inference(spm,[status(thm)],[c_0_47,c_0_22]) ).
cnf(c_0_49,plain,
( in(esk7_2(X1,X2),X1)
| subset(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_11]) ).
cnf(c_0_50,plain,
( subset(X1,X2)
| ~ in(esk7_2(X1,X2),X2) ),
inference(split_conjunct,[status(thm)],[c_0_11]) ).
cnf(c_0_51,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| subset(cartesian_product2(X1,esk11_0),X2)
| in(esk7_2(cartesian_product2(X1,esk11_0),X2),cartesian_product2(X1,esk12_0)) ),
inference(spm,[status(thm)],[c_0_48,c_0_49]) ).
cnf(c_0_52,plain,
( in(X1,X2)
| ~ in(ordered_pair(X3,X1),cartesian_product2(X4,X2)) ),
inference(split_conjunct,[status(thm)],[c_0_13]) ).
cnf(c_0_53,negated_conjecture,
( subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0))
| subset(cartesian_product2(X1,esk11_0),cartesian_product2(X1,esk12_0)) ),
inference(spm,[status(thm)],[c_0_50,c_0_51]) ).
cnf(c_0_54,plain,
( in(X1,X2)
| ~ in(unordered_pair(unordered_pair(X3,X1),singleton(X3)),cartesian_product2(X4,X2)) ),
inference(rw,[status(thm)],[c_0_52,c_0_20]) ).
cnf(c_0_55,negated_conjecture,
subset(cartesian_product2(esk10_0,esk11_0),cartesian_product2(esk10_0,esk12_0)),
inference(ef,[status(thm)],[c_0_53]) ).
cnf(c_0_56,plain,
( in(X1,X2)
| ~ in(unordered_pair(singleton(X3),unordered_pair(X3,X1)),cartesian_product2(X4,X2)) ),
inference(spm,[status(thm)],[c_0_54,c_0_35]) ).
cnf(c_0_57,negated_conjecture,
( in(X1,cartesian_product2(esk10_0,esk12_0))
| ~ in(X1,cartesian_product2(esk10_0,esk11_0)) ),
inference(spm,[status(thm)],[c_0_17,c_0_55]) ).
cnf(c_0_58,negated_conjecture,
( in(X1,esk12_0)
| ~ in(unordered_pair(singleton(X2),unordered_pair(X2,X1)),cartesian_product2(esk10_0,esk11_0)) ),
inference(spm,[status(thm)],[c_0_56,c_0_57]) ).
cnf(c_0_59,negated_conjecture,
( in(X1,esk12_0)
| ~ in(X1,esk11_0)
| ~ in(X2,esk10_0) ),
inference(spm,[status(thm)],[c_0_58,c_0_41]) ).
cnf(c_0_60,negated_conjecture,
( in(X1,esk12_0)
| ~ in(X1,esk11_0) ),
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_59,c_0_38]),c_0_39]) ).
cnf(c_0_61,negated_conjecture,
( subset(X1,esk12_0)
| ~ in(esk7_2(X1,esk12_0),esk11_0) ),
inference(spm,[status(thm)],[c_0_50,c_0_60]) ).
cnf(c_0_62,negated_conjecture,
~ subset(esk11_0,esk12_0),
inference(split_conjunct,[status(thm)],[c_0_12]) ).
cnf(c_0_63,negated_conjecture,
$false,
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_61,c_0_49]),c_0_62]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET964+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.13 % Command : java -jar /export/starexec/sandbox2/solver/bin/mcs_scs.jar %d %s
% 0.14/0.34 % Computer : n001.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sat Aug 26 12:03:01 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.56 start to proof: theBenchmark
% 0.20/0.72 % Version : CSE_E---1.5
% 0.20/0.72 % Problem : theBenchmark.p
% 0.20/0.72 % Proof found
% 0.20/0.72 % SZS status Theorem for theBenchmark.p
% 0.20/0.72 % SZS output start Proof
% See solution above
% 0.20/0.72 % Total time : 0.145000 s
% 0.20/0.72 % SZS output end Proof
% 0.20/0.72 % Total time : 0.148000 s
%------------------------------------------------------------------------------