TSTP Solution File: SEU053+1 by CSE_E---1.5
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- Process Solution
%------------------------------------------------------------------------------
% File : CSE_E---1.5
% Problem : SEU053+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -jar /export/starexec/sandbox/solver/bin/mcs_scs.jar %d %s
% Computer : n020.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 16:22:16 EDT 2023
% Result : Theorem 9.06s 9.11s
% Output : CNFRefutation 9.06s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 46
% Syntax : Number of formulae : 107 ( 32 unt; 32 typ; 0 def)
% Number of atoms : 186 ( 61 equ)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 183 ( 72 ~; 74 |; 21 &)
% ( 4 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 3 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 31 ( 22 >; 9 *; 0 +; 0 <<)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-2 aty)
% Number of functors : 22 ( 22 usr; 10 con; 0-2 aty)
% Number of variables : 100 ( 5 sgn; 57 !; 1 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
tff(decl_22,type,
in: ( $i * $i ) > $o ).
tff(decl_23,type,
empty: $i > $o ).
tff(decl_24,type,
function: $i > $o ).
tff(decl_25,type,
relation: $i > $o ).
tff(decl_26,type,
one_to_one: $i > $o ).
tff(decl_27,type,
set_intersection2: ( $i * $i ) > $i ).
tff(decl_28,type,
singleton: $i > $i ).
tff(decl_29,type,
disjoint: ( $i * $i ) > $o ).
tff(decl_30,type,
empty_set: $i ).
tff(decl_31,type,
relation_dom: $i > $i ).
tff(decl_32,type,
apply: ( $i * $i ) > $i ).
tff(decl_33,type,
element: ( $i * $i ) > $o ).
tff(decl_34,type,
relation_empty_yielding: $i > $o ).
tff(decl_35,type,
powerset: $i > $i ).
tff(decl_36,type,
disjoint_nonempty: ( $i * $i ) > $o ).
tff(decl_37,type,
subset: ( $i * $i ) > $o ).
tff(decl_38,type,
relation_image: ( $i * $i ) > $i ).
tff(decl_39,type,
esk1_2: ( $i * $i ) > $i ).
tff(decl_40,type,
esk2_1: $i > $i ).
tff(decl_41,type,
esk3_1: $i > $i ).
tff(decl_42,type,
esk4_1: $i > $i ).
tff(decl_43,type,
esk5_0: $i ).
tff(decl_44,type,
esk6_0: $i ).
tff(decl_45,type,
esk7_1: $i > $i ).
tff(decl_46,type,
esk8_0: $i ).
tff(decl_47,type,
esk9_0: $i ).
tff(decl_48,type,
esk10_0: $i ).
tff(decl_49,type,
esk11_1: $i > $i ).
tff(decl_50,type,
esk12_0: $i ).
tff(decl_51,type,
esk13_0: $i ).
tff(decl_52,type,
esk14_0: $i ).
tff(decl_53,type,
esk15_0: $i ).
fof(t122_funct_1,conjecture,
! [X1] :
( ( relation(X1)
& function(X1) )
=> ( ! [X2,X3] : relation_image(X1,set_intersection2(X2,X3)) = set_intersection2(relation_image(X1,X2),relation_image(X1,X3))
=> one_to_one(X1) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t122_funct_1) ).
fof(d8_funct_1,axiom,
! [X1] :
( ( relation(X1)
& function(X1) )
=> ( one_to_one(X1)
<=> ! [X2,X3] :
( ( in(X2,relation_dom(X1))
& in(X3,relation_dom(X1))
& apply(X1,X2) = apply(X1,X3) )
=> X2 = X3 ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d8_funct_1) ).
fof(t5_subset,axiom,
! [X1,X2,X3] :
~ ( in(X1,X2)
& element(X2,powerset(X3))
& empty(X3) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t5_subset) ).
fof(existence_m1_subset_1,axiom,
! [X1] :
? [X2] : element(X2,X1),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_m1_subset_1) ).
fof(t2_subset,axiom,
! [X1,X2] :
( element(X1,X2)
=> ( empty(X2)
| in(X1,X2) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t2_subset) ).
fof(d1_tarski,axiom,
! [X1,X2] :
( X2 = singleton(X1)
<=> ! [X3] :
( in(X3,X2)
<=> X3 = X1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1_tarski) ).
fof(t117_funct_1,axiom,
! [X1,X2] :
( ( relation(X2)
& function(X2) )
=> ( in(X1,relation_dom(X2))
=> relation_image(X2,singleton(X1)) = singleton(apply(X2,X1)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t117_funct_1) ).
fof(fc2_subset_1,axiom,
! [X1] : ~ empty(singleton(X1)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc2_subset_1) ).
fof(d7_xboole_0,axiom,
! [X1,X2] :
( disjoint(X1,X2)
<=> set_intersection2(X1,X2) = empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d7_xboole_0) ).
fof(t17_zfmisc_1,axiom,
! [X1,X2] :
( X1 != X2
=> disjoint(singleton(X1),singleton(X2)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_zfmisc_1) ).
fof(t149_relat_1,axiom,
! [X1] :
( relation(X1)
=> relation_image(X1,empty_set) = empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t149_relat_1) ).
fof(t6_boole,axiom,
! [X1] :
( empty(X1)
=> X1 = empty_set ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6_boole) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fc12_relat_1) ).
fof(idempotence_k3_xboole_0,axiom,
! [X1,X2] : set_intersection2(X1,X1) = X1,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',idempotence_k3_xboole_0) ).
fof(c_0_14,negated_conjecture,
~ ! [X1] :
( ( relation(X1)
& function(X1) )
=> ( ! [X2,X3] : relation_image(X1,set_intersection2(X2,X3)) = set_intersection2(relation_image(X1,X2),relation_image(X1,X3))
=> one_to_one(X1) ) ),
inference(assume_negation,[status(cth)],[t122_funct_1]) ).
fof(c_0_15,plain,
! [X20,X21,X22] :
( ( ~ one_to_one(X20)
| ~ in(X21,relation_dom(X20))
| ~ in(X22,relation_dom(X20))
| apply(X20,X21) != apply(X20,X22)
| X21 = X22
| ~ relation(X20)
| ~ function(X20) )
& ( in(esk2_1(X20),relation_dom(X20))
| one_to_one(X20)
| ~ relation(X20)
| ~ function(X20) )
& ( in(esk3_1(X20),relation_dom(X20))
| one_to_one(X20)
| ~ relation(X20)
| ~ function(X20) )
& ( apply(X20,esk2_1(X20)) = apply(X20,esk3_1(X20))
| one_to_one(X20)
| ~ relation(X20)
| ~ function(X20) )
& ( esk2_1(X20) != esk3_1(X20)
| one_to_one(X20)
| ~ relation(X20)
| ~ function(X20) ) ),
inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[d8_funct_1])])])])]) ).
fof(c_0_16,negated_conjecture,
! [X58,X59] :
( relation(esk15_0)
& function(esk15_0)
& relation_image(esk15_0,set_intersection2(X58,X59)) = set_intersection2(relation_image(esk15_0,X58),relation_image(esk15_0,X59))
& ~ one_to_one(esk15_0) ),
inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_14])])])]) ).
fof(c_0_17,plain,
! [X73,X74,X75] :
( ~ in(X73,X74)
| ~ element(X74,powerset(X75))
| ~ empty(X75) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t5_subset])]) ).
fof(c_0_18,plain,
! [X25] : element(esk4_1(X25),X25),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[existence_m1_subset_1])]) ).
fof(c_0_19,plain,
! [X66,X67] :
( ~ element(X66,X67)
| empty(X67)
| in(X66,X67) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t2_subset])]) ).
fof(c_0_20,plain,
! [X11,X12,X13,X14,X15,X16] :
( ( ~ in(X13,X12)
| X13 = X11
| X12 != singleton(X11) )
& ( X14 != X11
| in(X14,X12)
| X12 != singleton(X11) )
& ( ~ in(esk1_2(X15,X16),X16)
| esk1_2(X15,X16) != X15
| X16 = singleton(X15) )
& ( in(esk1_2(X15,X16),X16)
| esk1_2(X15,X16) = X15
| X16 = singleton(X15) ) ),
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)],[d1_tarski])])])])])]) ).
fof(c_0_21,plain,
! [X55,X56] :
( ~ relation(X56)
| ~ function(X56)
| ~ in(X55,relation_dom(X56))
| relation_image(X56,singleton(X55)) = singleton(apply(X56,X55)) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t117_funct_1])]) ).
cnf(c_0_22,plain,
( in(esk2_1(X1),relation_dom(X1))
| one_to_one(X1)
| ~ relation(X1)
| ~ function(X1) ),
inference(split_conjunct,[status(thm)],[c_0_15]) ).
cnf(c_0_23,negated_conjecture,
relation(esk15_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_24,negated_conjecture,
function(esk15_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_25,negated_conjecture,
~ one_to_one(esk15_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
fof(c_0_26,plain,
! [X1] : ~ empty(singleton(X1)),
inference(fof_simplification,[status(thm)],[fc2_subset_1]) ).
fof(c_0_27,plain,
! [X18,X19] :
( ( ~ disjoint(X18,X19)
| set_intersection2(X18,X19) = empty_set )
& ( set_intersection2(X18,X19) != empty_set
| disjoint(X18,X19) ) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[d7_xboole_0])]) ).
fof(c_0_28,plain,
! [X61,X62] :
( X61 = X62
| disjoint(singleton(X61),singleton(X62)) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t17_zfmisc_1])]) ).
fof(c_0_29,plain,
! [X60] :
( ~ relation(X60)
| relation_image(X60,empty_set) = empty_set ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t149_relat_1])]) ).
cnf(c_0_30,plain,
( ~ in(X1,X2)
| ~ element(X2,powerset(X3))
| ~ empty(X3) ),
inference(split_conjunct,[status(thm)],[c_0_17]) ).
cnf(c_0_31,plain,
element(esk4_1(X1),X1),
inference(split_conjunct,[status(thm)],[c_0_18]) ).
cnf(c_0_32,plain,
( empty(X2)
| in(X1,X2)
| ~ element(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_19]) ).
cnf(c_0_33,plain,
( X1 = X3
| ~ in(X1,X2)
| X2 != singleton(X3) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
cnf(c_0_34,plain,
( relation_image(X1,singleton(X2)) = singleton(apply(X1,X2))
| ~ relation(X1)
| ~ function(X1)
| ~ in(X2,relation_dom(X1)) ),
inference(split_conjunct,[status(thm)],[c_0_21]) ).
cnf(c_0_35,negated_conjecture,
in(esk2_1(esk15_0),relation_dom(esk15_0)),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_22,c_0_23]),c_0_24])]),c_0_25]) ).
fof(c_0_36,plain,
! [X30] : ~ empty(singleton(X30)),
inference(variable_rename,[status(thm)],[c_0_26]) ).
cnf(c_0_37,plain,
( set_intersection2(X1,X2) = empty_set
| ~ disjoint(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_27]) ).
cnf(c_0_38,plain,
( X1 = X2
| disjoint(singleton(X1),singleton(X2)) ),
inference(split_conjunct,[status(thm)],[c_0_28]) ).
cnf(c_0_39,plain,
( relation_image(X1,empty_set) = empty_set
| ~ relation(X1) ),
inference(split_conjunct,[status(thm)],[c_0_29]) ).
cnf(c_0_40,plain,
( in(esk3_1(X1),relation_dom(X1))
| one_to_one(X1)
| ~ relation(X1)
| ~ function(X1) ),
inference(split_conjunct,[status(thm)],[c_0_15]) ).
cnf(c_0_41,plain,
( apply(X1,esk2_1(X1)) = apply(X1,esk3_1(X1))
| one_to_one(X1)
| ~ relation(X1)
| ~ function(X1) ),
inference(split_conjunct,[status(thm)],[c_0_15]) ).
cnf(c_0_42,plain,
( ~ empty(X1)
| ~ in(X2,esk4_1(powerset(X1))) ),
inference(spm,[status(thm)],[c_0_30,c_0_31]) ).
cnf(c_0_43,plain,
( empty(X1)
| in(esk4_1(X1),X1) ),
inference(spm,[status(thm)],[c_0_32,c_0_31]) ).
cnf(c_0_44,plain,
( in(X1,X3)
| X1 != X2
| X3 != singleton(X2) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
cnf(c_0_45,plain,
( X1 = X2
| ~ in(X1,singleton(X2)) ),
inference(er,[status(thm)],[c_0_33]) ).
cnf(c_0_46,negated_conjecture,
singleton(apply(esk15_0,esk2_1(esk15_0))) = relation_image(esk15_0,singleton(esk2_1(esk15_0))),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_34,c_0_35]),c_0_23]),c_0_24])]) ).
cnf(c_0_47,plain,
~ empty(singleton(X1)),
inference(split_conjunct,[status(thm)],[c_0_36]) ).
cnf(c_0_48,negated_conjecture,
relation_image(esk15_0,set_intersection2(X1,X2)) = set_intersection2(relation_image(esk15_0,X1),relation_image(esk15_0,X2)),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_49,plain,
( set_intersection2(singleton(X1),singleton(X2)) = empty_set
| X1 = X2 ),
inference(spm,[status(thm)],[c_0_37,c_0_38]) ).
cnf(c_0_50,negated_conjecture,
relation_image(esk15_0,empty_set) = empty_set,
inference(spm,[status(thm)],[c_0_39,c_0_23]) ).
cnf(c_0_51,negated_conjecture,
in(esk3_1(esk15_0),relation_dom(esk15_0)),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_40,c_0_23]),c_0_24])]),c_0_25]) ).
cnf(c_0_52,negated_conjecture,
apply(esk15_0,esk3_1(esk15_0)) = apply(esk15_0,esk2_1(esk15_0)),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_41,c_0_23]),c_0_24])]),c_0_25]) ).
fof(c_0_53,plain,
! [X76] :
( ~ empty(X76)
| X76 = empty_set ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t6_boole])]) ).
cnf(c_0_54,plain,
( empty(esk4_1(powerset(X1)))
| ~ empty(X1) ),
inference(spm,[status(thm)],[c_0_42,c_0_43]) ).
cnf(c_0_55,plain,
empty(empty_set),
inference(split_conjunct,[status(thm)],[fc12_relat_1]) ).
cnf(c_0_56,plain,
in(X1,singleton(X1)),
inference(er,[status(thm)],[inference(er,[status(thm)],[c_0_44])]) ).
cnf(c_0_57,negated_conjecture,
( X1 = apply(esk15_0,esk2_1(esk15_0))
| ~ in(X1,relation_image(esk15_0,singleton(esk2_1(esk15_0)))) ),
inference(spm,[status(thm)],[c_0_45,c_0_46]) ).
cnf(c_0_58,negated_conjecture,
~ empty(relation_image(esk15_0,singleton(esk2_1(esk15_0)))),
inference(spm,[status(thm)],[c_0_47,c_0_46]) ).
fof(c_0_59,plain,
! [X33] : set_intersection2(X33,X33) = X33,
inference(variable_rename,[status(thm)],[inference(fof_simplification,[status(thm)],[idempotence_k3_xboole_0])]) ).
cnf(c_0_60,negated_conjecture,
( set_intersection2(relation_image(esk15_0,singleton(X1)),relation_image(esk15_0,singleton(X2))) = empty_set
| X1 = X2 ),
inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_48,c_0_49]),c_0_50]) ).
cnf(c_0_61,negated_conjecture,
relation_image(esk15_0,singleton(esk3_1(esk15_0))) = relation_image(esk15_0,singleton(esk2_1(esk15_0))),
inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_34,c_0_51]),c_0_23]),c_0_24])]),c_0_52]),c_0_46]) ).
cnf(c_0_62,plain,
( one_to_one(X1)
| esk2_1(X1) != esk3_1(X1)
| ~ relation(X1)
| ~ function(X1) ),
inference(split_conjunct,[status(thm)],[c_0_15]) ).
cnf(c_0_63,plain,
( X1 = empty_set
| ~ empty(X1) ),
inference(split_conjunct,[status(thm)],[c_0_53]) ).
cnf(c_0_64,plain,
empty(esk4_1(powerset(empty_set))),
inference(spm,[status(thm)],[c_0_54,c_0_55]) ).
cnf(c_0_65,negated_conjecture,
in(apply(esk15_0,esk2_1(esk15_0)),relation_image(esk15_0,singleton(esk2_1(esk15_0)))),
inference(spm,[status(thm)],[c_0_56,c_0_46]) ).
cnf(c_0_66,negated_conjecture,
apply(esk15_0,esk2_1(esk15_0)) = esk4_1(relation_image(esk15_0,singleton(esk2_1(esk15_0)))),
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_57,c_0_43]),c_0_58]) ).
cnf(c_0_67,plain,
set_intersection2(X1,X1) = X1,
inference(split_conjunct,[status(thm)],[c_0_59]) ).
cnf(c_0_68,negated_conjecture,
( set_intersection2(relation_image(esk15_0,singleton(X1)),relation_image(esk15_0,singleton(esk2_1(esk15_0)))) = empty_set
| X1 = esk3_1(esk15_0) ),
inference(spm,[status(thm)],[c_0_60,c_0_61]) ).
cnf(c_0_69,negated_conjecture,
esk3_1(esk15_0) != esk2_1(esk15_0),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_62,c_0_23]),c_0_24])]),c_0_25]) ).
cnf(c_0_70,plain,
esk4_1(powerset(empty_set)) = empty_set,
inference(spm,[status(thm)],[c_0_63,c_0_64]) ).
cnf(c_0_71,negated_conjecture,
in(esk4_1(relation_image(esk15_0,singleton(esk2_1(esk15_0)))),relation_image(esk15_0,singleton(esk2_1(esk15_0)))),
inference(rw,[status(thm)],[c_0_65,c_0_66]) ).
cnf(c_0_72,negated_conjecture,
relation_image(esk15_0,singleton(esk2_1(esk15_0))) = empty_set,
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_67,c_0_68]),c_0_69]) ).
cnf(c_0_73,plain,
~ in(X1,empty_set),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_42,c_0_70]),c_0_55])]) ).
cnf(c_0_74,negated_conjecture,
$false,
inference(sr,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_71,c_0_72]),c_0_72]),c_0_73]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU053+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.12 % Command : java -jar /export/starexec/sandbox/solver/bin/mcs_scs.jar %d %s
% 0.12/0.33 % Computer : n020.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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 23 13:49:30 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.56 start to proof: theBenchmark
% 9.06/9.11 % Version : CSE_E---1.5
% 9.06/9.11 % Problem : theBenchmark.p
% 9.06/9.11 % Proof found
% 9.06/9.11 % SZS status Theorem for theBenchmark.p
% 9.06/9.11 % SZS output start Proof
% See solution above
% 9.06/9.12 % Total time : 8.545000 s
% 9.06/9.12 % SZS output end Proof
% 9.06/9.12 % Total time : 8.549000 s
%------------------------------------------------------------------------------