%------------------------------------------------------------------------------
% File : CSE_E---1.7
% Problem : MGT026+1 : TPTP v9.2.1. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/lemma_parallel_prover %s --lemma-prover /export/starexec/sandbox/solver/bin/cse --final-prover /export/starexec/sandbox/solver/bin/eprover --proof-time %d --global-time-limit %d
% Computer : n031.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 May 5 04:47:10 PM UTC 2026
% Result : Theorem 1.27s 1.72s
% Output : CNFRefutation 1.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 11
% Syntax : Number of formulae : 61 ( 8 unt; 0 def)
% Number of atoms : 213 ( 21 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 257 ( 105 ~; 105 |; 32 &)
% ( 1 <=>; 14 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-4 aty)
% Number of functors : 9 ( 9 usr; 5 con; 0-2 aty)
% Number of variables : 104 ( 0 sgn 57 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(mp_greater_or_equal,axiom,
! [X5,X6] :
( greater_or_equal(X5,X6)
<=> ( greater(X5,X6)
| X5 = X6 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_greater_or_equal) ).
fof(mp_critical_point_after_EP,axiom,
! [X1] :
( environment(X1)
=> greater_or_equal(critical_point(X1),appear(efficient_producers,X1)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_critical_point_after_EP) ).
fof(prove_l8,conjecture,
! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4)
& greater(X4,critical_point(X1)) )
=> selection_favors(efficient_producers,first_movers,X4) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_l8) ).
fof(mp_greater_transitivity,axiom,
! [X5,X6,X7] :
( ( greater(X5,X6)
& greater(X6,X7) )
=> greater(X5,X7) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_greater_transitivity) ).
fof(t6,hypothesis,
! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4)
& greater_or_equal(X4,appear(efficient_producers,X1)) )
=> greater(cardinality_at_time(efficient_producers,X4),zero) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',t6) ).
fof(d1,hypothesis,
! [X1,X8] :
( ( environment(X1)
& X8 = critical_point(X1) )
=> ( ~ greater(growth_rate(efficient_producers,X8),growth_rate(first_movers,X8))
& ! [X4] :
( ( subpopulations(first_movers,efficient_producers,X1,X4)
& greater(X4,X8) )
=> greater(growth_rate(efficient_producers,X4),growth_rate(first_movers,X4)) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d1) ).
fof(mp2_favour_members,axiom,
! [X1,X2,X3,X4] :
( ( environment(X1)
& subpopulation(X2,X1,X4)
& subpopulation(X3,X1,X4)
& greater(cardinality_at_time(X2,X4),zero)
& cardinality_at_time(X3,X4) = zero )
=> selection_favors(X2,X3,X4) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp2_favour_members) ).
fof(mp_first_movers_exist,axiom,
! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4) )
=> greater_or_equal(cardinality_at_time(first_movers,X4),zero) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_first_movers_exist) ).
fof(mp_subpopulations,axiom,
! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4) )
=> ( subpopulation(first_movers,X1,X4)
& subpopulation(efficient_producers,X1,X4) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_subpopulations) ).
fof(mp_non_empty_fm_and_ep,axiom,
! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4)
& greater(cardinality_at_time(first_movers,X4),zero)
& greater(cardinality_at_time(efficient_producers,X4),zero) )
=> subpopulations(first_movers,efficient_producers,X1,X4) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp_non_empty_fm_and_ep) ).
fof(mp1_high_growth_rates,axiom,
! [X1,X2,X3,X4] :
( ( environment(X1)
& subpopulations(X2,X3,X1,X4)
& greater(growth_rate(X3,X4),growth_rate(X2,X4)) )
=> selection_favors(X3,X2,X4) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',mp1_high_growth_rates) ).
fof(c_0_11,plain,
! [X54,X55] :
( ( ~ greater_or_equal(X54,X55)
| greater(X54,X55)
| X54 = X55 )
& ( ~ greater(X54,X55)
| greater_or_equal(X54,X55) )
& ( X54 != X55
| greater_or_equal(X54,X55) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_greater_or_equal])])]) ).
fof(c_0_12,plain,
! [X50] :
( ~ environment(X50)
| greater_or_equal(critical_point(X50),appear(efficient_producers,X50)) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_critical_point_after_EP])]) ).
fof(c_0_13,negated_conjecture,
~ ! [X1,X4] :
( ( environment(X1)
& in_environment(X1,X4)
& greater(X4,critical_point(X1)) )
=> selection_favors(efficient_producers,first_movers,X4) ),
inference(assume_negation,[status(cth)],[prove_l8]) ).
cnf(c_0_14,plain,
( greater(X1,X2)
| X1 = X2
| ~ greater_or_equal(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_11]) ).
cnf(c_0_15,plain,
( greater_or_equal(critical_point(X1),appear(efficient_producers,X1))
| ~ environment(X1) ),
inference(split_conjunct,[status(thm)],[c_0_12]) ).
fof(c_0_16,negated_conjecture,
( environment(esk1_0)
& in_environment(esk1_0,esk2_0)
& greater(esk2_0,critical_point(esk1_0))
& ~ selection_favors(efficient_producers,first_movers,esk2_0) ),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_13])])]) ).
fof(c_0_17,plain,
! [X51,X52,X53] :
( ~ greater(X51,X52)
| ~ greater(X52,X53)
| greater(X51,X53) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_greater_transitivity])]) ).
cnf(c_0_18,plain,
( appear(efficient_producers,X1) = critical_point(X1)
| greater(critical_point(X1),appear(efficient_producers,X1))
| ~ environment(X1) ),
inference(spm,[status(thm)],[c_0_14,c_0_15]) ).
cnf(c_0_19,negated_conjecture,
environment(esk1_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
fof(c_0_20,hypothesis,
! [X59,X60] :
( ~ environment(X59)
| ~ in_environment(X59,X60)
| ~ greater_or_equal(X60,appear(efficient_producers,X59))
| greater(cardinality_at_time(efficient_producers,X60),zero) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[t6])]) ).
cnf(c_0_21,plain,
( greater(X1,X3)
| ~ greater(X1,X2)
| ~ greater(X2,X3) ),
inference(split_conjunct,[status(thm)],[c_0_17]) ).
cnf(c_0_22,negated_conjecture,
( appear(efficient_producers,esk1_0) = critical_point(esk1_0)
| greater(critical_point(esk1_0),appear(efficient_producers,esk1_0)) ),
inference(spm,[status(thm)],[c_0_18,c_0_19]) ).
cnf(c_0_23,hypothesis,
( greater(cardinality_at_time(efficient_producers,X2),zero)
| ~ environment(X1)
| ~ in_environment(X1,X2)
| ~ greater_or_equal(X2,appear(efficient_producers,X1)) ),
inference(split_conjunct,[status(thm)],[c_0_20]) ).
cnf(c_0_24,plain,
( greater_or_equal(X1,X2)
| ~ greater(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_11]) ).
cnf(c_0_25,negated_conjecture,
( appear(efficient_producers,esk1_0) = critical_point(esk1_0)
| greater(X1,appear(efficient_producers,esk1_0))
| ~ greater(X1,critical_point(esk1_0)) ),
inference(spm,[status(thm)],[c_0_21,c_0_22]) ).
cnf(c_0_26,negated_conjecture,
greater(esk2_0,critical_point(esk1_0)),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_27,hypothesis,
( greater(cardinality_at_time(efficient_producers,X1),zero)
| ~ in_environment(X2,X1)
| ~ greater(X1,appear(efficient_producers,X2))
| ~ environment(X2) ),
inference(spm,[status(thm)],[c_0_23,c_0_24]) ).
cnf(c_0_28,negated_conjecture,
( appear(efficient_producers,esk1_0) = critical_point(esk1_0)
| greater(esk2_0,appear(efficient_producers,esk1_0)) ),
inference(spm,[status(thm)],[c_0_25,c_0_26]) ).
cnf(c_0_29,negated_conjecture,
in_environment(esk1_0,esk2_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_30,negated_conjecture,
( appear(efficient_producers,esk1_0) = critical_point(esk1_0)
| greater(cardinality_at_time(efficient_producers,esk2_0),zero) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_27,c_0_28]),c_0_29]),c_0_19])]) ).
fof(c_0_31,hypothesis,
! [X1,X8] :
( ( environment(X1)
& X8 = critical_point(X1) )
=> ( ~ greater(growth_rate(efficient_producers,X8),growth_rate(first_movers,X8))
& ! [X4] :
( ( subpopulations(first_movers,efficient_producers,X1,X4)
& greater(X4,X8) )
=> greater(growth_rate(efficient_producers,X4),growth_rate(first_movers,X4)) ) ) ),
inference(fof_simplification,[status(thm)],[d1]) ).
fof(c_0_32,plain,
! [X40,X41,X42,X43] :
( ~ environment(X40)
| ~ subpopulation(X41,X40,X43)
| ~ subpopulation(X42,X40,X43)
| ~ greater(cardinality_at_time(X41,X43),zero)
| cardinality_at_time(X42,X43) != zero
| selection_favors(X41,X42,X43) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp2_favour_members])]) ).
cnf(c_0_33,hypothesis,
( greater(cardinality_at_time(efficient_producers,esk2_0),zero)
| greater(cardinality_at_time(efficient_producers,X1),zero)
| ~ in_environment(esk1_0,X1)
| ~ greater(X1,critical_point(esk1_0)) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_27,c_0_30]),c_0_19])]) ).
fof(c_0_34,hypothesis,
! [X56,X57,X58] :
( ( ~ greater(growth_rate(efficient_producers,X57),growth_rate(first_movers,X57))
| ~ environment(X56)
| X57 != critical_point(X56) )
& ( ~ subpopulations(first_movers,efficient_producers,X56,X58)
| ~ greater(X58,X57)
| greater(growth_rate(efficient_producers,X58),growth_rate(first_movers,X58))
| ~ environment(X56)
| X57 != critical_point(X56) ) ),
inference(distribute,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_31])])])]) ).
fof(c_0_35,plain,
! [X46,X47] :
( ~ environment(X46)
| ~ in_environment(X46,X47)
| greater_or_equal(cardinality_at_time(first_movers,X47),zero) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_first_movers_exist])]) ).
cnf(c_0_36,plain,
( selection_favors(X2,X4,X3)
| ~ environment(X1)
| ~ subpopulation(X2,X1,X3)
| ~ subpopulation(X4,X1,X3)
| ~ greater(cardinality_at_time(X2,X3),zero)
| cardinality_at_time(X4,X3) != zero ),
inference(split_conjunct,[status(thm)],[c_0_32]) ).
cnf(c_0_37,negated_conjecture,
greater(cardinality_at_time(efficient_producers,esk2_0),zero),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_33,c_0_26]),c_0_29])]) ).
fof(c_0_38,plain,
! [X48,X49] :
( ( subpopulation(first_movers,X48,X49)
| ~ environment(X48)
| ~ in_environment(X48,X49) )
& ( subpopulation(efficient_producers,X48,X49)
| ~ environment(X48)
| ~ in_environment(X48,X49) ) ),
inference(distribute,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_subpopulations])])]) ).
cnf(c_0_39,hypothesis,
( greater(growth_rate(efficient_producers,X2),growth_rate(first_movers,X2))
| ~ subpopulations(first_movers,efficient_producers,X1,X2)
| ~ greater(X2,X3)
| ~ environment(X1)
| X3 != critical_point(X1) ),
inference(split_conjunct,[status(thm)],[c_0_34]) ).
fof(c_0_40,plain,
! [X44,X45] :
( ~ environment(X44)
| ~ in_environment(X44,X45)
| ~ greater(cardinality_at_time(first_movers,X45),zero)
| ~ greater(cardinality_at_time(efficient_producers,X45),zero)
| subpopulations(first_movers,efficient_producers,X44,X45) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp_non_empty_fm_and_ep])]) ).
cnf(c_0_41,plain,
( greater_or_equal(cardinality_at_time(first_movers,X2),zero)
| ~ environment(X1)
| ~ in_environment(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_35]) ).
cnf(c_0_42,negated_conjecture,
( selection_favors(efficient_producers,X1,esk2_0)
| cardinality_at_time(X1,esk2_0) != zero
| ~ subpopulation(efficient_producers,X2,esk2_0)
| ~ subpopulation(X1,X2,esk2_0)
| ~ environment(X2) ),
inference(spm,[status(thm)],[c_0_36,c_0_37]) ).
cnf(c_0_43,plain,
( subpopulation(efficient_producers,X1,X2)
| ~ environment(X1)
| ~ in_environment(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_38]) ).
cnf(c_0_44,hypothesis,
( greater(growth_rate(efficient_producers,X1),growth_rate(first_movers,X1))
| ~ greater(X1,critical_point(X2))
| ~ subpopulations(first_movers,efficient_producers,X2,X1)
| ~ environment(X2) ),
inference(er,[status(thm)],[c_0_39]) ).
cnf(c_0_45,plain,
( subpopulations(first_movers,efficient_producers,X1,X2)
| ~ environment(X1)
| ~ in_environment(X1,X2)
| ~ greater(cardinality_at_time(first_movers,X2),zero)
| ~ greater(cardinality_at_time(efficient_producers,X2),zero) ),
inference(split_conjunct,[status(thm)],[c_0_40]) ).
cnf(c_0_46,negated_conjecture,
greater_or_equal(cardinality_at_time(first_movers,esk2_0),zero),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_41,c_0_29]),c_0_19])]) ).
cnf(c_0_47,negated_conjecture,
( selection_favors(efficient_producers,X1,esk2_0)
| cardinality_at_time(X1,esk2_0) != zero
| ~ in_environment(X2,esk2_0)
| ~ subpopulation(X1,X2,esk2_0)
| ~ environment(X2) ),
inference(spm,[status(thm)],[c_0_42,c_0_43]) ).
cnf(c_0_48,plain,
( subpopulation(first_movers,X1,X2)
| ~ environment(X1)
| ~ in_environment(X1,X2) ),
inference(split_conjunct,[status(thm)],[c_0_38]) ).
cnf(c_0_49,negated_conjecture,
~ selection_favors(efficient_producers,first_movers,esk2_0),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
cnf(c_0_50,hypothesis,
( greater(growth_rate(efficient_producers,X1),growth_rate(first_movers,X1))
| ~ in_environment(X2,X1)
| ~ greater(cardinality_at_time(first_movers,X1),zero)
| ~ greater(cardinality_at_time(efficient_producers,X1),zero)
| ~ greater(X1,critical_point(X2))
| ~ environment(X2) ),
inference(spm,[status(thm)],[c_0_44,c_0_45]) ).
cnf(c_0_51,negated_conjecture,
( cardinality_at_time(first_movers,esk2_0) = zero
| greater(cardinality_at_time(first_movers,esk2_0),zero) ),
inference(spm,[status(thm)],[c_0_14,c_0_46]) ).
cnf(c_0_52,negated_conjecture,
( cardinality_at_time(first_movers,esk2_0) != zero
| ~ in_environment(X1,esk2_0)
| ~ environment(X1) ),
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_47,c_0_48]),c_0_49]) ).
fof(c_0_53,plain,
! [X36,X37,X38,X39] :
( ~ environment(X36)
| ~ subpopulations(X37,X38,X36,X39)
| ~ greater(growth_rate(X38,X39),growth_rate(X37,X39))
| selection_favors(X38,X37,X39) ),
inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[mp1_high_growth_rates])]) ).
cnf(c_0_54,negated_conjecture,
( greater(growth_rate(efficient_producers,esk2_0),growth_rate(first_movers,esk2_0))
| ~ in_environment(X1,esk2_0)
| ~ greater(esk2_0,critical_point(X1))
| ~ environment(X1) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_50,c_0_51]),c_0_37])]),c_0_52]) ).
cnf(c_0_55,plain,
( selection_favors(X3,X2,X4)
| ~ environment(X1)
| ~ subpopulations(X2,X3,X1,X4)
| ~ greater(growth_rate(X3,X4),growth_rate(X2,X4)) ),
inference(split_conjunct,[status(thm)],[c_0_53]) ).
cnf(c_0_56,negated_conjecture,
greater(growth_rate(efficient_producers,esk2_0),growth_rate(first_movers,esk2_0)),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_54,c_0_26]),c_0_29]),c_0_19])]) ).
cnf(c_0_57,negated_conjecture,
( ~ subpopulations(first_movers,efficient_producers,X1,esk2_0)
| ~ environment(X1) ),
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_55,c_0_56]),c_0_49]) ).
cnf(c_0_58,negated_conjecture,
( ~ in_environment(X1,esk2_0)
| ~ greater(cardinality_at_time(first_movers,esk2_0),zero)
| ~ environment(X1) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_57,c_0_45]),c_0_37])]) ).
cnf(c_0_59,negated_conjecture,
( ~ in_environment(X1,esk2_0)
| ~ environment(X1) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_58,c_0_51]),c_0_52]) ).
cnf(c_0_60,negated_conjecture,
$false,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_59,c_0_29]),c_0_19])]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : MGT026+1 : TPTP v9.2.1. Released v2.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/lemma_parallel_prover %s --lemma-prover /export/starexec/sandbox/solver/bin/cse --final-prover /export/starexec/sandbox/solver/bin/eprover --proof-time %d --global-time-limit %d
% 0.14/0.33 % Computer : n031.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Mon May 4 23:29:23 EDT 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.34 % start to proof: theBenchmark
% 1.27/1.72 % Version : CSE_E---1.7
% 1.27/1.72 % Problem : theBenchmark.p
% 1.27/1.72 % SZS status Theorem for theBenchmark.p
% 1.27/1.72 % SZS output start CNFRefutation
% See solution above
% 1.27/1.72 % Total time : 1.371s
%------------------------------------------------------------------------------