%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : MGT026+1 : TPTP v9.3.1. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:02:10 AM UTC 2026
% Result : Theorem 0.10s 0.40s
% Output : CNFRefutation 0.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 30
% Number of leaves : 11
% Syntax : Number of formulae : 84 ( 16 unt; 0 def)
% Number of atoms : 231 ( 60 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 225 ( 78 ~; 106 |; 28 &)
% ( 1 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 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 : 77 ( 0 sgn 29 !; 2 ?)
% Comments :
%------------------------------------------------------------------------------
fof(mp1_high_growth_rates,axiom,
! [E,S1,S2,T] :
( ( greater(growth_rate(S2,T),growth_rate(S1,T))
& subpopulations(S1,S2,E,T)
& environment(E) )
=> selection_favors(S2,S1,T) ),
file('theBenchmark.p',mp1_high_growth_rates) ).
fof(mp2_favour_members,axiom,
! [E,S1,S2,T] :
( ( cardinality_at_time(S2,T) = zero
& greater(cardinality_at_time(S1,T),zero)
& subpopulation(S2,E,T)
& subpopulation(S1,E,T)
& environment(E) )
=> selection_favors(S1,S2,T) ),
file('theBenchmark.p',mp2_favour_members) ).
fof(mp_non_empty_fm_and_ep,axiom,
! [E,T] :
( ( greater(cardinality_at_time(efficient_producers,T),zero)
& greater(cardinality_at_time(first_movers,T),zero)
& in_environment(E,T)
& environment(E) )
=> subpopulations(first_movers,efficient_producers,E,T) ),
file('theBenchmark.p',mp_non_empty_fm_and_ep) ).
fof(mp_first_movers_exist,axiom,
! [E,T] :
( ( in_environment(E,T)
& environment(E) )
=> greater_or_equal(cardinality_at_time(first_movers,T),zero) ),
file('theBenchmark.p',mp_first_movers_exist) ).
fof(mp_subpopulations,axiom,
! [E,T] :
( ( in_environment(E,T)
& environment(E) )
=> ( subpopulation(efficient_producers,E,T)
& subpopulation(first_movers,E,T) ) ),
file('theBenchmark.p',mp_subpopulations) ).
fof(mp_critical_point_after_EP,axiom,
! [E] :
( environment(E)
=> greater_or_equal(critical_point(E),appear(efficient_producers,E)) ),
file('theBenchmark.p',mp_critical_point_after_EP) ).
fof(mp_greater_transitivity,axiom,
! [X,Y,Z] :
( ( greater(Y,Z)
& greater(X,Y) )
=> greater(X,Z) ),
file('theBenchmark.p',mp_greater_transitivity) ).
fof(mp_greater_or_equal,axiom,
! [X,Y] :
( greater_or_equal(X,Y)
<=> ( X = Y
| greater(X,Y) ) ),
file('theBenchmark.p',mp_greater_or_equal) ).
fof(d1,axiom,
! [E,Tc] :
( ( Tc = critical_point(E)
& environment(E) )
=> ( ! [T] :
( ( greater(T,Tc)
& subpopulations(first_movers,efficient_producers,E,T) )
=> greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T)) )
& ~ greater(growth_rate(efficient_producers,Tc),growth_rate(first_movers,Tc)) ) ),
file('theBenchmark.p',d1) ).
fof(t6,axiom,
! [E,T] :
( ( greater_or_equal(T,appear(efficient_producers,E))
& in_environment(E,T)
& environment(E) )
=> greater(cardinality_at_time(efficient_producers,T),zero) ),
file('theBenchmark.p',t6) ).
fof(prove_l8,conjecture,
! [E,T] :
( ( greater(T,critical_point(E))
& in_environment(E,T)
& environment(E) )
=> selection_favors(efficient_producers,first_movers,T) ),
file('theBenchmark.p',prove_l8) ).
fof(prove_l8_neg,negated_conjecture,
~ ! [E,T] :
( ( greater(T,critical_point(E))
& in_environment(E,T)
& environment(E) )
=> selection_favors(efficient_producers,first_movers,T) ),
inference(assume_negation,[status(cth)],[prove_l8]) ).
fof(nnf_11,plain,
? [E,T] :
( ~ selection_favors(efficient_producers,first_movers,T)
& greater(T,critical_point(E))
& in_environment(E,T)
& environment(E) ),
inference(fof_nnf,[status(thm)],[prove_l8_neg]) ).
fof(sk_prove_l8_neg,plain,
( ~ selection_favors(efficient_producers,first_movers,sK1)
& greater(sK1,critical_point(sK0))
& in_environment(sK0,sK1)
& environment(sK0) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(E,sK0),skolemize(T,sK1)],[nnf_11]) ).
cnf(c_12,negated_conjecture,
environment(sK0),
inference(split_conjunct,[status(thm)],[sk_prove_l8_neg]) ).
cnf(c_13,plain,
( greater(cardinality_at_time(efficient_producers,B),zero)
| ~ greater_or_equal(B,appear(efficient_producers,A))
| ~ in_environment(A,B)
| ~ environment(A) ),
inference(clausify,[status(thm)],[t6]) ).
cnf(c_14,plain,
( greater(growth_rate(efficient_producers,C),growth_rate(first_movers,C))
| ~ greater(C,B)
| ~ subpopulations(first_movers,efficient_producers,A,C)
| critical_point(A) != B
| ~ environment(A) ),
inference(clausify,[status(thm)],[d1]) ).
cnf(c_16,plain,
( greater_or_equal(critical_point(A),appear(efficient_producers,A))
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp_critical_point_after_EP]) ).
cnf(c_17,plain,
( subpopulation(efficient_producers,A,B)
| ~ in_environment(A,B)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp_subpopulations]) ).
cnf(c_18,plain,
( subpopulation(first_movers,A,B)
| ~ in_environment(A,B)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp_subpopulations]) ).
cnf(c_19,plain,
( greater_or_equal(cardinality_at_time(first_movers,B),zero)
| ~ in_environment(A,B)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp_first_movers_exist]) ).
cnf(c_20,plain,
( subpopulations(first_movers,efficient_producers,A,B)
| ~ greater(cardinality_at_time(efficient_producers,B),zero)
| ~ greater(cardinality_at_time(first_movers,B),zero)
| ~ in_environment(A,B)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp_non_empty_fm_and_ep]) ).
cnf(c_21,plain,
( selection_favors(B,D,C)
| cardinality_at_time(D,C) != zero
| ~ greater(cardinality_at_time(B,C),zero)
| ~ subpopulation(D,A,C)
| ~ subpopulation(B,A,C)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp2_favour_members]) ).
cnf(c_22,plain,
( selection_favors(C,B,D)
| ~ greater(growth_rate(C,D),growth_rate(B,D))
| ~ subpopulations(B,C,A,D)
| ~ environment(A) ),
inference(clausify,[status(thm)],[mp1_high_growth_rates]) ).
cnf(c_23,negated_conjecture,
( selection_favors(B,A,C)
| ~ greater(growth_rate(B,C),growth_rate(A,C))
| ~ subpopulations(A,B,sK0,C) ),
inference(resolve,[status(thm)],[c_12,c_22]) ).
cnf(c_24,negated_conjecture,
( selection_favors(A,C,B)
| cardinality_at_time(C,B) != zero
| ~ greater(cardinality_at_time(A,B),zero)
| ~ subpopulation(C,sK0,B)
| ~ subpopulation(A,sK0,B) ),
inference(resolve,[status(thm)],[c_12,c_21]) ).
cnf(c_25,negated_conjecture,
( subpopulations(first_movers,efficient_producers,sK0,A)
| ~ greater(cardinality_at_time(efficient_producers,A),zero)
| ~ greater(cardinality_at_time(first_movers,A),zero)
| ~ in_environment(sK0,A) ),
inference(resolve,[status(thm)],[c_12,c_20]) ).
cnf(c_26,negated_conjecture,
( greater_or_equal(cardinality_at_time(first_movers,A),zero)
| ~ in_environment(sK0,A) ),
inference(resolve,[status(thm)],[c_12,c_19]) ).
cnf(c_27,negated_conjecture,
( subpopulation(first_movers,sK0,A)
| ~ in_environment(sK0,A) ),
inference(resolve,[status(thm)],[c_12,c_18]) ).
cnf(c_28,negated_conjecture,
( subpopulation(efficient_producers,sK0,A)
| ~ in_environment(sK0,A) ),
inference(resolve,[status(thm)],[c_12,c_17]) ).
cnf(c_29,negated_conjecture,
greater_or_equal(critical_point(sK0),appear(efficient_producers,sK0)),
inference(resolve,[status(thm)],[c_12,c_16]) ).
cnf(c_31,negated_conjecture,
( greater(growth_rate(efficient_producers,B),growth_rate(first_movers,B))
| ~ greater(B,A)
| ~ subpopulations(first_movers,efficient_producers,sK0,B)
| critical_point(sK0) != A ),
inference(resolve,[status(thm)],[c_12,c_14]) ).
cnf(c_32,negated_conjecture,
( greater(cardinality_at_time(efficient_producers,A),zero)
| ~ greater_or_equal(A,appear(efficient_producers,sK0))
| ~ in_environment(sK0,A) ),
inference(resolve,[status(thm)],[c_12,c_13]) ).
cnf(c_33,negated_conjecture,
~ selection_favors(efficient_producers,first_movers,sK1),
inference(split_conjunct,[status(thm)],[sk_prove_l8_neg]) ).
cnf(c_34,negated_conjecture,
greater(sK1,critical_point(sK0)),
inference(split_conjunct,[status(thm)],[sk_prove_l8_neg]) ).
cnf(c_35,negated_conjecture,
in_environment(sK0,sK1),
inference(split_conjunct,[status(thm)],[sk_prove_l8_neg]) ).
cnf(c_37,plain,
( ~ greater(A,B)
| greater_or_equal(A,B) ),
inference(clausify,[status(thm)],[mp_greater_or_equal]) ).
cnf(c_38,plain,
( B = A
| greater(A,B)
| ~ greater_or_equal(A,B) ),
inference(clausify,[status(thm)],[mp_greater_or_equal]) ).
cnf(c_39,plain,
( greater(A,C)
| ~ greater(B,C)
| ~ greater(A,B) ),
inference(clausify,[status(thm)],[mp_greater_transitivity]) ).
cnf(c_42,plain,
( greater(cardinality_at_time(efficient_producers,sK1),zero)
| ~ greater_or_equal(sK1,appear(efficient_producers,sK0)) ),
inference(resolve,[status(thm)],[c_35,c_32]) ).
cnf(c_43,plain,
subpopulation(efficient_producers,sK0,sK1),
inference(resolve,[status(thm)],[c_35,c_28]) ).
cnf(c_44,plain,
subpopulation(first_movers,sK0,sK1),
inference(resolve,[status(thm)],[c_35,c_27]) ).
cnf(c_45,plain,
greater_or_equal(cardinality_at_time(first_movers,sK1),zero),
inference(resolve,[status(thm)],[c_35,c_26]) ).
cnf(c_46,plain,
( subpopulations(first_movers,efficient_producers,sK0,sK1)
| ~ greater(cardinality_at_time(efficient_producers,sK1),zero)
| ~ greater(cardinality_at_time(first_movers,sK1),zero) ),
inference(resolve,[status(thm)],[c_35,c_25]) ).
cnf(c_48,plain,
greater_or_equal(sK1,critical_point(sK0)),
inference(resolve,[status(thm)],[c_37,c_34]) ).
cnf(c_114,negated_conjecture,
( appear(efficient_producers,sK0) = critical_point(sK0)
| greater(critical_point(sK0),appear(efficient_producers,sK0)) ),
inference(resolve,[status(thm)],[c_38,c_29]) ).
cnf(c_49,negated_conjecture,
( critical_point(sK0) = appear(efficient_producers,sK0)
| greater(critical_point(sK0),appear(efficient_producers,sK0)) ),
inference(copy,[status(thm)],[c_114]) ).
cnf(c_50,plain,
( greater(sK1,A)
| ~ greater(critical_point(sK0),A) ),
inference(resolve,[status(thm)],[c_39,c_34]) ).
cnf(c_54,plain,
( selection_favors(efficient_producers,A,sK1)
| cardinality_at_time(A,sK1) != zero
| ~ greater(cardinality_at_time(efficient_producers,sK1),zero)
| ~ subpopulation(A,sK0,sK1) ),
inference(resolve,[status(thm)],[c_43,c_24]) ).
cnf(c_115,plain,
( zero = cardinality_at_time(first_movers,sK1)
| greater(cardinality_at_time(first_movers,sK1),zero) ),
inference(resolve,[status(thm)],[c_45,c_38]) ).
cnf(c_58,plain,
( cardinality_at_time(first_movers,sK1) = zero
| greater(cardinality_at_time(first_movers,sK1),zero) ),
inference(copy,[status(thm)],[c_115]) ).
cnf(c_59,plain,
( greater(sK1,appear(efficient_producers,sK0))
| critical_point(sK0) = appear(efficient_producers,sK0) ),
inference(resolve,[status(thm)],[c_49,c_50]) ).
cnf(c_62,plain,
( subpopulations(first_movers,efficient_producers,sK0,sK1)
| ~ greater(cardinality_at_time(efficient_producers,sK1),zero)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_58,c_46]) ).
cnf(c_67,plain,
( greater_or_equal(sK1,appear(efficient_producers,sK0))
| critical_point(sK0) = appear(efficient_producers,sK0) ),
inference(resolve,[status(thm)],[c_59,c_37]) ).
cnf(c_68,plain,
( greater(cardinality_at_time(efficient_producers,sK1),zero)
| critical_point(sK0) = appear(efficient_producers,sK0) ),
inference(resolve,[status(thm)],[c_67,c_42]) ).
cnf(c_116,plain,
( selection_favors(efficient_producers,first_movers,sK1)
| cardinality_at_time(first_movers,sK1) != zero
| ~ greater(cardinality_at_time(efficient_producers,sK1),zero) ),
inference(resolve,[status(thm)],[c_54,c_44]) ).
cnf(c_74,negated_conjecture,
( cardinality_at_time(first_movers,sK1) != zero
| ~ greater(cardinality_at_time(efficient_producers,sK1),zero) ),
inference(resolve,[status(thm)],[c_33,c_116]) ).
cnf(c_75,plain,
( critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) != zero ),
inference(resolve,[status(thm)],[c_74,c_68]) ).
cnf(c_77,plain,
( critical_point(sK0) = appear(efficient_producers,sK0)
| subpopulations(first_movers,efficient_producers,sK0,sK1)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_62,c_68]) ).
cnf(c_78,plain,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| ~ greater(sK1,A)
| critical_point(sK0) != A
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_77,c_31]) ).
cnf(c_117,negated_conjecture,
( selection_favors(efficient_producers,first_movers,sK1)
| ~ greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_77,c_23]) ).
cnf(c_79,negated_conjecture,
( ~ greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_33,c_117]) ).
cnf(c_118,negated_conjecture,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| critical_point(sK0) != critical_point(sK0)
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_78,c_34]) ).
cnf(c_80,negated_conjecture,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(copy,[status(thm)],[c_118]) ).
cnf(c_119,plain,
( critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_80,c_79]) ).
cnf(c_120,plain,
( critical_point(sK0) = appear(efficient_producers,sK0)
| critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(copy,[status(thm)],[c_119]) ).
cnf(c_81,plain,
( critical_point(sK0) = appear(efficient_producers,sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(copy,[status(thm)],[c_120]) ).
cnf(c_86,plain,
( greater_or_equal(sK1,appear(efficient_producers,sK0))
| cardinality_at_time(first_movers,sK1) = zero ),
inference(paramod,[status(thm)],[c_81,c_48]) ).
cnf(c_89,plain,
( greater(cardinality_at_time(efficient_producers,sK1),zero)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_86,c_42]) ).
cnf(c_121,plain,
( subpopulations(first_movers,efficient_producers,sK0,sK1)
| cardinality_at_time(first_movers,sK1) = zero
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_89,c_62]) ).
cnf(c_90,plain,
( subpopulations(first_movers,efficient_producers,sK0,sK1)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(copy,[status(thm)],[c_121]) ).
cnf(c_95,plain,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| ~ greater(sK1,A)
| critical_point(sK0) != A
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_90,c_31]) ).
cnf(c_122,negated_conjecture,
( selection_favors(efficient_producers,first_movers,sK1)
| ~ greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_90,c_23]) ).
cnf(c_96,negated_conjecture,
( ~ greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_33,c_122]) ).
cnf(c_123,negated_conjecture,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| critical_point(sK0) != critical_point(sK0)
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_95,c_34]) ).
cnf(c_99,negated_conjecture,
( greater(growth_rate(efficient_producers,sK1),growth_rate(first_movers,sK1))
| cardinality_at_time(first_movers,sK1) = zero ),
inference(copy,[status(thm)],[c_123]) ).
cnf(c_124,plain,
( cardinality_at_time(first_movers,sK1) = zero
| cardinality_at_time(first_movers,sK1) = zero ),
inference(resolve,[status(thm)],[c_99,c_96]) ).
cnf(c_100,plain,
cardinality_at_time(first_movers,sK1) = zero,
inference(copy,[status(thm)],[c_124]) ).
cnf(c_101,plain,
critical_point(sK0) = appear(efficient_producers,sK0),
inference(resolve,[status(thm)],[c_100,c_75]) ).
cnf(c_102,negated_conjecture,
~ greater(cardinality_at_time(efficient_producers,sK1),zero),
inference(resolve,[status(thm)],[c_100,c_74]) ).
cnf(c_105,plain,
~ greater_or_equal(sK1,appear(efficient_producers,sK0)),
inference(resolve,[status(thm)],[c_102,c_42]) ).
cnf(c_106,plain,
~ greater(sK1,appear(efficient_producers,sK0)),
inference(resolve,[status(thm)],[c_37,c_105]) ).
cnf(c_125,plain,
greater(sK1,appear(efficient_producers,sK0)),
inference(paramod,[status(thm)],[c_101,c_34]) ).
cnf(c_113,plain,
$false,
inference(resolve,[status(thm)],[c_106,c_125]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : MGT026+1 : TPTP v9.3.1. Released v2.0.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n011.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sat Sep 26 00:05:00 UTC 2026
% 0.10/0.38 % CPUTime :
% 0.10/0.38 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.10/0.38 % Process 3220472 was started by sandbox2 on n011,
% 0.10/0.38 % Sat Sep 26 00:05:00 2026
% 0.10/0.38 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.10/0.38
% 0.10/0.38 % From the command line: assign(max_seconds, 300).
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% 0.10/0.40 % SZS status Theorem for theBenchmark
% 0.10/0.40
% 0.10/0.40 % Proof 1 at 0.00 (+ 0.00) seconds.
% 0.10/0.40 % Length of proof is 69.
% 0.10/0.40 % Level of proof is 20.
% 0.10/0.40 % Maximum clause weight is 35.000.
% 0.10/0.40 % Given clauses 83.
% 0.10/0.40
% 0.10/0.40 % SZS output start CNFRefutation for theBenchmark
% See solution above
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