%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : MGT023+1 : TPTP v9.3.1. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n016.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 08:02:09 AM UTC 2026
% Result : Theorem 0.11s 5.39s
% Output : CNFRefutation 0.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 3
% Syntax : Number of formulae : 46 ( 8 unt; 0 def)
% Number of atoms : 171 ( 28 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 196 ( 71 ~; 93 |; 26 &)
% ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-4 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 44 ( 0 sgn 25 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(d1,axiom,
! [E,To] :
( ( ! [T] :
( ( greater(T,To)
& subpopulations(first_movers,efficient_producers,E,T) )
=> greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T)) )
& in_environment(E,To)
& ~ greater(growth_rate(efficient_producers,To),growth_rate(first_movers,To))
& environment(E) )
=> To = critical_point(E) ),
file('theBenchmark.p',d1) ).
fof(l12,axiom,
! [E] :
( ( stable(E)
& environment(E) )
=> ? [To] :
( ! [T] :
( ( greater(T,To)
& subpopulations(first_movers,efficient_producers,E,T) )
=> greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T)) )
& ~ greater(growth_rate(efficient_producers,To),growth_rate(first_movers,To))
& in_environment(E,To) ) ),
file('theBenchmark.p',l12) ).
fof(prove_l5,conjecture,
! [E] :
( ( stable(E)
& environment(E) )
=> in_environment(E,critical_point(E)) ),
file('theBenchmark.p',prove_l5) ).
fof(prove_l5_neg,negated_conjecture,
~ ! [E] :
( ( stable(E)
& environment(E) )
=> in_environment(E,critical_point(E)) ),
inference(assume_negation,[status(cth)],[prove_l5]) ).
fof(nnf_3,plain,
? [E] :
( ~ in_environment(E,critical_point(E))
& stable(E)
& environment(E) ),
inference(fof_nnf,[status(thm)],[prove_l5_neg]) ).
fof(sk_prove_l5_neg,plain,
( ~ in_environment(sK0,critical_point(sK0))
& stable(sK0)
& environment(sK0) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(E,sK0)],[nnf_3]) ).
cnf(c_4,negated_conjecture,
environment(sK0),
inference(split_conjunct,[status(thm)],[sk_prove_l5_neg]) ).
fof(nnf_2,plain,
! [E] :
( ? [To] :
( ! [T] :
( greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T))
| ~ greater(T,To)
| ~ subpopulations(first_movers,efficient_producers,E,T) )
& ~ greater(growth_rate(efficient_producers,To),growth_rate(first_movers,To))
& in_environment(E,To) )
| ~ stable(E)
| ~ environment(E) ),
inference(fof_nnf,[status(thm)],[l12]) ).
fof(sk_l12_sk,plain,
! [E] :
( ( ! [T] :
( greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T))
| ~ greater(T,sK1(E))
| ~ subpopulations(first_movers,efficient_producers,E,T) )
& ~ greater(growth_rate(efficient_producers,sK1(E)),growth_rate(first_movers,sK1(E)))
& in_environment(E,sK1(E)) )
| ~ stable(E)
| ~ environment(E) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(To,sK1(E))],[nnf_2]) ).
fof(sk_l12,plain,
( ! [VAR_0,VAR_1] :
( greater(growth_rate(efficient_producers,VAR_1),growth_rate(first_movers,VAR_1))
| ~ greater(VAR_1,sK1(VAR_0))
| ~ subpopulations(first_movers,efficient_producers,VAR_0,VAR_1)
| ~ stable(VAR_0)
| ~ environment(VAR_0) )
& ! [VAR_0] :
( ~ greater(growth_rate(efficient_producers,sK1(VAR_0)),growth_rate(first_movers,sK1(VAR_0)))
| ~ stable(VAR_0)
| ~ environment(VAR_0) )
& ! [VAR_0] :
( in_environment(VAR_0,sK1(VAR_0))
| ~ stable(VAR_0)
| ~ environment(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_l12_sk]) ).
cnf(c_5,plain,
( greater(growth_rate(efficient_producers,B),growth_rate(first_movers,B))
| ~ greater(B,sK1(A))
| ~ subpopulations(first_movers,efficient_producers,A,B)
| ~ stable(A)
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_l12]) ).
cnf(c_6,plain,
( ~ greater(growth_rate(efficient_producers,sK1(A)),growth_rate(first_movers,sK1(A)))
| ~ stable(A)
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_l12]) ).
cnf(c_7,plain,
( in_environment(A,sK1(A))
| ~ stable(A)
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_l12]) ).
fof(nnf_1,plain,
! [E,To] :
( critical_point(E) = To
| ? [T] :
( ~ greater(growth_rate(efficient_producers,T),growth_rate(first_movers,T))
& greater(T,To)
& subpopulations(first_movers,efficient_producers,E,T) )
| ~ in_environment(E,To)
| greater(growth_rate(efficient_producers,To),growth_rate(first_movers,To))
| ~ environment(E) ),
inference(fof_nnf,[status(thm)],[d1]) ).
fof(sk_d1_sk,plain,
! [E,To] :
( critical_point(E) = To
| ( ~ greater(growth_rate(efficient_producers,sK2(E,To)),growth_rate(first_movers,sK2(E,To)))
& greater(sK2(E,To),To)
& subpopulations(first_movers,efficient_producers,E,sK2(E,To)) )
| ~ in_environment(E,To)
| greater(growth_rate(efficient_producers,To),growth_rate(first_movers,To))
| ~ environment(E) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(T,sK2(E,To))],[nnf_1]) ).
fof(sk_d1,plain,
( ! [VAR_0,VAR_1] :
( critical_point(VAR_0) = VAR_1
| ~ greater(growth_rate(efficient_producers,sK2(VAR_0,VAR_1)),growth_rate(first_movers,sK2(VAR_0,VAR_1)))
| ~ in_environment(VAR_0,VAR_1)
| greater(growth_rate(efficient_producers,VAR_1),growth_rate(first_movers,VAR_1))
| ~ environment(VAR_0) )
& ! [VAR_0,VAR_1] :
( critical_point(VAR_0) = VAR_1
| greater(sK2(VAR_0,VAR_1),VAR_1)
| ~ in_environment(VAR_0,VAR_1)
| greater(growth_rate(efficient_producers,VAR_1),growth_rate(first_movers,VAR_1))
| ~ environment(VAR_0) )
& ! [VAR_0,VAR_1] :
( critical_point(VAR_0) = VAR_1
| subpopulations(first_movers,efficient_producers,VAR_0,sK2(VAR_0,VAR_1))
| ~ in_environment(VAR_0,VAR_1)
| greater(growth_rate(efficient_producers,VAR_1),growth_rate(first_movers,VAR_1))
| ~ environment(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_d1_sk]) ).
cnf(c_8,plain,
( critical_point(A) = B
| ~ greater(growth_rate(efficient_producers,sK2(A,B)),growth_rate(first_movers,sK2(A,B)))
| ~ in_environment(A,B)
| greater(growth_rate(efficient_producers,B),growth_rate(first_movers,B))
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_d1]) ).
cnf(c_9,plain,
( critical_point(A) = B
| greater(sK2(A,B),B)
| ~ in_environment(A,B)
| greater(growth_rate(efficient_producers,B),growth_rate(first_movers,B))
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_d1]) ).
cnf(c_10,plain,
( critical_point(A) = B
| subpopulations(first_movers,efficient_producers,A,sK2(A,B))
| ~ in_environment(A,B)
| greater(growth_rate(efficient_producers,B),growth_rate(first_movers,B))
| ~ environment(A) ),
inference(split_conjunct,[status(thm)],[sk_d1]) ).
cnf(c_11,negated_conjecture,
( critical_point(sK0) = A
| subpopulations(first_movers,efficient_producers,sK0,sK2(sK0,A))
| ~ in_environment(sK0,A)
| greater(growth_rate(efficient_producers,A),growth_rate(first_movers,A)) ),
inference(resolve,[status(thm)],[c_4,c_10]) ).
cnf(c_12,negated_conjecture,
( critical_point(sK0) = A
| greater(sK2(sK0,A),A)
| ~ in_environment(sK0,A)
| greater(growth_rate(efficient_producers,A),growth_rate(first_movers,A)) ),
inference(resolve,[status(thm)],[c_4,c_9]) ).
cnf(c_13,negated_conjecture,
( critical_point(sK0) = A
| ~ greater(growth_rate(efficient_producers,sK2(sK0,A)),growth_rate(first_movers,sK2(sK0,A)))
| ~ in_environment(sK0,A)
| greater(growth_rate(efficient_producers,A),growth_rate(first_movers,A)) ),
inference(resolve,[status(thm)],[c_4,c_8]) ).
cnf(c_14,negated_conjecture,
( in_environment(sK0,sK1(sK0))
| ~ stable(sK0) ),
inference(resolve,[status(thm)],[c_4,c_7]) ).
cnf(c_15,negated_conjecture,
( ~ greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0)))
| ~ stable(sK0) ),
inference(resolve,[status(thm)],[c_4,c_6]) ).
cnf(c_16,negated_conjecture,
( greater(growth_rate(efficient_producers,A),growth_rate(first_movers,A))
| ~ greater(A,sK1(sK0))
| ~ subpopulations(first_movers,efficient_producers,sK0,A)
| ~ stable(sK0) ),
inference(resolve,[status(thm)],[c_4,c_5]) ).
cnf(c_17,negated_conjecture,
~ in_environment(sK0,critical_point(sK0)),
inference(split_conjunct,[status(thm)],[sk_prove_l5_neg]) ).
cnf(c_18,negated_conjecture,
stable(sK0),
inference(split_conjunct,[status(thm)],[sk_prove_l5_neg]) ).
cnf(c_19,negated_conjecture,
( greater(growth_rate(efficient_producers,A),growth_rate(first_movers,A))
| ~ greater(A,sK1(sK0))
| ~ subpopulations(first_movers,efficient_producers,sK0,A) ),
inference(resolve,[status(thm)],[c_18,c_16]) ).
cnf(c_20,negated_conjecture,
~ greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))),
inference(resolve,[status(thm)],[c_18,c_15]) ).
cnf(c_21,negated_conjecture,
in_environment(sK0,sK1(sK0)),
inference(resolve,[status(thm)],[c_18,c_14]) ).
cnf(c_29,negated_conjecture,
( critical_point(sK0) = sK1(sK0)
| ~ greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0))))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(resolve,[status(thm)],[c_21,c_13]) ).
cnf(c_30,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| ~ greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0))))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(copy,[status(thm)],[c_29]) ).
cnf(c_22,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| ~ greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0)))) ),
inference(resolve,[status(thm)],[c_20,c_30]) ).
cnf(c_31,negated_conjecture,
( critical_point(sK0) = sK1(sK0)
| greater(sK2(sK0,sK1(sK0)),sK1(sK0))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(resolve,[status(thm)],[c_21,c_12]) ).
cnf(c_32,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| greater(sK2(sK0,sK1(sK0)),sK1(sK0))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(copy,[status(thm)],[c_31]) ).
cnf(c_23,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| greater(sK2(sK0,sK1(sK0)),sK1(sK0)) ),
inference(resolve,[status(thm)],[c_20,c_32]) ).
cnf(c_33,negated_conjecture,
( critical_point(sK0) = sK1(sK0)
| subpopulations(first_movers,efficient_producers,sK0,sK2(sK0,sK1(sK0)))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(resolve,[status(thm)],[c_21,c_11]) ).
cnf(c_34,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| subpopulations(first_movers,efficient_producers,sK0,sK2(sK0,sK1(sK0)))
| greater(growth_rate(efficient_producers,sK1(sK0)),growth_rate(first_movers,sK1(sK0))) ),
inference(copy,[status(thm)],[c_33]) ).
cnf(c_24,negated_conjecture,
( sK1(sK0) = critical_point(sK0)
| subpopulations(first_movers,efficient_producers,sK0,sK2(sK0,sK1(sK0))) ),
inference(resolve,[status(thm)],[c_20,c_34]) ).
cnf(c_25,plain,
( greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0))))
| ~ greater(sK2(sK0,sK1(sK0)),sK1(sK0))
| sK1(sK0) = critical_point(sK0) ),
inference(resolve,[status(thm)],[c_24,c_19]) ).
cnf(c_35,plain,
( sK1(sK0) = critical_point(sK0)
| greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0))))
| sK1(sK0) = critical_point(sK0) ),
inference(resolve,[status(thm)],[c_25,c_23]) ).
cnf(c_26,plain,
( greater(growth_rate(efficient_producers,sK2(sK0,sK1(sK0))),growth_rate(first_movers,sK2(sK0,sK1(sK0))))
| sK1(sK0) = critical_point(sK0) ),
inference(copy,[status(thm)],[c_35]) ).
cnf(c_36,plain,
( sK1(sK0) = critical_point(sK0)
| sK1(sK0) = critical_point(sK0) ),
inference(resolve,[status(thm)],[c_26,c_22]) ).
cnf(c_27,plain,
sK1(sK0) = critical_point(sK0),
inference(copy,[status(thm)],[c_36]) ).
cnf(c_37,negated_conjecture,
in_environment(sK0,critical_point(sK0)),
inference(paramod,[status(thm)],[c_27,c_21]) ).
cnf(c_28,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_17,c_37]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : MGT023+1 : TPTP v9.3.1. Released v2.0.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/5.36 % Computer : n016.cluster.edu
% 0.11/5.36 % Model : x86_64 x86_64
% 0.11/5.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/5.36 % Memory : 8046.5625MB
% 0.11/5.36 % OS : Linux 6.8.0-71-generic
% 0.11/5.36 % CPULimit : 300
% 0.11/5.36 % WCLimit : 300
% 0.11/5.36 % DateTime : Sat Sep 26 00:09:34 UTC 2026
% 0.11/5.37 % CPUTime :
% 0.11/5.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.11/5.37 % Process 3627086 was started by sandbox on n016,
% 0.11/5.37 % Sat Sep 26 00:09:34 2026
% 0.11/5.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.11/5.37
% 0.11/5.37 % From the command line: assign(max_seconds, 300).
% 0.11/5.39
% 0.11/5.39 % SZS status Theorem for theBenchmark
% 0.11/5.39
% 0.11/5.39 % Proof 1 at 0.00 (+ 0.00) seconds.
% 0.11/5.39 % Length of proof is 28.
% 0.11/5.39 % Level of proof is 8.
% 0.11/5.39 % Maximum clause weight is 25.000.
% 0.11/5.39 % Given clauses 13.
% 0.11/5.39
% 0.11/5.39 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------