%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : GEO084+1 : TPTP v9.3.1. Released v2.4.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 07:17:01 AM UTC 2026
% Result : Theorem 18.08s 3.00s
% Output : CNFRefutation 18.08s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 48 ( 19 unt; 0 def)
% Number of atoms : 110 ( 17 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 100 ( 38 ~; 34 |; 20 &)
% ( 3 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 4 con; 0-2 aty)
% Number of variables : 99 ( 2 sgn 52 !; 14 ?)
% Comments :
%------------------------------------------------------------------------------
fof(part_of_defn,axiom,
! [C,C1] :
( part_of(C1,C)
<=> ! [P] :
( incident_c(P,C1)
=> incident_c(P,C) ) ),
file('theBenchmark.p',part_of_defn) ).
fof(sum_defn,axiom,
! [C,C1,C2] :
( C = sum(C1,C2)
<=> ! [Q] :
( incident_c(Q,C)
<=> ( incident_c(Q,C2)
| incident_c(Q,C1) ) ) ),
file('theBenchmark.p',sum_defn) ).
fof(c3,axiom,
! [C] :
? [P] : inner_point(P,C),
file('theBenchmark.p',c3) ).
fof(c4,axiom,
! [C,P] :
( inner_point(P,C)
=> ? [C1,C2] :
( C = sum(C1,C2)
& meet(P,C1,C2) ) ),
file('theBenchmark.p',c4) ).
fof(c8,axiom,
! [C1,C2] :
( ? [P] : meet(P,C1,C2)
=> ? [C] : C = sum(C1,C2) ),
file('theBenchmark.p',c8) ).
fof(corollary_2_6_2,conjecture,
! [C1,C2,C3,P] :
( ( meet(P,C1,C2)
& part_of(C2,C3)
& part_of(C1,C3) )
=> part_of(sum(C1,C2),C3) ),
file('theBenchmark.p',corollary_2_6_2) ).
fof(corollary_2_6_2_neg,negated_conjecture,
~ ! [C1,C2,C3,P] :
( ( meet(P,C1,C2)
& part_of(C2,C3)
& part_of(C1,C3) )
=> part_of(sum(C1,C2),C3) ),
inference(assume_negation,[status(cth)],[corollary_2_6_2]) ).
fof(nnf_10,plain,
! [C] :
? [P] : inner_point(P,C),
inference(fof_nnf,[status(thm)],[c3]) ).
fof(sk_c3,plain,
! [C] : inner_point(sK0(C),C),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(P,sK0(C))],[nnf_10]) ).
cnf(c_21,plain,
inner_point(sK0(A),A),
inference(split_conjunct,[status(thm)],[sk_c3]) ).
fof(nnf_11,plain,
! [C,P] :
( ? [C1,C2] :
( sum(C1,C2) = C
& meet(P,C1,C2) )
| ~ inner_point(P,C) ),
inference(fof_nnf,[status(thm)],[c4]) ).
fof(sk_c4_sk,plain,
! [C,P] :
( ( sum(sK1(C,P),sK2(C,P)) = C
& meet(P,sK1(C,P),sK2(C,P)) )
| ~ inner_point(P,C) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(C1,sK1(C,P)),skolemize(C2,sK2(C,P))],[nnf_11]) ).
fof(sk_c4,plain,
( ! [VAR_0,VAR_1] :
( sum(sK1(VAR_1,VAR_0),sK2(VAR_1,VAR_0)) = VAR_1
| ~ inner_point(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( meet(VAR_0,sK1(VAR_1,VAR_0),sK2(VAR_1,VAR_0))
| ~ inner_point(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_c4_sk]) ).
cnf(c_22,plain,
( meet(A,sK1(B,A),sK2(B,A))
| ~ inner_point(A,B) ),
inference(split_conjunct,[status(thm)],[sk_c4]) ).
cnf(c_23,plain,
( sum(sK1(B,A),sK2(B,A)) = B
| ~ inner_point(A,B) ),
inference(split_conjunct,[status(thm)],[sk_c4]) ).
cnf(c_33,plain,
sum(sK1(A,sK0(A)),sK2(A,sK0(A))) = A,
inference(resolve,[status(thm)],[c_23,c_21]) ).
cnf(c_35,plain,
meet(sK0(A),sK1(A,sK0(A)),sK2(A,sK0(A))),
inference(resolve,[status(thm)],[c_22,c_21]) ).
fof(nnf_17,plain,
? [C1,C2,C3,P] :
( ~ part_of(sum(C1,C2),C3)
& meet(P,C1,C2)
& part_of(C2,C3)
& part_of(C1,C3) ),
inference(fof_nnf,[status(thm)],[corollary_2_6_2_neg]) ).
fof(sk_corollary_2_6_2_neg,plain,
( ~ part_of(sum(sK3,sK4),sK5)
& meet(sK8,sK3,sK4)
& part_of(sK4,sK5)
& part_of(sK3,sK5) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3,sK5,sK4,sK8]),skolemize(C1,sK3),skolemize(C2,sK4),skolemize(C3,sK5),skolemize(P,sK8)],[nnf_17]) ).
cnf(c_38,negated_conjecture,
~ part_of(sum(sK3,sK4),sK5),
inference(split_conjunct,[status(thm)],[sk_corollary_2_6_2_neg]) ).
cnf(c_40,negated_conjecture,
part_of(sK4,sK5),
inference(split_conjunct,[status(thm)],[sk_corollary_2_6_2_neg]) ).
cnf(c_41,negated_conjecture,
part_of(sK3,sK5),
inference(split_conjunct,[status(thm)],[sk_corollary_2_6_2_neg]) ).
fof(nnf_15,plain,
! [C1,C2] :
( ? [C] : sum(C1,C2) = C
| ! [P] : ~ meet(P,C1,C2) ),
inference(fof_nnf,[status(thm)],[c8]) ).
fof(sk_c8_sk,plain,
! [C1,C2] :
( sum(C1,C2) = sK6(C1,C2)
| ! [P] : ~ meet(P,C1,C2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK6]),skolemize(C,sK6(C1,C2))],[nnf_15]) ).
fof(sk_c8,plain,
! [VAR_0,VAR_1,VAR_2] :
( sum(VAR_1,VAR_2) = sK6(VAR_1,VAR_2)
| ~ meet(VAR_0,VAR_1,VAR_2) ),
inference(cnf_transformation,[status(thm)],[sk_c8_sk]) ).
cnf(c_44,plain,
( sum(B,C) = sK6(B,C)
| ~ meet(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_c8]) ).
cnf(c_45,plain,
( sK6(B,C) = sum(B,C)
| ~ meet(A,B,C) ),
inference(copy,[status(thm)],[c_44]) ).
cnf(c_70,plain,
( incident_c(D,B)
| incident_c(D,A)
| ~ incident_c(D,C)
| sum(A,B) != C ),
inference(clausify,[status(thm)],[sum_defn]) ).
fof(nnf_1,plain,
! [C,C1] :
( ( ? [P] :
( ~ incident_c(P,C)
& incident_c(P,C1) )
| part_of(C1,C) )
& ( ! [X0] :
( incident_c(X0,C)
| ~ incident_c(X0,C1) )
| ~ part_of(C1,C) ) ),
inference(fof_nnf,[status(thm)],[part_of_defn]) ).
fof(sk_part_of_defn_sk,plain,
! [C,C1] :
( ( ( ~ incident_c(sK7(C,C1),C)
& incident_c(sK7(C,C1),C1) )
| part_of(C1,C) )
& ( ! [X0] :
( incident_c(X0,C)
| ~ incident_c(X0,C1) )
| ~ part_of(C1,C) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK7]),skolemize(P,sK7(C,C1))],[nnf_1]) ).
fof(sk_part_of_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ incident_c(sK7(VAR_1,VAR_0),VAR_1)
| part_of(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( incident_c(sK7(VAR_1,VAR_0),VAR_0)
| part_of(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1,VAR_2] :
( incident_c(VAR_2,VAR_1)
| ~ incident_c(VAR_2,VAR_0)
| ~ part_of(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_part_of_defn_sk]) ).
cnf(c_71,plain,
( ~ incident_c(sK7(B,A),B)
| part_of(A,B) ),
inference(split_conjunct,[status(thm)],[sk_part_of_defn]) ).
cnf(c_72,plain,
( incident_c(sK7(B,A),A)
| part_of(A,B) ),
inference(split_conjunct,[status(thm)],[sk_part_of_defn]) ).
cnf(c_73,plain,
( incident_c(C,B)
| ~ incident_c(C,A)
| ~ part_of(A,B) ),
inference(split_conjunct,[status(thm)],[sk_part_of_defn]) ).
cnf(c_4607,plain,
sK6(sK1(A,sK0(A)),sK2(A,sK0(A))) = sum(sK1(A,sK0(A)),sK2(A,sK0(A))),
inference(resolve,[status(thm)],[c_45,c_35]) ).
cnf(c_96,plain,
sK6(sK1(A,sK0(A)),sK2(A,sK0(A))) = A,
inference(paramod,[status(thm)],[c_33,c_4607]) ).
cnf(c_303,negated_conjecture,
~ incident_c(sK7(sK5,sum(sK3,sK4)),sK5),
inference(resolve,[status(thm)],[c_71,c_38]) ).
cnf(c_308,plain,
incident_c(sK7(sK5,sum(sK3,sK4)),sum(sK3,sK4)),
inference(resolve,[status(thm)],[c_72,c_38]) ).
cnf(c_311,plain,
( incident_c(A,sK5)
| ~ incident_c(A,sK3) ),
inference(resolve,[status(thm)],[c_73,c_41]) ).
cnf(c_312,plain,
( incident_c(A,sK5)
| ~ incident_c(A,sK4) ),
inference(resolve,[status(thm)],[c_73,c_40]) ).
cnf(c_1884,plain,
~ incident_c(sK7(sK5,sum(sK3,sK4)),sK4),
inference(resolve,[status(thm)],[c_312,c_303]) ).
cnf(c_1885,plain,
~ incident_c(sK7(sK5,sum(sK3,sK4)),sK3),
inference(resolve,[status(thm)],[c_311,c_303]) ).
cnf(c_4608,plain,
A = sK6(sK1(A,sK0(A)),sK2(A,sK0(A))),
inference(copy,[status(thm)],[c_96]) ).
cnf(c_4609,plain,
( incident_c(A,C)
| incident_c(A,B)
| ~ incident_c(A,sK6(sK1(sum(B,C),sK0(sum(B,C))),sK2(sum(B,C),sK0(sum(B,C))))) ),
inference(resolve,[status(thm)],[c_70,c_4608]) ).
cnf(c_4610,plain,
( incident_c(sK7(sK5,sum(sK3,sK4)),A)
| ~ incident_c(sK7(sK5,sum(sK3,sK4)),sK6(sK1(sum(sK3,A),sK0(sum(sK3,A))),sK2(sum(sK3,A),sK0(sum(sK3,A))))) ),
inference(resolve,[status(thm)],[c_4609,c_1885]) ).
cnf(c_4611,plain,
~ incident_c(sK7(sK5,sum(sK3,sK4)),sK6(sK1(sum(sK3,sK4),sK0(sum(sK3,sK4))),sK2(sum(sK3,sK4),sK0(sum(sK3,sK4))))),
inference(resolve,[status(thm)],[c_4610,c_1884]) ).
cnf(c_4612,plain,
~ incident_c(sK7(sK5,sum(sK3,sK4)),sum(sK3,sK4)),
inference(paramod,[status(thm)],[c_96,c_4611]) ).
cnf(c_4606,plain,
$false,
inference(resolve,[status(thm)],[c_308,c_4612]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : GEO084+1 : TPTP v9.3.1. Released v2.4.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n005.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Fri Sep 25 12:41:18 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.37 % Process 3508611 was started by sandbox2 on n005,
% 0.09/0.37 % Fri Sep 25 12:41:18 2026
% 0.09/0.37 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.09/0.38
% 0.09/0.38 % From the command line: assign(max_seconds, 300).
% 18.08/3.00
% 18.08/3.00 % given #500 (T,wt=8): 2733 incident_c(f5(f13(c11,c12),c12,c12),c12). [resolve(2729,a,120,a)].
% 18.08/3.00
% 18.08/3.00 % SZS status Theorem for theBenchmark
% 18.08/3.00
% 18.08/3.00 % Proof 1 at 2.33 (+ 0.26) seconds.
% 18.08/3.00 % Length of proof is 28.
% 18.08/3.00 % Level of proof is 4.
% 18.08/3.00 % Maximum clause weight is 14.000.
% 18.08/3.00 % Given clauses 552.
% 18.08/3.00
% 18.08/3.00 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------