%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : GRP039-3 : TPTP v9.3.1. Released v1.0.0.
% 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 07:26:21 AM UTC 2026
% Result : Unsatisfiable 5.80s 1.41s
% Output : CNFRefutation 5.80s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 18
% Syntax : Number of formulae : 85 ( 37 unt; 0 def)
% Number of atoms : 166 ( 11 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 144 ( 63 ~; 81 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 5 con; 0-2 aty)
% Number of variables : 106 ( 0 sgn 31 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
fof(left_identity,axiom,
! [VAR_0] : product(identity,VAR_0,VAR_0),
file('theBenchmark.p',left_identity) ).
cnf(c_1,plain,
product(identity,A,A),
inference(clausify,[status(thm)],[left_identity]) ).
fof(right_identity,axiom,
! [VAR_0] : product(VAR_0,identity,VAR_0),
file('theBenchmark.p',right_identity) ).
cnf(c_2,plain,
product(A,identity,A),
inference(clausify,[status(thm)],[right_identity]) ).
fof(left_inverse,axiom,
! [VAR_0] : product(inverse(VAR_0),VAR_0,identity),
file('theBenchmark.p',left_inverse) ).
cnf(c_3,plain,
product(inverse(A),A,identity),
inference(clausify,[status(thm)],[left_inverse]) ).
fof(right_inverse,axiom,
! [VAR_0] : product(VAR_0,inverse(VAR_0),identity),
file('theBenchmark.p',right_inverse) ).
cnf(c_4,plain,
product(A,inverse(A),identity),
inference(clausify,[status(thm)],[right_inverse]) ).
fof(total_function1,axiom,
! [VAR_0,VAR_1] : product(VAR_0,VAR_1,multiply(VAR_0,VAR_1)),
file('theBenchmark.p',total_function1) ).
cnf(c_5,plain,
product(A,B,multiply(A,B)),
inference(clausify,[status(thm)],[total_function1]) ).
fof(total_function2,axiom,
! [VAR_0,VAR_1,VAR_2,VAR_3] :
( VAR_2 = VAR_3
| ~ product(VAR_0,VAR_1,VAR_3)
| ~ product(VAR_0,VAR_1,VAR_2) ),
file('theBenchmark.p',total_function2) ).
cnf(c_6,plain,
( C = D
| ~ product(A,B,D)
| ~ product(A,B,C) ),
inference(clausify,[status(thm)],[total_function2]) ).
fof(associativity1,axiom,
! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4,VAR_5] :
( product(VAR_0,VAR_4,VAR_5)
| ~ product(VAR_2,VAR_3,VAR_5)
| ~ product(VAR_1,VAR_3,VAR_4)
| ~ product(VAR_0,VAR_1,VAR_2) ),
file('theBenchmark.p',associativity1) ).
cnf(c_7,plain,
( product(A,E,F)
| ~ product(C,D,F)
| ~ product(B,D,E)
| ~ product(A,B,C) ),
inference(clausify,[status(thm)],[associativity1]) ).
fof(associativity2,axiom,
! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4,VAR_5] :
( product(VAR_2,VAR_3,VAR_5)
| ~ product(VAR_0,VAR_4,VAR_5)
| ~ product(VAR_1,VAR_3,VAR_4)
| ~ product(VAR_0,VAR_1,VAR_2) ),
file('theBenchmark.p',associativity2) ).
cnf(c_8,plain,
( product(C,D,F)
| ~ product(A,E,F)
| ~ product(B,D,E)
| ~ product(A,B,C) ),
inference(clausify,[status(thm)],[associativity2]) ).
fof(closure_of_product_and_inverse,axiom,
! [VAR_0,VAR_1,VAR_2] :
( subgroup_member(VAR_2)
| ~ product(VAR_0,inverse(VAR_1),VAR_2)
| ~ subgroup_member(VAR_1)
| ~ subgroup_member(VAR_0) ),
file('theBenchmark.p',closure_of_product_and_inverse) ).
cnf(c_9,plain,
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(clausify,[status(thm)],[closure_of_product_and_inverse]) ).
fof(inverse_is_self_cancelling,axiom,
! [VAR_0] : inverse(inverse(VAR_0)) = VAR_0,
file('theBenchmark.p',inverse_is_self_cancelling) ).
cnf(c_12,plain,
inverse(inverse(A)) = A,
inference(clausify,[status(thm)],[inverse_is_self_cancelling]) ).
fof(identity_is_in_subgroup,axiom,
subgroup_member(identity),
file('theBenchmark.p',identity_is_in_subgroup) ).
cnf(c_13,plain,
subgroup_member(identity),
inference(clausify,[status(thm)],[identity_is_in_subgroup]) ).
fof(subgroup_member_inverse_are_in_subgroup,axiom,
! [VAR_0] :
( subgroup_member(inverse(VAR_0))
| ~ subgroup_member(VAR_0) ),
file('theBenchmark.p',subgroup_member_inverse_are_in_subgroup) ).
cnf(c_14,plain,
( subgroup_member(inverse(A))
| ~ subgroup_member(A) ),
inference(clausify,[status(thm)],[subgroup_member_inverse_are_in_subgroup]) ).
fof(an_element_in_O2,axiom,
! [VAR_0,VAR_1] :
( subgroup_member(VAR_0)
| subgroup_member(VAR_1)
| subgroup_member(element_in_O2(VAR_0,VAR_1)) ),
file('theBenchmark.p',an_element_in_O2) ).
cnf(c_15,plain,
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ),
inference(clausify,[status(thm)],[an_element_in_O2]) ).
fof(property_of_O2,axiom,
! [VAR_0,VAR_1] :
( subgroup_member(VAR_0)
| subgroup_member(VAR_1)
| product(VAR_0,element_in_O2(VAR_0,VAR_1),VAR_1) ),
file('theBenchmark.p',property_of_O2) ).
cnf(c_16,plain,
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ),
inference(clausify,[status(thm)],[property_of_O2]) ).
fof(b_is_in_subgroup,axiom,
subgroup_member(b),
file('theBenchmark.p',b_is_in_subgroup) ).
cnf(c_17,plain,
subgroup_member(b),
inference(clausify,[status(thm)],[b_is_in_subgroup]) ).
fof(b_times_a_inverse_is_c,axiom,
product(b,inverse(a),c),
file('theBenchmark.p',b_times_a_inverse_is_c) ).
cnf(c_18,plain,
product(b,inverse(a),c),
inference(clausify,[status(thm)],[b_times_a_inverse_is_c]) ).
fof(a_times_c_is_d,axiom,
product(a,c,d),
file('theBenchmark.p',a_times_c_is_d) ).
cnf(c_19,plain,
product(a,c,d),
inference(clausify,[status(thm)],[a_times_c_is_d]) ).
fof(prove_d_is_in_subgroup,axiom,
~ subgroup_member(d),
file('theBenchmark.p',prove_d_is_in_subgroup) ).
cnf(c_20,plain,
~ subgroup_member(d),
inference(clausify,[status(thm)],[prove_d_is_in_subgroup]) ).
cnf(c_30,plain,
( multiply(A,B) = C
| ~ product(A,B,C) ),
inference(resolve,[status(thm)],[c_6,c_5]) ).
cnf(c_31,plain,
( identity = B
| ~ product(A,inverse(A),B) ),
inference(resolve,[status(thm)],[c_6,c_4]) ).
cnf(c_35,plain,
( B = A
| ~ product(A,identity,B) ),
inference(resolve,[status(thm)],[c_6,c_2]) ).
cnf(c_38,plain,
( product(A,C,D)
| ~ product(identity,B,D)
| ~ product(inverse(A),B,C) ),
inference(resolve,[status(thm)],[c_7,c_4]) ).
cnf(c_43,plain,
( product(A,identity,D)
| ~ product(C,inverse(B),D)
| ~ product(A,B,C) ),
inference(resolve,[status(thm)],[c_7,c_4]) ).
cnf(c_51,plain,
( product(A,D,C)
| ~ product(B,C,D)
| ~ product(A,B,identity) ),
inference(resolve,[status(thm)],[c_7,c_1]) ).
cnf(c_56,plain,
( product(identity,B,D)
| ~ product(A,C,D)
| ~ product(inverse(A),B,C) ),
inference(resolve,[status(thm)],[c_8,c_4]) ).
cnf(c_73,plain,
( subgroup_member(multiply(A,inverse(B)))
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(resolve,[status(thm)],[c_9,c_5]) ).
cnf(c_119,plain,
subgroup_member(inverse(b)),
inference(resolve,[status(thm)],[c_17,c_14]) ).
cnf(c_1662,plain,
( subgroup_member(c)
| ~ subgroup_member(a)
| ~ subgroup_member(b) ),
inference(resolve,[status(thm)],[c_18,c_9]) ).
cnf(c_122,plain,
( subgroup_member(c)
| ~ subgroup_member(a) ),
inference(resolve,[status(thm)],[c_17,c_1662]) ).
cnf(c_1663,plain,
identity = inverse(identity),
inference(resolve,[status(thm)],[c_31,c_1]) ).
cnf(c_213,plain,
inverse(identity) = identity,
inference(copy,[status(thm)],[c_1663]) ).
cnf(c_1664,plain,
( subgroup_member(B)
| ~ product(A,identity,B)
| ~ subgroup_member(identity)
| ~ subgroup_member(A) ),
inference(paramod,[status(thm)],[c_213,c_9]) ).
cnf(c_254,plain,
( subgroup_member(B)
| ~ product(A,identity,B)
| ~ subgroup_member(A) ),
inference(resolve,[status(thm)],[c_13,c_1664]) ).
cnf(c_1665,plain,
( product(A,identity,B)
| ~ product(identity,inverse(inverse(A)),B) ),
inference(resolve,[status(thm)],[c_38,c_4]) ).
cnf(c_337,plain,
( product(A,identity,B)
| ~ product(identity,A,B) ),
inference(paramod,[status(thm)],[c_12,c_1665]) ).
cnf(c_1666,plain,
( product(b,identity,A)
| ~ product(c,inverse(inverse(a)),A) ),
inference(resolve,[status(thm)],[c_43,c_18]) ).
cnf(c_484,plain,
( product(b,identity,A)
| ~ product(c,a,A) ),
inference(paramod,[status(thm)],[c_12,c_1666]) ).
cnf(c_633,plain,
product(b,identity,multiply(c,a)),
inference(resolve,[status(thm)],[c_484,c_5]) ).
cnf(c_647,plain,
multiply(c,a) = b,
inference(resolve,[status(thm)],[c_633,c_35]) ).
cnf(c_648,plain,
product(c,a,b),
inference(paramod,[status(thm)],[c_647,c_5]) ).
cnf(c_781,plain,
( product(A,b,a)
| ~ product(A,c,identity) ),
inference(resolve,[status(thm)],[c_51,c_648]) ).
cnf(c_782,plain,
( product(A,d,c)
| ~ product(A,a,identity) ),
inference(resolve,[status(thm)],[c_51,c_19]) ).
cnf(c_908,plain,
product(inverse(c),b,a),
inference(resolve,[status(thm)],[c_781,c_3]) ).
cnf(c_926,plain,
multiply(inverse(c),b) = a,
inference(resolve,[status(thm)],[c_908,c_30]) ).
cnf(c_965,plain,
product(inverse(a),d,c),
inference(resolve,[status(thm)],[c_782,c_3]) ).
cnf(c_1667,plain,
( product(identity,A,c)
| ~ product(inverse(inverse(a)),A,d) ),
inference(resolve,[status(thm)],[c_56,c_965]) ).
cnf(c_1102,plain,
( product(identity,A,c)
| ~ product(a,A,d) ),
inference(paramod,[status(thm)],[c_12,c_1667]) ).
cnf(c_1668,plain,
( subgroup_member(a)
| subgroup_member(d)
| product(identity,element_in_O2(a,d),c) ),
inference(resolve,[status(thm)],[c_1102,c_16]) ).
cnf(c_1213,plain,
( subgroup_member(a)
| product(identity,element_in_O2(a,d),c) ),
inference(resolve,[status(thm)],[c_20,c_1668]) ).
cnf(c_1246,plain,
( product(element_in_O2(a,d),identity,c)
| subgroup_member(a) ),
inference(resolve,[status(thm)],[c_1213,c_337]) ).
cnf(c_1378,plain,
( subgroup_member(c)
| ~ subgroup_member(element_in_O2(a,d))
| subgroup_member(a) ),
inference(resolve,[status(thm)],[c_1246,c_254]) ).
cnf(c_1669,plain,
( subgroup_member(a)
| subgroup_member(d)
| subgroup_member(c)
| subgroup_member(a) ),
inference(resolve,[status(thm)],[c_1378,c_15]) ).
cnf(c_1670,plain,
( subgroup_member(d)
| subgroup_member(c)
| subgroup_member(a) ),
inference(copy,[status(thm)],[c_1669]) ).
cnf(c_1497,plain,
( subgroup_member(c)
| subgroup_member(a) ),
inference(resolve,[status(thm)],[c_20,c_1670]) ).
cnf(c_1498,plain,
( subgroup_member(inverse(c))
| subgroup_member(a) ),
inference(resolve,[status(thm)],[c_1497,c_14]) ).
cnf(c_1671,plain,
( subgroup_member(multiply(A,inverse(inverse(b))))
| ~ subgroup_member(A) ),
inference(resolve,[status(thm)],[c_73,c_119]) ).
cnf(c_1570,plain,
( subgroup_member(multiply(A,b))
| ~ subgroup_member(A) ),
inference(paramod,[status(thm)],[c_12,c_1671]) ).
cnf(c_1672,plain,
( subgroup_member(a)
| subgroup_member(multiply(inverse(c),b)) ),
inference(resolve,[status(thm)],[c_1570,c_1498]) ).
cnf(c_1673,plain,
( subgroup_member(a)
| subgroup_member(a) ),
inference(paramod,[status(thm)],[c_926,c_1672]) ).
cnf(c_1577,plain,
subgroup_member(a),
inference(copy,[status(thm)],[c_1673]) ).
cnf(c_1590,plain,
subgroup_member(c),
inference(resolve,[status(thm)],[c_1577,c_122]) ).
cnf(c_1637,plain,
subgroup_member(inverse(c)),
inference(resolve,[status(thm)],[c_1590,c_14]) ).
cnf(c_1674,plain,
( subgroup_member(B)
| ~ product(a,inverse(A),B)
| ~ subgroup_member(A) ),
inference(resolve,[status(thm)],[c_9,c_1577]) ).
cnf(c_1675,plain,
( subgroup_member(A)
| ~ product(a,inverse(inverse(c)),A) ),
inference(resolve,[status(thm)],[c_1674,c_1637]) ).
cnf(c_1676,plain,
~ product(a,inverse(inverse(c)),d),
inference(resolve,[status(thm)],[c_1675,c_20]) ).
cnf(c_1677,plain,
~ product(a,c,d),
inference(paramod,[status(thm)],[c_12,c_1676]) ).
cnf(c_1661,plain,
$false,
inference(resolve,[status(thm)],[c_19,c_1677]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : GRP039-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.18/0.42 % Computer : n005.cluster.edu
% 0.18/0.42 % Model : x86_64 x86_64
% 0.18/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.42 % Memory : 8046.5625MB
% 0.18/0.42 % OS : Linux 6.8.0-71-generic
% 0.18/0.42 % CPULimit : 300
% 0.18/0.42 % WCLimit : 300
% 0.18/0.42 % DateTime : Fri Sep 25 14:19:02 UTC 2026
% 0.18/0.42 % CPUTime :
% 0.18/0.42 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.18/0.42 % Process 3675256 was started by sandbox2 on n005,
% 0.18/0.42 % Fri Sep 25 14:19:02 2026
% 0.18/0.42 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.18/0.43
% 0.18/0.43 % From the command line: assign(max_seconds, 300).
% 5.80/1.41
% 5.80/1.41 % SZS status Unsatisfiable for theBenchmark
% 5.80/1.41
% 5.80/1.41 % Proof 1 at 0.65 (+ 0.32) seconds.
% 5.80/1.41 % Length of proof is 51.
% 5.80/1.41 % Level of proof is 13.
% 5.80/1.41 % Maximum clause weight is 16.000.
% 5.80/1.41 % Given clauses 299.
% 5.80/1.41
% 5.80/1.41 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------