%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : KLE082+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n001.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 07:44:16 AM UTC 2026
% Result : Theorem 2.72s 0.75s
% Output : CNFRefutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 29
% Number of leaves : 14
% Syntax : Number of formulae : 109 ( 105 unt; 0 def)
% Number of atoms : 117 ( 116 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 15 ( 7 ~; 0 |; 6 &)
% ( 0 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 1 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 2 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-2 aty)
% Number of variables : 168 ( 18 sgn 30 !; 2 ?)
% Comments :
%------------------------------------------------------------------------------
fof(additive_commutativity,axiom,
! [A,B] : addition(A,B) = addition(B,A),
file('theBenchmark.p',additive_commutativity) ).
fof(additive_associativity,axiom,
! [C,B,A] : addition(A,addition(B,C)) = addition(addition(A,B),C),
file('theBenchmark.p',additive_associativity) ).
fof(additive_identity,axiom,
! [A] : addition(A,zero) = A,
file('theBenchmark.p',additive_identity) ).
fof(additive_idempotence,axiom,
! [A] : addition(A,A) = A,
file('theBenchmark.p',additive_idempotence) ).
fof(multiplicative_right_identity,axiom,
! [A] : multiplication(A,one) = A,
file('theBenchmark.p',multiplicative_right_identity) ).
fof(multiplicative_left_identity,axiom,
! [A] : multiplication(one,A) = A,
file('theBenchmark.p',multiplicative_left_identity) ).
fof(right_distributivity,axiom,
! [A,B,C] : multiplication(A,addition(B,C)) = addition(multiplication(A,B),multiplication(A,C)),
file('theBenchmark.p',right_distributivity) ).
fof(left_distributivity,axiom,
! [A,B,C] : multiplication(addition(A,B),C) = addition(multiplication(A,C),multiplication(B,C)),
file('theBenchmark.p',left_distributivity) ).
fof(left_annihilation,axiom,
! [A] : multiplication(zero,A) = zero,
file('theBenchmark.p',left_annihilation) ).
fof(domain1,axiom,
! [X0] : addition(X0,multiplication(domain(X0),X0)) = multiplication(domain(X0),X0),
file('theBenchmark.p',domain1) ).
fof(domain2,axiom,
! [X0,X1] : domain(multiplication(X0,X1)) = domain(multiplication(X0,domain(X1))),
file('theBenchmark.p',domain2) ).
fof(domain3,axiom,
! [X0] : addition(domain(X0),one) = one,
file('theBenchmark.p',domain3) ).
fof(domain5,axiom,
! [X0,X1] : domain(addition(X0,X1)) = addition(domain(X0),domain(X1)),
file('theBenchmark.p',domain5) ).
fof(goals,conjecture,
! [X0,X1] :
( ! [X2] :
( multiplication(domain(X2),antidomain(X2)) = zero
& addition(domain(X2),antidomain(X2)) = one )
=> addition(antidomain(multiplication(X0,X1)),antidomain(multiplication(X0,domain(X1)))) = antidomain(multiplication(X0,domain(X1))) ),
file('theBenchmark.p',goals) ).
fof(goals_neg,negated_conjecture,
~ ! [X0,X1] :
( ! [X2] :
( multiplication(domain(X2),antidomain(X2)) = zero
& addition(domain(X2),antidomain(X2)) = one )
=> addition(antidomain(multiplication(X0,X1)),antidomain(multiplication(X0,domain(X1)))) = antidomain(multiplication(X0,domain(X1))) ),
inference(assume_negation,[status(cth)],[goals]) ).
fof(nnf_17,plain,
? [X0,X1] :
( antidomain(multiplication(X0,domain(X1))) != addition(antidomain(multiplication(X0,X1)),antidomain(multiplication(X0,domain(X1))))
& ! [X2] :
( multiplication(domain(X2),antidomain(X2)) = zero
& addition(domain(X2),antidomain(X2)) = one ) ),
inference(fof_nnf,[status(thm)],[goals_neg]) ).
fof(sk_goals_neg,plain,
( antidomain(multiplication(sK0,domain(sK1))) != addition(antidomain(multiplication(sK0,sK1)),antidomain(multiplication(sK0,domain(sK1))))
& ! [X2] :
( multiplication(domain(X2),antidomain(X2)) = zero
& addition(domain(X2),antidomain(X2)) = one ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(X0,sK0),skolemize(X1,sK1)],[nnf_17]) ).
cnf(c_20,negated_conjecture,
antidomain(multiplication(sK0,domain(sK1))) != addition(antidomain(multiplication(sK0,sK1)),antidomain(multiplication(sK0,domain(sK1)))),
inference(split_conjunct,[status(thm)],[sk_goals_neg]) ).
cnf(c_21,negated_conjecture,
addition(antidomain(multiplication(sK0,sK1)),antidomain(multiplication(sK0,domain(sK1)))) != antidomain(multiplication(sK0,domain(sK1))),
inference(copy,[status(thm)],[c_20]) ).
cnf(c_22,negated_conjecture,
multiplication(domain(A),antidomain(A)) = zero,
inference(split_conjunct,[status(thm)],[sk_goals_neg]) ).
cnf(c_23,negated_conjecture,
addition(domain(A),antidomain(A)) = one,
inference(split_conjunct,[status(thm)],[sk_goals_neg]) ).
cnf(c_24,plain,
domain(addition(A,B)) = addition(domain(A),domain(B)),
inference(clausify,[status(thm)],[domain5]) ).
cnf(c_25,plain,
addition(domain(A),domain(B)) = domain(addition(A,B)),
inference(copy,[status(thm)],[c_24]) ).
cnf(c_27,plain,
addition(domain(A),one) = one,
inference(clausify,[status(thm)],[domain3]) ).
cnf(c_28,plain,
domain(multiplication(A,domain(B))) = domain(multiplication(A,B)),
inference(clausify,[status(thm)],[domain2]) ).
cnf(c_29,plain,
multiplication(domain(A),A) = addition(A,multiplication(domain(A),A)),
inference(clausify,[status(thm)],[domain1]) ).
cnf(c_30,plain,
addition(A,multiplication(domain(A),A)) = multiplication(domain(A),A),
inference(copy,[status(thm)],[c_29]) ).
cnf(c_31,plain,
multiplication(zero,A) = zero,
inference(clausify,[status(thm)],[left_annihilation]) ).
cnf(c_33,plain,
multiplication(addition(A,B),C) = addition(multiplication(A,C),multiplication(B,C)),
inference(clausify,[status(thm)],[left_distributivity]) ).
cnf(c_34,plain,
addition(multiplication(A,B),multiplication(C,B)) = multiplication(addition(A,C),B),
inference(copy,[status(thm)],[c_33]) ).
cnf(c_35,plain,
multiplication(A,addition(B,C)) = addition(multiplication(A,B),multiplication(A,C)),
inference(clausify,[status(thm)],[right_distributivity]) ).
cnf(c_36,plain,
addition(multiplication(A,B),multiplication(A,C)) = multiplication(A,addition(B,C)),
inference(copy,[status(thm)],[c_35]) ).
cnf(c_37,plain,
multiplication(one,A) = A,
inference(clausify,[status(thm)],[multiplicative_left_identity]) ).
cnf(c_38,plain,
multiplication(A,one) = A,
inference(clausify,[status(thm)],[multiplicative_right_identity]) ).
cnf(c_40,plain,
addition(A,A) = A,
inference(clausify,[status(thm)],[additive_idempotence]) ).
cnf(c_41,plain,
addition(A,zero) = A,
inference(clausify,[status(thm)],[additive_identity]) ).
cnf(c_42,plain,
addition(addition(A,B),C) = addition(A,addition(B,C)),
inference(clausify,[status(thm)],[additive_associativity]) ).
cnf(c_43,plain,
addition(A,B) = addition(B,A),
inference(clausify,[status(thm)],[additive_commutativity]) ).
cnf(c_44,plain,
addition(one,domain(A)) = one,
inference(paramod,[status(thm)],[c_43,c_27]) ).
cnf(c_50,plain,
addition(domain(multiplication(A,B)),antidomain(multiplication(A,domain(B)))) = one,
inference(paramod,[status(thm)],[c_28,c_23]) ).
cnf(c_1570,plain,
addition(zero,multiplication(A,antidomain(B))) = multiplication(addition(domain(B),A),antidomain(B)),
inference(paramod,[status(thm)],[c_22,c_34]) ).
cnf(c_55,plain,
multiplication(addition(domain(A),B),antidomain(A)) = addition(zero,multiplication(B,antidomain(A))),
inference(copy,[status(thm)],[c_1570]) ).
cnf(c_58,plain,
addition(zero,multiplication(A,B)) = multiplication(addition(zero,A),B),
inference(paramod,[status(thm)],[c_31,c_34]) ).
cnf(c_1571,plain,
addition(multiplication(A,B),zero) = multiplication(addition(A,zero),B),
inference(paramod,[status(thm)],[c_31,c_34]) ).
cnf(c_1572,plain,
addition(zero,multiplication(A,B)) = multiplication(addition(A,zero),B),
inference(paramod,[status(thm)],[c_43,c_1571]) ).
cnf(c_1573,plain,
multiplication(addition(zero,A),B) = multiplication(addition(A,zero),B),
inference(paramod,[status(thm)],[c_58,c_1572]) ).
cnf(c_59,plain,
multiplication(addition(zero,A),B) = multiplication(A,B),
inference(paramod,[status(thm)],[c_41,c_1573]) ).
cnf(c_1574,plain,
multiplication(addition(domain(A),B),antidomain(A)) = multiplication(addition(zero,B),antidomain(A)),
inference(paramod,[status(thm)],[c_58,c_55]) ).
cnf(c_61,plain,
multiplication(addition(domain(A),B),antidomain(A)) = multiplication(B,antidomain(A)),
inference(paramod,[status(thm)],[c_59,c_1574]) ).
cnf(c_1575,plain,
addition(one,domain(one)) = multiplication(domain(one),one),
inference(paramod,[status(thm)],[c_38,c_30]) ).
cnf(c_1576,plain,
one = multiplication(domain(one),one),
inference(paramod,[status(thm)],[c_44,c_1575]) ).
cnf(c_1577,plain,
one = domain(one),
inference(paramod,[status(thm)],[c_38,c_1576]) ).
cnf(c_71,plain,
domain(one) = one,
inference(copy,[status(thm)],[c_1577]) ).
cnf(c_1578,plain,
addition(A,multiplication(A,B)) = multiplication(A,addition(one,B)),
inference(paramod,[status(thm)],[c_38,c_36]) ).
cnf(c_72,plain,
multiplication(A,addition(one,B)) = addition(A,multiplication(A,B)),
inference(copy,[status(thm)],[c_1578]) ).
cnf(c_1579,plain,
addition(addition(A,domain(B)),antidomain(B)) = addition(A,one),
inference(paramod,[status(thm)],[c_23,c_42]) ).
cnf(c_78,plain,
addition(antidomain(A),addition(B,domain(A))) = addition(B,one),
inference(paramod,[status(thm)],[c_43,c_1579]) ).
cnf(c_1580,plain,
addition(addition(A,B),B) = addition(A,B),
inference(paramod,[status(thm)],[c_40,c_42]) ).
cnf(c_86,plain,
addition(A,addition(B,A)) = addition(B,A),
inference(paramod,[status(thm)],[c_43,c_1580]) ).
cnf(c_1581,plain,
addition(multiplication(one,A),multiplication(domain(B),A)) = multiplication(one,A),
inference(paramod,[status(thm)],[c_44,c_34]) ).
cnf(c_1582,plain,
addition(A,multiplication(domain(B),A)) = multiplication(one,A),
inference(paramod,[status(thm)],[c_37,c_1581]) ).
cnf(c_92,plain,
addition(A,multiplication(domain(B),A)) = A,
inference(paramod,[status(thm)],[c_37,c_1582]) ).
cnf(c_1583,plain,
A = multiplication(domain(A),A),
inference(paramod,[status(thm)],[c_92,c_30]) ).
cnf(c_95,plain,
multiplication(domain(A),A) = A,
inference(copy,[status(thm)],[c_1583]) ).
cnf(c_1584,plain,
addition(one,domain(A)) = domain(addition(one,A)),
inference(paramod,[status(thm)],[c_71,c_25]) ).
cnf(c_1585,plain,
one = domain(addition(one,A)),
inference(paramod,[status(thm)],[c_44,c_1584]) ).
cnf(c_104,plain,
domain(addition(one,A)) = one,
inference(copy,[status(thm)],[c_1585]) ).
cnf(c_1586,plain,
domain(multiplication(A,one)) = domain(multiplication(A,addition(one,B))),
inference(paramod,[status(thm)],[c_104,c_28]) ).
cnf(c_1587,plain,
domain(A) = domain(multiplication(A,addition(one,B))),
inference(paramod,[status(thm)],[c_38,c_1586]) ).
cnf(c_112,plain,
domain(multiplication(A,addition(one,B))) = domain(A),
inference(copy,[status(thm)],[c_1587]) ).
cnf(c_1588,plain,
addition(domain(A),antidomain(multiplication(one,domain(A)))) = one,
inference(paramod,[status(thm)],[c_37,c_50]) ).
cnf(c_115,plain,
addition(domain(A),antidomain(domain(A))) = one,
inference(paramod,[status(thm)],[c_37,c_1588]) ).
cnf(c_1589,plain,
addition(antidomain(A),one) = addition(domain(A),antidomain(A)),
inference(paramod,[status(thm)],[c_23,c_86]) ).
cnf(c_1590,plain,
addition(one,antidomain(A)) = addition(domain(A),antidomain(A)),
inference(paramod,[status(thm)],[c_43,c_1589]) ).
cnf(c_282,plain,
addition(one,antidomain(A)) = one,
inference(paramod,[status(thm)],[c_23,c_1590]) ).
cnf(c_1591,plain,
multiplication(one,antidomain(A)) = multiplication(antidomain(domain(A)),antidomain(A)),
inference(paramod,[status(thm)],[c_115,c_61]) ).
cnf(c_1592,plain,
antidomain(A) = multiplication(antidomain(domain(A)),antidomain(A)),
inference(paramod,[status(thm)],[c_37,c_1591]) ).
cnf(c_387,plain,
multiplication(antidomain(domain(A)),antidomain(A)) = antidomain(A),
inference(copy,[status(thm)],[c_1592]) ).
cnf(c_1593,plain,
multiplication(antidomain(domain(A)),addition(one,antidomain(A))) = addition(antidomain(domain(A)),antidomain(A)),
inference(paramod,[status(thm)],[c_387,c_72]) ).
cnf(c_1594,plain,
multiplication(antidomain(domain(A)),one) = addition(antidomain(domain(A)),antidomain(A)),
inference(paramod,[status(thm)],[c_282,c_1593]) ).
cnf(c_1595,plain,
antidomain(domain(A)) = addition(antidomain(domain(A)),antidomain(A)),
inference(paramod,[status(thm)],[c_38,c_1594]) ).
cnf(c_1596,plain,
antidomain(domain(A)) = addition(antidomain(A),antidomain(domain(A))),
inference(paramod,[status(thm)],[c_43,c_1595]) ).
cnf(c_411,plain,
addition(antidomain(A),antidomain(domain(A))) = antidomain(domain(A)),
inference(copy,[status(thm)],[c_1596]) ).
cnf(c_1597,plain,
addition(antidomain(multiplication(A,domain(B))),antidomain(domain(multiplication(A,B)))) = antidomain(domain(multiplication(A,domain(B)))),
inference(paramod,[status(thm)],[c_28,c_411]) ).
cnf(c_448,plain,
addition(antidomain(multiplication(A,domain(B))),antidomain(domain(multiplication(A,B)))) = antidomain(domain(multiplication(A,B))),
inference(paramod,[status(thm)],[c_28,c_1597]) ).
cnf(c_1598,plain,
addition(antidomain(A),domain(addition(B,A))) = addition(domain(B),one),
inference(paramod,[status(thm)],[c_25,c_78]) ).
cnf(c_1599,plain,
addition(antidomain(A),domain(addition(B,A))) = addition(one,domain(B)),
inference(paramod,[status(thm)],[c_43,c_1598]) ).
cnf(c_1314,plain,
addition(antidomain(A),domain(addition(B,A))) = one,
inference(paramod,[status(thm)],[c_44,c_1599]) ).
cnf(c_1600,plain,
addition(antidomain(multiplication(A,B)),domain(multiplication(A,addition(one,B)))) = one,
inference(paramod,[status(thm)],[c_72,c_1314]) ).
cnf(c_1601,plain,
addition(antidomain(multiplication(A,B)),domain(A)) = one,
inference(paramod,[status(thm)],[c_112,c_1600]) ).
cnf(c_1341,plain,
addition(domain(A),antidomain(multiplication(A,B))) = one,
inference(paramod,[status(thm)],[c_43,c_1601]) ).
cnf(c_1602,plain,
multiplication(one,antidomain(A)) = multiplication(antidomain(multiplication(A,B)),antidomain(A)),
inference(paramod,[status(thm)],[c_1341,c_61]) ).
cnf(c_1603,plain,
antidomain(A) = multiplication(antidomain(multiplication(A,B)),antidomain(A)),
inference(paramod,[status(thm)],[c_37,c_1602]) ).
cnf(c_1374,plain,
multiplication(antidomain(multiplication(A,B)),antidomain(A)) = antidomain(A),
inference(copy,[status(thm)],[c_1603]) ).
cnf(c_1453,plain,
multiplication(antidomain(A),antidomain(domain(A))) = antidomain(domain(A)),
inference(paramod,[status(thm)],[c_95,c_1374]) ).
cnf(c_1604,plain,
multiplication(antidomain(A),addition(one,antidomain(domain(A)))) = addition(antidomain(A),antidomain(domain(A))),
inference(paramod,[status(thm)],[c_1453,c_72]) ).
cnf(c_1605,plain,
multiplication(antidomain(A),one) = addition(antidomain(A),antidomain(domain(A))),
inference(paramod,[status(thm)],[c_282,c_1604]) ).
cnf(c_1606,plain,
antidomain(A) = addition(antidomain(A),antidomain(domain(A))),
inference(paramod,[status(thm)],[c_38,c_1605]) ).
cnf(c_1607,plain,
antidomain(A) = antidomain(domain(A)),
inference(paramod,[status(thm)],[c_411,c_1606]) ).
cnf(c_1493,plain,
antidomain(domain(A)) = antidomain(A),
inference(copy,[status(thm)],[c_1607]) ).
cnf(c_1608,plain,
addition(antidomain(multiplication(A,domain(B))),antidomain(multiplication(A,B))) = antidomain(domain(multiplication(A,B))),
inference(paramod,[status(thm)],[c_1493,c_448]) ).
cnf(c_1609,plain,
addition(antidomain(multiplication(A,B)),antidomain(multiplication(A,domain(B)))) = antidomain(domain(multiplication(A,B))),
inference(paramod,[status(thm)],[c_43,c_1608]) ).
cnf(c_1560,plain,
addition(antidomain(multiplication(A,B)),antidomain(multiplication(A,domain(B)))) = antidomain(multiplication(A,B)),
inference(paramod,[status(thm)],[c_1493,c_1609]) ).
cnf(c_1610,negated_conjecture,
antidomain(multiplication(sK0,sK1)) != antidomain(multiplication(sK0,domain(sK1))),
inference(paramod,[status(thm)],[c_1560,c_21]) ).
cnf(c_1567,negated_conjecture,
antidomain(multiplication(sK0,domain(sK1))) != antidomain(multiplication(sK0,sK1)),
inference(copy,[status(thm)],[c_1610]) ).
cnf(c_1611,plain,
antidomain(domain(multiplication(A,B))) = antidomain(multiplication(A,domain(B))),
inference(paramod,[status(thm)],[c_28,c_1493]) ).
cnf(c_1612,plain,
antidomain(multiplication(A,B)) = antidomain(multiplication(A,domain(B))),
inference(paramod,[status(thm)],[c_1493,c_1611]) ).
cnf(c_1568,plain,
antidomain(multiplication(A,domain(B))) = antidomain(multiplication(A,B)),
inference(copy,[status(thm)],[c_1612]) ).
cnf(c_1569,plain,
$false,
inference(resolve,[status(thm)],[c_1568,c_1567]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE082+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n001.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Fri Sep 25 19:25:40 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.08/0.35 % Process 4024387 was started by sandbox2 on n001,
% 0.08/0.35 % Fri Sep 25 19:25:40 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.08/0.36
% 0.08/0.36 % From the command line: assign(max_seconds, 300).
% 2.72/0.75
% 2.72/0.75 % SZS status Theorem for theBenchmark
% 2.72/0.75
% 2.72/0.75 % Proof 1 at 0.25 (+ 0.12) seconds.
% 2.72/0.75 % Length of proof is 63.
% 2.72/0.75 % Level of proof is 12.
% 2.72/0.75 % Maximum clause weight is 42.000.
% 2.72/0.75 % Given clauses 122.
% 2.72/0.75
% 2.72/0.75 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------