%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : KLE167+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n015.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 07:44:27 AM UTC 2026
% Result : Theorem 0.24s 0.54s
% Output : CNFRefutation 0.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 10
% Syntax : Number of formulae : 42 ( 31 unt; 0 def)
% Number of atoms : 59 ( 29 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 34 ( 17 ~; 8 |; 5 &)
% ( 1 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 4 con; 0-2 aty)
% Number of variables : 60 ( 4 sgn 21 !; 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(idempotence,axiom,
! [A] : addition(A,A) = A,
file('theBenchmark.p',idempotence) ).
fof(multiplicative_right_identity,axiom,
! [A] : multiplication(A,one) = A,
file('theBenchmark.p',multiplicative_right_identity) ).
fof(distributivity1,axiom,
! [A,B,C] : multiplication(A,addition(B,C)) = addition(multiplication(A,B),multiplication(A,C)),
file('theBenchmark.p',distributivity1) ).
fof(star_unfold1,axiom,
! [A] : addition(one,multiplication(A,star(A))) = star(A),
file('theBenchmark.p',star_unfold1) ).
fof(star_induction2,axiom,
! [A,B,C] :
( leq(addition(multiplication(C,A),B),C)
=> leq(multiplication(B,star(A)),C) ),
file('theBenchmark.p',star_induction2) ).
fof(order,axiom,
! [A,B] :
( leq(A,B)
<=> addition(A,B) = B ),
file('theBenchmark.p',order) ).
fof(goals,conjecture,
! [X0,X1] :
( leq(X0,multiplication(X0,star(X1)))
& ( multiplication(X0,X1) = zero
=> leq(multiplication(X0,star(X1)),X0) ) ),
file('theBenchmark.p',goals) ).
fof(goals_neg,negated_conjecture,
~ ! [X0,X1] :
( leq(X0,multiplication(X0,star(X1)))
& ( multiplication(X0,X1) = zero
=> leq(multiplication(X0,star(X1)),X0) ) ),
inference(assume_negation,[status(cth)],[goals]) ).
cnf(c_20,plain,
addition(A,B) = addition(B,A),
inference(clausify,[status(thm)],[additive_commutativity]) ).
cnf(c_21,plain,
addition(addition(A,B),C) = addition(A,addition(B,C)),
inference(clausify,[status(thm)],[additive_associativity]) ).
cnf(c_22,plain,
addition(A,zero) = A,
inference(clausify,[status(thm)],[additive_identity]) ).
cnf(c_23,plain,
addition(A,A) = A,
inference(clausify,[status(thm)],[idempotence]) ).
cnf(c_25,plain,
multiplication(A,one) = A,
inference(clausify,[status(thm)],[multiplicative_right_identity]) ).
cnf(c_27,plain,
multiplication(A,addition(B,C)) = addition(multiplication(A,B),multiplication(A,C)),
inference(clausify,[status(thm)],[distributivity1]) ).
cnf(c_28,plain,
addition(multiplication(A,B),multiplication(A,C)) = multiplication(A,addition(B,C)),
inference(copy,[status(thm)],[c_27]) ).
cnf(c_32,plain,
star(A) = addition(one,multiplication(A,star(A))),
inference(clausify,[status(thm)],[star_unfold1]) ).
cnf(c_33,plain,
addition(one,multiplication(A,star(A))) = star(A),
inference(copy,[status(thm)],[c_32]) ).
cnf(c_37,plain,
( leq(multiplication(C,star(B)),A)
| ~ leq(addition(multiplication(A,B),C),A) ),
inference(clausify,[status(thm)],[star_induction2]) ).
cnf(c_44,plain,
( addition(A,B) != B
| leq(A,B) ),
inference(clausify,[status(thm)],[order]) ).
fof(nnf_19,plain,
? [X0,X1] :
( ~ leq(X0,multiplication(X0,star(X1)))
| ( ~ leq(multiplication(X0,star(X1)),X0)
& multiplication(X0,X1) = zero ) ),
inference(fof_nnf,[status(thm)],[goals_neg]) ).
fof(sk_goals_neg_sk,plain,
( ~ leq(sK0,multiplication(sK0,star(sK1)))
| ( ~ leq(multiplication(sK0,star(sK1)),sK0)
& multiplication(sK0,sK1) = zero ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(X0,sK0),skolemize(X1,sK1)],[nnf_19]) ).
fof(sk_goals_neg,plain,
( ( ~ leq(sK0,multiplication(sK0,star(sK1)))
| ~ leq(multiplication(sK0,star(sK1)),sK0) )
& ( ~ leq(sK0,multiplication(sK0,star(sK1)))
| multiplication(sK0,sK1) = zero ) ),
inference(cnf_transformation,[status(thm)],[sk_goals_neg_sk]) ).
cnf(c_45,negated_conjecture,
( ~ leq(sK0,multiplication(sK0,star(sK1)))
| multiplication(sK0,sK1) = zero ),
inference(split_conjunct,[status(thm)],[sk_goals_neg]) ).
cnf(c_46,negated_conjecture,
( ~ leq(sK0,multiplication(sK0,star(sK1)))
| ~ leq(multiplication(sK0,star(sK1)),sK0) ),
inference(split_conjunct,[status(thm)],[sk_goals_neg]) ).
cnf(c_186,plain,
A = addition(zero,A),
inference(paramod,[status(thm)],[c_22,c_20]) ).
cnf(c_49,plain,
addition(zero,A) = A,
inference(copy,[status(thm)],[c_186]) ).
cnf(c_187,plain,
addition(A,B) = addition(A,addition(A,B)),
inference(paramod,[status(thm)],[c_23,c_21]) ).
cnf(c_50,plain,
addition(A,addition(A,B)) = addition(A,B),
inference(copy,[status(thm)],[c_187]) ).
cnf(c_78,plain,
leq(A,A),
inference(resolve,[status(thm)],[c_44,c_23]) ).
cnf(c_92,plain,
leq(A,addition(A,B)),
inference(resolve,[status(thm)],[c_44,c_50]) ).
cnf(c_100,plain,
leq(multiplication(A,B),multiplication(A,addition(B,C))),
inference(paramod,[status(thm)],[c_28,c_92]) ).
cnf(c_188,plain,
leq(multiplication(A,one),multiplication(A,star(B))),
inference(paramod,[status(thm)],[c_33,c_100]) ).
cnf(c_148,plain,
leq(A,multiplication(A,star(B))),
inference(paramod,[status(thm)],[c_25,c_188]) ).
cnf(c_154,negated_conjecture,
~ leq(multiplication(sK0,star(sK1)),sK0),
inference(resolve,[status(thm)],[c_148,c_46]) ).
cnf(c_155,negated_conjecture,
multiplication(sK0,sK1) = zero,
inference(resolve,[status(thm)],[c_148,c_45]) ).
cnf(c_189,negated_conjecture,
~ leq(addition(multiplication(sK0,sK1),sK0),sK0),
inference(resolve,[status(thm)],[c_37,c_154]) ).
cnf(c_190,negated_conjecture,
~ leq(addition(zero,sK0),sK0),
inference(paramod,[status(thm)],[c_155,c_189]) ).
cnf(c_191,negated_conjecture,
~ leq(sK0,sK0),
inference(paramod,[status(thm)],[c_49,c_190]) ).
cnf(c_185,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_78,c_191]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : KLE167+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.20/0.45 % Computer : n015.cluster.edu
% 0.20/0.45 % Model : x86_64 x86_64
% 0.20/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.20/0.45 % Memory : 8046.5625MB
% 0.20/0.45 % OS : Linux 6.8.0-71-generic
% 0.20/0.45 % CPULimit : 300
% 0.20/0.45 % WCLimit : 300
% 0.20/0.46 % DateTime : Fri Sep 25 19:33:12 UTC 2026
% 0.20/0.46 % CPUTime :
% 0.20/0.46 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.20/0.46 % Process 2066007 was started by sandbox on n015,
% 0.20/0.46 % Fri Sep 25 19:33:12 2026
% 0.20/0.46 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.20/0.47
% 0.20/0.47 % From the command line: assign(max_seconds, 300).
% 0.24/0.54
% 0.24/0.54 % SZS status Theorem for theBenchmark
% 0.24/0.54
% 0.24/0.54 % Proof 1 at 0.03 (+ 0.02) seconds.
% 0.24/0.54 % Length of proof is 32.
% 0.24/0.54 % Level of proof is 7.
% 0.24/0.54 % Maximum clause weight is 30.000.
% 0.24/0.54 % Given clauses 51.
% 0.24/0.54
% 0.24/0.54 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------