%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU098+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n018.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 08:45:01 AM UTC 2026
% Result : Theorem 18.27s 2.76s
% Output : CNFRefutation 18.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 12
% Syntax : Number of formulae : 64 ( 20 unt; 0 def)
% Number of atoms : 148 ( 15 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 131 ( 47 ~; 52 |; 19 &)
% ( 3 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-4 aty)
% Number of variables : 91 ( 10 sgn 24 !; 1 ?)
% Comments :
%------------------------------------------------------------------------------
fof(dt_k7_funct_3,axiom,
! [A,B] :
( function(first_projection(A,B))
& relation(first_projection(A,B)) ),
file('theBenchmark.p',dt_k7_funct_3) ).
fof(dt_k9_funct_3,axiom,
! [A,B] :
( relation_of2_as_subset(first_projection_as_func_of(A,B),cartesian_product2(A,B),A)
& quasi_total(first_projection_as_func_of(A,B),cartesian_product2(A,B),A)
& function(first_projection_as_func_of(A,B)) ),
file('theBenchmark.p',dt_k9_funct_3) ).
fof(fc13_finset_1,axiom,
! [A,B] :
( ( finite(B)
& function(A)
& relation(A) )
=> finite(relation_image(A,B)) ),
file('theBenchmark.p',fc13_finset_1) ).
fof(redefinition_k2_funct_2,axiom,
! [A,B,C,D] :
( ( relation_of2(C,A,B)
& quasi_total(C,A,B)
& function(C) )
=> function_image(A,B,C,D) = relation_image(C,D) ),
file('theBenchmark.p',redefinition_k2_funct_2) ).
fof(redefinition_k9_funct_3,axiom,
! [A,B] : first_projection_as_func_of(A,B) = first_projection(A,B),
file('theBenchmark.p',redefinition_k9_funct_3) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ),
file('theBenchmark.p',redefinition_m2_relset_1) ).
fof(t100_funct_3,axiom,
! [A] :
( ( function(A)
& relation(A) )
=> function_image(cartesian_product2(relation_dom(A),relation_rng(A)),relation_dom(A),first_projection_as_func_of(relation_dom(A),relation_rng(A)),A) = relation_dom(A) ),
file('theBenchmark.p',t100_funct_3) ).
fof(t13_finset_1,axiom,
! [A,B] :
( ( finite(B)
& subset(A,B) )
=> finite(A) ),
file('theBenchmark.p',t13_finset_1) ).
fof(t19_finset_1,axiom,
! [A,B] :
( ( finite(B)
& finite(A) )
=> finite(cartesian_product2(A,B)) ),
file('theBenchmark.p',t19_finset_1) ).
fof(t21_relat_1,axiom,
! [A] :
( relation(A)
=> subset(A,cartesian_product2(relation_dom(A),relation_rng(A))) ),
file('theBenchmark.p',t21_relat_1) ).
fof(t26_finset_1,axiom,
! [A] :
( ( function(A)
& relation(A) )
=> ( finite(relation_dom(A))
=> finite(relation_rng(A)) ) ),
file('theBenchmark.p',t26_finset_1) ).
fof(t29_finset_1,conjecture,
! [A] :
( ( function(A)
& relation(A) )
=> ( finite(relation_dom(A))
<=> finite(A) ) ),
file('theBenchmark.p',t29_finset_1) ).
fof(t29_finset_1_neg,negated_conjecture,
~ ! [A] :
( ( function(A)
& relation(A) )
=> ( finite(relation_dom(A))
<=> finite(A) ) ),
inference(assume_negation,[status(cth)],[t29_finset_1]) ).
cnf(c_114,plain,
function(first_projection(A,B)),
inference(clausify,[status(thm)],[dt_k7_funct_3]) ).
cnf(c_115,plain,
function(first_projection_as_func_of(A,B)),
inference(clausify,[status(thm)],[dt_k9_funct_3]) ).
cnf(c_116,plain,
( finite(relation_image(A,B))
| ~ finite(B)
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[fc13_finset_1]) ).
cnf(c_130,plain,
( relation_image(A,D) = function_image(B,C,A,D)
| ~ relation_of2(A,B,C)
| ~ quasi_total(A,B,C)
| ~ function(A) ),
inference(clausify,[status(thm)],[redefinition_k2_funct_2]) ).
cnf(c_131,plain,
( relation_dom(A) = function_image(cartesian_product2(relation_dom(A),relation_rng(A)),relation_dom(A),first_projection_as_func_of(relation_dom(A),relation_rng(A)),A)
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t100_funct_3]) ).
cnf(c_133,plain,
( finite(relation_rng(A))
| ~ finite(relation_dom(A))
| ~ function(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t26_finset_1]) ).
fof(nnf_78,plain,
? [A] :
( ( ~ finite(A)
| ~ finite(relation_dom(A)) )
& ( finite(A)
| finite(relation_dom(A)) )
& function(A)
& relation(A) ),
inference(fof_nnf,[status(thm)],[t29_finset_1_neg]) ).
fof(sk_t29_finset_1_neg,plain,
( ( ~ finite(sK0)
| ~ finite(relation_dom(sK0)) )
& ( finite(sK0)
| finite(relation_dom(sK0)) )
& function(sK0)
& relation(sK0) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(A,sK0)],[nnf_78]) ).
cnf(c_134,negated_conjecture,
function(sK0),
inference(split_conjunct,[status(thm)],[sk_t29_finset_1_neg]) ).
cnf(c_135,plain,
( relation_of2(A,B,C)
| ~ relation_of2_as_subset(A,B,C) ),
inference(clausify,[status(thm)],[redefinition_m2_relset_1]) ).
cnf(c_139,plain,
relation_of2_as_subset(first_projection_as_func_of(A,B),cartesian_product2(A,B),A),
inference(clausify,[status(thm)],[dt_k9_funct_3]) ).
cnf(c_140,plain,
( finite(A)
| ~ finite(B)
| ~ subset(A,B) ),
inference(clausify,[status(thm)],[t13_finset_1]) ).
cnf(c_142,plain,
( subset(A,cartesian_product2(relation_dom(A),relation_rng(A)))
| ~ relation(A) ),
inference(clausify,[status(thm)],[t21_relat_1]) ).
cnf(c_148,plain,
( finite(A)
| ~ finite(cartesian_product2(relation_dom(A),relation_rng(A)))
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_142,c_140]) ).
cnf(c_153,negated_conjecture,
( relation_dom(sK0) = function_image(cartesian_product2(relation_dom(sK0),relation_rng(sK0)),relation_dom(sK0),first_projection_as_func_of(relation_dom(sK0),relation_rng(sK0)),sK0)
| ~ relation(sK0) ),
inference(resolve,[status(thm)],[c_134,c_131]) ).
cnf(c_154,negated_conjecture,
( function_image(cartesian_product2(relation_dom(sK0),relation_rng(sK0)),relation_dom(sK0),first_projection_as_func_of(relation_dom(sK0),relation_rng(sK0)),sK0) = relation_dom(sK0)
| ~ relation(sK0) ),
inference(copy,[status(thm)],[c_153]) ).
cnf(c_156,negated_conjecture,
( finite(relation_rng(sK0))
| ~ finite(relation_dom(sK0))
| ~ relation(sK0) ),
inference(resolve,[status(thm)],[c_134,c_133]) ).
cnf(c_201,plain,
( relation_image(first_projection(A,B),E) = function_image(C,D,first_projection(A,B),E)
| ~ relation_of2(first_projection(A,B),C,D)
| ~ quasi_total(first_projection(A,B),C,D) ),
inference(resolve,[status(thm)],[c_130,c_114]) ).
cnf(c_202,plain,
( function_image(C,D,first_projection(A,B),E) = relation_image(first_projection(A,B),E)
| ~ relation_of2(first_projection(A,B),C,D)
| ~ quasi_total(first_projection(A,B),C,D) ),
inference(copy,[status(thm)],[c_201]) ).
cnf(c_265,plain,
( finite(relation_image(first_projection_as_func_of(A,B),C))
| ~ finite(C)
| ~ relation(first_projection_as_func_of(A,B)) ),
inference(resolve,[status(thm)],[c_116,c_115]) ).
cnf(c_269,negated_conjecture,
( ~ finite(sK0)
| ~ finite(relation_dom(sK0)) ),
inference(split_conjunct,[status(thm)],[sk_t29_finset_1_neg]) ).
cnf(c_270,negated_conjecture,
( finite(sK0)
| finite(relation_dom(sK0)) ),
inference(split_conjunct,[status(thm)],[sk_t29_finset_1_neg]) ).
cnf(c_271,negated_conjecture,
relation(sK0),
inference(split_conjunct,[status(thm)],[sk_t29_finset_1_neg]) ).
cnf(c_279,plain,
( finite(cartesian_product2(A,B))
| ~ finite(B)
| ~ finite(A) ),
inference(clausify,[status(thm)],[t19_finset_1]) ).
cnf(c_280,plain,
first_projection_as_func_of(A,B) = first_projection(A,B),
inference(clausify,[status(thm)],[redefinition_k9_funct_3]) ).
cnf(c_345,plain,
quasi_total(first_projection_as_func_of(A,B),cartesian_product2(A,B),A),
inference(clausify,[status(thm)],[dt_k9_funct_3]) ).
cnf(c_346,plain,
quasi_total(first_projection(A,B),cartesian_product2(A,B),A),
inference(paramod,[status(thm)],[c_280,c_345]) ).
cnf(c_347,plain,
relation(first_projection(A,B)),
inference(clausify,[status(thm)],[dt_k7_funct_3]) ).
cnf(c_359,plain,
relation_of2(first_projection_as_func_of(A,B),cartesian_product2(A,B),A),
inference(resolve,[status(thm)],[c_139,c_135]) ).
cnf(c_360,plain,
relation_of2(first_projection(A,B),cartesian_product2(A,B),A),
inference(paramod,[status(thm)],[c_280,c_359]) ).
cnf(c_361,negated_conjecture,
( finite(relation_rng(sK0))
| ~ finite(relation_dom(sK0)) ),
inference(resolve,[status(thm)],[c_271,c_156]) ).
cnf(c_18236,negated_conjecture,
( function_image(cartesian_product2(relation_dom(sK0),relation_rng(sK0)),relation_dom(sK0),first_projection(relation_dom(sK0),relation_rng(sK0)),sK0) = relation_dom(sK0)
| ~ relation(sK0) ),
inference(paramod,[status(thm)],[c_280,c_154]) ).
cnf(c_363,negated_conjecture,
function_image(cartesian_product2(relation_dom(sK0),relation_rng(sK0)),relation_dom(sK0),first_projection(relation_dom(sK0),relation_rng(sK0)),sK0) = relation_dom(sK0),
inference(resolve,[status(thm)],[c_271,c_18236]) ).
cnf(c_18237,plain,
( finite(relation_image(first_projection_as_func_of(A,B),C))
| ~ finite(C)
| ~ relation(first_projection(A,B)) ),
inference(paramod,[status(thm)],[c_280,c_265]) ).
cnf(c_18238,plain,
( finite(relation_image(first_projection(A,B),C))
| ~ finite(C)
| ~ relation(first_projection(A,B)) ),
inference(paramod,[status(thm)],[c_280,c_18237]) ).
cnf(c_366,plain,
( finite(relation_image(first_projection(B,C),A))
| ~ finite(A) ),
inference(resolve,[status(thm)],[c_347,c_18238]) ).
cnf(c_423,plain,
( finite(sK0)
| ~ finite(cartesian_product2(relation_dom(sK0),relation_rng(sK0))) ),
inference(resolve,[status(thm)],[c_271,c_148]) ).
cnf(c_427,plain,
( finite(sK0)
| finite(cartesian_product2(relation_dom(sK0),A))
| ~ finite(A) ),
inference(resolve,[status(thm)],[c_279,c_270]) ).
cnf(c_18239,plain,
( function_image(cartesian_product2(A,B),A,first_projection(A,B),C) = relation_image(first_projection(A,B),C)
| ~ relation_of2(first_projection(A,B),cartesian_product2(A,B),A) ),
inference(resolve,[status(thm)],[c_346,c_202]) ).
cnf(c_681,plain,
function_image(cartesian_product2(A,B),A,first_projection(A,B),C) = relation_image(first_projection(A,B),C),
inference(resolve,[status(thm)],[c_360,c_18239]) ).
cnf(c_693,negated_conjecture,
relation_image(first_projection(relation_dom(sK0),relation_rng(sK0)),sK0) = relation_dom(sK0),
inference(paramod,[status(thm)],[c_681,c_363]) ).
cnf(c_743,plain,
( finite(sK0)
| finite(relation_rng(sK0)) ),
inference(resolve,[status(thm)],[c_361,c_270]) ).
cnf(c_757,plain,
( finite(relation_dom(sK0))
| finite(relation_image(first_projection(A,B),sK0)) ),
inference(resolve,[status(thm)],[c_366,c_270]) ).
cnf(c_18240,plain,
( finite(sK0)
| finite(cartesian_product2(relation_dom(sK0),relation_rng(sK0)))
| finite(sK0) ),
inference(resolve,[status(thm)],[c_743,c_427]) ).
cnf(c_2783,plain,
( finite(cartesian_product2(relation_dom(sK0),relation_rng(sK0)))
| finite(sK0) ),
inference(copy,[status(thm)],[c_18240]) ).
cnf(c_18241,plain,
( finite(relation_dom(sK0))
| finite(relation_dom(sK0)) ),
inference(paramod,[status(thm)],[c_693,c_757]) ).
cnf(c_17588,plain,
finite(relation_dom(sK0)),
inference(copy,[status(thm)],[c_18241]) ).
cnf(c_17593,negated_conjecture,
~ finite(sK0),
inference(resolve,[status(thm)],[c_17588,c_269]) ).
cnf(c_18104,negated_conjecture,
finite(cartesian_product2(relation_dom(sK0),relation_rng(sK0))),
inference(resolve,[status(thm)],[c_17593,c_2783]) ).
cnf(c_18242,plain,
finite(sK0),
inference(resolve,[status(thm)],[c_18104,c_423]) ).
cnf(c_18235,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_17593,c_18242]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SEU098+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n018.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sat Sep 26 08:37:51 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.10/0.37 % Process 23687 was started by sandbox on n018,
% 0.10/0.37 % Sat Sep 26 08:37:51 2026
% 0.10/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.10/0.39
% 0.10/0.39 % From the command line: assign(max_seconds, 300).
% 18.27/2.76
% 18.27/2.76 % given #500 (T,wt=10): 2079 -ordinal(powerset(c2)) | ordinal(f7(c2)). [resolve(590,a,352,b)].
% 18.27/2.76
% 18.27/2.76 % SZS status Theorem for theBenchmark
% 18.27/2.76
% 18.27/2.76 % Proof 1 at 1.69 (+ 0.65) seconds.
% 18.27/2.76 % Length of proof is 54.
% 18.27/2.76 % Level of proof is 9.
% 18.27/2.76 % Maximum clause weight is 31.000.
% 18.27/2.76 % Given clauses 908.
% 18.27/2.76
% 18.27/2.76 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------