%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU174+2 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n012.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 08:45:19 AM UTC 2026
% Result : Theorem 2.93s 0.80s
% Output : CNFRefutation 2.93s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 8
% Syntax : Number of formulae : 45 ( 24 unt; 0 def)
% Number of atoms : 128 ( 39 equ)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 139 ( 56 ~; 54 |; 19 &)
% ( 3 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-3 aty)
% Number of variables : 67 ( 7 sgn 41 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(d8_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ! [C] :
( element(C,powerset(powerset(A)))
=> ( C = complements_of_subsets(A,B)
<=> ! [D] :
( element(D,powerset(A))
=> ( in(D,C)
<=> in(subset_complement(A,D),B) ) ) ) ) ),
file('theBenchmark.p',d8_setfam_1) ).
fof(involutiveness_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> complements_of_subsets(A,complements_of_subsets(A,B)) = B ),
file('theBenchmark.p',involutiveness_k7_setfam_1) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A),
file('theBenchmark.p',rc1_xboole_0) ).
fof(t2_xboole_1,axiom,
! [A] : subset(empty_set,A),
file('theBenchmark.p',t2_xboole_1) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ),
file('theBenchmark.p',t3_subset) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('theBenchmark.p',t6_boole) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( empty(B)
& in(A,B) ),
file('theBenchmark.p',t7_boole) ).
fof(t46_setfam_1,conjecture,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ~ ( complements_of_subsets(A,B) = empty_set
& B != empty_set ) ),
file('theBenchmark.p',t46_setfam_1) ).
fof(t46_setfam_1_neg,negated_conjecture,
~ ! [A,B] :
( element(B,powerset(powerset(A)))
=> ~ ( complements_of_subsets(A,B) = empty_set
& B != empty_set ) ),
inference(assume_negation,[status(cth)],[t46_setfam_1]) ).
fof(nnf_120,plain,
? [A,B] :
( complements_of_subsets(A,B) = empty_set
& empty_set != B
& element(B,powerset(powerset(A))) ),
inference(fof_nnf,[status(thm)],[t46_setfam_1_neg]) ).
fof(sk_t46_setfam_1_neg,plain,
( complements_of_subsets(sK0,sK1) = empty_set
& empty_set != sK1
& element(sK1,powerset(powerset(sK0))) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK0]),skolemize(A,sK0),skolemize(B,sK1)],[nnf_120]) ).
cnf(c_121,negated_conjecture,
complements_of_subsets(sK0,sK1) = empty_set,
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_122,negated_conjecture,
empty_set = complements_of_subsets(sK0,sK1),
inference(copy,[status(thm)],[c_121]) ).
cnf(c_123,negated_conjecture,
empty_set != sK1,
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_124,negated_conjecture,
complements_of_subsets(sK0,sK1) != sK1,
inference(paramod,[status(thm)],[c_122,c_123]) ).
cnf(c_125,negated_conjecture,
element(sK1,powerset(powerset(sK0))),
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_139,plain,
( ~ empty(B)
| ~ in(A,B) ),
inference(clausify,[status(thm)],[t7_boole]) ).
cnf(c_141,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
cnf(c_142,plain,
( complements_of_subsets(sK0,sK1) = A
| ~ empty(A) ),
inference(paramod,[status(thm)],[c_122,c_141]) ).
cnf(c_172,plain,
( ~ subset(A,B)
| element(A,powerset(B)) ),
inference(clausify,[status(thm)],[t3_subset]) ).
cnf(c_197,plain,
subset(empty_set,A),
inference(clausify,[status(thm)],[t2_xboole_1]) ).
cnf(c_198,plain,
subset(complements_of_subsets(sK0,sK1),A),
inference(paramod,[status(thm)],[c_122,c_197]) ).
fof(nnf_59,plain,
? [A] : empty(A),
inference(fof_nnf,[status(thm)],[rc1_xboole_0]) ).
fof(sk_rc1_xboole_0,plain,
empty(sK2),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(A,sK2)],[nnf_59]) ).
cnf(c_233,plain,
empty(sK2),
inference(split_conjunct,[status(thm)],[sk_rc1_xboole_0]) ).
cnf(c_256,plain,
( complements_of_subsets(B,complements_of_subsets(B,A)) = A
| ~ element(A,powerset(powerset(B))) ),
inference(clausify,[status(thm)],[involutiveness_k7_setfam_1]) ).
fof(nnf_21,plain,
! [A,B] :
( ! [C] :
( ( ( ? [D] :
( ( ~ in(subset_complement(A,D),B)
| ~ in(D,C) )
& ( in(subset_complement(A,D),B)
| in(D,C) )
& element(D,powerset(A)) )
| complements_of_subsets(A,B) = C )
& ( ! [X0] :
( ( ( ~ in(subset_complement(A,X0),B)
| in(X0,C) )
& ( in(subset_complement(A,X0),B)
| ~ in(X0,C) ) )
| ~ element(X0,powerset(A)) )
| complements_of_subsets(A,B) != C ) )
| ~ element(C,powerset(powerset(A))) )
| ~ element(B,powerset(powerset(A))) ),
inference(fof_nnf,[status(thm)],[d8_setfam_1]) ).
fof(sk_d8_setfam_1_sk,plain,
! [A,B] :
( ! [C] :
( ( ( ( ( ~ in(subset_complement(A,sK3(A,B,C)),B)
| ~ in(sK3(A,B,C),C) )
& ( in(subset_complement(A,sK3(A,B,C)),B)
| in(sK3(A,B,C),C) )
& element(sK3(A,B,C),powerset(A)) )
| complements_of_subsets(A,B) = C )
& ( ! [X0] :
( ( ( ~ in(subset_complement(A,X0),B)
| in(X0,C) )
& ( in(subset_complement(A,X0),B)
| ~ in(X0,C) ) )
| ~ element(X0,powerset(A)) )
| complements_of_subsets(A,B) != C ) )
| ~ element(C,powerset(powerset(A))) )
| ~ element(B,powerset(powerset(A))) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(D,sK3(A,B,C))],[nnf_21]) ).
fof(sk_d8_setfam_1,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( ~ in(subset_complement(VAR_1,sK3(VAR_1,VAR_0,VAR_2)),VAR_0)
| ~ in(sK3(VAR_1,VAR_0,VAR_2),VAR_2)
| complements_of_subsets(VAR_1,VAR_0) = VAR_2
| ~ element(VAR_2,powerset(powerset(VAR_1)))
| ~ element(VAR_0,powerset(powerset(VAR_1))) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(subset_complement(VAR_1,sK3(VAR_1,VAR_0,VAR_2)),VAR_0)
| in(sK3(VAR_1,VAR_0,VAR_2),VAR_2)
| complements_of_subsets(VAR_1,VAR_0) = VAR_2
| ~ element(VAR_2,powerset(powerset(VAR_1)))
| ~ element(VAR_0,powerset(powerset(VAR_1))) )
& ! [VAR_0,VAR_1,VAR_2] :
( element(sK3(VAR_1,VAR_0,VAR_2),powerset(VAR_1))
| complements_of_subsets(VAR_1,VAR_0) = VAR_2
| ~ element(VAR_2,powerset(powerset(VAR_1)))
| ~ element(VAR_0,powerset(powerset(VAR_1))) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( ~ in(subset_complement(VAR_1,VAR_3),VAR_0)
| in(VAR_3,VAR_2)
| ~ element(VAR_3,powerset(VAR_1))
| complements_of_subsets(VAR_1,VAR_0) != VAR_2
| ~ element(VAR_2,powerset(powerset(VAR_1)))
| ~ element(VAR_0,powerset(powerset(VAR_1))) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( in(subset_complement(VAR_1,VAR_3),VAR_0)
| ~ in(VAR_3,VAR_2)
| ~ element(VAR_3,powerset(VAR_1))
| complements_of_subsets(VAR_1,VAR_0) != VAR_2
| ~ element(VAR_2,powerset(powerset(VAR_1)))
| ~ element(VAR_0,powerset(powerset(VAR_1))) ) ),
inference(cnf_transformation,[status(thm)],[sk_d8_setfam_1_sk]) ).
cnf(c_274,plain,
( in(subset_complement(B,sK3(B,A,C)),A)
| in(sK3(B,A,C),C)
| complements_of_subsets(B,A) = C
| ~ element(C,powerset(powerset(B)))
| ~ element(A,powerset(powerset(B))) ),
inference(split_conjunct,[status(thm)],[sk_d8_setfam_1]) ).
cnf(c_373,plain,
( in(subset_complement(B,sK3(B,A,A)),A)
| in(sK3(B,A,A),A)
| complements_of_subsets(B,A) = A
| ~ element(A,powerset(powerset(B))) ),
inference(factor,[status(thm)],[c_274]) ).
cnf(c_492,plain,
element(complements_of_subsets(sK0,sK1),powerset(A)),
inference(resolve,[status(thm)],[c_198,c_172]) ).
cnf(c_620,plain,
complements_of_subsets(sK0,sK1) = sK2,
inference(resolve,[status(thm)],[c_233,c_142]) ).
cnf(c_621,plain,
~ in(A,sK2),
inference(resolve,[status(thm)],[c_233,c_139]) ).
cnf(c_639,plain,
element(sK2,powerset(A)),
inference(paramod,[status(thm)],[c_620,c_492]) ).
cnf(c_3665,negated_conjecture,
sK2 != sK1,
inference(paramod,[status(thm)],[c_620,c_124]) ).
cnf(c_664,negated_conjecture,
sK1 != sK2,
inference(copy,[status(thm)],[c_3665]) ).
cnf(c_3666,negated_conjecture,
complements_of_subsets(sK0,complements_of_subsets(sK0,sK1)) = sK1,
inference(resolve,[status(thm)],[c_256,c_125]) ).
cnf(c_715,negated_conjecture,
complements_of_subsets(sK0,sK2) = sK1,
inference(paramod,[status(thm)],[c_620,c_3666]) ).
cnf(c_3667,plain,
( in(subset_complement(A,sK3(A,sK2,sK2)),sK2)
| in(sK3(A,sK2,sK2),sK2)
| complements_of_subsets(A,sK2) = sK2 ),
inference(resolve,[status(thm)],[c_639,c_373]) ).
cnf(c_3668,plain,
( in(subset_complement(A,sK3(A,sK2,sK2)),sK2)
| complements_of_subsets(A,sK2) = sK2 ),
inference(resolve,[status(thm)],[c_621,c_3667]) ).
cnf(c_3656,plain,
complements_of_subsets(A,sK2) = sK2,
inference(resolve,[status(thm)],[c_621,c_3668]) ).
cnf(c_3669,negated_conjecture,
sK2 = sK1,
inference(paramod,[status(thm)],[c_3656,c_715]) ).
cnf(c_3670,negated_conjecture,
sK1 = sK2,
inference(copy,[status(thm)],[c_3669]) ).
cnf(c_3664,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_664,c_3670]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : SEU174+2 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.02 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.03/0.36 % Computer : n012.cluster.edu
% 0.03/0.36 % Model : x86_64 x86_64
% 0.03/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.03/0.36 % Memory : 8046.5625MB
% 0.03/0.36 % OS : Linux 6.8.0-71-generic
% 0.03/0.36 % CPULimit : 300
% 0.03/0.36 % WCLimit : 300
% 0.03/0.36 % DateTime : Sat Sep 26 08:44:20 UTC 2026
% 0.03/0.36 % CPUTime :
% 0.03/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.03/0.36 % Process 174544 was started by sandbox2 on n012,
% 0.03/0.36 % Sat Sep 26 08:44:20 2026
% 0.03/0.36 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.03/0.37
% 0.03/0.37 % From the command line: assign(max_seconds, 300).
% 2.93/0.80
% 2.93/0.80 % SZS status Theorem for theBenchmark
% 2.93/0.80
% 2.93/0.80 % Proof 1 at 0.32 (+ 0.05) seconds.
% 2.93/0.80 % Length of proof is 31.
% 2.93/0.80 % Level of proof is 7.
% 2.93/0.80 % Maximum clause weight is 48.000.
% 2.93/0.80 % Given clauses 250.
% 2.93/0.80
% 2.93/0.80 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------