%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU174+1 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n016.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 0.09s 0.41s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 9
% Syntax : Number of formulae : 53 ( 28 unt; 0 def)
% Number of atoms : 142 ( 41 equ)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 148 ( 59 ~; 56 |; 23 &)
% ( 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 : 9 ( 9 usr; 4 con; 0-3 aty)
% Number of variables : 81 ( 10 sgn 46 !; 7 ?)
% 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(rc2_subset_1,axiom,
! [A] :
? [B] :
( empty(B)
& element(B,powerset(A)) ),
file('theBenchmark.p',rc2_subset_1) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A),
file('theBenchmark.p',reflexivity_r1_tarski) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ),
file('theBenchmark.p',t3_subset) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( empty(C)
& element(B,powerset(C))
& in(A,B) ),
file('theBenchmark.p',t5_subset) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('theBenchmark.p',t6_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]) ).
cnf(c_30,plain,
subset(A,A),
inference(clausify,[status(thm)],[reflexivity_r1_tarski]) ).
cnf(c_32,plain,
( ~ subset(A,B)
| element(A,powerset(B)) ),
inference(clausify,[status(thm)],[t3_subset]) ).
cnf(c_35,plain,
element(A,powerset(A)),
inference(resolve,[status(thm)],[c_32,c_30]) ).
fof(nnf_26,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_26]) ).
cnf(c_39,negated_conjecture,
complements_of_subsets(sK0,sK1) = empty_set,
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_40,negated_conjecture,
empty_set = complements_of_subsets(sK0,sK1),
inference(copy,[status(thm)],[c_39]) ).
cnf(c_41,negated_conjecture,
empty_set != sK1,
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_42,negated_conjecture,
complements_of_subsets(sK0,sK1) != sK1,
inference(paramod,[status(thm)],[c_40,c_41]) ).
cnf(c_43,negated_conjecture,
element(sK1,powerset(powerset(sK0))),
inference(split_conjunct,[status(thm)],[sk_t46_setfam_1_neg]) ).
cnf(c_46,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
cnf(c_47,plain,
( complements_of_subsets(sK0,sK1) = A
| ~ empty(A) ),
inference(paramod,[status(thm)],[c_40,c_46]) ).
cnf(c_48,plain,
( ~ empty(C)
| ~ element(B,powerset(C))
| ~ in(A,B) ),
inference(clausify,[status(thm)],[t5_subset]) ).
fof(nnf_15,plain,
! [A] :
? [B] :
( empty(B)
& element(B,powerset(A)) ),
inference(fof_nnf,[status(thm)],[rc2_subset_1]) ).
fof(sk_rc2_subset_1,plain,
! [A] :
( empty(sK2(A))
& element(sK2(A),powerset(A)) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(B,sK2(A))],[nnf_15]) ).
cnf(c_53,plain,
empty(sK2(A)),
inference(split_conjunct,[status(thm)],[sk_rc2_subset_1]) ).
cnf(c_54,plain,
element(sK2(A),powerset(A)),
inference(split_conjunct,[status(thm)],[sk_rc2_subset_1]) ).
fof(nnf_14,plain,
? [A] : empty(A),
inference(fof_nnf,[status(thm)],[rc1_xboole_0]) ).
fof(sk_rc1_xboole_0,plain,
empty(sK3),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(A,sK3)],[nnf_14]) ).
cnf(c_55,plain,
empty(sK3),
inference(split_conjunct,[status(thm)],[sk_rc1_xboole_0]) ).
cnf(c_58,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_3,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,sK4(A,B,C)),B)
| ~ in(sK4(A,B,C),C) )
& ( in(subset_complement(A,sK4(A,B,C)),B)
| in(sK4(A,B,C),C) )
& element(sK4(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,[sK4]),skolemize(D,sK4(A,B,C))],[nnf_3]) ).
fof(sk_d8_setfam_1,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( ~ in(subset_complement(VAR_1,sK4(VAR_1,VAR_0,VAR_2)),VAR_0)
| ~ in(sK4(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,sK4(VAR_1,VAR_0,VAR_2)),VAR_0)
| in(sK4(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(sK4(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_67,plain,
( in(subset_complement(B,sK4(B,A,C)),A)
| in(sK4(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_73,plain,
( in(subset_complement(B,sK4(B,A,A)),A)
| in(sK4(B,A,A),A)
| complements_of_subsets(B,A) = A
| ~ element(A,powerset(powerset(B))) ),
inference(factor,[status(thm)],[c_67]) ).
cnf(c_180,plain,
complements_of_subsets(sK0,sK1) = sK2(A),
inference(resolve,[status(thm)],[c_53,c_47]) ).
cnf(c_88,plain,
sK2(A) = complements_of_subsets(sK0,sK1),
inference(copy,[status(thm)],[c_180]) ).
cnf(c_90,plain,
element(complements_of_subsets(sK0,sK1),powerset(A)),
inference(paramod,[status(thm)],[c_88,c_54]) ).
cnf(c_91,plain,
complements_of_subsets(sK0,sK1) = sK3,
inference(resolve,[status(thm)],[c_55,c_47]) ).
cnf(c_181,plain,
( ~ empty(B)
| ~ in(A,B) ),
inference(resolve,[status(thm)],[c_48,c_35]) ).
cnf(c_93,plain,
~ in(A,sK3),
inference(resolve,[status(thm)],[c_181,c_55]) ).
cnf(c_94,plain,
element(sK3,powerset(A)),
inference(paramod,[status(thm)],[c_91,c_90]) ).
cnf(c_182,negated_conjecture,
sK3 != sK1,
inference(paramod,[status(thm)],[c_91,c_42]) ).
cnf(c_96,negated_conjecture,
sK1 != sK3,
inference(copy,[status(thm)],[c_182]) ).
cnf(c_183,negated_conjecture,
complements_of_subsets(sK0,complements_of_subsets(sK0,sK1)) = sK1,
inference(resolve,[status(thm)],[c_58,c_43]) ).
cnf(c_101,negated_conjecture,
complements_of_subsets(sK0,sK3) = sK1,
inference(paramod,[status(thm)],[c_91,c_183]) ).
cnf(c_184,plain,
( in(subset_complement(A,sK4(A,sK3,sK3)),sK3)
| in(sK4(A,sK3,sK3),sK3)
| complements_of_subsets(A,sK3) = sK3 ),
inference(resolve,[status(thm)],[c_94,c_73]) ).
cnf(c_185,plain,
( in(subset_complement(A,sK4(A,sK3,sK3)),sK3)
| complements_of_subsets(A,sK3) = sK3 ),
inference(resolve,[status(thm)],[c_93,c_184]) ).
cnf(c_169,plain,
complements_of_subsets(A,sK3) = sK3,
inference(resolve,[status(thm)],[c_93,c_185]) ).
cnf(c_186,negated_conjecture,
sK3 = sK1,
inference(paramod,[status(thm)],[c_169,c_101]) ).
cnf(c_187,negated_conjecture,
sK1 = sK3,
inference(copy,[status(thm)],[c_186]) ).
cnf(c_179,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_96,c_187]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU174+1 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n016.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sat Sep 26 08:49:31 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.37 % Process 464330 was started by sandbox on n016,
% 0.09/0.37 % Sat Sep 26 08:49:31 2026
% 0.09/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.09/0.38
% 0.09/0.38 % From the command line: assign(max_seconds, 300).
% 0.09/0.41
% 0.09/0.41 % SZS status Theorem for theBenchmark
% 0.09/0.41
% 0.09/0.41 % Proof 1 at 0.02 (+ 0.00) seconds.
% 0.09/0.41 % Length of proof is 35.
% 0.09/0.41 % Level of proof is 8.
% 0.09/0.41 % Maximum clause weight is 29.000.
% 0.09/0.41 % Given clauses 47.
% 0.09/0.41
% 0.09/0.41 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------