%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SEU187+2 : 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 : n004.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 08:45:21 AM UTC 2026
% Result : Theorem 12.34s 7.10s
% Output : CNFRefutation 12.34s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 22
% Syntax : Number of formulae : 96 ( 51 unt; 0 def)
% Number of atoms : 209 ( 72 equ)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 192 ( 79 ~; 81 |; 15 &)
% ( 4 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-3 aty)
% Number of variables : 136 ( 6 sgn 56 !; 6 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A),
file('theBenchmark.p',commutativity_k2_tarski) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( subset(B,A)
& subset(A,B) ) ),
file('theBenchmark.p',d10_xboole_0) ).
fof(d1_xboole_0,axiom,
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ),
file('theBenchmark.p',d1_xboole_0) ).
fof(d4_subset_1,axiom,
! [A] : cast_to_subset(A) = A,
file('theBenchmark.p',d4_subset_1) ).
fof(d5_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] : in(ordered_pair(D,C),A) ) ) ),
file('theBenchmark.p',d5_relat_1) ).
fof(d5_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> subset_complement(A,B) = set_difference(A,B) ),
file('theBenchmark.p',d5_subset_1) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
file('theBenchmark.p',d5_tarski) ).
fof(dt_k2_subset_1,axiom,
! [A] : element(cast_to_subset(A),powerset(A)),
file('theBenchmark.p',dt_k2_subset_1) ).
fof(dt_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> element(subset_complement(A,B),powerset(A)) ),
file('theBenchmark.p',dt_k3_subset_1) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A),
file('theBenchmark.p',existence_m1_subset_1) ).
fof(fc4_relat_1,axiom,
( relation(empty_set)
& empty(empty_set) ),
file('theBenchmark.p',fc4_relat_1) ).
fof(t25_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(A,B)
=> ( subset(relation_rng(A),relation_rng(B))
& subset(relation_dom(A),relation_dom(B)) ) ) ) ),
file('theBenchmark.p',t25_relat_1) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set,
file('theBenchmark.p',t2_boole) ).
fof(t2_xboole_1,axiom,
! [A] : subset(empty_set,A),
file('theBenchmark.p',t2_xboole_1) ).
fof(t3_boole,axiom,
! [A] : set_difference(A,empty_set) = A,
file('theBenchmark.p',t3_boole) ).
fof(t48_xboole_1,axiom,
! [A,B] : set_difference(A,set_difference(A,B)) = set_intersection2(A,B),
file('theBenchmark.p',t48_xboole_1) ).
fof(t50_subset_1,axiom,
! [A] :
( A != empty_set
=> ! [B] :
( element(B,powerset(A))
=> ! [C] :
( element(C,A)
=> ( ~ in(C,B)
=> in(C,subset_complement(A,B)) ) ) ) ),
file('theBenchmark.p',t50_subset_1) ).
fof(t69_enumset1,axiom,
! [A] : unordered_pair(A,A) = singleton(A),
file('theBenchmark.p',t69_enumset1) ).
fof(dt_k4_relat_1,axiom,
! [A] :
( relation(A)
=> relation(relation_inverse(A)) ),
file('theBenchmark.p',dt_k4_relat_1) ).
fof(involutiveness_k4_relat_1,axiom,
! [A] :
( relation(A)
=> relation_inverse(relation_inverse(A)) = A ),
file('theBenchmark.p',involutiveness_k4_relat_1) ).
fof(t37_relat_1,axiom,
! [A] :
( relation(A)
=> ( relation_dom(A) = relation_rng(relation_inverse(A))
& relation_rng(A) = relation_dom(relation_inverse(A)) ) ),
file('theBenchmark.p',t37_relat_1) ).
fof(t60_relat_1,conjecture,
( relation_rng(empty_set) = empty_set
& relation_dom(empty_set) = empty_set ),
file('theBenchmark.p',t60_relat_1) ).
fof(t60_relat_1_neg,negated_conjecture,
~ ( relation_rng(empty_set) = empty_set
& relation_dom(empty_set) = empty_set ),
inference(assume_negation,[status(cth)],[t60_relat_1]) ).
cnf(c_175,plain,
( relation(relation_inverse(A))
| ~ relation(A) ),
inference(clausify,[status(thm)],[dt_k4_relat_1]) ).
cnf(c_176,plain,
( relation_inverse(relation_inverse(A)) = A
| ~ relation(A) ),
inference(clausify,[status(thm)],[involutiveness_k4_relat_1]) ).
cnf(c_186,plain,
( relation_rng(relation_inverse(A)) = relation_dom(A)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t37_relat_1]) ).
cnf(c_195,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_200,plain,
( ~ subset(A,B)
| ~ subset(B,A)
| A = B ),
inference(clausify,[status(thm)],[d10_xboole_0]) ).
cnf(c_217,plain,
( ~ in(B,A)
| empty_set != A ),
inference(clausify,[status(thm)],[d1_xboole_0]) ).
cnf(c_261,plain,
cast_to_subset(A) = A,
inference(clausify,[status(thm)],[d4_subset_1]) ).
fof(nnf_22,plain,
! [A] :
( ! [B] :
( ( ? [C] :
( ( ! [D] : ~ in(ordered_pair(D,C),A)
| ~ in(C,B) )
& ( ? [X0] : in(ordered_pair(X0,C),A)
| in(C,B) ) )
| relation_rng(A) = B )
& ( ! [X1] :
( ( ! [X3] : ~ in(ordered_pair(X3,X1),A)
| in(X1,B) )
& ( ? [X2] : in(ordered_pair(X2,X1),A)
| ~ in(X1,B) ) )
| relation_rng(A) != B ) )
| ~ relation(A) ),
inference(fof_nnf,[status(thm)],[d5_relat_1]) ).
fof(sk_d5_relat_1_sk,plain,
! [A] :
( ! [B] :
( ( ( ( ! [D] : ~ in(ordered_pair(D,sK3(A,B)),A)
| ~ in(sK3(A,B),B) )
& ( in(ordered_pair(sK4(A,B),sK3(A,B)),A)
| in(sK3(A,B),B) ) )
| relation_rng(A) = B )
& ( ! [X1] :
( ( ! [X3] : ~ in(ordered_pair(X3,X1),A)
| in(X1,B) )
& ( in(ordered_pair(sK0(A,B,X1),X1),A)
| ~ in(X1,B) ) )
| relation_rng(A) != B ) )
| ~ relation(A) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK3,sK4]),skolemize(X2,sK0(A,B,X1)),skolemize(C,sK3(A,B)),skolemize(X0,sK4(A,B))],[nnf_22]) ).
fof(sk_d5_relat_1,plain,
( ! [VAR_0,VAR_1,VAR_2] :
( ~ in(ordered_pair(VAR_2,sK3(VAR_0,VAR_1)),VAR_0)
| ~ in(sK3(VAR_0,VAR_1),VAR_1)
| relation_rng(VAR_0) = VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1] :
( in(ordered_pair(sK4(VAR_0,VAR_1),sK3(VAR_0,VAR_1)),VAR_0)
| in(sK3(VAR_0,VAR_1),VAR_1)
| relation_rng(VAR_0) = VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( ~ in(ordered_pair(VAR_3,VAR_2),VAR_0)
| in(VAR_2,VAR_1)
| relation_rng(VAR_0) != VAR_1
| ~ relation(VAR_0) )
& ! [VAR_0,VAR_1,VAR_2] :
( in(ordered_pair(sK0(VAR_0,VAR_1,VAR_2),VAR_2),VAR_0)
| ~ in(VAR_2,VAR_1)
| relation_rng(VAR_0) != VAR_1
| ~ relation(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_d5_relat_1_sk]) ).
cnf(c_274,plain,
( in(ordered_pair(sK0(A,B,C),C),A)
| ~ in(C,B)
| relation_rng(A) != B
| ~ relation(A) ),
inference(split_conjunct,[status(thm)],[sk_d5_relat_1]) ).
cnf(c_278,plain,
( subset_complement(B,A) = set_difference(B,A)
| ~ element(A,powerset(B)) ),
inference(clausify,[status(thm)],[d5_subset_1]) ).
cnf(c_279,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
cnf(c_280,plain,
ordered_pair(A,B) = unordered_pair(singleton(A),unordered_pair(A,B)),
inference(paramod,[status(thm)],[c_195,c_279]) ).
cnf(c_300,plain,
element(cast_to_subset(A),powerset(A)),
inference(clausify,[status(thm)],[dt_k2_subset_1]) ).
cnf(c_301,plain,
element(A,powerset(A)),
inference(paramod,[status(thm)],[c_261,c_300]) ).
cnf(c_302,plain,
( element(subset_complement(B,A),powerset(B))
| ~ element(A,powerset(B)) ),
inference(clausify,[status(thm)],[dt_k3_subset_1]) ).
fof(nnf_35,plain,
! [A] :
? [B] : element(B,A),
inference(fof_nnf,[status(thm)],[existence_m1_subset_1]) ).
fof(sk_existence_m1_subset_1,plain,
! [A] : element(sK1(A),A),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(B,sK1(A))],[nnf_35]) ).
cnf(c_308,plain,
element(sK1(A),A),
inference(split_conjunct,[status(thm)],[sk_existence_m1_subset_1]) ).
cnf(c_318,plain,
relation(empty_set),
inference(clausify,[status(thm)],[fc4_relat_1]) ).
cnf(c_382,plain,
( subset(relation_dom(A),relation_dom(B))
| ~ subset(A,B)
| ~ relation(B)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t25_relat_1]) ).
cnf(c_383,plain,
( subset(relation_rng(A),relation_rng(B))
| ~ subset(A,B)
| ~ relation(B)
| ~ relation(A) ),
inference(clausify,[status(thm)],[t25_relat_1]) ).
cnf(c_386,plain,
set_intersection2(A,empty_set) = empty_set,
inference(clausify,[status(thm)],[t2_boole]) ).
cnf(c_389,plain,
subset(empty_set,A),
inference(clausify,[status(thm)],[t2_xboole_1]) ).
cnf(c_400,plain,
set_difference(A,empty_set) = A,
inference(clausify,[status(thm)],[t3_boole]) ).
cnf(c_420,plain,
set_difference(A,set_difference(A,B)) = set_intersection2(A,B),
inference(clausify,[status(thm)],[t48_xboole_1]) ).
cnf(c_421,plain,
set_intersection2(A,B) = set_difference(A,set_difference(A,B)),
inference(copy,[status(thm)],[c_420]) ).
cnf(c_428,plain,
( in(C,subset_complement(A,B))
| in(C,B)
| ~ element(C,A)
| ~ element(B,powerset(A))
| empty_set = A ),
inference(clausify,[status(thm)],[t50_subset_1]) ).
cnf(c_436,plain,
singleton(A) = unordered_pair(A,A),
inference(clausify,[status(thm)],[t69_enumset1]) ).
cnf(c_451,negated_conjecture,
( relation_rng(empty_set) != empty_set
| relation_dom(empty_set) != empty_set ),
inference(clausify,[status(thm)],[t60_relat_1_neg]) ).
cnf(c_8294,plain,
( in(unordered_pair(singleton(sK0(A,B,C)),unordered_pair(sK0(A,B,C),C)),A)
| ~ in(C,B)
| relation_rng(A) != B
| ~ relation(A) ),
inference(paramod,[status(thm)],[c_280,c_274]) ).
cnf(c_8295,plain,
( in(unordered_pair(unordered_pair(sK0(A,B,C),sK0(A,B,C)),unordered_pair(sK0(A,B,C),C)),A)
| ~ in(C,B)
| relation_rng(A) != B
| ~ relation(A) ),
inference(paramod,[status(thm)],[c_436,c_8294]) ).
cnf(c_8296,plain,
( in(unordered_pair(unordered_pair(sK0(A,B,C),sK0(A,B,C)),unordered_pair(C,sK0(A,B,C))),A)
| ~ in(C,B)
| relation_rng(A) != B
| ~ relation(A) ),
inference(paramod,[status(thm)],[c_195,c_8295]) ).
cnf(c_486,plain,
( in(unordered_pair(unordered_pair(C,sK0(A,B,C)),unordered_pair(sK0(A,B,C),sK0(A,B,C))),A)
| ~ in(C,B)
| relation_rng(A) != B
| ~ relation(A) ),
inference(paramod,[status(thm)],[c_195,c_8296]) ).
cnf(c_8297,plain,
set_difference(A,set_difference(A,empty_set)) = empty_set,
inference(paramod,[status(thm)],[c_421,c_386]) ).
cnf(c_544,plain,
set_difference(A,A) = empty_set,
inference(paramod,[status(thm)],[c_400,c_8297]) ).
cnf(c_599,plain,
~ in(A,empty_set),
inference(xxres,[status(thm)],[c_217]) ).
cnf(c_604,plain,
relation_rng(relation_inverse(empty_set)) = relation_dom(empty_set),
inference(resolve,[status(thm)],[c_318,c_186]) ).
cnf(c_605,plain,
relation_inverse(relation_inverse(empty_set)) = empty_set,
inference(resolve,[status(thm)],[c_318,c_176]) ).
cnf(c_606,plain,
relation(relation_inverse(empty_set)),
inference(resolve,[status(thm)],[c_318,c_175]) ).
cnf(c_8298,plain,
relation_rng(relation_inverse(relation_inverse(empty_set))) = relation_dom(relation_inverse(empty_set)),
inference(resolve,[status(thm)],[c_606,c_186]) ).
cnf(c_8299,plain,
relation_rng(empty_set) = relation_dom(relation_inverse(empty_set)),
inference(paramod,[status(thm)],[c_605,c_8298]) ).
cnf(c_619,plain,
relation_dom(relation_inverse(empty_set)) = relation_rng(empty_set),
inference(copy,[status(thm)],[c_8299]) ).
cnf(c_8300,plain,
subset_complement(A,A) = set_difference(A,A),
inference(resolve,[status(thm)],[c_301,c_278]) ).
cnf(c_2394,plain,
subset_complement(A,A) = empty_set,
inference(paramod,[status(thm)],[c_544,c_8300]) ).
cnf(c_8301,plain,
element(subset_complement(A,A),powerset(A)),
inference(resolve,[status(thm)],[c_302,c_301]) ).
cnf(c_2400,plain,
element(empty_set,powerset(A)),
inference(paramod,[status(thm)],[c_2394,c_8301]) ).
cnf(c_8302,plain,
subset_complement(A,empty_set) = set_difference(A,empty_set),
inference(resolve,[status(thm)],[c_2400,c_278]) ).
cnf(c_2449,plain,
subset_complement(A,empty_set) = A,
inference(paramod,[status(thm)],[c_400,c_8302]) ).
cnf(c_8303,plain,
( subset(relation_dom(empty_set),relation_dom(A))
| ~ subset(empty_set,A)
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_382,c_318]) ).
cnf(c_3756,plain,
( subset(relation_dom(empty_set),relation_dom(A))
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_389,c_8303]) ).
cnf(c_8304,plain,
( subset(relation_rng(empty_set),relation_rng(A))
| ~ subset(empty_set,A)
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_383,c_318]) ).
cnf(c_3853,plain,
( subset(relation_rng(empty_set),relation_rng(A))
| ~ relation(A) ),
inference(resolve,[status(thm)],[c_389,c_8304]) ).
cnf(c_8305,plain,
subset(relation_dom(empty_set),relation_dom(relation_inverse(empty_set))),
inference(resolve,[status(thm)],[c_3756,c_606]) ).
cnf(c_4116,plain,
subset(relation_dom(empty_set),relation_rng(empty_set)),
inference(paramod,[status(thm)],[c_619,c_8305]) ).
cnf(c_8306,plain,
( ~ subset(relation_rng(empty_set),relation_dom(empty_set))
| relation_dom(empty_set) = relation_rng(empty_set) ),
inference(resolve,[status(thm)],[c_4116,c_200]) ).
cnf(c_4150,plain,
( ~ subset(relation_rng(empty_set),relation_dom(empty_set))
| relation_rng(empty_set) = relation_dom(empty_set) ),
inference(copy,[status(thm)],[c_8306]) ).
cnf(c_8307,plain,
subset(relation_rng(empty_set),relation_rng(relation_inverse(empty_set))),
inference(resolve,[status(thm)],[c_3853,c_606]) ).
cnf(c_4295,plain,
subset(relation_rng(empty_set),relation_dom(empty_set)),
inference(paramod,[status(thm)],[c_604,c_8307]) ).
cnf(c_4297,plain,
relation_rng(empty_set) = relation_dom(empty_set),
inference(resolve,[status(thm)],[c_4295,c_4150]) ).
cnf(c_8308,negated_conjecture,
( relation_dom(empty_set) != empty_set
| relation_dom(empty_set) != empty_set ),
inference(paramod,[status(thm)],[c_4297,c_451]) ).
cnf(c_4309,negated_conjecture,
relation_dom(empty_set) != empty_set,
inference(copy,[status(thm)],[c_8308]) ).
cnf(c_8309,plain,
( in(B,subset_complement(A,empty_set))
| in(B,empty_set)
| ~ element(B,A)
| empty_set = A ),
inference(resolve,[status(thm)],[c_428,c_2400]) ).
cnf(c_8310,plain,
( in(B,A)
| in(B,empty_set)
| ~ element(B,A)
| empty_set = A ),
inference(paramod,[status(thm)],[c_2449,c_8309]) ).
cnf(c_5443,plain,
( in(B,A)
| ~ element(B,A)
| empty_set = A ),
inference(resolve,[status(thm)],[c_599,c_8310]) ).
cnf(c_5578,plain,
( in(sK1(A),A)
| empty_set = A ),
inference(resolve,[status(thm)],[c_5443,c_308]) ).
cnf(c_8311,plain,
( in(unordered_pair(unordered_pair(A,sK0(empty_set,relation_dom(empty_set),A)),unordered_pair(sK0(empty_set,relation_dom(empty_set),A),sK0(empty_set,relation_dom(empty_set),A))),empty_set)
| ~ in(A,relation_dom(empty_set))
| ~ relation(empty_set) ),
inference(resolve,[status(thm)],[c_486,c_4297]) ).
cnf(c_8312,plain,
( in(unordered_pair(unordered_pair(A,sK0(empty_set,relation_dom(empty_set),A)),unordered_pair(sK0(empty_set,relation_dom(empty_set),A),sK0(empty_set,relation_dom(empty_set),A))),empty_set)
| ~ in(A,relation_dom(empty_set)) ),
inference(resolve,[status(thm)],[c_318,c_8311]) ).
cnf(c_8221,plain,
~ in(A,relation_dom(empty_set)),
inference(resolve,[status(thm)],[c_599,c_8312]) ).
cnf(c_8313,plain,
empty_set = relation_dom(empty_set),
inference(resolve,[status(thm)],[c_8221,c_5578]) ).
cnf(c_8314,plain,
relation_dom(empty_set) = empty_set,
inference(copy,[status(thm)],[c_8313]) ).
cnf(c_8293,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_4309,c_8314]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SEU187+2 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/5.38 % Computer : n004.cluster.edu
% 0.12/5.38 % Model : x86_64 x86_64
% 0.12/5.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/5.38 % Memory : 8046.5625MB
% 0.12/5.38 % OS : Linux 6.8.0-71-generic
% 0.12/5.38 % CPULimit : 300
% 0.12/5.38 % WCLimit : 300
% 0.12/5.38 % DateTime : Sat Sep 26 08:46:42 UTC 2026
% 0.12/5.38 % CPUTime :
% 0.12/5.39 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.12/5.39 % Process 1279387 was started by sandbox on n004,
% 0.12/5.39 % Sat Sep 26 08:46:42 2026
% 0.12/5.39 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.14/5.42
% 0.14/5.42 % From the command line: assign(max_seconds, 300).
% 0.14/5.42
% 0.14/5.42 % SInE: 139 to sos, 27 to usable, of 166 axioms (tolerance 2.00, depth 0 fixpoint).
% 12.34/7.10
% 12.34/7.10 % given #500 (F,wt=15): 4792 -disjoint(powerset(A),f45(A)). [resolve(4676,a,178,a)].
% 12.34/7.10
% 12.34/7.10 % SZS status Theorem for theBenchmark
% 12.34/7.10
% 12.34/7.10 % Proof 1 at 1.27 (+ 0.35) seconds.
% 12.34/7.10 % Length of proof is 69.
% 12.34/7.10 % Level of proof is 9.
% 12.34/7.10 % Maximum clause weight is 514.000.
% 12.34/7.10 % Given clauses 732.
% 12.34/7.10
% 12.34/7.10 % SZS output start CNFRefutation for theBenchmark
% See solution above
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