%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SEV506^1 : TPTP v9.3.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n014.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 : Wed Sep 30 08:37:23 AM UTC 2026
% Result : Theorem 11.64s 3.56s
% Output : Refutation 12.08s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 7
% Syntax : Number of formulae : 29 ( 18 unt; 0 typ; 3 def)
% Number of atoms : 165 ( 72 equ; 0 cnn)
% Maximal formula atoms : 3 ( 5 avg)
% Number of connectives : 381 ( 20 ~; 7 |; 18 &; 313 @)
% ( 0 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 11 usr; 6 con; 0-3 aty)
% Number of variables : 88 ( 8 ^; 77 !; 0 ?; 88 :)
% ( 0 !>; 0 ?*; 0 @-; 3 @+)
% Comments :
%------------------------------------------------------------------------------
thf('type/nums/num_type',type,
'type/nums/num': $tType ).
thf('const/sets/IN_decl',type,
'const/sets/IN':
!>[TA: $tType] : ( TA > ( TA > $o ) > $o ) ).
thf('const/sets/FINITE_decl',type,
'const/sets/FINITE':
!>[TA: $tType] : ( ( TA > $o ) > $o ) ).
thf('const/nums/NUMERAL_decl',type,
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
thf('const/nums/_0_decl',type,
'const/nums/_0': 'type/nums/num' ).
thf('const/iterate/nsum_decl',type,
'const/iterate/nsum':
!>[TA: $tType] : ( ( TA > $o ) > ( TA > 'type/nums/num' ) > 'type/nums/num' ) ).
thf('const/iterate/.._decl',type,
'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
thf('const/arith/<=_decl',type,
'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
thf(sk1_decl,type,
sk1: 'type/nums/num' > 'type/nums/num' ).
thf(sk2_decl,type,
sk2: 'type/nums/num' ).
thf(sk3_decl,type,
sk3: 'type/nums/num' ).
thf(sk10_decl,type,
sk10: 'type/nums/num' ).
thf(sk1_def,definition,
( sk1
= ( @+[A: 'type/nums/num' > 'type/nums/num'] :
~ ! [B: 'type/nums/num',C: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [D: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ B @ D )
& ( 'const/arith/<=' @ D @ C ) )
=> ( ( A @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: 'type/nums/num'] :
~ ! [B: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) @ sk1 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [C: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ A @ C )
& ( 'const/arith/<=' @ C @ B ) )
=> ( ( sk1 @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( @+[A: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ A ) @ sk1 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( ! [B: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ B )
& ( 'const/arith/<=' @ B @ A ) )
=> ( ( sk1 @ B )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ) ) ).
thf(5,axiom,
! [A: 'type/nums/num',B: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/FINITE_NUMSEG_') ).
thf(15,plain,
! [A: 'type/nums/num',B: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(16,plain,
! [B: 'type/nums/num',A: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) ),
inference(cnf,[status(esa)],[15]) ).
thf(4,axiom,
! [TA: $tType,A: TA > 'type/nums/num',B: TA > $o] :
( ( 'const/sets/FINITE' @ TA @ B )
=> ( ( ( 'const/iterate/nsum' @ TA @ B @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [C: TA] :
( ( 'const/sets/IN' @ TA @ C @ B )
=> ( ( A @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/NSUM_EQ_0_IFF_') ).
thf(12,plain,
! [TA: $tType,A: TA > 'type/nums/num',B: TA > $o] :
( ( 'const/sets/FINITE' @ TA @ B )
=> ( ( ( 'const/iterate/nsum' @ TA @ B @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [C: TA] :
( ( 'const/sets/IN' @ TA @ C @ B )
=> ( ( A @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(13,plain,
! [TA: $tType,B: TA > $o,A: TA > 'type/nums/num'] :
( ~ ( 'const/sets/FINITE' @ TA @ B )
| ( ( ( 'const/iterate/nsum' @ TA @ B @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [C: TA] :
( ( 'const/sets/IN' @ TA @ C @ B )
=> ( ( A @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ) ),
inference(cnf,[status(esa)],[12]) ).
thf(14,plain,
! [TA: $tType,B: TA > $o,A: TA > 'type/nums/num'] :
( ( ( ( 'const/iterate/nsum' @ TA @ B @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [C: TA] :
( ( 'const/sets/IN' @ TA @ C @ B )
=> ( ( A @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) )
| ~ ( 'const/sets/FINITE' @ TA @ B ) ),
inference(lifteq,[status(thm)],[13]) ).
thf(129,plain,
! [TA: $tType,D: TA > $o,C: TA > 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( ( 'const/iterate/nsum' @ TA @ D @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [E: TA] :
( ( 'const/sets/IN' @ TA @ E @ D )
=> ( ( C @ E )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) )
| ~ $true
| ( ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) )
!= ( 'const/sets/FINITE' @ TA @ D ) ) ),
inference(paramod_ordered,[status(thm)],[16,14]) ).
thf(130,plain,
! [TA: $tType,D: TA > $o,C: TA > 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( ( 'const/iterate/nsum' @ TA @ D @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [E: TA] :
( ( 'const/sets/IN' @ TA @ E @ D )
=> ( ( C @ E )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) )
| ( ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) )
!= ( 'const/sets/FINITE' @ TA @ D ) ) ),
inference(simp,[status(thm)],[129]) ).
thf(131,plain,
! [C: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [D: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ D @ ( 'const/iterate/..' @ A @ B ) )
=> ( ( C @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(pattern_uni,[status(thm)],[130:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( 'const/iterate/..' @ A @ B )),bind_type(TA,$thf( 'type/nums/num' ))]]) ).
thf(1,conjecture,
! [A: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [D: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ B @ D )
& ( 'const/arith/<=' @ D @ C ) )
=> ( ( A @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/NSUM_EQ_0_IFF_NUMSEG_') ).
thf(2,negated_conjecture,
~ ! [A: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [D: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ B @ D )
& ( 'const/arith/<=' @ D @ C ) )
=> ( ( A @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(6,plain,
~ ! [A: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ! [D: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ B @ D )
& ( 'const/arith/<=' @ D @ C ) )
=> ( ( A @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(7,plain,
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( ! [A: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(cnf,[status(esa)],[6]) ).
thf(8,plain,
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( ! [A: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(lifteq,[status(thm)],[7]) ).
thf(155,plain,
! [C: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( ! [D: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ D @ ( 'const/iterate/..' @ A @ B ) )
=> ( ( C @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
!= ( ! [D: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ D )
& ( 'const/arith/<=' @ D @ sk3 ) )
=> ( ( sk1 @ D )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) )
| ( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) @ C )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ),
inference(paramod_ordered,[status(thm)],[131,8]) ).
thf(156,plain,
( ( ! [A: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ A @ ( 'const/iterate/..' @ sk2 @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
!= ( ! [A: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(pattern_uni,[status(thm)],[155:[bind(A,$thf( sk2 )),bind(B,$thf( sk3 )),bind(C,$thf( sk1 ))]]) ).
thf(171,plain,
( ( ^ [A: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ A @ ( 'const/iterate/..' @ sk2 @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
!= ( ^ [A: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) )
=> ( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(simp,[status(thm)],[156]) ).
thf(194,plain,
( ( ( ^ [A: 'type/nums/num'] : ( 'const/sets/IN' @ 'type/nums/num' @ A @ ( 'const/iterate/..' @ sk2 @ sk3 ) ) )
!= ( ^ [A: 'type/nums/num'] :
( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) ) ) )
| ( ( ^ [A: 'type/nums/num'] :
( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
!= ( ^ [A: 'type/nums/num'] :
( ( sk1 @ A )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ),
inference(simp,[status(thm)],[171]) ).
thf(196,plain,
( ( ^ [A: 'type/nums/num'] : ( 'const/sets/IN' @ 'type/nums/num' @ A @ ( 'const/iterate/..' @ sk2 @ sk3 ) ) )
!= ( ^ [A: 'type/nums/num'] :
( ( 'const/arith/<=' @ sk2 @ A )
& ( 'const/arith/<=' @ A @ sk3 ) ) ) ),
inference(simp,[status(thm)],[194]) ).
thf(197,plain,
( ( 'const/sets/IN' @ 'type/nums/num' @ sk10 @ ( 'const/iterate/..' @ sk2 @ sk3 ) )
!= ( ( 'const/arith/<=' @ sk2 @ sk10 )
& ( 'const/arith/<=' @ sk10 @ sk3 ) ) ),
inference(func_ext,[status(esa)],[196]) ).
thf(3,axiom,
! [A: 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ C @ ( 'const/iterate/..' @ A @ B ) )
= ( ( 'const/arith/<=' @ A @ C )
& ( 'const/arith/<=' @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/IN_NUMSEG_') ).
thf(9,plain,
! [A: 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ C @ ( 'const/iterate/..' @ A @ B ) )
= ( ( 'const/arith/<=' @ A @ C )
& ( 'const/arith/<=' @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(10,plain,
! [C: 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ C @ ( 'const/iterate/..' @ A @ B ) )
= ( ( 'const/arith/<=' @ A @ C )
& ( 'const/arith/<=' @ C @ B ) ) ),
inference(cnf,[status(esa)],[9]) ).
thf(11,plain,
! [C: 'type/nums/num',B: 'type/nums/num',A: 'type/nums/num'] :
( ( 'const/sets/IN' @ 'type/nums/num' @ C @ ( 'const/iterate/..' @ A @ B ) )
= ( ( 'const/arith/<=' @ A @ C )
& ( 'const/arith/<=' @ C @ B ) ) ),
inference(lifteq,[status(thm)],[10]) ).
thf(403,plain,
$false,
inference(simplifyReflect,[status(thm)],[197,11]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEV506^1 : TPTP v9.3.1. Released v7.0.0.
% 0.00/0.09 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.23/0.28 % Computer : n014.cluster.edu
% 0.23/0.28 % Model : x86_64 x86_64
% 0.23/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.23/0.28 % Memory : 8046.5625MB
% 0.23/0.28 % OS : Linux 6.8.0-71-generic
% 0.23/0.28 % CPULimit : 300
% 0.23/0.28 % WCLimit : 300
% 0.23/0.28 % DateTime : Tue Sep 29 15:54:14 UTC 2026
% 0.23/0.29 % CPUTime :
% 0.23/0.29 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.90/0.70 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.30/0.83 % [INFO] Parsing done (126ms).
% 1.30/0.83 % [INFO] Running in sequential loop mode.
% 1.93/1.08 % [INFO] eprover registered as external prover.
% 1.93/1.09 % [INFO] Scanning for conjecture ...
% 2.19/1.18 % [INFO] Found a conjecture (or negated_conjecture) and 3 axioms. Running axiom selection ...
% 2.30/1.21 % [INFO] Axiom selection finished. Selected 3 axioms (removed 0 axioms).
% 2.30/1.22 % [INFO] Problem is higher-order (TPTP THF).
% 2.30/1.23 % [INFO] Type checking passed.
% 2.30/1.23 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 11.64/3.55 % [INFO] Killing All external provers ...
% 11.64/3.55 % Time passed: 3122ms (effective reasoning time: 2712ms)
% 11.64/3.55 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 11.64/3.56 % Axioms used in derivation (3): thm/iterate/FINITE_NUMSEG_, thm/iterate/NSUM_EQ_0_IFF_, thm/iterate/IN_NUMSEG_
% 11.64/3.56 % No. of inferences in proof: 26
% 11.64/3.56 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 3122 ms resp. 2712 ms w/o parsing
% 12.08/3.65 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 12.08/3.65 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------