%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUN051^1 : 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 : 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 : Wed Sep 30 08:19:50 AM UTC 2026
% Result : Theorem 4.74s 1.80s
% Output : Refutation 4.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 7
% Syntax : Number of formulae : 32 ( 20 unt; 0 typ; 3 def)
% Number of atoms : 95 ( 37 equ; 0 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 429 ( 18 ~; 7 |; 1 &; 395 @)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 10 usr; 5 con; 0-3 aty)
% Number of variables : 75 ( 18 ^; 54 !; 0 ?; 75 :)
% ( 0 !>; 0 ?*; 0 @-; 3 @+)
% Comments :
%------------------------------------------------------------------------------
thf('type/nums/num_type',type,
'type/nums/num': $tType ).
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' > 'type/nums/num' ).
thf('const/arith/-_decl',type,
'const/arith/-': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' ).
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(sk1_def,definition,
( sk1
= ( @+[A: 'type/nums/num' > 'type/nums/num'] :
~ ! [B: 'type/nums/num',C: 'type/nums/num'] :
( ( 'const/arith/<=' @ B @ C )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ C @ B ) )
@ ^ [D: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ D @ B ) ) ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: 'type/nums/num'] :
~ ! [B: 'type/nums/num'] :
( ( 'const/arith/<=' @ A @ B )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ B ) @ sk1 )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ B @ A ) )
@ ^ [C: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ C @ A ) ) ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( @+[A: 'type/nums/num'] :
~ ( ( 'const/arith/<=' @ sk2 @ A )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ A ) @ sk1 )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ A @ sk2 ) )
@ ^ [B: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ B @ sk2 ) ) ) ) ) ) ) ).
thf(1,conjecture,
! [A: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( 'const/arith/<=' @ B @ C )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ C @ B ) )
@ ^ [D: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ D @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/NSUM_OFFSET_0_') ).
thf(2,negated_conjecture,
~ ! [A: 'type/nums/num' > 'type/nums/num',B: 'type/nums/num',C: 'type/nums/num'] :
( ( 'const/arith/<=' @ B @ C )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ C @ B ) )
@ ^ [D: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ D @ B ) ) ) ) ),
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/arith/<=' @ B @ C )
=> ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ C ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ C @ B ) )
@ ^ [D: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ D @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(7,plain,
( ( 'const/arith/<=' @ sk2 @ sk3 )
& ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [A: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ A @ sk2 ) ) ) ) ),
inference(cnf,[status(esa)],[6]) ).
thf(8,plain,
'const/arith/<=' @ sk2 @ sk3,
inference(cnfConj,[status(thm)],[7]) ).
thf(3,axiom,
! [A: 'type/nums/num',B: 'type/nums/num'] :
( ( 'const/arith/<=' @ B @ A )
=> ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/arith/SUB_ADD_') ).
thf(11,plain,
! [A: 'type/nums/num',B: 'type/nums/num'] :
( ( 'const/arith/<=' @ B @ A )
=> ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(12,plain,
! [B: 'type/nums/num',A: 'type/nums/num'] :
( ~ ( 'const/arith/<=' @ B @ A )
| ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A ) ),
inference(cnf,[status(esa)],[11]) ).
thf(13,plain,
! [B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A )
| ~ ( 'const/arith/<=' @ B @ A ) ),
inference(lifteq,[status(thm)],[12]) ).
thf(97,plain,
! [B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A )
| ~ $true
| ( ( 'const/arith/<=' @ sk2 @ sk3 )
!= ( 'const/arith/<=' @ B @ A ) ) ),
inference(paramod_ordered,[status(thm)],[8,13]) ).
thf(98,plain,
! [B: 'type/nums/num',A: 'type/nums/num'] :
( ( ( 'const/arith/+' @ ( 'const/arith/-' @ A @ B ) @ B )
= A )
| ( ( 'const/arith/<=' @ sk2 @ sk3 )
!= ( 'const/arith/<=' @ B @ A ) ) ),
inference(simp,[status(thm)],[97]) ).
thf(99,plain,
( ( 'const/arith/+' @ ( 'const/arith/-' @ sk3 @ sk2 ) @ sk2 )
= sk3 ),
inference(pattern_uni,[status(thm)],[98:[bind(A,$thf( sk3 )),bind(B,$thf( sk2 ))]]) ).
thf(5,axiom,
! [A: 'type/nums/num',B: 'type/nums/num' > 'type/nums/num',C: 'type/nums/num',D: 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ C @ A ) @ ( 'const/arith/+' @ D @ A ) ) @ B )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ C @ D )
@ ^ [E: 'type/nums/num'] : ( B @ ( 'const/arith/+' @ E @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/iterate/NSUM_OFFSET_') ).
thf(17,plain,
! [A: 'type/nums/num',B: 'type/nums/num' > 'type/nums/num',C: 'type/nums/num',D: 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ C @ A ) @ ( 'const/arith/+' @ D @ A ) ) @ B )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ C @ D )
@ ^ [E: 'type/nums/num'] : ( B @ ( 'const/arith/+' @ E @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(18,plain,
! [D: 'type/nums/num',C: 'type/nums/num',B: 'type/nums/num' > 'type/nums/num',A: 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ C @ A ) @ ( 'const/arith/+' @ D @ A ) ) @ B )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ C @ D )
@ ^ [E: 'type/nums/num'] : ( B @ ( 'const/arith/+' @ E @ A ) ) ) ),
inference(cnf,[status(esa)],[17]) ).
thf(19,plain,
! [D: 'type/nums/num',C: 'type/nums/num',B: 'type/nums/num' > 'type/nums/num',A: 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ C @ A ) @ ( 'const/arith/+' @ D @ A ) ) @ B )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ C @ D )
@ ^ [E: 'type/nums/num'] : ( B @ ( 'const/arith/+' @ E @ A ) ) ) ),
inference(lifteq,[status(thm)],[18]) ).
thf(119,plain,
! [D: 'type/nums/num',C: 'type/nums/num',B: 'type/nums/num' > 'type/nums/num',A: 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ C @ A ) @ sk3 ) @ B )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ C @ D )
@ ^ [E: 'type/nums/num'] : ( B @ ( 'const/arith/+' @ E @ A ) ) ) )
| ( ( 'const/arith/+' @ ( 'const/arith/-' @ sk3 @ sk2 ) @ sk2 )
!= ( 'const/arith/+' @ D @ A ) ) ),
inference(paramod_ordered,[status(thm)],[99,19]) ).
thf(120,plain,
! [B: 'type/nums/num',A: 'type/nums/num' > 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ B @ sk2 ) @ sk3 ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [C: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ C @ sk2 ) ) ) ),
inference(pattern_uni,[status(thm)],[119:[bind(A,$thf( sk2 )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( 'const/arith/-' @ sk3 @ sk2 ))]]) ).
thf(126,plain,
! [B: 'type/nums/num',A: 'type/nums/num' > 'type/nums/num'] :
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ B @ sk2 ) @ sk3 ) @ A )
= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [C: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ C @ sk2 ) ) ) ),
inference(simp,[status(thm)],[120]) ).
thf(9,plain,
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [A: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ A @ sk2 ) ) ) ),
inference(cnfConj,[status(thm)],[7]) ).
thf(10,plain,
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [A: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ A @ sk2 ) ) )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 ) ),
inference(lifteq,[status(thm)],[9]) ).
thf(163,plain,
! [B: 'type/nums/num',A: 'type/nums/num' > 'type/nums/num'] :
( ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ B @ sk2 ) @ sk3 ) @ A )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 ) )
| ( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ B @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [C: 'type/nums/num'] : ( A @ ( 'const/arith/+' @ C @ sk2 ) ) )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ ( 'const/arith/-' @ sk3 @ sk2 ) )
@ ^ [C: 'type/nums/num'] : ( sk1 @ ( 'const/arith/+' @ C @ sk2 ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[126,10]) ).
thf(206,plain,
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ ( 'const/arith/+' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ sk2 ) @ sk3 ) @ sk1 )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 ) ),
inference(pre_uni,[status(thm)],[163:[bind(A,$thf( sk1 )),bind(B,$thf( 'const/nums/NUMERAL' @ 'const/nums/_0' ))]]) ).
thf(4,axiom,
! [A: 'type/nums/num'] :
( ( 'const/arith/+' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thm/arith/ADD_CLAUSES_0') ).
thf(14,plain,
! [A: 'type/nums/num'] :
( ( 'const/arith/+' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(15,plain,
! [A: 'type/nums/num'] :
( ( 'const/arith/+' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A )
= A ),
inference(cnf,[status(esa)],[14]) ).
thf(16,plain,
! [A: 'type/nums/num'] :
( ( 'const/arith/+' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) @ A )
= A ),
inference(lifteq,[status(thm)],[15]) ).
thf(239,plain,
( ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 )
!= ( 'const/iterate/nsum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sk2 @ sk3 ) @ sk1 ) ),
inference(rewrite,[status(thm)],[206,16]) ).
thf(240,plain,
$false,
inference(simp,[status(thm)],[239]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : NUN051^1 : TPTP v9.3.1. Released v7.0.0.
% 0.00/0.04 % 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
% 0.04/0.13 % Computer : n012.cluster.edu
% 0.04/0.13 % Model : x86_64 x86_64
% 0.04/0.13 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.04/0.13 % Memory : 8046.5625MB
% 0.04/0.13 % OS : Linux 6.8.0-71-generic
% 0.04/0.13 % CPULimit : 300
% 0.04/0.13 % WCLimit : 300
% 0.04/0.13 % DateTime : Tue Sep 29 13:07:03 UTC 2026
% 0.04/0.13 % CPUTime :
% 0.04/0.13 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
% 0.38/0.42 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.54/0.53 % [INFO] Parsing done (103ms).
% 0.73/0.53 % [INFO] Running in sequential loop mode.
% 0.99/0.72 % [INFO] eprover registered as external prover.
% 0.99/0.73 % [INFO] Scanning for conjecture ...
% 0.99/0.78 % [INFO] Found a conjecture (or negated_conjecture) and 3 axioms. Running axiom selection ...
% 1.19/0.80 % [INFO] Axiom selection finished. Selected 3 axioms (removed 0 axioms).
% 1.19/0.80 % [INFO] Problem is higher-order (TPTP THF).
% 1.19/0.81 % [INFO] Type checking passed.
% 1.19/0.81 % [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 ...
% 4.25/1.79 % [INFO] Killing All external provers ...
% 4.74/1.80 % Time passed: 1591ms (effective reasoning time: 1259ms)
% 4.74/1.80 % 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)>
% 4.74/1.80 % Axioms used in derivation (3): thm/arith/SUB_ADD_, thm/iterate/NSUM_OFFSET_, thm/arith/ADD_CLAUSES_0
% 4.74/1.80 % No. of inferences in proof: 29
% 4.74/1.80 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 1591 ms resp. 1259 ms w/o parsing
% 4.74/1.83 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 4.74/1.83 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------