%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV514^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:41:25 AM UTC 2026
% Result : Theorem 1.31s 1.28s
% Output : Refutation 1.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 37 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 317 ( 88 equ; 0 cnn)
% Maximal formula atoms : 8 ( 8 avg)
% Number of connectives : 625 ( 48 ~; 35 |; 20 &; 505 @)
% ( 1 <=>; 16 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 13 usr; 5 con; 0-3 aty)
% Number of variables : 83 ( 0 ^; 72 !; 11 ?; 83 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_6,type,
'type/nums/num': $tType ).
thf(type_def_7,type,
'type/realax/real': $tType ).
thf(func_def_0,type,
'type/realax/real': $tType ).
thf(func_def_1,type,
'type/nums/num': $tType ).
thf(func_def_2,type,
'const/sets/IN':
!>[X0: $tType] : ( X0 > ( X0 > $o ) > $o ) ).
thf(func_def_3,type,
'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
thf(func_def_4,type,
'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
thf(func_def_5,type,
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
thf(func_def_6,type,
'const/nums/_0': 'type/nums/num' ).
thf(func_def_7,type,
'const/iterate/sum':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > 'type/realax/real' ) > 'type/realax/real' ) ).
thf(func_def_8,type,
'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
thf(func_def_9,type,
'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
thf(func_def_13,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > 'type/realax/real' ) > X0 ) ).
thf(func_def_14,type,
sK1: 'type/nums/num' ).
thf(func_def_15,type,
sK2: 'type/nums/num' ).
thf(func_def_16,type,
sK3: 'type/nums/num' > 'type/realax/real' ).
thf(func_def_18,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f39,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f38,f35]) ).
thf(f35,plain,
( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f32]) ).
thf(f32,plain,
( ( $true != $true )
| ( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
= $true ) ),
inference(superposition,[status(thm)],[f22,f31]) ).
thf(f31,plain,
( ( 'const/sets/IN' @ 'type/nums/num' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ ( 'const/iterate/..' @ sK1 @ sK2 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f30]) ).
thf(f30,plain,
( ( $true != $true )
| ( ( 'const/sets/IN' @ 'type/nums/num' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ ( 'const/iterate/..' @ sK1 @ sK2 ) )
= $true ) ),
inference(superposition,[status(thm)],[f27,f25]) ).
thf(f25,plain,
! [X0: $tType,X2: X0 > $o,X1: X0 > 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
= $true )
| ( ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X2 @ X1 ) @ X2 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ( ( ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X2 @ X1 ) @ X2 )
= $true )
& ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ ( sK0 @ X0 @ X2 @ X1 ) ) )
!= $true ) )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f12,f17]) ).
thf(f17,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ? [X3: X0] :
( ( ( 'const/sets/IN' @ X0 @ X3 @ X2 )
= $true )
& ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ X3 ) )
!= $true ) )
=> ( ( ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X2 @ X1 ) @ X2 )
= $true )
& ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ ( sK0 @ X0 @ X2 @ X1 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f12,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ? [X3: X0] :
( ( ( 'const/sets/IN' @ X0 @ X3 @ X2 )
= $true )
& ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ X3 ) )
!= $true ) )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ! [X3: X0] :
( ( ( 'const/sets/IN' @ X0 @ X3 @ X2 )
= $true )
=> ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ X3 ) )
= $true ) )
=> ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ! [X3: X0] :
( ( 'const/sets/IN' @ X0 @ X3 @ X2 )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ X3 ) ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ! [X3: X0] :
( ( 'const/sets/IN' @ X0 @ X3 @ X2 )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ X3 ) ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/SUM_POS_LE_') ).
thf(f27,plain,
( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true )
& ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sK1 @ X3 )
!= $true )
| ( ( 'const/arith/<=' @ X3 @ sK2 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( sK3 @ X3 ) )
= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2,sK3])],[f14,f19]) ).
thf(f19,plain,
( ? [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
!= $true )
& ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X3 )
!= $true )
| ( ( 'const/arith/<=' @ X3 @ X1 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) )
= $true ) ) )
=> ( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true )
& ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ sK1 @ X3 )
!= $true )
| ( ( 'const/arith/<=' @ X3 @ sK2 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( sK3 @ X3 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f14,plain,
? [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
!= $true )
& ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X3 )
!= $true )
| ( ( 'const/arith/<=' @ X3 @ X1 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) )
= $true ) ) ),
inference(flattening,[status(thm)],[f13]) ).
thf(f13,plain,
? [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
!= $true )
& ! [X3: 'type/nums/num'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) )
= $true )
| ( ( 'const/arith/<=' @ X3 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X3 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ! [X3: 'type/nums/num'] :
( ( ( ( 'const/arith/<=' @ X3 @ X1 )
= $true )
& ( ( 'const/arith/<=' @ X0 @ X3 )
= $true ) )
=> ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) )
= $true ) )
=> ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
~ ! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X3 )
& ( 'const/arith/<=' @ X3 @ X1 ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X3 )
& ( 'const/arith/<=' @ X3 @ X1 ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num' > 'type/realax/real'] :
( ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X3 )
& ( 'const/arith/<=' @ X3 @ X1 ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X2 @ X3 ) ) )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/SUM_POS_LE_NUMSEG_') ).
thf(f22,plain,
! [X2: 'type/nums/num',X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X2 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( ( ( 'const/arith/<=' @ X2 @ X1 )
= $true )
& ( ( 'const/arith/<=' @ X0 @ X2 )
= $true ) )
| ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
!= $true ) )
& ( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
= $true )
| ( ( 'const/arith/<=' @ X2 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X2 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f15]) ).
thf(f15,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( ( ( 'const/arith/<=' @ X2 @ X1 )
= $true )
& ( ( 'const/arith/<=' @ X0 @ X2 )
= $true ) )
| ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
!= $true ) )
& ( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
= $true )
| ( ( 'const/arith/<=' @ X2 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X2 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( ( 'const/arith/<=' @ X2 @ X1 )
= $true )
& ( ( 'const/arith/<=' @ X0 @ X2 )
= $true ) )
<=> ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
= ( 'const/arith/<=' @ X2 @ X1 ) )
& ( 'const/arith/<=' @ X0 @ X2 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
= ( 'const/arith/<=' @ X2 @ X1 ) )
& ( 'const/arith/<=' @ X0 @ X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/IN_NUMSEG_') ).
thf(f38,plain,
( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f37,f34]) ).
thf(f34,plain,
( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ sK2 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f33]) ).
thf(f33,plain,
( ( $true != $true )
| ( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ sK2 )
= $true ) ),
inference(superposition,[status(thm)],[f23,f31]) ).
thf(f23,plain,
! [X2: 'type/nums/num',X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/sets/IN' @ 'type/nums/num' @ X2 @ ( 'const/iterate/..' @ X0 @ X1 ) )
!= $true )
| ( ( 'const/arith/<=' @ X2 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f37,plain,
( ( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ sK2 )
!= $true )
| ( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f36]) ).
thf(f36,plain,
( ( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) )
!= $true )
| ( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK3 ) @ sK2 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f27,f29]) ).
thf(f29,plain,
! [X0: 'type/nums/num' > $o] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ X0 @ sK3 ) )
= $true )
| ( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ X0 @ sK3 ) )
!= $true )
| ( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ X0 @ sK3 ) @ sK2 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f28]) ).
thf(f28,plain,
! [X0: 'type/nums/num' > $o] :
( ( ( 'const/arith/<=' @ ( sK0 @ 'type/nums/num' @ X0 @ sK3 ) @ sK2 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ X0 @ sK3 ) )
= $true )
| ( $true != $true )
| ( ( 'const/arith/<=' @ sK1 @ ( sK0 @ 'type/nums/num' @ X0 @ sK3 ) )
!= $true ) ),
inference(superposition,[status(thm)],[f24,f26]) ).
thf(f26,plain,
! [X3: 'type/nums/num'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( sK3 @ X3 ) )
= $true )
| ( ( 'const/arith/<=' @ sK1 @ X3 )
!= $true )
| ( ( 'const/arith/<=' @ X3 @ sK2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f24,plain,
! [X0: $tType,X2: X0 > $o,X1: X0 > 'type/realax/real'] :
( ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( X1 @ ( sK0 @ X0 @ X2 @ X1 ) ) )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEV514^1 : TPTP v9.2.1. Released v7.0.0.
% 0.10/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n020.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 12:48:27 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.44 ---- Original DTF file ---
% 0.16/0.44 thf(spec,logic,$$dhol).
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 % File : SEV514^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.44 % Domain : Analysis
% 0.16/0.44 % Problem : SUM_POS_LE_NUMSEG
% 0.16/0.44 % Version : Especial.
% 0.16/0.44 % English :
% 0.16/0.44
% 0.16/0.44 % Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% 0.16/0.44 % Source : [Kal16]
% 0.16/0.44 % Names : SUM_POS_LE_NUMSEG_.p [Kal16]
% 0.16/0.44
% 0.16/0.44 % Status : Theorem
% 0.16/0.44 % Rating : 0.00 v7.1.0
% 0.16/0.44 % Syntax : Number of formulae : 13 ( 1 unt; 10 typ; 0 def)
% 0.16/0.44 % Number of atoms : 13 ( 1 equ; 0 cnn)
% 0.16/0.44 % Maximal formula atoms : 5 ( 4 avg)
% 0.16/0.44 % Number of connectives : 48 ( 0 ~; 0 |; 2 &; 42 @)
% 0.16/0.44 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.16/0.44 % Maximal formula depth : 11 ( 9 avg)
% 0.16/0.44 % Number of types : 3 ( 2 usr)
% 0.16/0.44 % Number of type conns : 19 ( 19 >; 0 *; 0 +; 0 <<)
% 0.16/0.44 % Number of symbols : 9 ( 8 usr; 1 con; 0-3 aty)
% 0.16/0.44 % Number of variables : 13 ( 0 ^; 11 !; 0 ?; 13 :)
% 0.16/0.44 % ( 2 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.44 % SPC : TH1_THM_EQU_NAR
% 0.16/0.44
% 0.16/0.44 % Comments : Exported from core HOL Light.
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 thf('thf_type_type/realax/real',type,
% 0.16/0.44 'type/realax/real': $tType ).
% 0.16/0.44
% 0.16/0.44 thf('thf_type_type/nums/num',type,
% 0.16/0.44 'type/nums/num': $tType ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/sets/IN',type,
% 0.16/0.44 'const/sets/IN':
% 0.16/0.44 !>[A: $tType] : ( A > ( A > $o ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/realax/real_of_num',type,
% 0.16/0.44 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/realax/real_le',type,
% 0.16/0.44 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/nums/NUMERAL',type,
% 0.16/0.44 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/nums/_0',type,
% 0.16/0.44 'const/nums/_0': 'type/nums/num' ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/iterate/sum',type,
% 0.16/0.44 'const/iterate/sum':
% 0.16/0.44 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/iterate/..',type,
% 0.16/0.44 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 0.16/0.44
% 0.16/0.44 thf('thf_const_const/arith/<=',type,
% 0.16/0.44 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
% 0.16/0.44
% 0.16/0.44 thf('thm/iterate/SUM_POS_LE_',axiom,
% 0.16/0.44 ! [A: $tType,A0: A > 'type/realax/real',A1: A > $o] :
% 0.16/0.44 ( ! [A2: A] :
% 0.16/0.44 ( ( 'const/sets/IN' @ A @ A2 @ A1 )
% 0.16/0.44 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( A0 @ A2 ) ) )
% 0.16/0.44 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ A @ A1 @ A0 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf('thm/iterate/IN_NUMSEG_',axiom,
% 0.16/0.44 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 0.16/0.44 ( ( 'const/sets/IN' @ 'type/nums/num' @ A1 @ ( 'const/iterate/..' @ A @ A0 ) )
% 0.16/0.44 = ( ( 'const/arith/<=' @ A @ A1 )
% 0.16/0.44 & ( 'const/arith/<=' @ A1 @ A0 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf('thm/iterate/SUM_POS_LE_NUMSEG_',conjecture,
% 0.16/0.44 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num' > 'type/realax/real'] :
% 0.16/0.44 ( ! [A2: 'type/nums/num'] :
% 0.16/0.44 ( ( ( 'const/arith/<=' @ A @ A2 )
% 0.16/0.44 & ( 'const/arith/<=' @ A2 @ A0 ) )
% 0.16/0.44 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( A1 @ A2 ) ) )
% 0.16/0.44 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ A0 ) @ A1 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 ------------------------
% 1.31/1.26 ---- Embedded in THF ---
% 1.31/1.26 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.31/1.26 %%% Generated on Tue Feb 24 12:48:28 EST 2026
% 1.31/1.26 %%% using '$$dhol' embedding, version 1.3.0.
% 1.31/1.26 %%% Logic specification used:
% 1.31/1.26 %%% thf(spec, logic, $$dhol).
% 1.31/1.26
% 1.31/1.26 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2DTF2953.p
% See solution above
% 1.31/1.26 ------------------------
% 1.31/1.27 ---- Cleaned THF ---
% 1.31/1.27 thf('thf_type_type/realax/real',type,
% 1.31/1.27 'type/realax/real': $tType ).
% 1.31/1.27
% 1.31/1.27 thf('thf_type_type/nums/num',type,
% 1.31/1.27 'type/nums/num': $tType ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/sets/IN',type,
% 1.31/1.27 'const/sets/IN':
% 1.31/1.27 !>[A: $tType] : ( A > ( A > $o ) > $o ) ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/realax/real_of_num',type,
% 1.31/1.27 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/realax/real_le',type,
% 1.31/1.27 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/nums/NUMERAL',type,
% 1.31/1.27 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/nums/_0',type,
% 1.31/1.27 'const/nums/_0': 'type/nums/num' ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/iterate/sum',type,
% 1.31/1.27 'const/iterate/sum':
% 1.31/1.27 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/iterate/..',type,
% 1.31/1.27 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 1.31/1.27
% 1.31/1.27 thf('thf_const_const/arith/<=',type,
% 1.31/1.27 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
% 1.31/1.27
% 1.31/1.27 thf('thm/iterate/SUM_POS_LE_',axiom,
% 1.31/1.27 ! [A: $tType,A0: A > 'type/realax/real',A1: A > $o] :
% 1.31/1.27 ( ! [A2: A] :
% 1.31/1.27 ( ( 'const/sets/IN' @ A @ A2 @ A1 )
% 1.31/1.27 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( A0 @ A2 ) ) )
% 1.31/1.27 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ A @ A1 @ A0 ) ) ) ).
% 1.31/1.27
% 1.31/1.27 thf('thm/iterate/IN_NUMSEG_',axiom,
% 1.31/1.27 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 1.31/1.27 ( ( 'const/sets/IN' @ 'type/nums/num' @ A1 @ ( 'const/iterate/..' @ A @ A0 ) )
% 1.31/1.27 = ( ( 'const/arith/<=' @ A @ A1 )
% 1.31/1.27 & ( 'const/arith/<=' @ A1 @ A0 ) ) ) ).
% 1.31/1.27
% 1.31/1.27 thf('thm/iterate/SUM_POS_LE_NUMSEG_',conjecture,
% 1.31/1.27 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num' > 'type/realax/real'] :
% 1.31/1.27 ( ! [A2: 'type/nums/num'] :
% 1.31/1.27 ( ( ( 'const/arith/<=' @ A @ A2 )
% 1.31/1.27 & ( 'const/arith/<=' @ A2 @ A0 ) )
% 1.31/1.27 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( A1 @ A2 ) ) )
% 1.31/1.27 => ( 'const/realax/real_le' @ ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ A0 ) @ A1 ) ) ) ).
% 1.31/1.27 ------------------------
% 1.31/1.28 % (3069)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/2Mi)
% 1.31/1.28 % (3072)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/18Mi)
% 1.31/1.28 % (3070)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/2Mi)
% 1.31/1.28 % (3068)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/27Mi)
% 1.31/1.28 % (3067)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/4Mi)
% 1.31/1.28 % (3071)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/275Mi)
% 1.31/1.28 % (3066)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2953 for (3000ds/183Mi)
% 1.31/1.28 % (3069)Instruction limit reached!
% 1.31/1.28 % (3069)------------------------------
% 1.31/1.28 % (3069)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.31/1.28 % (3069)Termination reason: Unknown
% 1.31/1.28 % (3069)Termination phase: Preprocessing 3
% 1.31/1.28
% 1.31/1.28 % (3069)Memory used [KB]: 895
% 1.31/1.28 % (3069)Time elapsed: 0.002 s
% 1.31/1.28 % (3069)Instructions burned: 2 (million)
% 1.31/1.28 % (3069)------------------------------
% 1.31/1.28 % (3069)------------------------------
% 1.31/1.28 % (3070)Instruction limit reached!
% 1.31/1.28 % (3070)------------------------------
% 1.31/1.28 % (3070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.31/1.28 % (3070)Termination reason: Unknown
% 1.31/1.28 % (3070)Termination phase: Property scanning
% 1.31/1.28
% 1.31/1.28 % (3070)Memory used [KB]: 1023
% 1.31/1.28 % (3070)Time elapsed: 0.002 s
% 1.31/1.28 % (3070)Instructions burned: 2 (million)
% 1.31/1.28 % (3070)------------------------------
% 1.31/1.28 % (3070)------------------------------
% 1.31/1.28 % (3067)Instruction limit reached!
% 1.31/1.28 % (3067)------------------------------
% 1.31/1.28 % (3067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.31/1.28 % (3067)Termination reason: Unknown
% 1.31/1.28 % (3067)Termination phase: Saturation
% 1.31/1.28
% 1.31/1.28 % (3067)Memory used [KB]: 5500
% 1.31/1.28 % (3067)Time elapsed: 0.003 s
% 1.31/1.28 % (3067)Instructions burned: 4 (million)
% 1.31/1.28 % (3067)------------------------------
% 1.31/1.28 % (3067)------------------------------
% 1.31/1.28 % (3071)First to succeed.
% 1.31/1.28 % (3071)Refutation found. Thanks to Tanya!
% 1.31/1.28 % SZS status Theorem for DTF2THF_2953
% 1.31/1.28 % SZS output start Proof for DTF2THF_2953
% 1.31/1.28 thf(type_def_6, type, 'type/nums/num': $tType).
% 1.31/1.28 thf(type_def_7, type, 'type/realax/real': $tType).
% 1.31/1.28 thf(func_def_0, type, 'type/realax/real': $tType).
% 1.31/1.28 thf(func_def_1, type, 'type/nums/num': $tType).
% 1.31/1.28 thf(func_def_2, type, 'const/sets/IN': !>[X0: $tType]:(X0 > (X0 > $o) > $o)).
% 1.31/1.28 thf(func_def_3, type, 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real').
% 1.31/1.28 thf(func_def_4, type, 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o).
% 1.31/1.28 thf(func_def_5, type, 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num').
% 1.31/1.28 thf(func_def_6, type, 'const/nums/_0': 'type/nums/num').
% 1.31/1.28 thf(func_def_7, type, 'const/iterate/sum': !>[X0: $tType]:((X0 > $o) > (X0 > 'type/realax/real') > 'type/realax/real')).
% 1.31/1.28 thf(func_def_8, type, 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o).
% 1.31/1.28 thf(func_def_9, type, 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o).
% 1.31/1.28 thf(func_def_13, type, sK0: !>[X0: $tType]:((X0 > $o) > (X0 > 'type/realax/real') > X0)).
% 1.31/1.28 thf(func_def_14, type, sK1: 'type/nums/num').
% 1.31/1.28 thf(func_def_15, type, sK2: 'type/nums/num').
% 1.31/1.28 thf(func_def_16, type, sK3: 'type/nums/num' > 'type/realax/real').
% 1.31/1.28 thf(func_def_18, type, ph5: !>[X0: $tType]:(X0)).
% 1.31/1.28 thf(f39,plain,(
% 1.31/1.28 $false),
% 1.31/1.28 inference(subsumption_resolution,[status(thm)],[f38,f35])).
% 1.31/1.28 thf(f35,plain,(
% 1.31/1.28 (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) = $true)),
% 1.31/1.28 inference(trivial_inequality_removal,[status(thm)],[f32])).
% 1.31/1.28 thf(f32,plain,(
% 1.31/1.28 ($true != $true) | (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) = $true)),
% 1.31/1.28 inference(superposition,[status(thm)],[f22,f31])).
% 1.31/1.28 thf(f31,plain,(
% 1.31/1.28 (('const/sets/IN' @ 'type/nums/num' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ ('const/iterate/..' @ sK1 @ sK2)) = $true)),
% 1.31/1.28 inference(trivial_inequality_removal,[status(thm)],[f30])).
% 1.31/1.28 thf(f30,plain,(
% 1.31/1.28 ($true != $true) | (('const/sets/IN' @ 'type/nums/num' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ ('const/iterate/..' @ sK1 @ sK2)) = $true)),
% 1.31/1.28 inference(superposition,[status(thm)],[f27,f25])).
% 1.31/1.28 thf(f25,plain,(
% 1.31/1.28 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0 > 'type/realax/real'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)) = $true) | (('const/sets/IN' @ X0 @ (sK0 @ X0 @ X2 @ X1) @ X2) = $true)) )),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f18])).
% 1.31/1.28 thf(f18,plain,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (((('const/sets/IN' @ X0 @ (sK0 @ X0 @ X2 @ X1) @ X2) = $true) & (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ (sK0 @ X0 @ X2 @ X1))) != $true)) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)) = $true))),
% 1.31/1.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f12,f17])).
% 1.31/1.28 thf(f17,plain,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (? [X3 : X0] : ((('const/sets/IN' @ X0 @ X3 @ X2) = $true) & (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ X3)) != $true)) => ((('const/sets/IN' @ X0 @ (sK0 @ X0 @ X2 @ X1) @ X2) = $true) & (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ (sK0 @ X0 @ X2 @ X1))) != $true)))),
% 1.31/1.28 introduced(definition,[],[choice_axiom])).
% 1.31/1.28 thf(f12,plain,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (? [X3 : X0] : ((('const/sets/IN' @ X0 @ X3 @ X2) = $true) & (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ X3)) != $true)) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)) = $true))),
% 1.31/1.28 inference(ennf_transformation,[status(thm)],[f9])).
% 1.31/1.28 thf(f9,plain,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (! [X3 : X0] : ((('const/sets/IN' @ X0 @ X3 @ X2) = $true) => (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ X3)) = $true)) => (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)) = $true))),
% 1.31/1.28 inference(fool_elimination,[status(thm)],[f8])).
% 1.31/1.28 thf(f8,plain,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (! [X3 : X0] : (('const/sets/IN' @ X0 @ X3 @ X2) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ X3))) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)))),
% 1.31/1.28 inference(rectify,[status(thm)],[f1])).
% 1.31/1.28 thf(f1,axiom,(
% 1.31/1.28 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (! [X3 : X0] : (('const/sets/IN' @ X0 @ X3 @ X2) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ X3))) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)))),
% 1.31/1.28 file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/SUM_POS_LE_')).
% 1.31/1.28 thf(f27,plain,(
% 1.31/1.28 (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true)),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f20])).
% 1.31/1.28 thf(f20,plain,(
% 1.31/1.28 (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true) & ! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ sK1 @ X3) != $true) | (('const/arith/<=' @ X3 @ sK2) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (sK3 @ X3)) = $true))),
% 1.31/1.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2,sK3])],[f14,f19])).
% 1.31/1.28 thf(f19,plain,(
% 1.31/1.28 ? [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) != $true) & ! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X3) != $true) | (('const/arith/<=' @ X3 @ X1) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3)) = $true))) => ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true) & ! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ sK1 @ X3) != $true) | (('const/arith/<=' @ X3 @ sK2) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (sK3 @ X3)) = $true)))),
% 1.31/1.28 introduced(definition,[],[choice_axiom])).
% 1.31/1.28 thf(f14,plain,(
% 1.31/1.28 ? [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) != $true) & ! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X3) != $true) | (('const/arith/<=' @ X3 @ X1) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3)) = $true)))),
% 1.31/1.28 inference(flattening,[status(thm)],[f13])).
% 1.31/1.28 thf(f13,plain,(
% 1.31/1.28 ? [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) != $true) & ! [X3 : 'type/nums/num'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3)) = $true) | ((('const/arith/<=' @ X3 @ X1) != $true) | (('const/arith/<=' @ X0 @ X3) != $true))))),
% 1.31/1.28 inference(ennf_transformation,[status(thm)],[f7])).
% 1.31/1.28 thf(f7,plain,(
% 1.31/1.28 ~! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : (! [X3 : 'type/nums/num'] : (((('const/arith/<=' @ X3 @ X1) = $true) & (('const/arith/<=' @ X0 @ X3) = $true)) => (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3)) = $true)) => (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) = $true))),
% 1.31/1.28 inference(fool_elimination,[status(thm)],[f6])).
% 1.31/1.28 thf(f6,plain,(
% 1.31/1.28 ~! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : (! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X3) & ('const/arith/<=' @ X3 @ X1)) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3))) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)))),
% 1.31/1.28 inference(rectify,[status(thm)],[f4])).
% 1.31/1.28 thf(f4,negated_conjecture,(
% 1.31/1.28 ~! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : (! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X3) & ('const/arith/<=' @ X3 @ X1)) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3))) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)))),
% 1.31/1.28 inference(negated_conjecture,[status(cth)],[f3])).
% 1.31/1.28 thf(f3,conjecture,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num' > 'type/realax/real'] : (! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X3) & ('const/arith/<=' @ X3 @ X1)) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X2 @ X3))) => ('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)))),
% 1.31/1.28 file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/SUM_POS_LE_NUMSEG_')).
% 1.31/1.28 thf(f22,plain,(
% 1.31/1.28 ( ! [X2 : 'type/nums/num',X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) != $true) | (('const/arith/<=' @ X0 @ X2) = $true)) )),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f16])).
% 1.31/1.28 thf(f16,plain,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((((('const/arith/<=' @ X2 @ X1) = $true) & (('const/arith/<=' @ X0 @ X2) = $true)) | (('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) != $true)) & ((('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) = $true) | (('const/arith/<=' @ X2 @ X1) != $true) | (('const/arith/<=' @ X0 @ X2) != $true)))),
% 1.31/1.28 inference(flattening,[status(thm)],[f15])).
% 1.31/1.28 thf(f15,plain,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((((('const/arith/<=' @ X2 @ X1) = $true) & (('const/arith/<=' @ X0 @ X2) = $true)) | (('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) != $true)) & ((('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) = $true) | ((('const/arith/<=' @ X2 @ X1) != $true) | (('const/arith/<=' @ X0 @ X2) != $true))))),
% 1.31/1.28 inference(nnf_transformation,[status(thm)],[f11])).
% 1.31/1.28 thf(f11,plain,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (((('const/arith/<=' @ X2 @ X1) = $true) & (('const/arith/<=' @ X0 @ X2) = $true)) <=> (('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) = $true))),
% 1.31/1.28 inference(fool_elimination,[status(thm)],[f10])).
% 1.31/1.28 thf(f10,plain,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) = ('const/arith/<=' @ X2 @ X1) & ('const/arith/<=' @ X0 @ X2))),
% 1.31/1.28 inference(rectify,[status(thm)],[f2])).
% 1.31/1.28 thf(f2,axiom,(
% 1.31/1.28 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) = ('const/arith/<=' @ X2 @ X1) & ('const/arith/<=' @ X0 @ X2))),
% 1.31/1.28 file('/export/starexec/sandbox/tmp/tmp.kB77H2eNa8/DTF2THF_2953.p','thm/iterate/IN_NUMSEG_')).
% 1.31/1.28 thf(f38,plain,(
% 1.31/1.28 (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true)),
% 1.31/1.28 inference(subsumption_resolution,[status(thm)],[f37,f34])).
% 1.31/1.28 thf(f34,plain,(
% 1.31/1.28 (('const/arith/<=' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ sK2) = $true)),
% 1.31/1.28 inference(trivial_inequality_removal,[status(thm)],[f33])).
% 1.31/1.28 thf(f33,plain,(
% 1.31/1.28 ($true != $true) | (('const/arith/<=' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ sK2) = $true)),
% 1.31/1.28 inference(superposition,[status(thm)],[f23,f31])).
% 1.31/1.28 thf(f23,plain,(
% 1.31/1.28 ( ! [X2 : 'type/nums/num',X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/sets/IN' @ 'type/nums/num' @ X2 @ ('const/iterate/..' @ X0 @ X1)) != $true) | (('const/arith/<=' @ X2 @ X1) = $true)) )),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f16])).
% 1.31/1.28 thf(f37,plain,(
% 1.31/1.28 (('const/arith/<=' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ sK2) != $true) | (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true)),
% 1.31/1.28 inference(trivial_inequality_removal,[status(thm)],[f36])).
% 1.31/1.28 thf(f36,plain,(
% 1.31/1.28 (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3)) != $true) | (('const/arith/<=' @ (sK0 @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK3) @ sK2) != $true) | ($true != $true)),
% 1.31/1.28 inference(superposition,[status(thm)],[f27,f29])).
% 1.31/1.28 thf(f29,plain,(
% 1.31/1.28 ( ! [X0 : 'type/nums/num' > $o] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ X0 @ sK3)) = $true) | (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ X0 @ sK3)) != $true) | (('const/arith/<=' @ (sK0 @ 'type/nums/num' @ X0 @ sK3) @ sK2) != $true)) )),
% 1.31/1.28 inference(trivial_inequality_removal,[status(thm)],[f28])).
% 1.31/1.28 thf(f28,plain,(
% 1.31/1.28 ( ! [X0 : 'type/nums/num' > $o] : ((('const/arith/<=' @ (sK0 @ 'type/nums/num' @ X0 @ sK3) @ sK2) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ 'type/nums/num' @ X0 @ sK3)) = $true) | ($true != $true) | (('const/arith/<=' @ sK1 @ (sK0 @ 'type/nums/num' @ X0 @ sK3)) != $true)) )),
% 1.31/1.28 inference(superposition,[status(thm)],[f24,f26])).
% 1.31/1.28 thf(f26,plain,(
% 1.31/1.28 ( ! [X3 : 'type/nums/num'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (sK3 @ X3)) = $true) | (('const/arith/<=' @ sK1 @ X3) != $true) | (('const/arith/<=' @ X3 @ sK2) != $true)) )),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f20])).
% 1.31/1.28 thf(f24,plain,(
% 1.31/1.28 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0 > 'type/realax/real'] : ((('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ (X1 @ (sK0 @ X0 @ X2 @ X1))) != $true) | (('const/realax/real_le' @ ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) @ ('const/iterate/sum' @ X0 @ X2 @ X1)) = $true)) )),
% 1.31/1.28 inference(cnf_transformation,[status(thm)],[f18])).
% 1.31/1.28 % SZS output end Proof for DTF2THF_2953
% 1.31/1.28 % (3071)------------------------------
% 1.31/1.28 % (3071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.31/1.28 % (3071)Termination reason: Refutation
% 1.31/1.28
% 1.31/1.28 % (3071)Memory used [KB]: 5500
% 1.31/1.28 % (3071)Time elapsed: 0.004 s
% 1.31/1.28 % (3071)Instructions burned: 4 (million)
% 1.31/1.28 % (3071)------------------------------
% 1.31/1.28 % (3071)------------------------------
% 1.31/1.28 % (3064)Success in time 0.012 s
%------------------------------------------------------------------------------