%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ANA107^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n028.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 08:38:53 AM UTC 2026
% Result : Theorem 1.68s 1.33s
% Output : Refutation 1.68s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 4
% Syntax : Number of formulae : 21 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 90 ( 18 equ; 0 cnn)
% Maximal formula atoms : 3 ( 4 avg)
% Number of connectives : 265 ( 12 ~; 3 |; 0 &; 246 @)
% ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<)
% Number of symbols : 13 ( 10 usr; 4 con; 0-3 aty)
% Number of variables : 63 ( 16 ^; 41 !; 6 ?; 63 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_6,type,
'type/realax/real': $tType ).
thf(type_def_7,type,
'type/nums/num': $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/FINITE':
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_3,type,
'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
thf(func_def_4,type,
'const/realax/real_abs': 'type/realax/real' > 'type/realax/real' ).
thf(func_def_5,type,
'const/iterate/sum':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > 'type/realax/real' ) > 'type/realax/real' ) ).
thf(func_def_6,type,
'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
thf(func_def_8,type,
vEPSILON:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_13,type,
sK0: 'type/nums/num' > 'type/realax/real' ).
thf(func_def_14,type,
sK1: 'type/nums/num' ).
thf(func_def_15,type,
sK2: 'type/nums/num' ).
thf(func_def_17,type,
ph4:
!>[X0: $tType] : X0 ).
thf(f22,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f21]) ).
thf(f21,plain,
$true != $true,
inference(superposition,[status(thm)],[f17,f20]) ).
thf(f20,plain,
! [X2: 'type/nums/num' > 'type/realax/real',X0: 'type/nums/num',X1: 'type/nums/num'] :
( $true
= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X2 @ Y0 ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f19]) ).
thf(f19,plain,
! [X2: 'type/nums/num' > 'type/realax/real',X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( $true != $true )
| ( $true
= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) @ X2 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X2 @ Y0 ) ) ) ) ) ),
inference(superposition,[status(thm)],[f16,f18]) ).
thf(f18,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: 'type/nums/num',X1: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ X0 @ X1 ) ),
file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/FINITE_NUMSEG_') ).
thf(f16,plain,
! [X0: $tType,X2: X0 > $o,X1: X0 > 'type/realax/real'] :
( ( ( 'const/sets/FINITE' @ X0 @ X2 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
@ ( 'const/iterate/sum' @ X0 @ X2
@ ^ [Y0: X0] : ( 'const/realax/real_abs' @ ( X1 @ Y0 ) ) ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ( ( 'const/sets/FINITE' @ X0 @ X2 )
!= $true )
| ( ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
@ ( 'const/iterate/sum' @ X0 @ X2
@ ^ [Y0: X0] : ( 'const/realax/real_abs' @ ( X1 @ Y0 ) ) ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ( ( 'const/sets/FINITE' @ X0 @ X2 )
= $true )
=> ( ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
@ ( 'const/iterate/sum' @ X0 @ X2
@ ^ [Y0: X0] : ( 'const/realax/real_abs' @ ( X1 @ Y0 ) ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ( 'const/sets/FINITE' @ X0 @ X2 )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
@ ( 'const/iterate/sum' @ X0 @ X2
@ ^ [X3: X0] : ( 'const/realax/real_abs' @ ( X1 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $tType,X1: X0 > 'type/realax/real',X2: X0 > $o] :
( ( 'const/sets/FINITE' @ X0 @ X2 )
=> ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ X0 @ X2 @ X1 ) )
@ ( 'const/iterate/sum' @ X0 @ X2
@ ^ [X3: X0] : ( 'const/realax/real_abs' @ ( X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/SUM_ABS_') ).
thf(f17,plain,
( $true
!= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( sK0 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( $true
!= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( sK0 @ Y0 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f12,f14]) ).
thf(f14,plain,
( ? [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( $true
!= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ Y0 ) ) ) ) )
=> ( $true
!= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( sK0 @ Y0 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f12,plain,
? [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( $true
!= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ Y0 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( $true
= ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [Y0: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [X3: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [X3: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ X3 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
@ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 )
@ ^ [X3: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( X0 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/SUM_ABS_NUMSEG_') ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ANA107^1 : TPTP v9.2.1. Released v7.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n028.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 03:50:27 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ANA107^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.47 % Domain : Analysis
% 0.29/0.47 % Problem : SUM_ABS_NUMSEG
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Kal16]
% 0.29/0.47 % Names : SUM_ABS_NUMSEG_.p [Kal16]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.00 v7.1.0
% 0.29/0.47 % Syntax : Number of formulae : 10 ( 0 unt; 7 typ; 0 def)
% 0.29/0.47 % Number of atoms : 7 ( 0 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 3 ( 2 avg)
% 0.29/0.47 % Number of connectives : 33 ( 0 ~; 0 |; 0 &; 32 @)
% 0.29/0.47 % ( 0 <=>; 1 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 11 ( 9 avg)
% 0.29/0.47 % Number of types : 3 ( 2 usr)
% 0.29/0.47 % Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 5 ( 5 usr; 0 con; 1-3 aty)
% 0.29/0.47 % Number of variables : 12 ( 2 ^; 8 !; 0 ?; 12 :)
% 0.29/0.47 % ( 2 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.47 % SPC : TH1_THM_NEQ_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : Exported from core HOL Light.
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 thf('thf_type_type/realax/real',type,
% 0.29/0.47 'type/realax/real': $tType ).
% 0.29/0.47
% 0.29/0.47 thf('thf_type_type/nums/num',type,
% 0.29/0.47 'type/nums/num': $tType ).
% 0.29/0.47
% 0.29/0.47 thf('thf_const_const/sets/FINITE',type,
% 0.29/0.47 'const/sets/FINITE':
% 0.29/0.47 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf('thf_const_const/realax/real_le',type,
% 0.29/0.47 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
% 0.29/0.47
% 0.29/0.47 thf('thf_const_const/realax/real_abs',type,
% 0.29/0.47 'const/realax/real_abs': 'type/realax/real' > 'type/realax/real' ).
% 0.29/0.47
% 0.29/0.47 thf('thf_const_const/iterate/sum',type,
% 0.29/0.47 'const/iterate/sum':
% 0.29/0.47 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 0.29/0.47
% 0.29/0.47 thf('thf_const_const/iterate/..',type,
% 0.29/0.47 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 0.29/0.47
% 0.29/0.47 thf('thm/iterate/SUM_ABS_',axiom,
% 0.29/0.47 ! [A: $tType,A0: A > 'type/realax/real',A1: A > $o] :
% 0.29/0.47 ( ( 'const/sets/FINITE' @ A @ A1 )
% 0.29/0.47 => ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ A @ A1 @ A0 ) )
% 0.29/0.47 @ ( 'const/iterate/sum' @ A @ A1
% 0.29/0.47 @ ^ [A2: A] : ( 'const/realax/real_abs' @ ( A0 @ A2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf('thm/iterate/FINITE_NUMSEG_',axiom,
% 0.29/0.47 ! [A: 'type/nums/num',A0: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ A0 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf('thm/iterate/SUM_ABS_NUMSEG_',conjecture,
% 0.29/0.47 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 0.29/0.47 ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A ) )
% 0.29/0.47 @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 )
% 0.29/0.47 @ ^ [A2: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( A @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 1.50/1.30 ---- Embedded in THF ---
% 1.68/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.68/1.30 %%% Generated on Tue Feb 24 03:50:27 EST 2026
% 1.68/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 1.68/1.30 %%% Logic specification used:
% 1.68/1.30 %%% thf(spec, logic, $$dhol).
% 1.68/1.30
% 1.68/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2DTF16199.p
% See solution above
% 1.68/1.30 ------------------------
% 1.68/1.31 ---- Cleaned THF ---
% 1.68/1.31 thf('thf_type_type/realax/real',type,
% 1.68/1.31 'type/realax/real': $tType ).
% 1.68/1.31
% 1.68/1.31 thf('thf_type_type/nums/num',type,
% 1.68/1.31 'type/nums/num': $tType ).
% 1.68/1.31
% 1.68/1.31 thf('thf_const_const/sets/FINITE',type,
% 1.68/1.31 'const/sets/FINITE':
% 1.68/1.31 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 1.68/1.31
% 1.68/1.31 thf('thf_const_const/realax/real_le',type,
% 1.68/1.31 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o ).
% 1.68/1.31
% 1.68/1.31 thf('thf_const_const/realax/real_abs',type,
% 1.68/1.31 'const/realax/real_abs': 'type/realax/real' > 'type/realax/real' ).
% 1.68/1.31
% 1.68/1.31 thf('thf_const_const/iterate/sum',type,
% 1.68/1.31 'const/iterate/sum':
% 1.68/1.31 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 1.68/1.31
% 1.68/1.31 thf('thf_const_const/iterate/..',type,
% 1.68/1.31 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 1.68/1.31
% 1.68/1.31 thf('thm/iterate/SUM_ABS_',axiom,
% 1.68/1.31 ! [A: $tType,A0: A > 'type/realax/real',A1: A > $o] :
% 1.68/1.31 ( ( 'const/sets/FINITE' @ A @ A1 )
% 1.68/1.31 => ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ A @ A1 @ A0 ) )
% 1.68/1.31 @ ( 'const/iterate/sum' @ A @ A1
% 1.68/1.31 @ ^ [A2: A] : ( 'const/realax/real_abs' @ ( A0 @ A2 ) ) ) ) ) ).
% 1.68/1.31
% 1.68/1.31 thf('thm/iterate/FINITE_NUMSEG_',axiom,
% 1.68/1.31 ! [A: 'type/nums/num',A0: 'type/nums/num'] : ( 'const/sets/FINITE' @ 'type/nums/num' @ ( 'const/iterate/..' @ A @ A0 ) ) ).
% 1.68/1.31
% 1.68/1.31 thf('thm/iterate/SUM_ABS_NUMSEG_',conjecture,
% 1.68/1.31 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 1.68/1.31 ( 'const/realax/real_le' @ ( 'const/realax/real_abs' @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A ) )
% 1.68/1.31 @ ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 )
% 1.68/1.31 @ ^ [A2: 'type/nums/num'] : ( 'const/realax/real_abs' @ ( A @ A2 ) ) ) ) ).
% 1.68/1.31 ------------------------
% 1.68/1.32 % (16474)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_16199 for (2999ds/27Mi)
% 1.68/1.32 % (16478)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16199 for (2999ds/18Mi)
% 1.68/1.33 % (16474)First to succeed.
% 1.68/1.33 % (16474)Refutation found. Thanks to Tanya!
% 1.68/1.33 % SZS status Theorem for DTF2THF_16199
% 1.68/1.33 % SZS output start Proof for DTF2THF_16199
% 1.68/1.33 thf(type_def_6, type, 'type/realax/real': $tType).
% 1.68/1.33 thf(type_def_7, type, 'type/nums/num': $tType).
% 1.68/1.33 thf(func_def_0, type, 'type/realax/real': $tType).
% 1.68/1.33 thf(func_def_1, type, 'type/nums/num': $tType).
% 1.68/1.33 thf(func_def_2, type, 'const/sets/FINITE': !>[X0: $tType]:((X0 > $o) > $o)).
% 1.68/1.33 thf(func_def_3, type, 'const/realax/real_le': 'type/realax/real' > 'type/realax/real' > $o).
% 1.68/1.33 thf(func_def_4, type, 'const/realax/real_abs': 'type/realax/real' > 'type/realax/real').
% 1.68/1.33 thf(func_def_5, type, 'const/iterate/sum': !>[X0: $tType]:((X0 > $o) > (X0 > 'type/realax/real') > 'type/realax/real')).
% 1.68/1.33 thf(func_def_6, type, 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o).
% 1.68/1.33 thf(func_def_8, type, vEPSILON: !>[X0: $tType]:((X0 > $o) > X0)).
% 1.68/1.33 thf(func_def_13, type, sK0: 'type/nums/num' > 'type/realax/real').
% 1.68/1.33 thf(func_def_14, type, sK1: 'type/nums/num').
% 1.68/1.33 thf(func_def_15, type, sK2: 'type/nums/num').
% 1.68/1.33 thf(func_def_17, type, ph4: !>[X0: $tType]:(X0)).
% 1.68/1.33 thf(f22,plain,(
% 1.68/1.33 $false),
% 1.68/1.33 inference(trivial_inequality_removal,[status(thm)],[f21])).
% 1.68/1.33 thf(f21,plain,(
% 1.68/1.33 ($true != $true)),
% 1.68/1.33 inference(superposition,[status(thm)],[f17,f20])).
% 1.68/1.33 thf(f20,plain,(
% 1.68/1.33 ( ! [X2 : 'type/nums/num' > 'type/realax/real',X0 : 'type/nums/num',X1 : 'type/nums/num'] : (($true = ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (X2 @ Y0))))))) )),
% 1.68/1.33 inference(trivial_inequality_removal,[status(thm)],[f19])).
% 1.68/1.33 thf(f19,plain,(
% 1.68/1.33 ( ! [X2 : 'type/nums/num' > 'type/realax/real',X0 : 'type/nums/num',X1 : 'type/nums/num'] : (($true != $true) | ($true = ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ X2)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (X2 @ Y0))))))) )),
% 1.68/1.33 inference(superposition,[status(thm)],[f16,f18])).
% 1.68/1.33 thf(f18,plain,(
% 1.68/1.33 ( ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/sets/FINITE' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1)) = $true)) )),
% 1.68/1.33 inference(cnf_transformation,[status(thm)],[f7])).
% 1.68/1.33 thf(f7,plain,(
% 1.68/1.33 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : (('const/sets/FINITE' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1)) = $true)),
% 1.68/1.33 inference(fool_elimination,[status(thm)],[f6])).
% 1.68/1.33 thf(f6,plain,(
% 1.68/1.33 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : ('const/sets/FINITE' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1))),
% 1.68/1.33 inference(rectify,[status(thm)],[f2])).
% 1.68/1.33 thf(f2,axiom,(
% 1.68/1.33 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : ('const/sets/FINITE' @ 'type/nums/num' @ ('const/iterate/..' @ X0 @ X1))),
% 1.68/1.33 file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/FINITE_NUMSEG_')).
% 1.68/1.33 thf(f16,plain,(
% 1.68/1.33 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0 > 'type/realax/real'] : ((('const/sets/FINITE' @ X0 @ X2) != $true) | (('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ X0 @ X2 @ X1)) @ ('const/iterate/sum' @ X0 @ X2 @ (^[Y0 : X0]: ('const/realax/real_abs' @ (X1 @ Y0))))) = $true)) )),
% 1.68/1.33 inference(cnf_transformation,[status(thm)],[f13])).
% 1.68/1.33 thf(f13,plain,(
% 1.68/1.33 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : ((('const/sets/FINITE' @ X0 @ X2) != $true) | (('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ X0 @ X2 @ X1)) @ ('const/iterate/sum' @ X0 @ X2 @ (^[Y0 : X0]: ('const/realax/real_abs' @ (X1 @ Y0))))) = $true))),
% 1.68/1.33 inference(ennf_transformation,[status(thm)],[f9])).
% 1.68/1.33 thf(f9,plain,(
% 1.68/1.33 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : ((('const/sets/FINITE' @ X0 @ X2) = $true) => (('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ X0 @ X2 @ X1)) @ ('const/iterate/sum' @ X0 @ X2 @ (^[Y0 : X0]: ('const/realax/real_abs' @ (X1 @ Y0))))) = $true))),
% 1.68/1.33 inference(fool_elimination,[status(thm)],[f8])).
% 1.68/1.33 thf(f8,plain,(
% 1.68/1.33 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (('const/sets/FINITE' @ X0 @ X2) => ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ X0 @ X2 @ X1)) @ ('const/iterate/sum' @ X0 @ X2 @ (^[X3 : X0] : ('const/realax/real_abs' @ (X1 @ X3))))))),
% 1.68/1.33 inference(rectify,[status(thm)],[f1])).
% 1.68/1.33 thf(f1,axiom,(
% 1.68/1.33 ! [X0 : $tType,X1 : X0 > 'type/realax/real',X2 : X0 > $o] : (('const/sets/FINITE' @ X0 @ X2) => ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ X0 @ X2 @ X1)) @ ('const/iterate/sum' @ X0 @ X2 @ (^[X3 : X0] : ('const/realax/real_abs' @ (X1 @ X3))))))),
% 1.68/1.33 file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/SUM_ABS_')).
% 1.68/1.33 thf(f17,plain,(
% 1.68/1.33 ($true != ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (sK0 @ Y0))))))),
% 1.68/1.33 inference(cnf_transformation,[status(thm)],[f15])).
% 1.68/1.33 thf(f15,plain,(
% 1.68/1.33 ($true != ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (sK0 @ Y0))))))),
% 1.68/1.33 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f12,f14])).
% 1.68/1.33 thf(f14,plain,(
% 1.68/1.33 ? [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ($true != ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (X0 @ Y0)))))) => ($true != ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (sK0 @ Y0))))))),
% 1.68/1.33 introduced(definition,[],[choice_axiom])).
% 1.68/1.33 thf(f12,plain,(
% 1.68/1.33 ? [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ($true != ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (X0 @ Y0))))))),
% 1.68/1.33 inference(ennf_transformation,[status(thm)],[f11])).
% 1.68/1.33 thf(f11,plain,(
% 1.68/1.33 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ($true = ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[Y0 : 'type/nums/num']: ('const/realax/real_abs' @ (X0 @ Y0))))))),
% 1.68/1.33 inference(fool_elimination,[status(thm)],[f10])).
% 1.68/1.33 thf(f10,plain,(
% 1.68/1.33 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[X3 : 'type/nums/num'] : ('const/realax/real_abs' @ (X0 @ X3)))))),
% 1.68/1.33 inference(rectify,[status(thm)],[f4])).
% 1.68/1.33 thf(f4,negated_conjecture,(
% 1.68/1.33 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[X3 : 'type/nums/num'] : ('const/realax/real_abs' @ (X0 @ X3)))))),
% 1.68/1.33 inference(negated_conjecture,[status(cth)],[f3])).
% 1.68/1.33 thf(f3,conjecture,(
% 1.68/1.33 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ('const/realax/real_le' @ ('const/realax/real_abs' @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) @ ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ (^[X3 : 'type/nums/num'] : ('const/realax/real_abs' @ (X0 @ X3)))))),
% 1.68/1.33 file('/export/starexec/sandbox2/tmp/tmp.eAR2YrsIYj/DTF2THF_16199.p','thm/iterate/SUM_ABS_NUMSEG_')).
% 1.68/1.33 % SZS output end Proof for DTF2THF_16199
% 1.68/1.33 % (16474)------------------------------
% 1.68/1.33 % (16474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.33 % (16474)Termination reason: Refutation
% 1.68/1.33
% 1.68/1.33 % (16474)Memory used [KB]: 5500
% 1.68/1.33 % (16474)Time elapsed: 0.004 s
% 1.68/1.33 % (16474)Instructions burned: 3 (million)
% 1.68/1.33 % (16474)------------------------------
% 1.68/1.33 % (16474)------------------------------
% 1.68/1.33 % (16471)Success in time 0.015 s
%------------------------------------------------------------------------------