%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ANA108^1 : 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 : n019.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 1.69s 1.49s
% Output : Refutation 1.69s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 41 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 235 ( 86 equ; 0 cnn)
% Maximal formula atoms : 6 ( 5 avg)
% Number of connectives : 434 ( 40 ~; 21 |; 21 &; 335 @)
% ( 2 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 13 usr; 5 con; 0-3 aty)
% Number of variables : 83 ( 0 ^; 74 !; 9 ?; 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/trivia/I':
!>[X0: $tType] : ( X0 > X0 ) ).
thf(func_def_3,type,
'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
thf(func_def_4,type,
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
thf(func_def_5,type,
'const/nums/_0': 'type/nums/num' ).
thf(func_def_6,type,
'const/iterate/sum':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > 'type/realax/real' ) > 'type/realax/real' ) ).
thf(func_def_7,type,
'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
thf(func_def_8,type,
'const/arith/<=': '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: '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_16,type,
sK3: 'type/nums/num' > 'type/nums/num' > ( 'type/nums/num' > 'type/realax/real' ) > 'type/nums/num' ).
thf(func_def_18,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f49,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f47,f42]) ).
thf(f42,plain,
( $true
= ( 'const/arith/<=' @ ( sK3 @ sK2 @ sK1 @ sK0 ) @ sK2 ) ),
inference(trivial_inequality_removal,[status(thm)],[f38]) ).
thf(f38,plain,
( ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
| ( $true
= ( 'const/arith/<=' @ ( sK3 @ sK2 @ sK1 @ sK0 ) @ sK2 ) ) ),
inference(superposition,[status(thm)],[f30,f32]) ).
thf(f32,plain,
! [X2: 'type/nums/num',X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num'] :
( ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
| ( $true
= ( 'const/arith/<=' @ ( sK3 @ X2 @ X1 @ X0 ) @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f26,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( $true
= ( 'const/arith/<=' @ X1 @ ( sK3 @ X2 @ X1 @ X0 ) ) )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( X0 @ ( sK3 @ X2 @ X1 @ X0 ) ) )
& ( $true
= ( 'const/arith/<=' @ ( sK3 @ X2 @ X1 @ X0 ) @ X2 ) ) )
| ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f18,f25]) ).
thf(f25,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ? [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X3 )
= $true )
& ( ( X0 @ X3 )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
& ( ( 'const/arith/<=' @ X3 @ X2 )
= $true ) )
=> ( ( $true
= ( 'const/arith/<=' @ X1 @ ( sK3 @ X2 @ X1 @ X0 ) ) )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( X0 @ ( sK3 @ X2 @ X1 @ X0 ) ) )
& ( $true
= ( 'const/arith/<=' @ ( sK3 @ X2 @ X1 @ X0 ) @ X2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ? [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X3 )
= $true )
& ( ( X0 @ X3 )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
& ( ( 'const/arith/<=' @ X3 @ X2 )
= $true ) )
| ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(flattening,[status(thm)],[f17]) ).
thf(f17,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
| ? [X3: 'type/nums/num'] :
( ( ( X0 @ X3 )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) )
& ( ( 'const/arith/<=' @ X3 @ X2 )
= $true )
& ( ( 'const/arith/<=' @ X1 @ X3 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ! [X3: 'type/nums/num'] :
( ( ( ( 'const/arith/<=' @ X3 @ X2 )
= $true )
& ( ( 'const/arith/<=' @ X1 @ X3 )
= $true ) )
=> ( ( X0 @ X3 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X3 )
& ( 'const/arith/<=' @ X3 @ X2 ) )
=> ( ( X0 @ X3 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ! [X3: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X3 )
& ( 'const/arith/<=' @ X3 @ X2 ) )
=> ( ( X0 @ X3 )
= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/iterate/SUM_EQ_0_NUMSEG_') ).
thf(f30,plain,
( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( $true
= ( 'const/arith/<' @ sK2 @ sK1 ) )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f21,f23]) ).
thf(f23,plain,
( ? [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<' @ X2 @ X1 )
= $true )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) )
=> ( ( $true
= ( 'const/arith/<' @ sK2 @ sK1 ) )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ sK1 @ sK2 ) @ sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
? [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<' @ X2 @ X1 )
= $true )
& ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<' @ X2 @ X1 )
= $true )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( 'const/arith/<' @ X2 @ X1 )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( 'const/arith/<' @ X2 @ X1 )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
! [X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( 'const/arith/<' @ X2 @ X1 )
=> ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/iterate/SUM_TRIV_NUMSEG_') ).
thf(f47,plain,
( $true
!= ( 'const/arith/<=' @ ( sK3 @ sK2 @ sK1 @ sK0 ) @ sK2 ) ),
inference(trivial_inequality_removal,[status(thm)],[f45]) ).
thf(f45,plain,
( ( $true
!= ( 'const/arith/<=' @ ( sK3 @ sK2 @ sK1 @ sK0 ) @ sK2 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f44,f41]) ).
thf(f41,plain,
( $true
= ( 'const/arith/<=' @ sK1 @ ( sK3 @ sK2 @ sK1 @ sK0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f39]) ).
thf(f39,plain,
( ( $true
= ( 'const/arith/<=' @ sK1 @ ( sK3 @ sK2 @ sK1 @ sK0 ) ) )
| ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
!= ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ),
inference(superposition,[status(thm)],[f30,f34]) ).
thf(f34,plain,
! [X2: 'type/nums/num',X0: 'type/nums/num' > 'type/realax/real',X1: 'type/nums/num'] :
( ( ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ X1 @ X2 ) @ X0 ) )
| ( $true
= ( 'const/arith/<=' @ X1 @ ( sK3 @ X2 @ X1 @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f44,plain,
! [X0: 'type/nums/num'] :
( ( $true
!= ( 'const/arith/<=' @ sK1 @ X0 ) )
| ( $true
!= ( 'const/arith/<=' @ X0 @ sK2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f43]) ).
thf(f43,plain,
! [X0: 'type/nums/num'] :
( ( $true
!= ( 'const/arith/<=' @ X0 @ sK2 ) )
| ( $true
!= ( 'const/arith/<=' @ sK1 @ X0 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f36,f29]) ).
thf(f29,plain,
! [X2: 'type/nums/num',X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X2 )
= $true )
| ( ( 'const/arith/<=' @ X0 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X1 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f20,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X2 )
= $true )
| ( ( 'const/arith/<=' @ X1 @ X2 )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X1 )
!= $true ) ),
inference(flattening,[status(thm)],[f19]) ).
thf(f19,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X0 @ X2 )
= $true )
| ( ( 'const/arith/<=' @ X1 @ X2 )
!= $true )
| ( ( 'const/arith/<=' @ X0 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( ( 'const/arith/<=' @ X1 @ X2 )
= $true )
& ( ( 'const/arith/<=' @ X0 @ X1 )
= $true ) )
=> ( ( 'const/arith/<=' @ X0 @ X2 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X2 )
& ( 'const/arith/<=' @ X0 @ X1 ) )
=> ( 'const/arith/<=' @ X0 @ X2 ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: 'type/nums/num',X1: 'type/nums/num',X2: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X2 )
& ( 'const/arith/<=' @ X0 @ X1 ) )
=> ( 'const/arith/<=' @ X0 @ X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/arith/LE_TRANS_') ).
thf(f36,plain,
( ( 'const/arith/<=' @ sK1 @ sK2 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f35]) ).
thf(f35,plain,
( ( ( 'const/arith/<=' @ sK1 @ sK2 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f27,f31]) ).
thf(f31,plain,
( $true
= ( 'const/arith/<' @ sK2 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f27,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/arith/<' @ X0 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X1 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( ( 'const/arith/<=' @ X1 @ X0 )
= $true )
| ( ( 'const/arith/<' @ X0 @ X1 )
= $true ) )
& ( ( ( 'const/arith/<' @ X0 @ X1 )
!= $true )
| ( ( 'const/arith/<=' @ X1 @ X0 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/arith/<=' @ X1 @ X0 )
= $true )
<=> ( ( 'const/arith/<' @ X0 @ X1 )
!= $true ) ),
inference(flattening,[status(thm)],[f15]) ).
thf(f15,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ( 'const/arith/<' @ X0 @ X1 )
!= $true )
<=> ( ( 'const/arith/<=' @ X1 @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ~ ( 'const/arith/<' @ X0 @ X1 ) )
= ( 'const/arith/<=' @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: 'type/nums/num',X1: 'type/nums/num'] :
( ( ~ ( 'const/arith/<' @ X0 @ X1 ) )
= ( 'const/arith/<=' @ X1 @ X0 ) ),
file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/arith/NOT_LT_') ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ANA108^1 : TPTP v9.2.1. Released v7.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 03:50:24 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.33/0.49 ---- Original DTF file ---
% 0.33/0.49 thf(spec,logic,$$dhol).
% 0.33/0.49 %------------------------------------------------------------------------------
% 0.33/0.49 % File : ANA108^1 : TPTP v9.2.1. Released v7.0.0.
% 0.33/0.49 % Domain : Analysis
% 0.33/0.49 % Problem : SUM_TRIV_NUMSEG
% 0.33/0.49 % Version : Especial.
% 0.33/0.49 % English :
% 0.33/0.49
% 0.33/0.49 % Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% 0.33/0.49 % Source : [Kal16]
% 0.33/0.49 % Names : SUM_TRIV_NUMSEG_.p [Kal16]
% 0.33/0.49
% 0.33/0.49 % Status : Theorem
% 0.33/0.49 % Rating : 0.00 v7.1.0
% 0.33/0.49 % Syntax : Number of formulae : 15 ( 2 unt; 10 typ; 0 def)
% 0.33/0.49 % Number of atoms : 15 ( 5 equ; 0 cnn)
% 0.33/0.49 % Maximal formula atoms : 4 ( 3 avg)
% 0.33/0.49 % Number of connectives : 42 ( 1 ~; 0 |; 2 &; 35 @)
% 0.33/0.49 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.33/0.49 % Maximal formula depth : 10 ( 6 avg)
% 0.33/0.49 % Number of types : 3 ( 2 usr)
% 0.33/0.49 % Number of type conns : 16 ( 16 >; 0 *; 0 +; 0 <<)
% 0.33/0.49 % Number of symbols : 9 ( 8 usr; 1 con; 0-3 aty)
% 0.33/0.49 % Number of variables : 16 ( 0 ^; 14 !; 0 ?; 16 :)
% 0.33/0.49 % ( 2 !>; 0 ?*; 0 @-; 0 @+)
% 0.33/0.49 % SPC : TH1_THM_EQU_NAR
% 0.33/0.49
% 0.33/0.49 % Comments : Exported from core HOL Light.
% 0.33/0.49 %------------------------------------------------------------------------------
% 0.33/0.49 thf('thf_type_type/realax/real',type,
% 0.33/0.49 'type/realax/real': $tType ).
% 0.33/0.49
% 0.33/0.49 thf('thf_type_type/nums/num',type,
% 0.33/0.49 'type/nums/num': $tType ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/trivia/I',type,
% 0.33/0.49 'const/trivia/I':
% 0.33/0.49 !>[A: $tType] : ( A > A ) ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/realax/real_of_num',type,
% 0.33/0.49 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/nums/NUMERAL',type,
% 0.33/0.49 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/nums/_0',type,
% 0.33/0.49 'const/nums/_0': 'type/nums/num' ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/iterate/sum',type,
% 0.33/0.49 'const/iterate/sum':
% 0.33/0.49 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/iterate/..',type,
% 0.33/0.49 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/arith/<=',type,
% 0.33/0.49 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
% 0.33/0.49
% 0.33/0.49 thf('thf_const_const/arith/<',type,
% 0.33/0.49 'const/arith/<': 'type/nums/num' > 'type/nums/num' > $o ).
% 0.33/0.49
% 0.33/0.49 thf('thm/trivia/I_THM_',axiom,
% 0.33/0.49 ! [A: $tType,A0: A] :
% 0.33/0.49 ( ( 'const/trivia/I' @ A @ A0 )
% 0.33/0.49 = A0 ) ).
% 0.33/0.49
% 0.33/0.49 thf('thm/arith/NOT_LT_',axiom,
% 0.33/0.49 ! [A: 'type/nums/num',A0: 'type/nums/num'] :
% 0.33/0.49 ( ( ~ ( 'const/arith/<' @ A @ A0 ) )
% 0.33/0.49 = ( 'const/arith/<=' @ A0 @ A ) ) ).
% 0.33/0.49
% 0.33/0.49 thf('thm/arith/LE_TRANS_',axiom,
% 0.33/0.49 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 0.33/0.49 ( ( ( 'const/arith/<=' @ A @ A0 )
% 0.33/0.49 & ( 'const/arith/<=' @ A0 @ A1 ) )
% 0.33/0.49 => ( 'const/arith/<=' @ A @ A1 ) ) ).
% 0.33/0.49
% 0.33/0.49 thf('thm/iterate/SUM_EQ_0_NUMSEG_',axiom,
% 0.33/0.49 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 0.33/0.49 ( ! [A2: 'type/nums/num'] :
% 0.33/0.49 ( ( ( 'const/arith/<=' @ A0 @ A2 )
% 0.33/0.49 & ( 'const/arith/<=' @ A2 @ A1 ) )
% 0.33/0.49 => ( ( A @ A2 )
% 0.33/0.49 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
% 0.33/0.49 => ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A )
% 0.33/0.49 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 thf('thm/iterate/SUM_TRIV_NUMSEG_',conjecture,
% 0.33/0.49 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 0.33/0.49 ( ( 'const/arith/<' @ A1 @ A0 )
% 0.33/0.49 => ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A )
% 0.33/0.49 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ).
% 0.33/0.49
% 0.33/0.49 %------------------------------------------------------------------------------
% 0.33/0.49 ------------------------
% 1.69/1.45 ---- Embedded in THF ---
% 1.69/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.69/1.45 %%% Generated on Tue Feb 24 03:50:25 EST 2026
% 1.69/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 1.69/1.45 %%% Logic specification used:
% 1.69/1.45 %%% thf(spec, logic, $$dhol).
% 1.69/1.45
% 1.69/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2DTF22648.p
% See solution above
% 1.69/1.45 ------------------------
% 1.69/1.46 ---- Cleaned THF ---
% 1.69/1.46 thf('thf_type_type/realax/real',type,
% 1.69/1.46 'type/realax/real': $tType ).
% 1.69/1.46
% 1.69/1.46 thf('thf_type_type/nums/num',type,
% 1.69/1.46 'type/nums/num': $tType ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/trivia/I',type,
% 1.69/1.46 'const/trivia/I':
% 1.69/1.46 !>[A: $tType] : ( A > A ) ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/realax/real_of_num',type,
% 1.69/1.46 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real' ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/nums/NUMERAL',type,
% 1.69/1.46 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/nums/_0',type,
% 1.69/1.46 'const/nums/_0': 'type/nums/num' ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/iterate/sum',type,
% 1.69/1.46 'const/iterate/sum':
% 1.69/1.46 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/realax/real' ) > 'type/realax/real' ) ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/iterate/..',type,
% 1.69/1.46 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/arith/<=',type,
% 1.69/1.46 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o ).
% 1.69/1.46
% 1.69/1.46 thf('thf_const_const/arith/<',type,
% 1.69/1.46 'const/arith/<': 'type/nums/num' > 'type/nums/num' > $o ).
% 1.69/1.46
% 1.69/1.46 thf('thm/trivia/I_THM_',axiom,
% 1.69/1.46 ! [A: $tType,A0: A] :
% 1.69/1.46 ( ( 'const/trivia/I' @ A @ A0 )
% 1.69/1.46 = A0 ) ).
% 1.69/1.46
% 1.69/1.46 thf('thm/arith/NOT_LT_',axiom,
% 1.69/1.46 ! [A: 'type/nums/num',A0: 'type/nums/num'] :
% 1.69/1.46 ( ( ~ ( 'const/arith/<' @ A @ A0 ) )
% 1.69/1.46 = ( 'const/arith/<=' @ A0 @ A ) ) ).
% 1.69/1.46
% 1.69/1.46 thf('thm/arith/LE_TRANS_',axiom,
% 1.69/1.46 ! [A: 'type/nums/num',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 1.69/1.46 ( ( ( 'const/arith/<=' @ A @ A0 )
% 1.69/1.46 & ( 'const/arith/<=' @ A0 @ A1 ) )
% 1.69/1.46 => ( 'const/arith/<=' @ A @ A1 ) ) ).
% 1.69/1.46
% 1.69/1.46 thf('thm/iterate/SUM_EQ_0_NUMSEG_',axiom,
% 1.69/1.46 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 1.69/1.46 ( ! [A2: 'type/nums/num'] :
% 1.69/1.46 ( ( ( 'const/arith/<=' @ A0 @ A2 )
% 1.69/1.46 & ( 'const/arith/<=' @ A2 @ A1 ) )
% 1.69/1.46 => ( ( A @ A2 )
% 1.69/1.46 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
% 1.69/1.46 => ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A )
% 1.69/1.46 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ).
% 1.69/1.46
% 1.69/1.46 thf('thm/iterate/SUM_TRIV_NUMSEG_',conjecture,
% 1.69/1.46 ! [A: 'type/nums/num' > 'type/realax/real',A0: 'type/nums/num',A1: 'type/nums/num'] :
% 1.69/1.46 ( ( 'const/arith/<' @ A1 @ A0 )
% 1.69/1.46 => ( ( 'const/iterate/sum' @ 'type/nums/num' @ ( 'const/iterate/..' @ A0 @ A1 ) @ A )
% 1.69/1.46 = ( 'const/realax/real_of_num' @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) ) ).
% 1.69/1.46 ------------------------
% 1.69/1.48 % (22778)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22648 for (2999ds/183Mi)
% 1.69/1.48 % (22784)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22648 for (2999ds/18Mi)
% 1.69/1.48 % (22783)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_22648 for (2999ds/275Mi)
% 1.69/1.48 % (22779)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_22648 for (2999ds/4Mi)
% 1.69/1.48 % (22782)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_22648 for (2999ds/2Mi)
% 1.69/1.48 % (22780)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_22648 for (2999ds/27Mi)
% 1.69/1.48 % (22781)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22648 for (2999ds/2Mi)
% 1.69/1.49 % (22782)Instruction limit reached!
% 1.69/1.49 % (22782)------------------------------
% 1.69/1.49 % (22782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.49 % (22782)Termination reason: Unknown
% 1.69/1.49 % (22782)Termination phase: Property scanning
% 1.69/1.49
% 1.69/1.49 % (22782)Memory used [KB]: 895
% 1.69/1.49 % (22782)Time elapsed: 0.003 s
% 1.69/1.49 % (22782)Instructions burned: 2 (million)
% 1.69/1.49 % (22782)------------------------------
% 1.69/1.49 % (22782)------------------------------
% 1.69/1.49 % (22781)Instruction limit reached!
% 1.69/1.49 % (22781)------------------------------
% 1.69/1.49 % (22781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.49 % (22781)Termination reason: Unknown
% 1.69/1.49 % (22781)Termination phase: Saturation
% 1.69/1.49
% 1.69/1.49 % (22781)Memory used [KB]: 5500
% 1.69/1.49 % (22781)Time elapsed: 0.004 s
% 1.69/1.49 % (22781)Instructions burned: 3 (million)
% 1.69/1.49 % (22781)------------------------------
% 1.69/1.49 % (22781)------------------------------
% 1.69/1.49 % (22779)Instruction limit reached!
% 1.69/1.49 % (22779)------------------------------
% 1.69/1.49 % (22779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.49 % (22779)Termination reason: Unknown
% 1.69/1.49 % (22779)Termination phase: Saturation
% 1.69/1.49
% 1.69/1.49 % (22779)Memory used [KB]: 5500
% 1.69/1.49 % (22779)Time elapsed: 0.004 s
% 1.69/1.49 % (22779)Instructions burned: 5 (million)
% 1.69/1.49 % (22779)------------------------------
% 1.69/1.49 % (22779)------------------------------
% 1.69/1.49 % (22783)First to succeed.
% 1.69/1.49 % (22783)Refutation found. Thanks to Tanya!
% 1.69/1.49 % SZS status Theorem for DTF2THF_22648
% 1.69/1.49 % SZS output start Proof for DTF2THF_22648
% 1.69/1.49 thf(type_def_6, type, 'type/nums/num': $tType).
% 1.69/1.49 thf(type_def_7, type, 'type/realax/real': $tType).
% 1.69/1.49 thf(func_def_0, type, 'type/realax/real': $tType).
% 1.69/1.49 thf(func_def_1, type, 'type/nums/num': $tType).
% 1.69/1.49 thf(func_def_2, type, 'const/trivia/I': !>[X0: $tType]:(X0 > X0)).
% 1.69/1.49 thf(func_def_3, type, 'const/realax/real_of_num': 'type/nums/num' > 'type/realax/real').
% 1.69/1.49 thf(func_def_4, type, 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num').
% 1.69/1.49 thf(func_def_5, type, 'const/nums/_0': 'type/nums/num').
% 1.69/1.49 thf(func_def_6, type, 'const/iterate/sum': !>[X0: $tType]:((X0 > $o) > (X0 > 'type/realax/real') > 'type/realax/real')).
% 1.69/1.49 thf(func_def_7, type, 'const/iterate/..': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' > $o).
% 1.69/1.49 thf(func_def_8, type, 'const/arith/<=': 'type/nums/num' > 'type/nums/num' > $o).
% 1.69/1.49 thf(func_def_9, type, 'const/arith/<': 'type/nums/num' > 'type/nums/num' > $o).
% 1.69/1.49 thf(func_def_13, type, sK0: 'type/nums/num' > 'type/realax/real').
% 1.69/1.49 thf(func_def_14, type, sK1: 'type/nums/num').
% 1.69/1.49 thf(func_def_15, type, sK2: 'type/nums/num').
% 1.69/1.49 thf(func_def_16, type, sK3: 'type/nums/num' > 'type/nums/num' > ('type/nums/num' > 'type/realax/real') > 'type/nums/num').
% 1.69/1.49 thf(func_def_18, type, ph5: !>[X0: $tType]:(X0)).
% 1.69/1.49 thf(f49,plain,(
% 1.69/1.49 $false),
% 1.69/1.49 inference(subsumption_resolution,[status(thm)],[f47,f42])).
% 1.69/1.49 thf(f42,plain,(
% 1.69/1.49 ($true = ('const/arith/<=' @ (sK3 @ sK2 @ sK1 @ sK0) @ sK2))),
% 1.69/1.49 inference(trivial_inequality_removal,[status(thm)],[f38])).
% 1.69/1.49 thf(f38,plain,(
% 1.69/1.49 (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0'))) | ($true = ('const/arith/<=' @ (sK3 @ sK2 @ sK1 @ sK0) @ sK2))),
% 1.69/1.49 inference(superposition,[status(thm)],[f30,f32])).
% 1.69/1.49 thf(f32,plain,(
% 1.69/1.49 ( ! [X2 : 'type/nums/num',X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num'] : ((('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) | ($true = ('const/arith/<=' @ (sK3 @ X2 @ X1 @ X0) @ X2))) )),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f26])).
% 1.69/1.49 thf(f26,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((($true = ('const/arith/<=' @ X1 @ (sK3 @ X2 @ X1 @ X0))) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != (X0 @ (sK3 @ X2 @ X1 @ X0))) & ($true = ('const/arith/<=' @ (sK3 @ X2 @ X1 @ X0) @ X2))) | (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f18,f25])).
% 1.69/1.49 thf(f25,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (? [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X3) = $true) & ((X0 @ X3) != ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0'))) & (('const/arith/<=' @ X3 @ X2) = $true)) => (($true = ('const/arith/<=' @ X1 @ (sK3 @ X2 @ X1 @ X0))) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != (X0 @ (sK3 @ X2 @ X1 @ X0))) & ($true = ('const/arith/<=' @ (sK3 @ X2 @ X1 @ X0) @ X2))))),
% 1.69/1.49 introduced(definition,[],[choice_axiom])).
% 1.69/1.49 thf(f18,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (? [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X3) = $true) & ((X0 @ X3) != ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0'))) & (('const/arith/<=' @ X3 @ X2) = $true)) | (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(flattening,[status(thm)],[f17])).
% 1.69/1.49 thf(f17,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) | ? [X3 : 'type/nums/num'] : (((X0 @ X3) != ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0'))) & ((('const/arith/<=' @ X3 @ X2) = $true) & (('const/arith/<=' @ X1 @ X3) = $true))))),
% 1.69/1.49 inference(ennf_transformation,[status(thm)],[f11])).
% 1.69/1.49 thf(f11,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (! [X3 : 'type/nums/num'] : (((('const/arith/<=' @ X3 @ X2) = $true) & (('const/arith/<=' @ X1 @ X3) = $true)) => ((X0 @ X3) = ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')))) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(fool_elimination,[status(thm)],[f10])).
% 1.69/1.49 thf(f10,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X3) & ('const/arith/<=' @ X3 @ X2)) => ((X0 @ X3) = ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')))) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(rectify,[status(thm)],[f4])).
% 1.69/1.49 thf(f4,axiom,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (! [X3 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X3) & ('const/arith/<=' @ X3 @ X2)) => ((X0 @ X3) = ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')))) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/iterate/SUM_EQ_0_NUMSEG_')).
% 1.69/1.49 thf(f30,plain,(
% 1.69/1.49 (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0))),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.69/1.49 thf(f24,plain,(
% 1.69/1.49 ($true = ('const/arith/<' @ sK2 @ sK1)) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0))),
% 1.69/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f21,f23])).
% 1.69/1.49 thf(f23,plain,(
% 1.69/1.49 ? [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<' @ X2 @ X1) = $true) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0))) => (($true = ('const/arith/<' @ sK2 @ sK1)) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ sK1 @ sK2) @ sK0)))),
% 1.69/1.49 introduced(definition,[],[choice_axiom])).
% 1.69/1.49 thf(f21,plain,(
% 1.69/1.49 ? [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<' @ X2 @ X1) = $true) & (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(ennf_transformation,[status(thm)],[f9])).
% 1.69/1.49 thf(f9,plain,(
% 1.69/1.49 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<' @ X2 @ X1) = $true) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(fool_elimination,[status(thm)],[f8])).
% 1.69/1.49 thf(f8,plain,(
% 1.69/1.49 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (('const/arith/<' @ X2 @ X1) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(rectify,[status(thm)],[f6])).
% 1.69/1.49 thf(f6,negated_conjecture,(
% 1.69/1.49 ~! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (('const/arith/<' @ X2 @ X1) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 inference(negated_conjecture,[status(cth)],[f5])).
% 1.69/1.49 thf(f5,conjecture,(
% 1.69/1.49 ! [X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (('const/arith/<' @ X2 @ X1) => (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)))),
% 1.69/1.49 file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/iterate/SUM_TRIV_NUMSEG_')).
% 1.69/1.49 thf(f47,plain,(
% 1.69/1.49 ($true != ('const/arith/<=' @ (sK3 @ sK2 @ sK1 @ sK0) @ sK2))),
% 1.69/1.49 inference(trivial_inequality_removal,[status(thm)],[f45])).
% 1.69/1.49 thf(f45,plain,(
% 1.69/1.49 ($true != ('const/arith/<=' @ (sK3 @ sK2 @ sK1 @ sK0) @ sK2)) | ($true != $true)),
% 1.69/1.49 inference(superposition,[status(thm)],[f44,f41])).
% 1.69/1.49 thf(f41,plain,(
% 1.69/1.49 ($true = ('const/arith/<=' @ sK1 @ (sK3 @ sK2 @ sK1 @ sK0)))),
% 1.69/1.49 inference(trivial_inequality_removal,[status(thm)],[f39])).
% 1.69/1.49 thf(f39,plain,(
% 1.69/1.49 ($true = ('const/arith/<=' @ sK1 @ (sK3 @ sK2 @ sK1 @ sK0))) | (('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) != ('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')))),
% 1.69/1.49 inference(superposition,[status(thm)],[f30,f34])).
% 1.69/1.49 thf(f34,plain,(
% 1.69/1.49 ( ! [X2 : 'type/nums/num',X0 : 'type/nums/num' > 'type/realax/real',X1 : 'type/nums/num'] : ((('const/realax/real_of_num' @ ('const/nums/NUMERAL' @ 'const/nums/_0')) = ('const/iterate/sum' @ 'type/nums/num' @ ('const/iterate/..' @ X1 @ X2) @ X0)) | ($true = ('const/arith/<=' @ X1 @ (sK3 @ X2 @ X1 @ X0)))) )),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f26])).
% 1.69/1.49 thf(f44,plain,(
% 1.69/1.49 ( ! [X0 : 'type/nums/num'] : (($true != ('const/arith/<=' @ sK1 @ X0)) | ($true != ('const/arith/<=' @ X0 @ sK2))) )),
% 1.69/1.49 inference(trivial_inequality_removal,[status(thm)],[f43])).
% 1.69/1.49 thf(f43,plain,(
% 1.69/1.49 ( ! [X0 : 'type/nums/num'] : (($true != ('const/arith/<=' @ X0 @ sK2)) | ($true != ('const/arith/<=' @ sK1 @ X0)) | ($true != $true)) )),
% 1.69/1.49 inference(superposition,[status(thm)],[f36,f29])).
% 1.69/1.49 thf(f29,plain,(
% 1.69/1.49 ( ! [X2 : 'type/nums/num',X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X2) = $true) | (('const/arith/<=' @ X0 @ X1) != $true) | (('const/arith/<=' @ X1 @ X2) != $true)) )),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f20])).
% 1.69/1.49 thf(f20,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X2) = $true) | (('const/arith/<=' @ X1 @ X2) != $true) | (('const/arith/<=' @ X0 @ X1) != $true))),
% 1.69/1.49 inference(flattening,[status(thm)],[f19])).
% 1.69/1.49 thf(f19,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<=' @ X0 @ X2) = $true) | ((('const/arith/<=' @ X1 @ X2) != $true) | (('const/arith/<=' @ X0 @ X1) != $true)))),
% 1.69/1.49 inference(ennf_transformation,[status(thm)],[f13])).
% 1.69/1.49 thf(f13,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : (((('const/arith/<=' @ X1 @ X2) = $true) & (('const/arith/<=' @ X0 @ X1) = $true)) => (('const/arith/<=' @ X0 @ X2) = $true))),
% 1.69/1.49 inference(fool_elimination,[status(thm)],[f12])).
% 1.69/1.49 thf(f12,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X2) & ('const/arith/<=' @ X0 @ X1)) => ('const/arith/<=' @ X0 @ X2))),
% 1.69/1.49 inference(rectify,[status(thm)],[f3])).
% 1.69/1.49 thf(f3,axiom,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num',X2 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X2) & ('const/arith/<=' @ X0 @ X1)) => ('const/arith/<=' @ X0 @ X2))),
% 1.69/1.49 file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/arith/LE_TRANS_')).
% 1.69/1.49 thf(f36,plain,(
% 1.69/1.49 (('const/arith/<=' @ sK1 @ sK2) != $true)),
% 1.69/1.49 inference(trivial_inequality_removal,[status(thm)],[f35])).
% 1.69/1.49 thf(f35,plain,(
% 1.69/1.49 (('const/arith/<=' @ sK1 @ sK2) != $true) | ($true != $true)),
% 1.69/1.49 inference(superposition,[status(thm)],[f27,f31])).
% 1.69/1.49 thf(f31,plain,(
% 1.69/1.49 ($true = ('const/arith/<' @ sK2 @ sK1))),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.69/1.49 thf(f27,plain,(
% 1.69/1.49 ( ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/arith/<' @ X0 @ X1) != $true) | (('const/arith/<=' @ X1 @ X0) != $true)) )),
% 1.69/1.49 inference(cnf_transformation,[status(thm)],[f22])).
% 1.69/1.49 thf(f22,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : (((('const/arith/<=' @ X1 @ X0) = $true) | (('const/arith/<' @ X0 @ X1) = $true)) & ((('const/arith/<' @ X0 @ X1) != $true) | (('const/arith/<=' @ X1 @ X0) != $true)))),
% 1.69/1.49 inference(nnf_transformation,[status(thm)],[f16])).
% 1.69/1.49 thf(f16,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : ((('const/arith/<=' @ X1 @ X0) = $true) <=> (('const/arith/<' @ X0 @ X1) != $true))),
% 1.69/1.49 inference(flattening,[status(thm)],[f15])).
% 1.69/1.49 thf(f15,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : (~(('const/arith/<' @ X0 @ X1) = $true) <=> (('const/arith/<=' @ X1 @ X0) = $true))),
% 1.69/1.49 inference(fool_elimination,[status(thm)],[f14])).
% 1.69/1.49 thf(f14,plain,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : (~('const/arith/<' @ X0 @ X1) = ('const/arith/<=' @ X1 @ X0))),
% 1.69/1.49 inference(rectify,[status(thm)],[f2])).
% 1.69/1.49 thf(f2,axiom,(
% 1.69/1.49 ! [X0 : 'type/nums/num',X1 : 'type/nums/num'] : (~('const/arith/<' @ X0 @ X1) = ('const/arith/<=' @ X1 @ X0))),
% 1.69/1.49 file('/export/starexec/sandbox/tmp/tmp.zfqTwh5XnT/DTF2THF_22648.p','thm/arith/NOT_LT_')).
% 1.69/1.49 % SZS output end Proof for DTF2THF_22648
% 1.69/1.49 % (22783)------------------------------
% 1.69/1.49 % (22783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.49 % (22783)Termination reason: Refutation
% 1.69/1.49
% 1.69/1.49 % (22783)Memory used [KB]: 5500
% 1.69/1.49 % (22783)Time elapsed: 0.007 s
% 1.69/1.49 % (22783)Instructions burned: 5 (million)
% 1.69/1.49 % (22783)------------------------------
% 1.69/1.49 % (22783)------------------------------
% 1.69/1.49 % (22777)Success in time 0.02 s
%------------------------------------------------------------------------------