%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV505^1 : 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 : n015.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 09:41:24 AM UTC 2026
% Result : Theorem 1.70s 1.31s
% Output : Refutation 1.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 7
% Syntax : Number of formulae : 44 ( 17 unt; 0 typ; 0 def)
% Number of atoms : 419 ( 124 equ; 0 cnn)
% Maximal formula atoms : 10 ( 9 avg)
% Number of connectives : 988 ( 64 ~; 47 |; 34 &; 824 @)
% ( 0 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 66 ( 66 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 16 usr; 3 con; 0-6 aty)
% Number of variables : 100 ( 0 ^; 85 !; 15 ?; 100 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_6,type,
'type/nums/num': $tType ).
thf(type_def_7,type,
sK1: $tType ).
thf(func_def_0,type,
'type/nums/num': $tType ).
thf(func_def_1,type,
'const/sets/UNION':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > $o ) > X0 > $o ) ).
thf(func_def_2,type,
'const/sets/INTER':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > $o ) > X0 > $o ) ).
thf(func_def_3,type,
'const/sets/IN':
!>[X0: $tType] : ( X0 > ( X0 > $o ) > $o ) ).
thf(func_def_4,type,
'const/sets/FINITE':
!>[X0: $tType] : ( ( X0 > $o ) > $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/nsum':
!>[X0: $tType] : ( ( X0 > $o ) > ( X0 > 'type/nums/num' ) > 'type/nums/num' ) ).
thf(func_def_8,type,
'const/iterate/neutral':
!>[X0: $tType] : ( ( X0 > X0 > X0 ) > X0 ) ).
thf(func_def_9,type,
'const/iterate/monoidal':
!>[X0: $tType] : ( ( X0 > X0 > X0 ) > $o ) ).
thf(func_def_10,type,
'const/iterate/iterate':
!>[X0: $tType,X1: $tType] : ( ( X1 > X1 > X1 ) > ( X0 > $o ) > ( X0 > X1 ) > X1 ) ).
thf(func_def_11,type,
'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' ).
thf(func_def_15,type,
sK0:
!>[X0: $tType,X1: $tType] : ( ( X0 > $o ) > ( X0 > $o ) > ( X0 > X1 ) > ( X1 > X1 > X1 ) > X0 ) ).
thf(func_def_16,type,
sK2: sK1 > 'type/nums/num' ).
thf(func_def_17,type,
sK3: sK1 > $o ).
thf(func_def_18,type,
sK4: sK1 > $o ).
thf(func_def_20,type,
ph6:
!>[X0: $tType] : X0 ).
thf(f45,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f44,f42]) ).
thf(f42,plain,
( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ),
inference(forward_demodulation,[status(thm)],[f41,f30]) ).
thf(f30,plain,
( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ),
inference(cnf_transformation,[status(thm)],[f4]) ).
thf(f4,axiom,
( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ),
file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/NEUTRAL_ADD_') ).
thf(f41,plain,
( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ),
inference(subsumption_resolution,[status(thm)],[f40,f27]) ).
thf(f27,plain,
( ( 'const/sets/FINITE' @ sK1 @ sK4 )
= $true ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f21,plain,
( ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ sK1 @ sK3 @ sK2 ) @ ( 'const/iterate/nsum' @ sK1 @ sK4 @ sK2 ) )
!= ( 'const/iterate/nsum' @ sK1 @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 ) )
& ! [X4: sK1] :
( ( $true
!= ( 'const/sets/IN' @ sK1 @ X4 @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( sK2 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
= $true )
& ( $true
= ( 'const/sets/FINITE' @ sK1 @ sK3 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2,sK3,sK4])],[f17,f20]) ).
thf(f20,plain,
( ? [X0: $tType,X1: X0 > 'type/nums/num',X2: X0 > $o,X3: X0 > $o] :
( ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X2 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) )
!= ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X2 @ X3 ) @ X1 ) )
& ! [X4: X0] :
( ( $true
!= ( 'const/sets/IN' @ X0 @ X4 @ ( 'const/sets/INTER' @ X0 @ X2 @ X3 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ X0 @ X3 )
= $true )
& ( $true
= ( 'const/sets/FINITE' @ X0 @ X2 ) ) )
=> ( ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ sK1 @ sK3 @ sK2 ) @ ( 'const/iterate/nsum' @ sK1 @ sK4 @ sK2 ) )
!= ( 'const/iterate/nsum' @ sK1 @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 ) )
& ! [X4: sK1] :
( ( $true
!= ( 'const/sets/IN' @ sK1 @ X4 @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( sK2 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
= $true )
& ( $true
= ( 'const/sets/FINITE' @ sK1 @ sK3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f17,plain,
? [X0: $tType,X1: X0 > 'type/nums/num',X2: X0 > $o,X3: X0 > $o] :
( ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X2 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) )
!= ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X2 @ X3 ) @ X1 ) )
& ! [X4: X0] :
( ( $true
!= ( 'const/sets/IN' @ X0 @ X4 @ ( 'const/sets/INTER' @ X0 @ X2 @ X3 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ X0 @ X3 )
= $true )
& ( $true
= ( 'const/sets/FINITE' @ X0 @ X2 ) ) ),
inference(flattening,[status(thm)],[f16]) ).
thf(f16,plain,
? [X0: $tType,X1: X0 > 'type/nums/num',X2: X0 > $o,X3: X0 > $o] :
( ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X2 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) )
!= ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X2 @ X3 ) @ X1 ) )
& ( $true
= ( 'const/sets/FINITE' @ X0 @ X2 ) )
& ! [X4: X0] :
( ( $true
!= ( 'const/sets/IN' @ X0 @ X4 @ ( 'const/sets/INTER' @ X0 @ X2 @ X3 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ X0 @ X3 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
~ ! [X0: $tType,X1: X0 > 'type/nums/num',X2: X0 > $o,X3: X0 > $o] :
( ( ( $true
= ( 'const/sets/FINITE' @ X0 @ X2 ) )
& ! [X4: X0] :
( ( $true
= ( 'const/sets/IN' @ X0 @ X4 @ ( 'const/sets/INTER' @ X0 @ X2 @ X3 ) ) )
=> ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X4 ) ) )
& ( ( 'const/sets/FINITE' @ X0 @ X3 )
= $true ) )
=> ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X2 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) )
= ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X2 @ X3 ) @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
~ ! [X0: $tType,X1: X0 > 'type/nums/num',X2: X0 > $o,X3: X0 > $o] :
( ( ! [X4: X0] :
( ( 'const/sets/IN' @ X0 @ X4 @ ( 'const/sets/INTER' @ X0 @ X2 @ X3 ) )
=> ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X4 ) ) )
& ( 'const/sets/FINITE' @ X0 @ X2 )
& ( 'const/sets/FINITE' @ X0 @ X3 ) )
=> ( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X2 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) )
= ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X2 @ X3 ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ! [X0: $tType,X1: X0 > 'type/nums/num',X3: X0 > $o,X4: X0 > $o] :
( ( ! [X5: X0] :
( ( 'const/sets/IN' @ X0 @ X5 @ ( 'const/sets/INTER' @ X0 @ X3 @ X4 ) )
=> ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X5 ) ) )
& ( 'const/sets/FINITE' @ X0 @ X3 )
& ( 'const/sets/FINITE' @ X0 @ X4 ) )
=> ( ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X3 @ X4 ) @ X1 )
= ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X4 @ X1 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
! [X0: $tType,X1: X0 > 'type/nums/num',X3: X0 > $o,X4: X0 > $o] :
( ( ! [X5: X0] :
( ( 'const/sets/IN' @ X0 @ X5 @ ( 'const/sets/INTER' @ X0 @ X3 @ X4 ) )
=> ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( X1 @ X5 ) ) )
& ( 'const/sets/FINITE' @ X0 @ X3 )
& ( 'const/sets/FINITE' @ X0 @ X4 ) )
=> ( ( 'const/iterate/nsum' @ X0 @ ( 'const/sets/UNION' @ X0 @ X3 @ X4 ) @ X1 )
= ( 'const/arith/+' @ ( 'const/iterate/nsum' @ X0 @ X3 @ X1 ) @ ( 'const/iterate/nsum' @ X0 @ X4 @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/NSUM_UNION_NONZERO_') ).
thf(f40,plain,
( ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f39,f26]) ).
thf(f26,plain,
( $true
= ( 'const/sets/FINITE' @ sK1 @ sK3 ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f39,plain,
( ( $true
!= ( 'const/sets/FINITE' @ sK1 @ sK3 ) )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true )
| ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f34,f22]) ).
thf(f22,plain,
( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
= $true ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
= $true ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+',
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+',
file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/MONOIDAL_ADD_') ).
thf(f34,plain,
( ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true )
| ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) )
| ( $true
!= ( 'const/sets/FINITE' @ sK1 @ sK3 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f33]) ).
thf(f33,plain,
( ( ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 )
!= ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 ) )
| ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true )
| ( $true
!= ( 'const/sets/FINITE' @ sK1 @ sK3 ) )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true )
| ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
!= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ) ),
inference(superposition,[status(thm)],[f31,f24]) ).
thf(f24,plain,
! [X1: $tType,X0: $tType,X2: X1 > X1 > X1,X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) )
| ( ( 'const/sets/FINITE' @ X0 @ X4 )
!= $true )
| ( ( 'const/sets/FINITE' @ X0 @ X5 )
!= $true )
| ( ( 'const/iterate/monoidal' @ X1 @ X2 )
!= $true )
| ( ( X3 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
! [X0: $tType,X1: $tType,X2: X1 > X1 > X1] :
( ( ( 'const/iterate/monoidal' @ X1 @ X2 )
!= $true )
| ! [X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( $true
= ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) ) )
& ( ( X3 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
| ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) )
| ( ( 'const/sets/FINITE' @ X0 @ X4 )
!= $true )
| ( ( 'const/sets/FINITE' @ X0 @ X5 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f15,f18]) ).
thf(f18,plain,
! [X1: $tType,X0: $tType,X2: X1 > X1 > X1,X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ? [X6: X0] :
( ( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
= $true )
& ( ( X3 @ X6 )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
=> ( ( $true
= ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) ) )
& ( ( X3 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f15,plain,
! [X0: $tType,X1: $tType,X2: X1 > X1 > X1] :
( ( ( 'const/iterate/monoidal' @ X1 @ X2 )
!= $true )
| ! [X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ? [X6: X0] :
( ( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
= $true )
& ( ( X3 @ X6 )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
| ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) )
| ( ( 'const/sets/FINITE' @ X0 @ X4 )
!= $true )
| ( ( 'const/sets/FINITE' @ X0 @ X5 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f14]) ).
thf(f14,plain,
! [X0: $tType,X1: $tType,X2: X1 > X1 > X1] :
( ! [X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) )
| ( ( 'const/sets/FINITE' @ X0 @ X4 )
!= $true )
| ? [X6: X0] :
( ( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
= $true )
& ( ( X3 @ X6 )
!= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
| ( ( 'const/sets/FINITE' @ X0 @ X5 )
!= $true ) )
| ( ( 'const/iterate/monoidal' @ X1 @ X2 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
! [X0: $tType,X1: $tType,X2: X1 > X1 > X1] :
( ( ( 'const/iterate/monoidal' @ X1 @ X2 )
= $true )
=> ! [X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( ( 'const/sets/FINITE' @ X0 @ X4 )
= $true )
& ! [X6: X0] :
( ( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
= $true )
=> ( ( X3 @ X6 )
= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
& ( ( 'const/sets/FINITE' @ X0 @ X5 )
= $true ) )
=> ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: $tType,X1: $tType,X2: X1 > X1 > X1] :
( ( 'const/iterate/monoidal' @ X1 @ X2 )
=> ! [X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( 'const/sets/FINITE' @ X0 @ X5 )
& ! [X6: X0] :
( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
=> ( ( X3 @ X6 )
= ( 'const/iterate/neutral' @ X1 @ X2 ) ) )
& ( 'const/sets/FINITE' @ X0 @ X4 ) )
=> ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $tType,X2: $tType,X1: X2 > X2 > X2] :
( ( 'const/iterate/monoidal' @ X2 @ X1 )
=> ! [X3: X0 > X2,X4: X0 > $o,X5: X0 > $o] :
( ( ( 'const/sets/FINITE' @ X0 @ X5 )
& ! [X6: X0] :
( ( 'const/sets/IN' @ X0 @ X6 @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) )
=> ( ( X3 @ X6 )
= ( 'const/iterate/neutral' @ X2 @ X1 ) ) )
& ( 'const/sets/FINITE' @ X0 @ X4 ) )
=> ( ( 'const/iterate/iterate' @ X0 @ X2 @ X1 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 )
= ( X1 @ ( 'const/iterate/iterate' @ X0 @ X2 @ X1 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X2 @ X1 @ X5 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/ITERATE_UNION_NONZERO_') ).
thf(f31,plain,
( ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 )
!= ( 'const/arith/+' @ ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ sK3 @ sK2 ) @ ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ sK4 @ sK2 ) ) ),
inference(definition_unfolding,[status(thm)],[f29,f23,f23,f23]) ).
thf(f23,plain,
! [X0: $tType] :
( 'const/iterate/nsum'
@ ( X0
= ( 'const/iterate/iterate' @ X0 @ 'type/nums/num' @ 'const/arith/+' ) ) ),
inference(cnf_transformation,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: $tType] :
( 'const/iterate/nsum'
@ ( X0
= ( 'const/iterate/iterate' @ X0 @ 'type/nums/num' @ 'const/arith/+' ) ) ),
file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/nsum_') ).
thf(f29,plain,
( ( 'const/arith/+' @ ( 'const/iterate/nsum' @ sK1 @ sK3 @ sK2 ) @ ( 'const/iterate/nsum' @ sK1 @ sK4 @ sK2 ) )
!= ( 'const/iterate/nsum' @ sK1 @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f44,plain,
( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f43]) ).
thf(f43,plain,
( ( $true != $true )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( sK2 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) ) ) ),
inference(superposition,[status(thm)],[f28,f38]) ).
thf(f38,plain,
( $true
= ( 'const/sets/IN' @ sK1 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) ),
inference(subsumption_resolution,[status(thm)],[f37,f22]) ).
thf(f37,plain,
( ( $true
= ( 'const/sets/IN' @ sK1 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f36,f27]) ).
thf(f36,plain,
( ( $true
= ( 'const/sets/IN' @ sK1 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true )
| ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f35,f26]) ).
thf(f35,plain,
( ( $true
!= ( 'const/sets/FINITE' @ sK1 @ sK3 ) )
| ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true )
| ( $true
= ( 'const/sets/IN' @ sK1 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f32]) ).
thf(f32,plain,
( ( $true
= ( 'const/sets/IN' @ sK1 @ ( sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+' ) @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( $true
!= ( 'const/sets/FINITE' @ sK1 @ sK3 ) )
| ( ( 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' )
!= $true )
| ( ( 'const/sets/FINITE' @ sK1 @ sK4 )
!= $true )
| ( ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 )
!= ( 'const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ( 'const/sets/UNION' @ sK1 @ sK3 @ sK4 ) @ sK2 ) ) ),
inference(superposition,[status(thm)],[f31,f25]) ).
thf(f25,plain,
! [X1: $tType,X0: $tType,X2: X1 > X1 > X1,X3: X0 > X1,X4: X0 > $o,X5: X0 > $o] :
( ( ( X2 @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3 ) @ ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3 ) )
= ( 'const/iterate/iterate' @ X0 @ X1 @ X2 @ ( 'const/sets/UNION' @ X0 @ X4 @ X5 ) @ X3 ) )
| ( ( 'const/sets/FINITE' @ X0 @ X4 )
!= $true )
| ( ( 'const/sets/FINITE' @ X0 @ X5 )
!= $true )
| ( $true
= ( 'const/sets/IN' @ X0 @ ( sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2 ) @ ( 'const/sets/INTER' @ X0 @ X4 @ X5 ) ) )
| ( ( 'const/iterate/monoidal' @ X1 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f28,plain,
! [X4: sK1] :
( ( $true
!= ( 'const/sets/IN' @ sK1 @ X4 @ ( 'const/sets/INTER' @ sK1 @ sK3 @ sK4 ) ) )
| ( ( 'const/nums/NUMERAL' @ 'const/nums/_0' )
= ( sK2 @ X4 ) ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEV505^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.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 12:48:31 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : SEV505^1 : TPTP v9.2.1. Released v7.0.0.
% 0.27/0.47 % Domain : Analysis
% 0.27/0.47 % Problem : NSUM_UNION_NONZERO
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English :
% 0.27/0.47
% 0.27/0.47 % Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% 0.27/0.47 % Source : [Kal16]
% 0.27/0.47 % Names : NSUM_UNION_NONZERO_.p [Kal16]
% 0.27/0.47
% 0.27/0.47 % Status : Theorem
% 0.27/0.47 % Rating : 0.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% 0.27/0.47 % Syntax : Number of formulae : 17 ( 3 unt; 12 typ; 1 def)
% 0.27/0.47 % Number of atoms : 18 ( 6 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 7 ( 3 avg)
% 0.27/0.47 % Number of connectives : 79 ( 0 ~; 0 |; 4 &; 70 @)
% 0.27/0.47 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 17 ( 7 avg)
% 0.27/0.47 % Number of types : 2 ( 1 usr)
% 0.27/0.47 % Number of type conns : 43 ( 43 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 12 ( 11 usr; 1 con; 0-5 aty)
% 0.27/0.47 % Number of variables : 22 ( 0 ^; 13 !; 0 ?; 22 :)
% 0.27/0.47 % ( 9 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.47 % SPC : TH1_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : Exported from core HOL Light.
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 thf('thf_type_type/nums/num',type,
% 0.27/0.47 'type/nums/num': $tType ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/sets/UNION',type,
% 0.27/0.47 'const/sets/UNION':
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/sets/INTER',type,
% 0.27/0.47 'const/sets/INTER':
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/sets/IN',type,
% 0.27/0.47 'const/sets/IN':
% 0.27/0.47 !>[A: $tType] : ( A > ( A > $o ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/sets/FINITE',type,
% 0.27/0.47 'const/sets/FINITE':
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/nums/NUMERAL',type,
% 0.27/0.47 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/nums/_0',type,
% 0.27/0.47 'const/nums/_0': 'type/nums/num' ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/iterate/nsum',type,
% 0.27/0.47 'const/iterate/nsum':
% 0.27/0.47 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/nums/num' ) > 'type/nums/num' ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/iterate/neutral',type,
% 0.27/0.47 'const/iterate/neutral':
% 0.27/0.47 !>[A: $tType] : ( ( A > A > A ) > A ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/iterate/monoidal',type,
% 0.27/0.47 'const/iterate/monoidal':
% 0.27/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/iterate/iterate',type,
% 0.27/0.47 'const/iterate/iterate':
% 0.27/0.47 !>[A: $tType,A0: $tType] : ( ( A0 > A0 > A0 ) > ( A > $o ) > ( A > A0 ) > A0 ) ).
% 0.27/0.47
% 0.27/0.47 thf('thf_const_const/arith/+',type,
% 0.27/0.47 'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' ).
% 0.27/0.47
% 0.27/0.47 thf('thm/iterate/ITERATE_UNION_NONZERO_',axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,A0: B > B > B] :
% 0.27/0.47 ( ( 'const/iterate/monoidal' @ B @ A0 )
% 0.27/0.47 => ! [A1: A > B,A2: A > $o,A3: A > $o] :
% 0.27/0.47 ( ( ( 'const/sets/FINITE' @ A @ A2 )
% 0.27/0.47 & ( 'const/sets/FINITE' @ A @ A3 )
% 0.27/0.47 & ! [A4: A] :
% 0.27/0.47 ( ( 'const/sets/IN' @ A @ A4 @ ( 'const/sets/INTER' @ A @ A2 @ A3 ) )
% 0.27/0.47 => ( ( A1 @ A4 )
% 0.27/0.47 = ( 'const/iterate/neutral' @ B @ A0 ) ) ) )
% 0.27/0.47 => ( ( 'const/iterate/iterate' @ A @ B @ A0 @ ( 'const/sets/UNION' @ A @ A2 @ A3 ) @ A1 )
% 0.27/0.47 = ( A0 @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A2 @ A1 ) @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf('thm/iterate/MONOIDAL_ADD_',axiom,
% 0.27/0.47 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' ).
% 0.27/0.47
% 0.27/0.47 thf('thm/iterate/nsum_',definition,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( 'const/iterate/nsum' @ A )
% 0.27/0.47 = ( 'const/iterate/iterate' @ A @ 'type/nums/num' @ 'const/arith/+' ) ) ).
% 0.27/0.47
% 0.27/0.47 thf('thm/iterate/NEUTRAL_ADD_',axiom,
% 0.27/0.47 ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
% 0.27/0.47 = ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
% 0.27/0.47
% 0.27/0.47 thf('thm/iterate/NSUM_UNION_NONZERO_',conjecture,
% 0.27/0.47 ! [A: $tType,A0: A > 'type/nums/num',A1: A > $o,A2: A > $o] :
% 0.27/0.47 ( ( ( 'const/sets/FINITE' @ A @ A1 )
% 0.27/0.47 & ( 'const/sets/FINITE' @ A @ A2 )
% 0.27/0.47 & ! [A3: A] :
% 0.27/0.47 ( ( 'const/sets/IN' @ A @ A3 @ ( 'const/sets/INTER' @ A @ A1 @ A2 ) )
% 0.27/0.47 => ( ( A0 @ A3 )
% 0.27/0.47 = ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
% 0.27/0.47 => ( ( 'const/iterate/nsum' @ A @ ( 'const/sets/UNION' @ A @ A1 @ A2 ) @ A0 )
% 0.27/0.47 = ( 'const/arith/+' @ ( 'const/iterate/nsum' @ A @ A1 @ A0 ) @ ( 'const/iterate/nsum' @ A @ A2 @ A0 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 1.52/1.27 ---- Embedded in THF ---
% 1.52/1.28 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.52/1.28 %%% Generated on Tue Feb 24 12:48:32 EST 2026
% 1.52/1.28 %%% using '$$dhol' embedding, version 1.3.0.
% 1.52/1.28 %%% Logic specification used:
% 1.52/1.28 %%% thf(spec, logic, $$dhol).
% 1.52/1.28
% 1.52/1.28 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2DTF8919.p
% See solution above
% 1.52/1.28 ------------------------
% 1.52/1.29 ---- Cleaned THF ---
% 1.52/1.29 thf('thf_type_type/nums/num',type,
% 1.52/1.29 'type/nums/num': $tType ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/sets/UNION',type,
% 1.52/1.29 'const/sets/UNION':
% 1.52/1.29 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/sets/INTER',type,
% 1.52/1.29 'const/sets/INTER':
% 1.52/1.29 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/sets/IN',type,
% 1.52/1.29 'const/sets/IN':
% 1.52/1.29 !>[A: $tType] : ( A > ( A > $o ) > $o ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/sets/FINITE',type,
% 1.52/1.29 'const/sets/FINITE':
% 1.52/1.29 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/nums/NUMERAL',type,
% 1.52/1.29 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/nums/_0',type,
% 1.52/1.29 'const/nums/_0': 'type/nums/num' ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/iterate/nsum',type,
% 1.52/1.29 'const/iterate/nsum':
% 1.52/1.29 !>[A: $tType] : ( ( A > $o ) > ( A > 'type/nums/num' ) > 'type/nums/num' ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/iterate/neutral',type,
% 1.52/1.29 'const/iterate/neutral':
% 1.52/1.29 !>[A: $tType] : ( ( A > A > A ) > A ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/iterate/monoidal',type,
% 1.52/1.29 'const/iterate/monoidal':
% 1.52/1.29 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/iterate/iterate',type,
% 1.52/1.29 'const/iterate/iterate':
% 1.52/1.29 !>[A: $tType,A0: $tType] : ( ( A0 > A0 > A0 ) > ( A > $o ) > ( A > A0 ) > A0 ) ).
% 1.52/1.29
% 1.52/1.29 thf('thf_const_const/arith/+',type,
% 1.52/1.29 'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num' ).
% 1.52/1.29
% 1.52/1.29 thf('thm/iterate/ITERATE_UNION_NONZERO_',axiom,
% 1.52/1.29 ! [A: $tType,B: $tType,A0: B > B > B] :
% 1.52/1.29 ( ( 'const/iterate/monoidal' @ B @ A0 )
% 1.52/1.29 => ! [A1: A > B,A2: A > $o,A3: A > $o] :
% 1.52/1.29 ( ( ( 'const/sets/FINITE' @ A @ A2 )
% 1.52/1.29 & ( 'const/sets/FINITE' @ A @ A3 )
% 1.52/1.29 & ! [A4: A] :
% 1.52/1.29 ( ( 'const/sets/IN' @ A @ A4 @ ( 'const/sets/INTER' @ A @ A2 @ A3 ) )
% 1.52/1.29 => ( ( A1 @ A4 )
% 1.52/1.29 = ( 'const/iterate/neutral' @ B @ A0 ) ) ) )
% 1.52/1.29 => ( ( 'const/iterate/iterate' @ A @ B @ A0 @ ( 'const/sets/UNION' @ A @ A2 @ A3 ) @ A1 )
% 1.52/1.29 = ( A0 @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A2 @ A1 ) @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 ) ) ) ) ) ).
% 1.52/1.29
% 1.52/1.29 thf('thm/iterate/MONOIDAL_ADD_',axiom,
% 1.52/1.29 'const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+' ).
% 1.52/1.29
% 1.52/1.29 thf('thm/iterate/nsum_',definition,
% 1.52/1.29 ! [A: $tType] :
% 1.52/1.29 ( ( 'const/iterate/nsum' @ A )
% 1.52/1.29 = ( 'const/iterate/iterate' @ A @ 'type/nums/num' @ 'const/arith/+' ) ) ).
% 1.52/1.29
% 1.52/1.29 thf('thm/iterate/NEUTRAL_ADD_',axiom,
% 1.52/1.29 ( ( 'const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+' )
% 1.52/1.29 = ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
% 1.52/1.29
% 1.52/1.29 thf('thm/iterate/NSUM_UNION_NONZERO_',conjecture,
% 1.52/1.29 ! [A: $tType,A0: A > 'type/nums/num',A1: A > $o,A2: A > $o] :
% 1.52/1.29 ( ( ( 'const/sets/FINITE' @ A @ A1 )
% 1.52/1.29 & ( 'const/sets/FINITE' @ A @ A2 )
% 1.52/1.29 & ! [A3: A] :
% 1.52/1.29 ( ( 'const/sets/IN' @ A @ A3 @ ( 'const/sets/INTER' @ A @ A1 @ A2 ) )
% 1.52/1.29 => ( ( A0 @ A3 )
% 1.52/1.29 = ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ) )
% 1.52/1.29 => ( ( 'const/iterate/nsum' @ A @ ( 'const/sets/UNION' @ A @ A1 @ A2 ) @ A0 )
% 1.52/1.29 = ( 'const/arith/+' @ ( 'const/iterate/nsum' @ A @ A1 @ A0 ) @ ( 'const/iterate/nsum' @ A @ A2 @ A0 ) ) ) ) ).
% 1.52/1.29 ------------------------
% 1.70/1.30 % (9021)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_8919 for (2999ds/275Mi)
% 1.70/1.30 % (9021)First to succeed.
% 1.70/1.31 % (9021)Refutation found. Thanks to Tanya!
% 1.70/1.31 % SZS status Theorem for DTF2THF_8919
% 1.70/1.31 % SZS output start Proof for DTF2THF_8919
% 1.70/1.31 thf(type_def_6, type, 'type/nums/num': $tType).
% 1.70/1.31 thf(type_def_7, type, sK1: $tType).
% 1.70/1.31 thf(func_def_0, type, 'type/nums/num': $tType).
% 1.70/1.31 thf(func_def_1, type, 'const/sets/UNION': !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0 > $o)).
% 1.70/1.31 thf(func_def_2, type, 'const/sets/INTER': !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0 > $o)).
% 1.70/1.31 thf(func_def_3, type, 'const/sets/IN': !>[X0: $tType]:(X0 > (X0 > $o) > $o)).
% 1.70/1.31 thf(func_def_4, type, 'const/sets/FINITE': !>[X0: $tType]:((X0 > $o) > $o)).
% 1.70/1.31 thf(func_def_5, type, 'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num').
% 1.70/1.31 thf(func_def_6, type, 'const/nums/_0': 'type/nums/num').
% 1.70/1.31 thf(func_def_7, type, 'const/iterate/nsum': !>[X0: $tType]:((X0 > $o) > (X0 > 'type/nums/num') > 'type/nums/num')).
% 1.70/1.31 thf(func_def_8, type, 'const/iterate/neutral': !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 1.70/1.31 thf(func_def_9, type, 'const/iterate/monoidal': !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 1.70/1.31 thf(func_def_10, type, 'const/iterate/iterate': !>[X0: $tType, X1: $tType]:((X1 > X1 > X1) > (X0 > $o) > (X0 > X1) > X1)).
% 1.70/1.31 thf(func_def_11, type, 'const/arith/+': 'type/nums/num' > 'type/nums/num' > 'type/nums/num').
% 1.70/1.31 thf(func_def_15, type, sK0: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > $o) > (X0 > X1) > (X1 > X1 > X1) > X0)).
% 1.70/1.31 thf(func_def_16, type, sK2: sK1 > 'type/nums/num').
% 1.70/1.31 thf(func_def_17, type, sK3: sK1 > $o).
% 1.70/1.31 thf(func_def_18, type, sK4: sK1 > $o).
% 1.70/1.31 thf(func_def_20, type, ph6: !>[X0: $tType]:(X0)).
% 1.70/1.31 thf(f45,plain,(
% 1.70/1.31 $false),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f44,f42])).
% 1.70/1.31 thf(f42,plain,(
% 1.70/1.31 (('const/nums/NUMERAL' @ 'const/nums/_0') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(forward_demodulation,[status(thm)],[f41,f30])).
% 1.70/1.31 thf(f30,plain,(
% 1.70/1.31 (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') = ('const/nums/NUMERAL' @ 'const/nums/_0'))),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f4])).
% 1.70/1.31 thf(f4,axiom,(
% 1.70/1.31 (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') = ('const/nums/NUMERAL' @ 'const/nums/_0'))),
% 1.70/1.31 file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/NEUTRAL_ADD_')).
% 1.70/1.31 thf(f41,plain,(
% 1.70/1.31 (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f40,f27])).
% 1.70/1.31 thf(f27,plain,(
% 1.70/1.31 (('const/sets/FINITE' @ sK1 @ sK4) = $true)),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f21])).
% 1.70/1.31 thf(f21,plain,(
% 1.70/1.31 (('const/arith/+' @ ('const/iterate/nsum' @ sK1 @ sK3 @ sK2) @ ('const/iterate/nsum' @ sK1 @ sK4 @ sK2)) != ('const/iterate/nsum' @ sK1 @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2)) & ! [X4 : sK1] : (($true != ('const/sets/IN' @ sK1 @ X4 @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (sK2 @ X4))) & (('const/sets/FINITE' @ sK1 @ sK4) = $true) & ($true = ('const/sets/FINITE' @ sK1 @ sK3))),
% 1.70/1.31 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2,sK3,sK4])],[f17,f20])).
% 1.70/1.31 thf(f20,plain,(
% 1.70/1.31 ? [X0 : $tType,X1 : X0 > 'type/nums/num',X2 : X0 > $o,X3 : X0 > $o] : ((('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X2 @ X1) @ ('const/iterate/nsum' @ X0 @ X3 @ X1)) != ('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X2 @ X3) @ X1)) & ! [X4 : X0] : (($true != ('const/sets/IN' @ X0 @ X4 @ ('const/sets/INTER' @ X0 @ X2 @ X3))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X4))) & (('const/sets/FINITE' @ X0 @ X3) = $true) & ($true = ('const/sets/FINITE' @ X0 @ X2))) => ((('const/arith/+' @ ('const/iterate/nsum' @ sK1 @ sK3 @ sK2) @ ('const/iterate/nsum' @ sK1 @ sK4 @ sK2)) != ('const/iterate/nsum' @ sK1 @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2)) & ! [X4 : sK1] : (($true != ('const/sets/IN' @ sK1 @ X4 @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (sK2 @ X4))) & (('const/sets/FINITE' @ sK1 @ sK4) = $true) & ($true = ('const/sets/FINITE' @ sK1 @ sK3)))),
% 1.70/1.31 introduced(definition,[],[choice_axiom])).
% 1.70/1.31 thf(f17,plain,(
% 1.70/1.31 ? [X0 : $tType,X1 : X0 > 'type/nums/num',X2 : X0 > $o,X3 : X0 > $o] : ((('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X2 @ X1) @ ('const/iterate/nsum' @ X0 @ X3 @ X1)) != ('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X2 @ X3) @ X1)) & ! [X4 : X0] : (($true != ('const/sets/IN' @ X0 @ X4 @ ('const/sets/INTER' @ X0 @ X2 @ X3))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X4))) & (('const/sets/FINITE' @ X0 @ X3) = $true) & ($true = ('const/sets/FINITE' @ X0 @ X2)))),
% 1.70/1.31 inference(flattening,[status(thm)],[f16])).
% 1.70/1.31 thf(f16,plain,(
% 1.70/1.31 ? [X0 : $tType,X1 : X0 > 'type/nums/num',X2 : X0 > $o,X3 : X0 > $o] : ((('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X2 @ X1) @ ('const/iterate/nsum' @ X0 @ X3 @ X1)) != ('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X2 @ X3) @ X1)) & (($true = ('const/sets/FINITE' @ X0 @ X2)) & ! [X4 : X0] : (($true != ('const/sets/IN' @ X0 @ X4 @ ('const/sets/INTER' @ X0 @ X2 @ X3))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X4))) & (('const/sets/FINITE' @ X0 @ X3) = $true)))),
% 1.70/1.31 inference(ennf_transformation,[status(thm)],[f13])).
% 1.70/1.31 thf(f13,plain,(
% 1.70/1.31 ~! [X0 : $tType,X1 : X0 > 'type/nums/num',X2 : X0 > $o,X3 : X0 > $o] : ((($true = ('const/sets/FINITE' @ X0 @ X2)) & ! [X4 : X0] : (($true = ('const/sets/IN' @ X0 @ X4 @ ('const/sets/INTER' @ X0 @ X2 @ X3))) => (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X4))) & (('const/sets/FINITE' @ X0 @ X3) = $true)) => (('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X2 @ X1) @ ('const/iterate/nsum' @ X0 @ X3 @ X1)) = ('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X2 @ X3) @ X1)))),
% 1.70/1.31 inference(fool_elimination,[status(thm)],[f12])).
% 1.70/1.31 thf(f12,plain,(
% 1.70/1.31 ~! [X0 : $tType,X1 : X0 > 'type/nums/num',X2 : X0 > $o,X3 : X0 > $o] : ((! [X4 : X0] : (('const/sets/IN' @ X0 @ X4 @ ('const/sets/INTER' @ X0 @ X2 @ X3)) => (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X4))) & ('const/sets/FINITE' @ X0 @ X2) & ('const/sets/FINITE' @ X0 @ X3)) => (('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X2 @ X1) @ ('const/iterate/nsum' @ X0 @ X3 @ X1)) = ('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X2 @ X3) @ X1)))),
% 1.70/1.31 inference(rectify,[status(thm)],[f6])).
% 1.70/1.31 thf(f6,negated_conjecture,(
% 1.70/1.31 ~! [X0 : $tType,X1 : X0 > 'type/nums/num',X3 : X0 > $o,X4 : X0 > $o] : ((! [X5 : X0] : (('const/sets/IN' @ X0 @ X5 @ ('const/sets/INTER' @ X0 @ X3 @ X4)) => (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X5))) & ('const/sets/FINITE' @ X0 @ X3) & ('const/sets/FINITE' @ X0 @ X4)) => (('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X3 @ X4) @ X1) = ('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X3 @ X1) @ ('const/iterate/nsum' @ X0 @ X4 @ X1))))),
% 1.70/1.31 inference(negated_conjecture,[status(cth)],[f5])).
% 1.70/1.31 thf(f5,conjecture,(
% 1.70/1.31 ! [X0 : $tType,X1 : X0 > 'type/nums/num',X3 : X0 > $o,X4 : X0 > $o] : ((! [X5 : X0] : (('const/sets/IN' @ X0 @ X5 @ ('const/sets/INTER' @ X0 @ X3 @ X4)) => (('const/nums/NUMERAL' @ 'const/nums/_0') = (X1 @ X5))) & ('const/sets/FINITE' @ X0 @ X3) & ('const/sets/FINITE' @ X0 @ X4)) => (('const/iterate/nsum' @ X0 @ ('const/sets/UNION' @ X0 @ X3 @ X4) @ X1) = ('const/arith/+' @ ('const/iterate/nsum' @ X0 @ X3 @ X1) @ ('const/iterate/nsum' @ X0 @ X4 @ X1))))),
% 1.70/1.31 file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/NSUM_UNION_NONZERO_')).
% 1.70/1.31 thf(f40,plain,(
% 1.70/1.31 (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+'))) | (('const/sets/FINITE' @ sK1 @ sK4) != $true)),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f39,f26])).
% 1.70/1.31 thf(f26,plain,(
% 1.70/1.31 ($true = ('const/sets/FINITE' @ sK1 @ sK3))),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f21])).
% 1.70/1.31 thf(f39,plain,(
% 1.70/1.31 ($true != ('const/sets/FINITE' @ sK1 @ sK3)) | (('const/sets/FINITE' @ sK1 @ sK4) != $true) | (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f34,f22])).
% 1.70/1.31 thf(f22,plain,(
% 1.70/1.31 (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') = $true)),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f9])).
% 1.70/1.31 thf(f9,plain,(
% 1.70/1.31 (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') = $true)),
% 1.70/1.31 inference(fool_elimination,[status(thm)],[f8])).
% 1.70/1.31 thf(f8,plain,(
% 1.70/1.31 ('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+')),
% 1.70/1.31 inference(rectify,[status(thm)],[f2])).
% 1.70/1.31 thf(f2,axiom,(
% 1.70/1.31 ('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+')),
% 1.70/1.31 file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/MONOIDAL_ADD_')).
% 1.70/1.31 thf(f34,plain,(
% 1.70/1.31 (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true) | (('const/sets/FINITE' @ sK1 @ sK4) != $true) | (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+'))) | ($true != ('const/sets/FINITE' @ sK1 @ sK3))),
% 1.70/1.31 inference(trivial_inequality_removal,[status(thm)],[f33])).
% 1.70/1.31 thf(f33,plain,(
% 1.70/1.31 (('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2) != ('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2)) | (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true) | ($true != ('const/sets/FINITE' @ sK1 @ sK3)) | (('const/sets/FINITE' @ sK1 @ sK4) != $true) | (('const/iterate/neutral' @ 'type/nums/num' @ 'const/arith/+') != (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(superposition,[status(thm)],[f31,f24])).
% 1.70/1.31 thf(f24,plain,(
% 1.70/1.31 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X1 > X1,X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3)) | (('const/sets/FINITE' @ X0 @ X4) != $true) | (('const/sets/FINITE' @ X0 @ X5) != $true) | (('const/iterate/monoidal' @ X1 @ X2) != $true) | ((X3 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2)) != ('const/iterate/neutral' @ X1 @ X2))) )),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.70/1.31 thf(f19,plain,(
% 1.70/1.31 ! [X0 : $tType,X1 : $tType,X2 : X1 > X1 > X1] : ((('const/iterate/monoidal' @ X1 @ X2) != $true) | ! [X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : ((($true = ('const/sets/IN' @ X0 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2) @ ('const/sets/INTER' @ X0 @ X4 @ X5))) & ((X3 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2)) != ('const/iterate/neutral' @ X1 @ X2))) | ((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3)) | (('const/sets/FINITE' @ X0 @ X4) != $true) | (('const/sets/FINITE' @ X0 @ X5) != $true)))),
% 1.70/1.31 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f15,f18])).
% 1.70/1.31 thf(f18,plain,(
% 1.70/1.31 ! [X1 : $tType,X0 : $tType,X2 : X1 > X1 > X1,X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (? [X6 : X0] : ((('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) = $true) & ((X3 @ X6) != ('const/iterate/neutral' @ X1 @ X2))) => (($true = ('const/sets/IN' @ X0 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2) @ ('const/sets/INTER' @ X0 @ X4 @ X5))) & ((X3 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2)) != ('const/iterate/neutral' @ X1 @ X2))))),
% 1.70/1.31 introduced(definition,[],[choice_axiom])).
% 1.70/1.31 thf(f15,plain,(
% 1.70/1.31 ! [X0 : $tType,X1 : $tType,X2 : X1 > X1 > X1] : ((('const/iterate/monoidal' @ X1 @ X2) != $true) | ! [X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (? [X6 : X0] : ((('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) = $true) & ((X3 @ X6) != ('const/iterate/neutral' @ X1 @ X2))) | ((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3)) | (('const/sets/FINITE' @ X0 @ X4) != $true) | (('const/sets/FINITE' @ X0 @ X5) != $true)))),
% 1.70/1.31 inference(flattening,[status(thm)],[f14])).
% 1.70/1.31 thf(f14,plain,(
% 1.70/1.31 ! [X0 : $tType,X1 : $tType,X2 : X1 > X1 > X1] : (! [X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3)) | ((('const/sets/FINITE' @ X0 @ X4) != $true) | ? [X6 : X0] : ((('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) = $true) & ((X3 @ X6) != ('const/iterate/neutral' @ X1 @ X2))) | (('const/sets/FINITE' @ X0 @ X5) != $true))) | (('const/iterate/monoidal' @ X1 @ X2) != $true))),
% 1.70/1.31 inference(ennf_transformation,[status(thm)],[f11])).
% 1.70/1.31 thf(f11,plain,(
% 1.70/1.31 ! [X0 : $tType,X1 : $tType,X2 : X1 > X1 > X1] : ((('const/iterate/monoidal' @ X1 @ X2) = $true) => ! [X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (((('const/sets/FINITE' @ X0 @ X4) = $true) & ! [X6 : X0] : ((('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) = $true) => ((X3 @ X6) = ('const/iterate/neutral' @ X1 @ X2))) & (('const/sets/FINITE' @ X0 @ X5) = $true)) => ((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3))))),
% 1.70/1.31 inference(fool_elimination,[status(thm)],[f10])).
% 1.70/1.31 thf(f10,plain,(
% 1.70/1.31 ! [X0 : $tType,X1 : $tType,X2 : X1 > X1 > X1] : (('const/iterate/monoidal' @ X1 @ X2) => ! [X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : ((('const/sets/FINITE' @ X0 @ X5) & ! [X6 : X0] : (('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) => ((X3 @ X6) = ('const/iterate/neutral' @ X1 @ X2))) & ('const/sets/FINITE' @ X0 @ X4)) => ((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3))))),
% 1.70/1.31 inference(rectify,[status(thm)],[f1])).
% 1.70/1.31 thf(f1,axiom,(
% 1.70/1.31 ! [X0 : $tType,X2 : $tType,X1 : X2 > X2 > X2] : (('const/iterate/monoidal' @ X2 @ X1) => ! [X3 : X0 > X2,X4 : X0 > $o,X5 : X0 > $o] : ((('const/sets/FINITE' @ X0 @ X5) & ! [X6 : X0] : (('const/sets/IN' @ X0 @ X6 @ ('const/sets/INTER' @ X0 @ X4 @ X5)) => ((X3 @ X6) = ('const/iterate/neutral' @ X2 @ X1))) & ('const/sets/FINITE' @ X0 @ X4)) => (('const/iterate/iterate' @ X0 @ X2 @ X1 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3) = (X1 @ ('const/iterate/iterate' @ X0 @ X2 @ X1 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X2 @ X1 @ X5 @ X3)))))),
% 1.70/1.31 file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/ITERATE_UNION_NONZERO_')).
% 1.70/1.31 thf(f31,plain,(
% 1.70/1.31 (('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2) != ('const/arith/+' @ ('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ sK3 @ sK2) @ ('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ sK4 @ sK2)))),
% 1.70/1.31 inference(definition_unfolding,[status(thm)],[f29,f23,f23,f23])).
% 1.70/1.31 thf(f23,plain,(
% 1.70/1.31 ( ! [X0 : $tType] : (('const/iterate/nsum' @ X0 = ('const/iterate/iterate' @ X0 @ 'type/nums/num' @ 'const/arith/+'))) )),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f3])).
% 1.70/1.31 thf(f3,axiom,(
% 1.70/1.31 ! [X0 : $tType] : ('const/iterate/nsum' @ X0 = ('const/iterate/iterate' @ X0 @ 'type/nums/num' @ 'const/arith/+'))),
% 1.70/1.31 file('/export/starexec/sandbox/tmp/tmp.o4ebysSUjy/DTF2THF_8919.p','thm/iterate/nsum_')).
% 1.70/1.31 thf(f29,plain,(
% 1.70/1.31 (('const/arith/+' @ ('const/iterate/nsum' @ sK1 @ sK3 @ sK2) @ ('const/iterate/nsum' @ sK1 @ sK4 @ sK2)) != ('const/iterate/nsum' @ sK1 @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2))),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f21])).
% 1.70/1.31 thf(f44,plain,(
% 1.70/1.31 (('const/nums/NUMERAL' @ 'const/nums/_0') = (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(trivial_inequality_removal,[status(thm)],[f43])).
% 1.70/1.31 thf(f43,plain,(
% 1.70/1.31 ($true != $true) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (sK2 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+')))),
% 1.70/1.31 inference(superposition,[status(thm)],[f28,f38])).
% 1.70/1.31 thf(f38,plain,(
% 1.70/1.31 ($true = ('const/sets/IN' @ sK1 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+') @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4)))),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f37,f22])).
% 1.70/1.31 thf(f37,plain,(
% 1.70/1.31 ($true = ('const/sets/IN' @ sK1 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+') @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true)),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f36,f27])).
% 1.70/1.31 thf(f36,plain,(
% 1.70/1.31 ($true = ('const/sets/IN' @ sK1 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+') @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/sets/FINITE' @ sK1 @ sK4) != $true) | (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true)),
% 1.70/1.31 inference(subsumption_resolution,[status(thm)],[f35,f26])).
% 1.70/1.31 thf(f35,plain,(
% 1.70/1.31 ($true != ('const/sets/FINITE' @ sK1 @ sK3)) | (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true) | ($true = ('const/sets/IN' @ sK1 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+') @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/sets/FINITE' @ sK1 @ sK4) != $true)),
% 1.70/1.31 inference(trivial_inequality_removal,[status(thm)],[f32])).
% 1.70/1.31 thf(f32,plain,(
% 1.70/1.31 ($true = ('const/sets/IN' @ sK1 @ (sK0 @ sK1 @ 'type/nums/num' @ sK4 @ sK3 @ sK2 @ 'const/arith/+') @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | ($true != ('const/sets/FINITE' @ sK1 @ sK3)) | (('const/iterate/monoidal' @ 'type/nums/num' @ 'const/arith/+') != $true) | (('const/sets/FINITE' @ sK1 @ sK4) != $true) | (('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2) != ('const/iterate/iterate' @ sK1 @ 'type/nums/num' @ 'const/arith/+' @ ('const/sets/UNION' @ sK1 @ sK3 @ sK4) @ sK2))),
% 1.70/1.31 inference(superposition,[status(thm)],[f31,f25])).
% 1.70/1.31 thf(f25,plain,(
% 1.70/1.31 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X1 > X1,X3 : X0 > X1,X4 : X0 > $o,X5 : X0 > $o] : (((X2 @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X4 @ X3) @ ('const/iterate/iterate' @ X0 @ X1 @ X2 @ X5 @ X3)) = ('const/iterate/iterate' @ X0 @ X1 @ X2 @ ('const/sets/UNION' @ X0 @ X4 @ X5) @ X3)) | (('const/sets/FINITE' @ X0 @ X4) != $true) | (('const/sets/FINITE' @ X0 @ X5) != $true) | ($true = ('const/sets/IN' @ X0 @ (sK0 @ X0 @ X1 @ X5 @ X4 @ X3 @ X2) @ ('const/sets/INTER' @ X0 @ X4 @ X5))) | (('const/iterate/monoidal' @ X1 @ X2) != $true)) )),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.70/1.31 thf(f28,plain,(
% 1.70/1.31 ( ! [X4 : sK1] : (($true != ('const/sets/IN' @ sK1 @ X4 @ ('const/sets/INTER' @ sK1 @ sK3 @ sK4))) | (('const/nums/NUMERAL' @ 'const/nums/_0') = (sK2 @ X4))) )),
% 1.70/1.31 inference(cnf_transformation,[status(thm)],[f21])).
% 1.70/1.31 % SZS output end Proof for DTF2THF_8919
% 1.70/1.31 % (9021)------------------------------
% 1.70/1.31 % (9021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.70/1.31 % (9021)Termination reason: Refutation
% 1.70/1.31
% 1.70/1.31 % (9021)Memory used [KB]: 5628
% 1.70/1.31 % (9021)Time elapsed: 0.006 s
% 1.70/1.31 % (9021)Instructions burned: 10 (million)
% 1.70/1.31 % (9021)------------------------------
% 1.70/1.31 % (9021)------------------------------
% 1.70/1.31 % (9015)Success in time 0.003 s
%------------------------------------------------------------------------------