%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : MSC020^5 : TPTP v9.2.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/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 09:21:41 AM UTC 2026
% Result : Theorem 1.66s 1.38s
% Output : Refutation 1.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 5
% Syntax : Number of formulae : 35 ( 5 unt; 0 typ; 2 def)
% Number of atoms : 442 ( 113 equ; 0 cnn)
% Maximal formula atoms : 13 ( 12 avg)
% Number of connectives : 661 ( 66 ~; 57 |; 48 &; 466 @)
% ( 2 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 22 ( 22 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 11 usr; 7 con; 0-2 aty)
% Number of variables : 189 ( 88 ^; 85 !; 16 ?; 189 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
cHALF: $i > $i > $o ).
thf(func_def_1,type,
cDOUBLE: $i > $i > $o ).
thf(func_def_2,type,
cS: $i > $i ).
thf(func_def_9,type,
sK2: ( $i > $i > $o ) > $i ).
thf(func_def_10,type,
sK3: ( $i > $i > $o ) > $i ).
thf(func_def_13,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f234,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f140,f224,f233]) ).
thf(f233,plain,
~ spl4_7,
inference(avatar_contradiction_clause,[status(thm)],[f232]) ).
thf(f232,plain,
( $false
| ~ spl4_7 ),
inference(subsumption_resolution,[status(thm)],[f228,f16]) ).
thf(f16,plain,
( $true
= ( cDOUBLE @ sK1 @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( ( cHALF @ ( cS @ c0 ) @ c0 )
= $true )
& ( ( cHALF @ sK0 @ sK1 )
!= $true )
& ( $true
= ( cDOUBLE @ sK1 @ sK0 ) )
& ( ( cHALF @ c0 @ c0 )
= $true )
& ! [X2: $i,X3: $i] :
( ( ( cHALF @ ( cS @ ( cS @ X3 ) ) @ ( cS @ X2 ) )
= $true )
| ( $true
!= ( cHALF @ X3 @ X2 ) ) )
& ! [X4: $i,X5: $i > $i > $o,X6: $i] :
( ( ( X5 @ c0 @ c0 )
!= $true )
| ( ( $true
!= ( X5 @ ( cS @ ( sK2 @ X5 ) ) @ ( cS @ ( cS @ ( sK3 @ X5 ) ) ) ) )
& ( ( X5 @ ( sK2 @ X5 ) @ ( sK3 @ X5 ) )
= $true ) )
| ( ( cDOUBLE @ X4 @ X6 )
!= $true )
| ( ( X5 @ X4 @ X6 )
= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f8,f10,f9]) ).
thf(f9,plain,
( ? [X0: $i,X1: $i] :
( ( ( cHALF @ X0 @ X1 )
!= $true )
& ( ( cDOUBLE @ X1 @ X0 )
= $true ) )
=> ( ( ( cHALF @ sK0 @ sK1 )
!= $true )
& ( $true
= ( cDOUBLE @ sK1 @ sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f10,plain,
! [X5: $i > $i > $o] :
( ? [X7: $i,X8: $i] :
( ( ( X5 @ ( cS @ X7 ) @ ( cS @ ( cS @ X8 ) ) )
!= $true )
& ( $true
= ( X5 @ X7 @ X8 ) ) )
=> ( ( $true
!= ( X5 @ ( cS @ ( sK2 @ X5 ) ) @ ( cS @ ( cS @ ( sK3 @ X5 ) ) ) ) )
& ( ( X5 @ ( sK2 @ X5 ) @ ( sK3 @ X5 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f8,plain,
( ( ( cHALF @ ( cS @ c0 ) @ c0 )
= $true )
& ? [X0: $i,X1: $i] :
( ( ( cHALF @ X0 @ X1 )
!= $true )
& ( ( cDOUBLE @ X1 @ X0 )
= $true ) )
& ( ( cHALF @ c0 @ c0 )
= $true )
& ! [X2: $i,X3: $i] :
( ( ( cHALF @ ( cS @ ( cS @ X3 ) ) @ ( cS @ X2 ) )
= $true )
| ( $true
!= ( cHALF @ X3 @ X2 ) ) )
& ! [X4: $i,X5: $i > $i > $o,X6: $i] :
( ( ( X5 @ c0 @ c0 )
!= $true )
| ? [X7: $i,X8: $i] :
( ( ( X5 @ ( cS @ X7 ) @ ( cS @ ( cS @ X8 ) ) )
!= $true )
& ( $true
= ( X5 @ X7 @ X8 ) ) )
| ( ( cDOUBLE @ X4 @ X6 )
!= $true )
| ( ( X5 @ X4 @ X6 )
= $true ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,plain,
( ( ( cHALF @ ( cS @ c0 ) @ c0 )
= $true )
& ? [X8: $i,X7: $i] :
( ( ( cHALF @ X8 @ X7 )
!= $true )
& ( $true
= ( cDOUBLE @ X7 @ X8 ) ) )
& ( ( cHALF @ c0 @ c0 )
= $true )
& ! [X1: $i,X0: $i] :
( ( ( cHALF @ ( cS @ ( cS @ X0 ) ) @ ( cS @ X1 ) )
= $true )
| ( ( cHALF @ X0 @ X1 )
!= $true ) )
& ! [X3: $i,X2: $i > $i > $o,X4: $i] :
( ( $true
!= ( X2 @ c0 @ c0 ) )
| ? [X6: $i,X5: $i] :
( ( ( X2 @ ( cS @ X6 ) @ ( cS @ ( cS @ X5 ) ) )
!= $true )
& ( ( X2 @ X6 @ X5 )
= $true ) )
| ( ( cDOUBLE @ X3 @ X4 )
!= $true )
| ( ( X2 @ X3 @ X4 )
= $true ) ) ),
inference(flattening,[status(thm)],[f6]) ).
thf(f6,plain,
( ? [X8: $i,X7: $i] :
( ( ( cHALF @ X8 @ X7 )
!= $true )
& ( $true
= ( cDOUBLE @ X7 @ X8 ) ) )
& ! [X4: $i,X2: $i > $i > $o,X3: $i] :
( ( ( X2 @ X3 @ X4 )
= $true )
| ? [X6: $i,X5: $i] :
( ( ( X2 @ ( cS @ X6 ) @ ( cS @ ( cS @ X5 ) ) )
!= $true )
& ( ( X2 @ X6 @ X5 )
= $true ) )
| ( $true
!= ( X2 @ c0 @ c0 ) )
| ( ( cDOUBLE @ X3 @ X4 )
!= $true ) )
& ( ( cHALF @ ( cS @ c0 ) @ c0 )
= $true )
& ! [X1: $i,X0: $i] :
( ( ( cHALF @ ( cS @ ( cS @ X0 ) ) @ ( cS @ X1 ) )
= $true )
| ( ( cHALF @ X0 @ X1 )
!= $true ) )
& ( ( cHALF @ c0 @ c0 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f5]) ).
thf(f5,plain,
~ ( ( ! [X4: $i,X2: $i > $i > $o,X3: $i] :
( ( ! [X6: $i,X5: $i] :
( ( ( X2 @ X6 @ X5 )
= $true )
=> ( ( X2 @ ( cS @ X6 ) @ ( cS @ ( cS @ X5 ) ) )
= $true ) )
& ( $true
= ( X2 @ c0 @ c0 ) )
& ( ( cDOUBLE @ X3 @ X4 )
= $true ) )
=> ( ( X2 @ X3 @ X4 )
= $true ) )
& ( ( cHALF @ ( cS @ c0 ) @ c0 )
= $true )
& ! [X1: $i,X0: $i] :
( ( ( cHALF @ X0 @ X1 )
= $true )
=> ( ( cHALF @ ( cS @ ( cS @ X0 ) ) @ ( cS @ X1 ) )
= $true ) )
& ( ( cHALF @ c0 @ c0 )
= $true ) )
=> ! [X8: $i,X7: $i] :
( ( $true
= ( cDOUBLE @ X7 @ X8 ) )
=> ( ( cHALF @ X8 @ X7 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f4]) ).
thf(f4,plain,
~ ( ( ! [X0: $i,X1: $i] :
( ( cHALF @ X0 @ X1 )
=> ( cHALF @ ( cS @ ( cS @ X0 ) ) @ ( cS @ X1 ) ) )
& ( cHALF @ c0 @ c0 )
& ( cHALF @ ( cS @ c0 ) @ c0 )
& ! [X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( X2 @ c0 @ c0 )
& ! [X5: $i,X6: $i] :
( ( X2 @ X6 @ X5 )
=> ( X2 @ ( cS @ X6 ) @ ( cS @ ( cS @ X5 ) ) ) )
& ( cDOUBLE @ X3 @ X4 ) )
=> ( X2 @ X3 @ X4 ) ) )
=> ! [X7: $i,X8: $i] :
( ( cDOUBLE @ X7 @ X8 )
=> ( cHALF @ X8 @ X7 ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,negated_conjecture,
~ ( ( ! [X3: $i,X4: $i] :
( ( cHALF @ X3 @ X4 )
=> ( cHALF @ ( cS @ ( cS @ X3 ) ) @ ( cS @ X4 ) ) )
& ( cHALF @ c0 @ c0 )
& ( cHALF @ ( cS @ c0 ) @ c0 )
& ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ c0 @ c0 )
& ! [X4: $i,X3: $i] :
( ( X0 @ X3 @ X4 )
=> ( X0 @ ( cS @ X3 ) @ ( cS @ ( cS @ X4 ) ) ) )
& ( cDOUBLE @ X1 @ X2 ) )
=> ( X0 @ X1 @ X2 ) ) )
=> ! [X1: $i,X2: $i] :
( ( cDOUBLE @ X1 @ X2 )
=> ( cHALF @ X2 @ X1 ) ) ),
inference(negated_conjecture,[status(cth)],[f1]) ).
thf(f1,conjecture,
( ( ! [X3: $i,X4: $i] :
( ( cHALF @ X3 @ X4 )
=> ( cHALF @ ( cS @ ( cS @ X3 ) ) @ ( cS @ X4 ) ) )
& ( cHALF @ c0 @ c0 )
& ( cHALF @ ( cS @ c0 ) @ c0 )
& ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ c0 @ c0 )
& ! [X4: $i,X3: $i] :
( ( X0 @ X3 @ X4 )
=> ( X0 @ ( cS @ X3 ) @ ( cS @ ( cS @ X4 ) ) ) )
& ( cDOUBLE @ X1 @ X2 ) )
=> ( X0 @ X1 @ X2 ) ) )
=> ! [X1: $i,X2: $i] :
( ( cDOUBLE @ X1 @ X2 )
=> ( cHALF @ X2 @ X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.WxY7ZJK8eR/DTF2THF_1034.p',cTHM301) ).
thf(f228,plain,
( ( $true
!= ( cDOUBLE @ sK1 @ sK0 ) )
| ~ spl4_7 ),
inference(trivial_inequality_removal,[status(thm)],[f226]) ).
thf(f226,plain,
( ( $true != $true )
| ( $true
!= ( cDOUBLE @ sK1 @ sK0 ) )
| ~ spl4_7 ),
inference(superposition,[status(thm)],[f17,f135]) ).
thf(f135,plain,
( ! [X0: $i,X1: $i] :
( ( ( cHALF @ X1 @ X0 )
= $true )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true ) )
| ~ spl4_7 ),
inference(avatar_component_clause,[status(thm)],[f134]) ).
thf(f134,definition,
( spl4_7
<=> ! [X0: $i,X1: $i] :
( ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( cHALF @ X1 @ X0 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl4_7])],[avatar_definition]) ).
thf(f17,plain,
( ( cHALF @ sK0 @ sK1 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f224,plain,
( spl4_7
| spl4_8 ),
inference(avatar_split_clause,[status(thm)],[f223,f137,f134]) ).
thf(f137,definition,
( spl4_8
<=> ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl4_8])],[avatar_definition]) ).
thf(f223,plain,
( ! [X0: $i,X1: $i] :
( ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( cHALF @ X1 @ X0 )
= $true ) )
| spl4_8 ),
inference(subsumption_resolution,[status(thm)],[f222,f15]) ).
thf(f15,plain,
( ( cHALF @ c0 @ c0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f222,plain,
( ! [X0: $i,X1: $i] :
( ( ( cHALF @ c0 @ c0 )
!= $true )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( cHALF @ X1 @ X0 )
= $true ) )
| spl4_8 ),
inference(beta_eta_normalization,[status(thm)],[f221]) ).
thf(f221,plain,
( ! [X0: $i,X1: $i] :
( ( ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ c0
@ c0 )
!= $true )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( $true
= ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ X0
@ X1 ) ) )
| spl4_8 ),
inference(trivial_inequality_removal,[status(thm)],[f220]) ).
thf(f220,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ X0
@ X1 ) )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( $true != $true )
| ( ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ c0
@ c0 )
!= $true ) )
| spl4_8 ),
inference(superposition,[status(thm)],[f139,f12]) ).
thf(f12,plain,
! [X6: $i,X4: $i,X5: $i > $i > $o] :
( ( ( X5 @ ( sK2 @ X5 ) @ ( sK3 @ X5 ) )
= $true )
| ( ( X5 @ X4 @ X6 )
= $true )
| ( ( X5 @ c0 @ c0 )
!= $true )
| ( ( cDOUBLE @ X4 @ X6 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f139,plain,
( ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) ) )
!= $true )
| spl4_8 ),
inference(avatar_component_clause,[status(thm)],[f137]) ).
thf(f140,plain,
( spl4_7
| ~ spl4_8 ),
inference(avatar_split_clause,[status(thm)],[f132,f137,f134]) ).
thf(f132,plain,
! [X0: $i,X1: $i] :
( ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) ) )
!= $true )
| ( ( cHALF @ X1 @ X0 )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f79,f15]) ).
thf(f79,plain,
! [X0: $i,X1: $i] :
( ( ( cHALF @ c0 @ c0 )
!= $true )
| ( ( cHALF @ X1 @ X0 )
= $true )
| ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] : ( cHALF @ Y1 @ Y0 ) ) )
!= $true )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f78]) ).
thf(f78,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ X0
@ X1 ) )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ c0
@ c0 )
!= $true )
| ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f74]) ).
thf(f74,plain,
! [X0: $i,X1: $i] :
( ( ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ c0
@ c0 )
!= $true )
| ( ( cDOUBLE @ X0 @ X1 )
!= $true )
| ( ( cHALF
@ ( sK3
@ ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) ) )
@ ( sK2
@ ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) ) ) )
!= $true )
| ( $true != $true )
| ( $true
= ( ^ [Y0: $i,Y1: $i] :
( cHALF
@ ( ^ [Y2: $i,Y3: $i] : Y3
@ Y0
@ Y1 )
@ ( ^ [Y2: $i,Y3: $i] : Y2
@ Y0
@ Y1 ) )
@ X0
@ X1 ) ) ),
inference(superposition,[status(thm)],[f13,f14]) ).
thf(f14,plain,
! [X2: $i,X3: $i] :
( ( ( cHALF @ ( cS @ ( cS @ X3 ) ) @ ( cS @ X2 ) )
= $true )
| ( $true
!= ( cHALF @ X3 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f13,plain,
! [X6: $i,X4: $i,X5: $i > $i > $o] :
( ( $true
!= ( X5 @ ( cS @ ( sK2 @ X5 ) ) @ ( cS @ ( cS @ ( sK3 @ X5 ) ) ) ) )
| ( ( X5 @ c0 @ c0 )
!= $true )
| ( ( X5 @ X4 @ X6 )
= $true )
| ( ( cDOUBLE @ X4 @ X6 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f11]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : MSC020^5 : TPTP v9.2.1. Released v4.0.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/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 10:42:09 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : MSC020^5 : TPTP v9.2.1. Released v4.0.0.
% 0.28/0.46 % Domain : Miscellaneous
% 0.28/0.46 % Problem : TPS problem THM301
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English : Relation between HALF and DOUBLE functions.
% 0.28/0.46
% 0.28/0.46 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Bro09]
% 0.28/0.46 % Names : tps_0274 [Bro09]
% 0.28/0.46 % : THM301 [TPS]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.25 v9.0.0, 0.33 v8.2.0, 0.36 v8.1.0, 0.42 v7.5.0, 0.33 v7.4.0, 0.44 v7.3.0, 0.50 v7.1.0, 0.43 v7.0.0, 0.50 v6.4.0, 0.57 v6.3.0, 0.33 v6.0.0, 0.17 v5.5.0, 0.20 v5.4.0, 0.75 v5.3.0, 1.00 v5.2.0, 0.75 v5.0.0, 0.50 v4.1.0, 0.67 v4.0.0
% 0.28/0.46 % Syntax : Number of formulae : 5 ( 0 unt; 4 typ; 0 def)
% 0.28/0.46 % Number of atoms : 7 ( 0 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 7 ( 7 avg)
% 0.28/0.46 % Number of connectives : 39 ( 0 ~; 0 |; 5 &; 29 @)
% 0.28/0.46 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 15 ( 15 avg)
% 0.28/0.46 % Number of types : 2 ( 0 usr)
% 0.28/0.46 % Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 4 ( 4 usr; 1 con; 0-2 aty)
% 0.28/0.46 % Number of variables : 9 ( 0 ^; 9 !; 0 ?; 9 :)
% 0.28/0.46 % SPC : TH0_THM_NEQ_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.28/0.46 % project in the Department of Mathematical Sciences at Carnegie
% 0.28/0.46 % Mellon University. Distributed under the Creative Commons copyleft
% 0.28/0.46 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.28/0.46 % :
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 thf(cHALF,type,
% 0.28/0.46 cHALF: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(cDOUBLE,type,
% 0.28/0.46 cDOUBLE: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(cS,type,
% 0.28/0.46 cS: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(c0,type,
% 0.28/0.46 c0: $i ).
% 0.28/0.46
% 0.28/0.46 thf(cTHM301,conjecture,
% 0.28/0.46 ( ( ! [Q: $i > $i > $o,Xu: $i,Xv: $i] :
% 0.28/0.46 ( ( ( cDOUBLE @ Xu @ Xv )
% 0.28/0.46 & ( Q @ c0 @ c0 )
% 0.28/0.46 & ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Q @ Xx @ Xy )
% 0.28/0.46 => ( Q @ ( cS @ Xx ) @ ( cS @ ( cS @ Xy ) ) ) ) )
% 0.28/0.46 => ( Q @ Xu @ Xv ) )
% 0.28/0.46 & ( cHALF @ c0 @ c0 )
% 0.28/0.46 & ( cHALF @ ( cS @ c0 ) @ c0 )
% 0.28/0.46 & ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( cHALF @ Xx @ Xy )
% 0.28/0.46 => ( cHALF @ ( cS @ ( cS @ Xx ) ) @ ( cS @ Xy ) ) ) )
% 0.28/0.46 => ! [Xu: $i,Xv: $i] :
% 0.28/0.46 ( ( cDOUBLE @ Xu @ Xv )
% 0.28/0.46 => ( cHALF @ Xv @ Xu ) ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 1.66/1.32 ---- Embedded in THF ---
% 1.66/1.32 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.66/1.32 %%% Generated on Tue Feb 24 10:42:10 EST 2026
% 1.66/1.32 %%% using '$$dhol' embedding, version 1.3.0.
% 1.66/1.32 %%% Logic specification used:
% 1.66/1.32 %%% thf(spec, logic, $$dhol).
% 1.66/1.32
% 1.66/1.32 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.WxY7ZJK8eR/DTF2DTF1034.p
% See solution above
% 1.66/1.32 ------------------------
% 1.66/1.33 ---- Cleaned THF ---
% 1.66/1.33 thf(cHALF,type,
% 1.66/1.33 cHALF: $i > $i > $o ).
% 1.66/1.33
% 1.66/1.33 thf(cDOUBLE,type,
% 1.66/1.33 cDOUBLE: $i > $i > $o ).
% 1.66/1.33
% 1.66/1.33 thf(cS,type,
% 1.66/1.33 cS: $i > $i ).
% 1.66/1.33
% 1.66/1.33 thf(c0,type,
% 1.66/1.33 c0: $i ).
% 1.66/1.33
% 1.66/1.33 thf(cTHM301,conjecture,
% 1.66/1.33 ( ( ! [Q: $i > $i > $o,Xu: $i,Xv: $i] :
% 1.66/1.33 ( ( ( cDOUBLE @ Xu @ Xv )
% 1.66/1.33 & ( Q @ c0 @ c0 )
% 1.66/1.33 & ! [Xx: $i,Xy: $i] :
% 1.66/1.33 ( ( Q @ Xx @ Xy )
% 1.66/1.33 => ( Q @ ( cS @ Xx ) @ ( cS @ ( cS @ Xy ) ) ) ) )
% 1.66/1.33 => ( Q @ Xu @ Xv ) )
% 1.66/1.33 & ( cHALF @ c0 @ c0 )
% 1.66/1.33 & ( cHALF @ ( cS @ c0 ) @ c0 )
% 1.66/1.33 & ! [Xx: $i,Xy: $i] :
% 1.66/1.33 ( ( cHALF @ Xx @ Xy )
% 1.66/1.33 => ( cHALF @ ( cS @ ( cS @ Xx ) ) @ ( cS @ Xy ) ) ) )
% 1.66/1.33 => ! [Xu: $i,Xv: $i] :
% 1.66/1.33 ( ( cDOUBLE @ Xu @ Xv )
% 1.66/1.33 => ( cHALF @ Xv @ Xu ) ) ) ).
% 1.66/1.33 ------------------------
% 1.66/1.35 % (1272)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_1034 for (2999ds/27Mi)
% 1.66/1.35 % (1270)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/183Mi)
% 1.66/1.35 % (1271)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_1034 for (2999ds/4Mi)
% 1.66/1.35 % (1273)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/2Mi)
% 1.66/1.35 % (1275)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_1034 for (2999ds/275Mi)
% 1.66/1.35 % (1276)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/18Mi)
% 1.66/1.35 % (1274)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_1034 for (2999ds/2Mi)
% 1.66/1.35 % (1273)Instruction limit reached!
% 1.66/1.35 % (1273)------------------------------
% 1.66/1.35 % (1273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.35 % (1273)Termination reason: Unknown
% 1.66/1.35 % (1273)Termination phase: Preprocessing 3
% 1.66/1.35
% 1.66/1.35 % (1273)Memory used [KB]: 895
% 1.66/1.35 % (1273)Time elapsed: 0.003 s
% 1.66/1.35 % (1273)Instructions burned: 2 (million)
% 1.66/1.35 % (1273)------------------------------
% 1.66/1.35 % (1273)------------------------------
% 1.66/1.36 % (1274)Instruction limit reached!
% 1.66/1.36 % (1274)------------------------------
% 1.66/1.36 % (1274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.36 % (1274)Termination reason: Unknown
% 1.66/1.36 % (1274)Termination phase: Saturation
% 1.66/1.36
% 1.66/1.36 % (1274)Memory used [KB]: 5500
% 1.66/1.36 % (1274)Time elapsed: 0.003 s
% 1.66/1.36 % (1274)Instructions burned: 3 (million)
% 1.66/1.36 % (1274)------------------------------
% 1.66/1.36 % (1274)------------------------------
% 1.66/1.36 % (1271)Instruction limit reached!
% 1.66/1.36 % (1271)------------------------------
% 1.66/1.36 % (1271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.36 % (1271)Termination reason: Unknown
% 1.66/1.36 % (1271)Termination phase: Saturation
% 1.66/1.36
% 1.66/1.36 % (1271)Memory used [KB]: 5500
% 1.66/1.36 % (1271)Time elapsed: 0.005 s
% 1.66/1.36 % (1271)Instructions burned: 4 (million)
% 1.66/1.36 % (1271)------------------------------
% 1.66/1.36 % (1271)------------------------------
% 1.66/1.36 % (1272)Instruction limit reached!
% 1.66/1.36 % (1272)------------------------------
% 1.66/1.36 % (1272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.36 % (1272)Termination reason: Unknown
% 1.66/1.36 % (1272)Termination phase: Saturation
% 1.66/1.36
% 1.66/1.36 % (1272)Memory used [KB]: 5628
% 1.66/1.36 % (1272)Time elapsed: 0.011 s
% 1.66/1.36 % (1272)Instructions burned: 27 (million)
% 1.66/1.36 % (1272)------------------------------
% 1.66/1.36 % (1272)------------------------------
% 1.66/1.36 % (1276)Instruction limit reached!
% 1.66/1.36 % (1276)------------------------------
% 1.66/1.36 % (1276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.36 % (1276)Termination reason: Unknown
% 1.66/1.36 % (1276)Termination phase: Saturation
% 1.66/1.36
% 1.66/1.36 % (1276)Memory used [KB]: 5628
% 1.66/1.36 % (1276)Time elapsed: 0.013 s
% 1.66/1.36 % (1276)Instructions burned: 18 (million)
% 1.66/1.36 % (1276)------------------------------
% 1.66/1.36 % (1276)------------------------------
% 1.66/1.37 % (1280)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/3Mi)
% 1.66/1.37 % (1280)Instruction limit reached!
% 1.66/1.37 % (1280)------------------------------
% 1.66/1.37 % (1280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.37 % (1280)Termination reason: Unknown
% 1.66/1.37 % (1280)Termination phase: Saturation
% 1.66/1.37
% 1.66/1.37 % (1280)Memory used [KB]: 5500
% 1.66/1.37 % (1280)Time elapsed: 0.003 s
% 1.66/1.37 % (1280)Instructions burned: 3 (million)
% 1.66/1.37 % (1280)------------------------------
% 1.66/1.37 % (1280)------------------------------
% 1.66/1.37 % (1277)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/3Mi)
% 1.66/1.37 % (1279)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/15Mi)
% 1.66/1.37 % (1275)Refutation not found, incomplete strategy
% 1.66/1.37 % (1275)------------------------------
% 1.66/1.37 % (1275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.37 % (1275)Termination reason: Refutation not found, incomplete strategy
% 1.66/1.37
% 1.66/1.37
% 1.66/1.37 % (1275)Memory used [KB]: 5628
% 1.66/1.37 % (1275)Time elapsed: 0.021 s
% 1.66/1.37 % (1275)Instructions burned: 30 (million)
% 1.66/1.37 % (1275)------------------------------
% 1.66/1.37 % (1275)------------------------------
% 1.66/1.37 % (1278)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/37Mi)
% 1.66/1.37 % (1277)Instruction limit reached!
% 1.66/1.37 % (1277)------------------------------
% 1.66/1.37 % (1277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.37 % (1277)Termination reason: Unknown
% 1.66/1.37 % (1277)Termination phase: Saturation
% 1.66/1.37
% 1.66/1.37 % (1277)Memory used [KB]: 5500
% 1.66/1.37 % (1277)Time elapsed: 0.004 s
% 1.66/1.37 % (1277)Instructions burned: 3 (million)
% 1.66/1.37 % (1277)------------------------------
% 1.66/1.37 % (1277)------------------------------
% 1.66/1.38 % (1270)First to succeed.
% 1.66/1.38 % (1282)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_1034 for (2999ds/7Mi)
% 1.66/1.38 % (1270)Refutation found. Thanks to Tanya!
% 1.66/1.38 % SZS status Theorem for DTF2THF_1034
% 1.66/1.38 % SZS output start Proof for DTF2THF_1034
% 1.66/1.38 thf(func_def_0, type, cHALF: $i > $i > $o).
% 1.66/1.38 thf(func_def_1, type, cDOUBLE: $i > $i > $o).
% 1.66/1.38 thf(func_def_2, type, cS: $i > $i).
% 1.66/1.38 thf(func_def_9, type, sK2: ($i > $i > $o) > $i).
% 1.66/1.38 thf(func_def_10, type, sK3: ($i > $i > $o) > $i).
% 1.66/1.38 thf(func_def_13, type, ph5: !>[X0: $tType]:(X0)).
% 1.66/1.38 thf(f234,plain,(
% 1.66/1.38 $false),
% 1.66/1.38 inference(avatar_sat_refutation,[status(thm)],[f140,f224,f233])).
% 1.66/1.38 thf(f233,plain,(
% 1.66/1.38 ~spl4_7),
% 1.66/1.38 inference(avatar_contradiction_clause,[status(thm)],[f232])).
% 1.66/1.38 thf(f232,plain,(
% 1.66/1.38 $false | ~spl4_7),
% 1.66/1.38 inference(subsumption_resolution,[status(thm)],[f228,f16])).
% 1.66/1.38 thf(f16,plain,(
% 1.66/1.38 ($true = (cDOUBLE @ sK1 @ sK0))),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 thf(f11,plain,(
% 1.66/1.38 ((cHALF @ (cS @ c0) @ c0) = $true) & (((cHALF @ sK0 @ sK1) != $true) & ($true = (cDOUBLE @ sK1 @ sK0))) & ((cHALF @ c0 @ c0) = $true) & ! [X2 : $i,X3 : $i] : (((cHALF @ (cS @ (cS @ X3)) @ (cS @ X2)) = $true) | ($true != (cHALF @ X3 @ X2))) & ! [X4 : $i,X5 : $i > $i > $o,X6 : $i] : (((X5 @ c0 @ c0) != $true) | (($true != (X5 @ (cS @ (sK2 @ X5)) @ (cS @ (cS @ (sK3 @ X5))))) & ((X5 @ (sK2 @ X5) @ (sK3 @ X5)) = $true)) | ((cDOUBLE @ X4 @ X6) != $true) | ((X5 @ X4 @ X6) = $true))),
% 1.66/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f8,f10,f9])).
% 1.66/1.38 thf(f9,plain,(
% 1.66/1.38 ? [X0 : $i,X1 : $i] : (((cHALF @ X0 @ X1) != $true) & ((cDOUBLE @ X1 @ X0) = $true)) => (((cHALF @ sK0 @ sK1) != $true) & ($true = (cDOUBLE @ sK1 @ sK0)))),
% 1.66/1.38 introduced(definition,[],[choice_axiom])).
% 1.66/1.38 thf(f10,plain,(
% 1.66/1.38 ! [X5 : $i > $i > $o] : (? [X7 : $i,X8 : $i] : (((X5 @ (cS @ X7) @ (cS @ (cS @ X8))) != $true) & ($true = (X5 @ X7 @ X8))) => (($true != (X5 @ (cS @ (sK2 @ X5)) @ (cS @ (cS @ (sK3 @ X5))))) & ((X5 @ (sK2 @ X5) @ (sK3 @ X5)) = $true)))),
% 1.66/1.38 introduced(definition,[],[choice_axiom])).
% 1.66/1.38 thf(f8,plain,(
% 1.66/1.38 ((cHALF @ (cS @ c0) @ c0) = $true) & ? [X0 : $i,X1 : $i] : (((cHALF @ X0 @ X1) != $true) & ((cDOUBLE @ X1 @ X0) = $true)) & ((cHALF @ c0 @ c0) = $true) & ! [X2 : $i,X3 : $i] : (((cHALF @ (cS @ (cS @ X3)) @ (cS @ X2)) = $true) | ($true != (cHALF @ X3 @ X2))) & ! [X4 : $i,X5 : $i > $i > $o,X6 : $i] : (((X5 @ c0 @ c0) != $true) | ? [X7 : $i,X8 : $i] : (((X5 @ (cS @ X7) @ (cS @ (cS @ X8))) != $true) & ($true = (X5 @ X7 @ X8))) | ((cDOUBLE @ X4 @ X6) != $true) | ((X5 @ X4 @ X6) = $true))),
% 1.66/1.38 inference(rectify,[status(thm)],[f7])).
% 1.66/1.38 thf(f7,plain,(
% 1.66/1.38 ((cHALF @ (cS @ c0) @ c0) = $true) & ? [X8 : $i,X7 : $i] : (((cHALF @ X8 @ X7) != $true) & ($true = (cDOUBLE @ X7 @ X8))) & ((cHALF @ c0 @ c0) = $true) & ! [X1 : $i,X0 : $i] : (((cHALF @ (cS @ (cS @ X0)) @ (cS @ X1)) = $true) | ((cHALF @ X0 @ X1) != $true)) & ! [X3 : $i,X2 : $i > $i > $o,X4 : $i] : (($true != (X2 @ c0 @ c0)) | ? [X6 : $i,X5 : $i] : (((X2 @ (cS @ X6) @ (cS @ (cS @ X5))) != $true) & ((X2 @ X6 @ X5) = $true)) | ((cDOUBLE @ X3 @ X4) != $true) | ((X2 @ X3 @ X4) = $true))),
% 1.66/1.38 inference(flattening,[status(thm)],[f6])).
% 1.66/1.38 thf(f6,plain,(
% 1.66/1.38 ? [X8 : $i,X7 : $i] : (((cHALF @ X8 @ X7) != $true) & ($true = (cDOUBLE @ X7 @ X8))) & (! [X4 : $i,X2 : $i > $i > $o,X3 : $i] : (((X2 @ X3 @ X4) = $true) | (? [X6 : $i,X5 : $i] : (((X2 @ (cS @ X6) @ (cS @ (cS @ X5))) != $true) & ((X2 @ X6 @ X5) = $true)) | ($true != (X2 @ c0 @ c0)) | ((cDOUBLE @ X3 @ X4) != $true))) & ((cHALF @ (cS @ c0) @ c0) = $true) & ! [X1 : $i,X0 : $i] : (((cHALF @ (cS @ (cS @ X0)) @ (cS @ X1)) = $true) | ((cHALF @ X0 @ X1) != $true)) & ((cHALF @ c0 @ c0) = $true))),
% 1.66/1.38 inference(ennf_transformation,[status(thm)],[f5])).
% 1.66/1.38 thf(f5,plain,(
% 1.66/1.38 ~((! [X4 : $i,X2 : $i > $i > $o,X3 : $i] : ((! [X6 : $i,X5 : $i] : (((X2 @ X6 @ X5) = $true) => ((X2 @ (cS @ X6) @ (cS @ (cS @ X5))) = $true)) & ($true = (X2 @ c0 @ c0)) & ((cDOUBLE @ X3 @ X4) = $true)) => ((X2 @ X3 @ X4) = $true)) & ((cHALF @ (cS @ c0) @ c0) = $true) & ! [X1 : $i,X0 : $i] : (((cHALF @ X0 @ X1) = $true) => ((cHALF @ (cS @ (cS @ X0)) @ (cS @ X1)) = $true)) & ((cHALF @ c0 @ c0) = $true)) => ! [X8 : $i,X7 : $i] : (($true = (cDOUBLE @ X7 @ X8)) => ((cHALF @ X8 @ X7) = $true)))),
% 1.66/1.38 inference(fool_elimination,[status(thm)],[f4])).
% 1.66/1.38 thf(f4,plain,(
% 1.66/1.38 ~((! [X0 : $i,X1 : $i] : ((cHALF @ X0 @ X1) => (cHALF @ (cS @ (cS @ X0)) @ (cS @ X1))) & (cHALF @ c0 @ c0) & (cHALF @ (cS @ c0) @ c0) & ! [X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((X2 @ c0 @ c0) & ! [X5 : $i,X6 : $i] : ((X2 @ X6 @ X5) => (X2 @ (cS @ X6) @ (cS @ (cS @ X5)))) & (cDOUBLE @ X3 @ X4)) => (X2 @ X3 @ X4))) => ! [X7 : $i,X8 : $i] : ((cDOUBLE @ X7 @ X8) => (cHALF @ X8 @ X7)))),
% 1.66/1.38 inference(rectify,[status(thm)],[f2])).
% 1.66/1.38 thf(f2,negated_conjecture,(
% 1.66/1.38 ~((! [X3 : $i,X4 : $i] : ((cHALF @ X3 @ X4) => (cHALF @ (cS @ (cS @ X3)) @ (cS @ X4))) & (cHALF @ c0 @ c0) & (cHALF @ (cS @ c0) @ c0) & ! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : (((X0 @ c0 @ c0) & ! [X4 : $i,X3 : $i] : ((X0 @ X3 @ X4) => (X0 @ (cS @ X3) @ (cS @ (cS @ X4)))) & (cDOUBLE @ X1 @ X2)) => (X0 @ X1 @ X2))) => ! [X1 : $i,X2 : $i] : ((cDOUBLE @ X1 @ X2) => (cHALF @ X2 @ X1)))),
% 1.66/1.38 inference(negated_conjecture,[status(cth)],[f1])).
% 1.66/1.38 thf(f1,conjecture,(
% 1.66/1.38 (! [X3 : $i,X4 : $i] : ((cHALF @ X3 @ X4) => (cHALF @ (cS @ (cS @ X3)) @ (cS @ X4))) & (cHALF @ c0 @ c0) & (cHALF @ (cS @ c0) @ c0) & ! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : (((X0 @ c0 @ c0) & ! [X4 : $i,X3 : $i] : ((X0 @ X3 @ X4) => (X0 @ (cS @ X3) @ (cS @ (cS @ X4)))) & (cDOUBLE @ X1 @ X2)) => (X0 @ X1 @ X2))) => ! [X1 : $i,X2 : $i] : ((cDOUBLE @ X1 @ X2) => (cHALF @ X2 @ X1))),
% 1.66/1.38 file('/export/starexec/sandbox2/tmp/tmp.WxY7ZJK8eR/DTF2THF_1034.p',cTHM301)).
% 1.66/1.38 thf(f228,plain,(
% 1.66/1.38 ($true != (cDOUBLE @ sK1 @ sK0)) | ~spl4_7),
% 1.66/1.38 inference(trivial_inequality_removal,[status(thm)],[f226])).
% 1.66/1.38 thf(f226,plain,(
% 1.66/1.38 ($true != $true) | ($true != (cDOUBLE @ sK1 @ sK0)) | ~spl4_7),
% 1.66/1.38 inference(superposition,[status(thm)],[f17,f135])).
% 1.66/1.38 thf(f135,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (((cHALF @ X1 @ X0) = $true) | ((cDOUBLE @ X0 @ X1) != $true)) ) | ~spl4_7),
% 1.66/1.38 inference(avatar_component_clause,[status(thm)],[f134])).
% 1.66/1.38 thf(f134,definition,(
% 1.66/1.38 spl4_7 <=> ! [X0 : $i,X1 : $i] : (((cDOUBLE @ X0 @ X1) != $true) | ((cHALF @ X1 @ X0) = $true))),
% 1.66/1.38 introduced(definition,[new_symbols(naming,[spl4_7])],[avatar_definition])).
% 1.66/1.38 thf(f17,plain,(
% 1.66/1.38 ((cHALF @ sK0 @ sK1) != $true)),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 thf(f224,plain,(
% 1.66/1.38 spl4_7 | spl4_8),
% 1.66/1.38 inference(avatar_split_clause,[status(thm)],[f223,f137,f134])).
% 1.66/1.38 thf(f137,definition,(
% 1.66/1.38 spl4_8 <=> ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0)))))) = $true)),
% 1.66/1.38 introduced(definition,[new_symbols(naming,[spl4_8])],[avatar_definition])).
% 1.66/1.38 thf(f223,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (((cDOUBLE @ X0 @ X1) != $true) | ((cHALF @ X1 @ X0) = $true)) ) | spl4_8),
% 1.66/1.38 inference(subsumption_resolution,[status(thm)],[f222,f15])).
% 1.66/1.38 thf(f15,plain,(
% 1.66/1.38 ((cHALF @ c0 @ c0) = $true)),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 thf(f222,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (((cHALF @ c0 @ c0) != $true) | ((cDOUBLE @ X0 @ X1) != $true) | ((cHALF @ X1 @ X0) = $true)) ) | spl4_8),
% 1.66/1.38 inference(beta_eta_normalization,[status(thm)],[f221])).
% 1.66/1.38 thf(f221,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : ((((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ c0 @ c0) != $true) | ((cDOUBLE @ X0 @ X1) != $true) | ($true = ((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ X0 @ X1))) ) | spl4_8),
% 1.66/1.38 inference(trivial_inequality_removal,[status(thm)],[f220])).
% 1.66/1.38 thf(f220,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ X0 @ X1)) | ((cDOUBLE @ X0 @ X1) != $true) | ($true != $true) | (((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ c0 @ c0) != $true)) ) | spl4_8),
% 1.66/1.38 inference(superposition,[status(thm)],[f139,f12])).
% 1.66/1.38 thf(f12,plain,(
% 1.66/1.38 ( ! [X6 : $i,X4 : $i,X5 : $i > $i > $o] : (((X5 @ (sK2 @ X5) @ (sK3 @ X5)) = $true) | ((X5 @ X4 @ X6) = $true) | ((X5 @ c0 @ c0) != $true) | ((cDOUBLE @ X4 @ X6) != $true)) )),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 thf(f139,plain,(
% 1.66/1.38 ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0)))))) != $true) | spl4_8),
% 1.66/1.38 inference(avatar_component_clause,[status(thm)],[f137])).
% 1.66/1.38 thf(f140,plain,(
% 1.66/1.38 spl4_7 | ~spl4_8),
% 1.66/1.38 inference(avatar_split_clause,[status(thm)],[f132,f137,f134])).
% 1.66/1.38 thf(f132,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (((cDOUBLE @ X0 @ X1) != $true) | ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0)))))) != $true) | ((cHALF @ X1 @ X0) = $true)) )),
% 1.66/1.38 inference(subsumption_resolution,[status(thm)],[f79,f15])).
% 1.66/1.38 thf(f79,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (((cHALF @ c0 @ c0) != $true) | ((cHALF @ X1 @ X0) = $true) | ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ Y1 @ Y0)))))) != $true) | ((cDOUBLE @ X0 @ X1) != $true)) )),
% 1.66/1.38 inference(beta_eta_normalization,[status(thm)],[f78])).
% 1.66/1.38 thf(f78,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ X0 @ X1)) | ((cDOUBLE @ X0 @ X1) != $true) | (((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ c0 @ c0) != $true) | ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1)))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))))) != $true)) )),
% 1.66/1.38 inference(trivial_inequality_removal,[status(thm)],[f74])).
% 1.66/1.38 thf(f74,plain,(
% 1.66/1.38 ( ! [X0 : $i,X1 : $i] : ((((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ c0 @ c0) != $true) | ((cDOUBLE @ X0 @ X1) != $true) | ((cHALF @ (sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1)))))) @ (sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))))) != $true) | ($true != $true) | ($true = ((^[Y0 : $i]: ((^[Y1 : $i]: (cHALF @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y3)))) @ Y0 @ Y1) @ ((^[Y2 : $i]: ((^[Y3 : $i]: (Y2)))) @ Y0 @ Y1))))) @ X0 @ X1))) )),
% 1.66/1.38 inference(superposition,[status(thm)],[f13,f14])).
% 1.66/1.38 thf(f14,plain,(
% 1.66/1.38 ( ! [X2 : $i,X3 : $i] : (((cHALF @ (cS @ (cS @ X3)) @ (cS @ X2)) = $true) | ($true != (cHALF @ X3 @ X2))) )),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 thf(f13,plain,(
% 1.66/1.38 ( ! [X6 : $i,X4 : $i,X5 : $i > $i > $o] : (($true != (X5 @ (cS @ (sK2 @ X5)) @ (cS @ (cS @ (sK3 @ X5))))) | ((X5 @ c0 @ c0) != $true) | ((X5 @ X4 @ X6) = $true) | ((cDOUBLE @ X4 @ X6) != $true)) )),
% 1.66/1.38 inference(cnf_transformation,[status(thm)],[f11])).
% 1.66/1.38 % SZS output end Proof for DTF2THF_1034
% 1.66/1.38 % (1270)------------------------------
% 1.66/1.38 % (1270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.38 % (1270)Termination reason: Refutation
% 1.66/1.38
% 1.66/1.38 % (1270)Memory used [KB]: 5756
% 1.66/1.38 % (1270)Time elapsed: 0.027 s
% 1.66/1.38 % (1270)Instructions burned: 39 (million)
% 1.66/1.38 % (1270)------------------------------
% 1.66/1.38 % (1270)------------------------------
% 1.66/1.38 % (1269)Success in time 0.039 s
%------------------------------------------------------------------------------