%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU787^2 : TPTP v9.2.1. Released v3.7.0.
% 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 09:37:12 AM UTC 2026
% Result : Theorem 1.53s 1.25s
% Output : Refutation 1.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 32
% Number of leaves : 13
% Syntax : Number of formulae : 73 ( 21 unt; 0 typ; 6 def)
% Number of atoms : 1024 ( 230 equ; 0 cnn)
% Maximal formula atoms : 20 ( 14 avg)
% Number of connectives : 1412 ( 62 ~; 128 |; 72 &; 985 @)
% ( 6 <=>; 118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 17 usr; 16 con; 0-2 aty)
% ( 41 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 176 ( 46 ^; 92 !; 38 ?; 176 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
binunion: $i > $i > $i ).
thf(func_def_2,type,
kpair: $i > $i > $i ).
thf(func_def_3,type,
breln1: $i > $i > $o ).
thf(f102,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f52,f53,f58,f82,f88,f95,f101]) ).
thf(f101,plain,
~ spl6_4,
inference(avatar_contradiction_clause,[status(thm)],[f100]) ).
thf(f100,plain,
( $false
| ~ spl6_4 ),
inference(trivial_inequality_removal,[status(thm)],[f96]) ).
thf(f96,plain,
( ( $true = $false )
| ~ spl6_4 ),
inference(superposition,[status(thm)],[f72,f21]) ).
thf(f21,plain,
( ( in @ sK4 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( ( breln1unionE = $true )
& ( ( in @ ( kpair @ sK3 @ sK4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ( ( sK5 = $true )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true ) )
& ( sK5 != $true )
& ( ( sK5 = $true )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true ) )
& ( ( in @ sK4 @ sK1 )
= $true )
& ( ( in @ sK3 @ sK1 )
= $true )
& ( ( breln1 @ sK1 @ sK2 )
= $true )
& ( ( breln1 @ sK1 @ sK0 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4,sK5])],[f10,f15,f14,f13,f12,f11]) ).
thf(f11,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true ) ) )
& ( ( in @ X4 @ X1 )
= $true ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK0 @ X2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
& ( ( breln1 @ sK1 @ X2 )
= $true ) )
& ( ( breln1 @ sK1 @ sK0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f12,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK0 @ X2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
& ( ( breln1 @ sK1 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
& ( ( breln1 @ sK1 @ sK2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f13,plain,
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
=> ( ? [X4: $i] :
( ( ( in @ ( kpair @ sK3 @ X4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ X4 ) @ sK2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
& ( ( in @ sK3 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f14,plain,
( ? [X4: $i] :
( ( ( in @ ( kpair @ sK3 @ X4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ X4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ X4 ) @ sK2 )
!= $true ) ) )
& ( $true
= ( in @ X4 @ sK1 ) ) )
=> ( ( ( in @ ( kpair @ sK3 @ sK4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true ) ) )
& ( ( in @ sK4 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f15,plain,
( ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true ) ) )
=> ( ( ( sK5 = $true )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true ) )
& ( sK5 != $true )
& ( ( sK5 = $true )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f10,plain,
( ( breln1unionE = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
= $true )
& ? [X5: $o] :
( ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true ) ) )
& ( ( in @ X4 @ X1 )
= $true ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) ) ),
inference(flattening,[status(thm)],[f9]) ).
thf(f9,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ? [X5: $o] :
( ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true ) )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true ) ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
= $true )
& ( ( in @ X4 @ X1 )
= $true ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
& ( breln1unionE = $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
~ ( ( breln1unionE = $true )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X1 @ X0 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ! [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
= $true )
=> ! [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
=> ( $true = X5 ) )
=> ( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
=> ( $true = X5 ) )
=> ( $true = X5 ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ( breln1unionE
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X1 @ X0 )
=> ! [X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
=> ! [X5: $o] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
=> X5 )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> X5 )
=> X5 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ( breln1unionE
=> ! [X1: $i,X0: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
=> ! [X5: $o] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
=> X5 )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> X5 )
=> X5 ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( breln1unionE
=> ! [X1: $i,X0: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
=> ! [X5: $o] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
=> X5 )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> X5 )
=> X5 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.M5IsfpftDM/DTF2THF_22206.p',breln1unionEcases) ).
thf(f72,plain,
( ( ( in @ sK4 @ sK1 )
= $false )
| ~ spl6_4 ),
inference(avatar_component_clause,[status(thm)],[f70]) ).
thf(f70,definition,
( spl6_4
<=> ( ( in @ sK4 @ sK1 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl6_4])],[avatar_definition]) ).
thf(f95,plain,
~ spl6_6,
inference(avatar_contradiction_clause,[status(thm)],[f94]) ).
thf(f94,plain,
( $false
| ~ spl6_6 ),
inference(trivial_inequality_removal,[status(thm)],[f90]) ).
thf(f90,plain,
( ( $true = $false )
| ~ spl6_6 ),
inference(superposition,[status(thm)],[f80,f20]) ).
thf(f20,plain,
( ( in @ sK3 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f80,plain,
( ( ( in @ sK3 @ sK1 )
= $false )
| ~ spl6_6 ),
inference(avatar_component_clause,[status(thm)],[f78]) ).
thf(f78,definition,
( spl6_6
<=> ( ( in @ sK3 @ sK1 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl6_6])],[avatar_definition]) ).
thf(f88,plain,
~ spl6_5,
inference(avatar_contradiction_clause,[status(thm)],[f87]) ).
thf(f87,plain,
( $false
| ~ spl6_5 ),
inference(trivial_inequality_removal,[status(thm)],[f83]) ).
thf(f83,plain,
( ( $true = $false )
| ~ spl6_5 ),
inference(superposition,[status(thm)],[f76,f18]) ).
thf(f18,plain,
( ( breln1 @ sK1 @ sK0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f76,plain,
( ( ( breln1 @ sK1 @ sK0 )
= $false )
| ~ spl6_5 ),
inference(avatar_component_clause,[status(thm)],[f74]) ).
thf(f74,definition,
( spl6_5
<=> ( ( breln1 @ sK1 @ sK0 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl6_5])],[avatar_definition]) ).
thf(f82,plain,
( spl6_5
| spl6_6
| spl6_4
| spl6_1
| spl6_3 ),
inference(avatar_split_clause,[status(thm)],[f67,f55,f45,f70,f78,f74]) ).
thf(f45,definition,
( spl6_1
<=> ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl6_1])],[avatar_definition]) ).
thf(f55,definition,
( spl6_3
<=> ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl6_3])],[avatar_definition]) ).
thf(f67,plain,
( ( ( in @ sK3 @ sK1 )
= $false )
| ( ( in @ sK4 @ sK1 )
= $false )
| ( ( breln1 @ sK1 @ sK0 )
= $false )
| spl6_1
| spl6_3 ),
inference(trivial_inequality_removal,[status(thm)],[f66]) ).
thf(f66,plain,
( ( $true = $false )
| ( ( in @ sK3 @ sK1 )
= $false )
| ( ( in @ sK4 @ sK1 )
= $false )
| ( ( breln1 @ sK1 @ sK0 )
= $false )
| spl6_1
| spl6_3 ),
inference(superposition,[status(thm)],[f19,f64]) ).
thf(f64,plain,
( ! [X0: $i] :
( ( $false
= ( breln1 @ X0 @ sK2 ) )
| ( ( in @ sK4 @ X0 )
= $false )
| ( ( breln1 @ X0 @ sK0 )
= $false )
| ( ( in @ sK3 @ X0 )
= $false ) )
| spl6_1
| spl6_3 ),
inference(subsumption_resolution,[status(thm)],[f63,f57]) ).
thf(f57,plain,
( ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true )
| spl6_3 ),
inference(avatar_component_clause,[status(thm)],[f55]) ).
thf(f63,plain,
( ! [X0: $i] :
( ( $false
= ( breln1 @ X0 @ sK2 ) )
| ( ( breln1 @ X0 @ sK0 )
= $false )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
= $true )
| ( ( in @ sK3 @ X0 )
= $false )
| ( ( in @ sK4 @ X0 )
= $false ) )
| spl6_1 ),
inference(subsumption_resolution,[status(thm)],[f62,f47]) ).
thf(f47,plain,
( ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true )
| spl6_1 ),
inference(avatar_component_clause,[status(thm)],[f45]) ).
thf(f62,plain,
! [X0: $i] :
( ( ( in @ sK3 @ X0 )
= $false )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
= $true )
| ( ( in @ sK4 @ X0 )
= $false )
| ( $false
= ( breln1 @ X0 @ sK2 ) )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
= $true )
| ( ( breln1 @ X0 @ sK0 )
= $false ) ),
inference(trivial_inequality_removal,[status(thm)],[f60]) ).
thf(f60,plain,
! [X0: $i] :
( ( ( in @ sK3 @ X0 )
= $false )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
= $true )
| ( ( breln1 @ X0 @ sK0 )
= $false )
| ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
= $true )
| ( $true = $false )
| ( ( in @ sK4 @ X0 )
= $false )
| ( $false
= ( breln1 @ X0 @ sK2 ) ) ),
inference(superposition,[status(thm)],[f25,f43]) ).
thf(f43,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i,X5: $i] :
( ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X1 @ X3 ) )
= $false )
| ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
= $true )
| ( ( in @ X5 @ X2 )
= $false )
| ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( in @ X4 @ X2 )
= $false )
| ( ( in @ ( kpair @ X4 @ X5 ) @ X1 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f42]) ).
thf(f42,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i,X5: $i] :
( ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
| ( in @ ( kpair @ X4 @ X5 ) @ X1 ) )
= $true )
| ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X1 @ X3 ) )
= $false )
| ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( in @ X4 @ X2 )
= $false )
| ( ( in @ X5 @ X2 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f41]) ).
thf(f41,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i,X5: $i] :
( ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( in @ X4 @ X2 )
= $false )
| ( ( in @ X5 @ X2 )
= $false )
| ( ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
| ( in @ ( kpair @ X4 @ X5 ) @ X1 ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f40]) ).
thf(f40,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i,X5: $i] :
( ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
| ( in @ ( kpair @ X4 @ X5 ) @ X1 ) ) ) )
= $true )
| ( ( in @ X4 @ X2 )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f39]) ).
thf(f39,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i,X5: $i] :
( ( ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ X3 )
| ( in @ ( kpair @ X4 @ Y0 ) @ X1 ) ) ) )
@ X5 )
= $true )
| ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( in @ X4 @ X2 )
= $false )
| ( ( breln1 @ X2 @ X3 )
= $false ) ),
inference(pi_clausification,[status(thm)],[f38]) ).
thf(f38,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ X3 )
| ( in @ ( kpair @ X4 @ Y0 ) @ X1 ) ) ) ) )
= $true )
| ( ( in @ X4 @ X2 )
= $false )
| ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( breln1 @ X2 @ X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f37]) ).
thf(f37,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( ( in @ X4 @ X2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ X4 @ Y0 ) @ X3 )
| ( in @ ( kpair @ X4 @ Y0 ) @ X1 ) ) ) ) ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f36]) ).
thf(f36,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ X3 )
| ( in @ ( kpair @ Y0 @ Y1 ) @ X1 ) ) ) ) ) )
@ X4 )
= $true )
| ( ( breln1 @ X2 @ X1 )
= $false ) ),
inference(pi_clausification,[status(thm)],[f35]) ).
thf(f35,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( breln1 @ X2 @ X3 )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ X3 )
| ( in @ ( kpair @ Y0 @ Y1 ) @ X1 ) ) ) ) ) ) )
= $true )
| ( ( breln1 @ X2 @ X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f34]) ).
thf(f34,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( ( breln1 @ X2 @ X3 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ ( binunion @ X1 @ X3 ) )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ X3 )
| ( in @ ( kpair @ Y0 @ Y1 ) @ X1 ) ) ) ) ) ) ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f33]) ).
thf(f33,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( ^ [Y0: $i] :
( ( breln1 @ X2 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ ( binunion @ X1 @ Y0 ) )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
| ( in @ ( kpair @ Y1 @ Y2 ) @ X1 ) ) ) ) ) ) ) )
@ X3 )
= $true ) ),
inference(pi_clausification,[status(thm)],[f32]) ).
thf(f32,plain,
! [X2: $i,X1: $i] :
( ( ( breln1 @ X2 @ X1 )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ( breln1 @ X2 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ ( binunion @ X1 @ Y0 ) )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
| ( in @ ( kpair @ Y1 @ Y2 ) @ X1 ) ) ) ) ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f31]) ).
thf(f31,plain,
! [X2: $i,X1: $i] :
( ( ( breln1 @ X2 @ X1 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( breln1 @ X2 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ ( binunion @ X1 @ Y0 ) )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
| ( in @ ( kpair @ Y1 @ Y2 ) @ X1 ) ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f30]) ).
thf(f30,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( ( breln1 @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ ( binunion @ X1 @ Y1 ) )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
| ( in @ ( kpair @ Y2 @ Y3 ) @ X1 ) ) ) ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f29]) ).
thf(f29,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( breln1 @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ ( binunion @ X1 @ Y1 ) )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
| ( in @ ( kpair @ Y2 @ Y3 ) @ X1 ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f28]) ).
thf(f28,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( breln1 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ ( binunion @ Y0 @ Y2 ) )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
| ( in @ ( kpair @ Y3 @ Y4 ) @ Y0 ) ) ) ) ) ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f27]) ).
thf(f27,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( breln1 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ ( binunion @ Y0 @ Y2 ) )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
| ( in @ ( kpair @ Y3 @ Y4 ) @ Y0 ) ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f26,f17]) ).
thf(f17,plain,
( breln1unionE
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( breln1 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ ( binunion @ Y0 @ Y2 ) )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
| ( in @ ( kpair @ Y3 @ Y4 ) @ Y0 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f6]) ).
thf(f6,plain,
( breln1unionE
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( breln1 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( breln1 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ ( binunion @ Y0 @ Y2 ) )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
| ( in @ ( kpair @ Y3 @ Y4 ) @ Y0 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
( breln1unionE
= ( ! [X0: $i,X1: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
| ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( breln1unionE
= ( ! [X0: $i,X1: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
| ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.M5IsfpftDM/DTF2THF_22206.p',breln1unionE) ).
thf(f26,plain,
breln1unionE = $true,
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f25,plain,
( ( in @ ( kpair @ sK3 @ sK4 ) @ ( binunion @ sK0 @ sK2 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f19,plain,
( ( breln1 @ sK1 @ sK2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f58,plain,
( ~ spl6_3
| spl6_2 ),
inference(avatar_split_clause,[status(thm)],[f24,f49,f55]) ).
thf(f49,definition,
( spl6_2
<=> ( sK5 = $true ) ),
introduced(definition,[new_symbols(naming,[spl6_2])],[avatar_definition]) ).
thf(f24,plain,
( ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK0 )
!= $true )
| ( sK5 = $true ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f53,plain,
~ spl6_2,
inference(avatar_split_clause,[status(thm)],[f23,f49]) ).
thf(f23,plain,
sK5 != $true,
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f52,plain,
( ~ spl6_1
| spl6_2 ),
inference(avatar_split_clause,[status(thm)],[f22,f49,f45]) ).
thf(f22,plain,
( ( ( in @ ( kpair @ sK3 @ sK4 ) @ sK2 )
!= $true )
| ( sK5 = $true ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU787^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n015.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:34:16 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : SEU787^2 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.45 % Domain : Set Theory
% 0.29/0.45 % Problem : Binary Relations on a Set
% 0.29/0.45 % Version : Especial > Reduced > Especial.
% 0.29/0.45 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S -> (! x:i.in x A ->
% 0.29/0.45 % (! y:i.in y A -> in (kpair x y) (binunion R S) ->
% 0.29/0.45 % (! phi:o.(in (kpair x y) R -> phi) ->
% 0.29/0.45 % (in (kpair x y) S -> phi) -> phi)))))
% 0.29/0.45
% 0.29/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.45 % Source : [Bro08]
% 0.29/0.45 % Names : ZFC289l [Bro08]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.29/0.45 % Syntax : Number of formulae : 7 ( 1 unt; 5 typ; 1 def)
% 0.29/0.45 % Number of atoms : 17 ( 1 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 8 ( 8 avg)
% 0.29/0.45 % Number of connectives : 60 ( 0 ~; 1 |; 0 &; 44 @)
% 0.29/0.45 % ( 0 <=>; 15 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 20 ( 11 avg)
% 0.29/0.45 % Number of types : 2 ( 0 usr)
% 0.29/0.45 % Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 6 ( 5 usr; 1 con; 0-2 aty)
% 0.29/0.45 % Number of variables : 11 ( 0 ^; 11 !; 0 ?; 11 :)
% 0.29/0.45 % SPC : TH0_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : http://mathgate.info/detsetitem.php?id=352
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 thf(in_type,type,
% 0.29/0.45 in: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(binunion_type,type,
% 0.29/0.45 binunion: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(kpair_type,type,
% 0.29/0.45 kpair: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(breln1_type,type,
% 0.29/0.45 breln1: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1unionE_type,type,
% 0.29/0.45 breln1unionE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(breln1unionE,definition,
% 0.29/0.45 ( breln1unionE
% 0.29/0.45 = ( ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [S: $i] :
% 0.29/0.45 ( ( breln1 @ A @ S )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(breln1unionEcases,conjecture,
% 0.29/0.45 ( breln1unionE
% 0.29/0.45 => ! [A: $i,R: $i] :
% 0.29/0.45 ( ( breln1 @ A @ R )
% 0.29/0.45 => ! [S: $i] :
% 0.29/0.45 ( ( breln1 @ A @ S )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => Xphi ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 ------------------------
% 1.53/1.21 ---- Embedded in THF ---
% 1.53/1.22 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.53/1.22 %%% Generated on Tue Feb 24 12:34:17 EST 2026
% 1.53/1.22 %%% using '$$dhol' embedding, version 1.3.0.
% 1.53/1.22 %%% Logic specification used:
% 1.53/1.22 %%% thf(spec, logic, $$dhol).
% 1.53/1.22
% 1.53/1.22 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.M5IsfpftDM/DTF2DTF22206.p
% See solution above
% 1.53/1.22 ------------------------
% 1.53/1.23 ---- Cleaned THF ---
% 1.53/1.23 thf(in_type,type,
% 1.53/1.23 in: $i > $i > $o ).
% 1.53/1.23
% 1.53/1.23 thf(binunion_type,type,
% 1.53/1.23 binunion: $i > $i > $i ).
% 1.53/1.23
% 1.53/1.23 thf(kpair_type,type,
% 1.53/1.23 kpair: $i > $i > $i ).
% 1.53/1.23
% 1.53/1.23 thf(breln1_type,type,
% 1.53/1.23 breln1: $i > $i > $o ).
% 1.53/1.23
% 1.53/1.23 thf(breln1unionE_type,type,
% 1.53/1.23 breln1unionE: $o ).
% 1.53/1.23
% 1.53/1.23 thf(breln1unionE,definition,
% 1.53/1.23 ( breln1unionE
% 1.53/1.23 = ( ! [A: $i,R: $i] :
% 1.53/1.23 ( ( breln1 @ A @ R )
% 1.53/1.23 => ! [S: $i] :
% 1.53/1.23 ( ( breln1 @ A @ S )
% 1.53/1.23 => ! [Xx: $i] :
% 1.53/1.23 ( ( in @ Xx @ A )
% 1.53/1.23 => ! [Xy: $i] :
% 1.53/1.23 ( ( in @ Xy @ A )
% 1.53/1.23 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 1.53/1.23 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 1.53/1.23 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 1.53/1.23
% 1.53/1.23 thf(breln1unionEcases,conjecture,
% 1.53/1.23 ( breln1unionE
% 1.53/1.23 => ! [A: $i,R: $i] :
% 1.53/1.23 ( ( breln1 @ A @ R )
% 1.53/1.23 => ! [S: $i] :
% 1.53/1.23 ( ( breln1 @ A @ S )
% 1.53/1.23 => ! [Xx: $i] :
% 1.53/1.23 ( ( in @ Xx @ A )
% 1.53/1.23 => ! [Xy: $i] :
% 1.53/1.23 ( ( in @ Xy @ A )
% 1.53/1.23 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 1.53/1.23 => ! [Xphi: $o] :
% 1.53/1.23 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 1.53/1.23 => Xphi )
% 1.53/1.23 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 1.53/1.23 => Xphi )
% 1.53/1.23 => Xphi ) ) ) ) ) ) ) ) ).
% 1.53/1.23 ------------------------
% 1.53/1.24 % (22493)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22206 for (2999ds/2Mi)
% 1.53/1.24 % (22490)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22206 for (2999ds/183Mi)
% 1.53/1.24 % (22493)Instruction limit reached!
% 1.53/1.24 % (22493)------------------------------
% 1.53/1.24 % (22493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.24 % (22493)Termination reason: Unknown
% 1.53/1.24 % (22493)Termination phase: Preprocessing 3
% 1.53/1.24
% 1.53/1.24 % (22493)Memory used [KB]: 1023
% 1.53/1.24 % (22493)Time elapsed: 0.002 s
% 1.53/1.24 % (22493)Instructions burned: 3 (million)
% 1.53/1.24 % (22493)------------------------------
% 1.53/1.24 % (22493)------------------------------
% 1.53/1.25 % (22490)First to succeed.
% 1.53/1.25 % (22490)Refutation found. Thanks to Tanya!
% 1.53/1.25 % SZS status Theorem for DTF2THF_22206
% 1.53/1.25 % SZS output start Proof for DTF2THF_22206
% 1.53/1.25 thf(func_def_0, type, in: $i > $i > $o).
% 1.53/1.25 thf(func_def_1, type, binunion: $i > $i > $i).
% 1.53/1.25 thf(func_def_2, type, kpair: $i > $i > $i).
% 1.53/1.25 thf(func_def_3, type, breln1: $i > $i > $o).
% 1.53/1.25 thf(f102,plain,(
% 1.53/1.25 $false),
% 1.53/1.25 inference(avatar_sat_refutation,[status(thm)],[f52,f53,f58,f82,f88,f95,f101])).
% 1.53/1.25 thf(f101,plain,(
% 1.53/1.25 ~spl6_4),
% 1.53/1.25 inference(avatar_contradiction_clause,[status(thm)],[f100])).
% 1.53/1.25 thf(f100,plain,(
% 1.53/1.25 $false | ~spl6_4),
% 1.53/1.25 inference(trivial_inequality_removal,[status(thm)],[f96])).
% 1.53/1.25 thf(f96,plain,(
% 1.53/1.25 ($true = $false) | ~spl6_4),
% 1.53/1.25 inference(superposition,[status(thm)],[f72,f21])).
% 1.53/1.25 thf(f21,plain,(
% 1.53/1.25 ((in @ sK4 @ sK1) = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f16,plain,(
% 1.53/1.25 (breln1unionE = $true) & ((((((in @ (kpair @ sK3 @ sK4) @ (binunion @ sK0 @ sK2)) = $true) & (((sK5 = $true) | ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true)) & (sK5 != $true) & ((sK5 = $true) | ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true))) & ((in @ sK4 @ sK1) = $true)) & ((in @ sK3 @ sK1) = $true)) & ((breln1 @ sK1 @ sK2) = $true)) & ((breln1 @ sK1 @ sK0) = $true))),
% 1.53/1.25 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4,sK5])],[f10,f15,f14,f13,f12,f11])).
% 1.53/1.25 thf(f11,plain,(
% 1.53/1.25 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ X0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X2) != $true))) & ((in @ X4 @ X1) = $true)) & ((in @ X3 @ X1) = $true)) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) => (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK0 @ X2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X2) != $true))) & ($true = (in @ X4 @ sK1))) & ((in @ X3 @ sK1) = $true)) & ((breln1 @ sK1 @ X2) = $true)) & ((breln1 @ sK1 @ sK0) = $true))),
% 1.53/1.25 introduced(definition,[],[choice_axiom])).
% 1.53/1.25 thf(f12,plain,(
% 1.53/1.25 ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK0 @ X2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X2) != $true))) & ($true = (in @ X4 @ sK1))) & ((in @ X3 @ sK1) = $true)) & ((breln1 @ sK1 @ X2) = $true)) => (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK0 @ sK2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK2) != $true))) & ($true = (in @ X4 @ sK1))) & ((in @ X3 @ sK1) = $true)) & ((breln1 @ sK1 @ sK2) = $true))),
% 1.53/1.25 introduced(definition,[],[choice_axiom])).
% 1.53/1.25 thf(f13,plain,(
% 1.53/1.25 ? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK0 @ sK2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK2) != $true))) & ($true = (in @ X4 @ sK1))) & ((in @ X3 @ sK1) = $true)) => (? [X4 : $i] : (((in @ (kpair @ sK3 @ X4) @ (binunion @ sK0 @ sK2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ sK3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK3 @ X4) @ sK2) != $true))) & ($true = (in @ X4 @ sK1))) & ((in @ sK3 @ sK1) = $true))),
% 1.53/1.25 introduced(definition,[],[choice_axiom])).
% 1.53/1.25 thf(f14,plain,(
% 1.53/1.25 ? [X4 : $i] : (((in @ (kpair @ sK3 @ X4) @ (binunion @ sK0 @ sK2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ sK3 @ X4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK3 @ X4) @ sK2) != $true))) & ($true = (in @ X4 @ sK1))) => (((in @ (kpair @ sK3 @ sK4) @ (binunion @ sK0 @ sK2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true))) & ((in @ sK4 @ sK1) = $true))),
% 1.53/1.25 introduced(definition,[],[choice_axiom])).
% 1.53/1.25 thf(f15,plain,(
% 1.53/1.25 ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true))) => (((sK5 = $true) | ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true)) & (sK5 != $true) & ((sK5 = $true) | ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true)))),
% 1.53/1.25 introduced(definition,[],[choice_axiom])).
% 1.53/1.25 thf(f10,plain,(
% 1.53/1.25 (breln1unionE = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) = $true) & ? [X5 : $o] : ((($true = X5) | ((in @ (kpair @ X3 @ X4) @ X0) != $true)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X2) != $true))) & ((in @ X4 @ X1) = $true)) & ((in @ X3 @ X1) = $true)) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true))),
% 1.53/1.25 inference(flattening,[status(thm)],[f9])).
% 1.53/1.25 thf(f9,plain,(
% 1.53/1.25 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((? [X5 : $o] : ((($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X2) != $true))) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ X0) != $true))) & ((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) = $true)) & ((in @ X4 @ X1) = $true)) & ((in @ X3 @ X1) = $true)) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) & (breln1unionE = $true)),
% 1.53/1.25 inference(ennf_transformation,[status(thm)],[f8])).
% 1.53/1.25 thf(f8,plain,(
% 1.53/1.25 ~((breln1unionE = $true) => ! [X0 : $i,X1 : $i] : (((breln1 @ X1 @ X0) = $true) => ! [X2 : $i] : (((breln1 @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ! [X4 : $i] : (((in @ X4 @ X1) = $true) => (((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) = $true) => ! [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ X0) = $true) => ($true = X5)) => ((((in @ (kpair @ X3 @ X4) @ X2) = $true) => ($true = X5)) => ($true = X5)))))))))),
% 1.53/1.25 inference(fool_elimination,[status(thm)],[f7])).
% 1.53/1.25 thf(f7,plain,(
% 1.53/1.25 ~(breln1unionE => ! [X0 : $i,X1 : $i] : ((breln1 @ X1 @ X0) => ! [X2 : $i] : ((breln1 @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => ! [X4 : $i] : ((in @ X4 @ X1) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) => ! [X5 : $o] : (((in @ (kpair @ X3 @ X4) @ X0) => X5) => (((in @ (kpair @ X3 @ X4) @ X2) => X5) => X5))))))))),
% 1.53/1.25 inference(rectify,[status(thm)],[f3])).
% 1.53/1.25 thf(f3,negated_conjecture,(
% 1.53/1.25 ~(breln1unionE => ! [X1 : $i,X0 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) => ! [X5 : $o] : (((in @ (kpair @ X3 @ X4) @ X1) => X5) => (((in @ (kpair @ X3 @ X4) @ X2) => X5) => X5))))))))),
% 1.53/1.25 inference(negated_conjecture,[status(cth)],[f2])).
% 1.53/1.25 thf(f2,conjecture,(
% 1.53/1.25 breln1unionE => ! [X1 : $i,X0 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) => ! [X5 : $o] : (((in @ (kpair @ X3 @ X4) @ X1) => X5) => (((in @ (kpair @ X3 @ X4) @ X2) => X5) => X5)))))))),
% 1.53/1.25 file('/export/starexec/sandbox/tmp/tmp.M5IsfpftDM/DTF2THF_22206.p',breln1unionEcases)).
% 1.53/1.25 thf(f72,plain,(
% 1.53/1.25 ((in @ sK4 @ sK1) = $false) | ~spl6_4),
% 1.53/1.25 inference(avatar_component_clause,[status(thm)],[f70])).
% 1.53/1.25 thf(f70,definition,(
% 1.53/1.25 spl6_4 <=> ((in @ sK4 @ sK1) = $false)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_4])],[avatar_definition])).
% 1.53/1.25 thf(f95,plain,(
% 1.53/1.25 ~spl6_6),
% 1.53/1.25 inference(avatar_contradiction_clause,[status(thm)],[f94])).
% 1.53/1.25 thf(f94,plain,(
% 1.53/1.25 $false | ~spl6_6),
% 1.53/1.25 inference(trivial_inequality_removal,[status(thm)],[f90])).
% 1.53/1.25 thf(f90,plain,(
% 1.53/1.25 ($true = $false) | ~spl6_6),
% 1.53/1.25 inference(superposition,[status(thm)],[f80,f20])).
% 1.53/1.25 thf(f20,plain,(
% 1.53/1.25 ((in @ sK3 @ sK1) = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f80,plain,(
% 1.53/1.25 ((in @ sK3 @ sK1) = $false) | ~spl6_6),
% 1.53/1.25 inference(avatar_component_clause,[status(thm)],[f78])).
% 1.53/1.25 thf(f78,definition,(
% 1.53/1.25 spl6_6 <=> ((in @ sK3 @ sK1) = $false)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_6])],[avatar_definition])).
% 1.53/1.25 thf(f88,plain,(
% 1.53/1.25 ~spl6_5),
% 1.53/1.25 inference(avatar_contradiction_clause,[status(thm)],[f87])).
% 1.53/1.25 thf(f87,plain,(
% 1.53/1.25 $false | ~spl6_5),
% 1.53/1.25 inference(trivial_inequality_removal,[status(thm)],[f83])).
% 1.53/1.25 thf(f83,plain,(
% 1.53/1.25 ($true = $false) | ~spl6_5),
% 1.53/1.25 inference(superposition,[status(thm)],[f76,f18])).
% 1.53/1.25 thf(f18,plain,(
% 1.53/1.25 ((breln1 @ sK1 @ sK0) = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f76,plain,(
% 1.53/1.25 ((breln1 @ sK1 @ sK0) = $false) | ~spl6_5),
% 1.53/1.25 inference(avatar_component_clause,[status(thm)],[f74])).
% 1.53/1.25 thf(f74,definition,(
% 1.53/1.25 spl6_5 <=> ((breln1 @ sK1 @ sK0) = $false)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_5])],[avatar_definition])).
% 1.53/1.25 thf(f82,plain,(
% 1.53/1.25 spl6_5 | spl6_6 | spl6_4 | spl6_1 | spl6_3),
% 1.53/1.25 inference(avatar_split_clause,[status(thm)],[f67,f55,f45,f70,f78,f74])).
% 1.53/1.25 thf(f45,definition,(
% 1.53/1.25 spl6_1 <=> ((in @ (kpair @ sK3 @ sK4) @ sK2) = $true)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_1])],[avatar_definition])).
% 1.53/1.25 thf(f55,definition,(
% 1.53/1.25 spl6_3 <=> ((in @ (kpair @ sK3 @ sK4) @ sK0) = $true)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_3])],[avatar_definition])).
% 1.53/1.25 thf(f67,plain,(
% 1.53/1.25 ((in @ sK3 @ sK1) = $false) | ((in @ sK4 @ sK1) = $false) | ((breln1 @ sK1 @ sK0) = $false) | (spl6_1 | spl6_3)),
% 1.53/1.25 inference(trivial_inequality_removal,[status(thm)],[f66])).
% 1.53/1.25 thf(f66,plain,(
% 1.53/1.25 ($true = $false) | ((in @ sK3 @ sK1) = $false) | ((in @ sK4 @ sK1) = $false) | ((breln1 @ sK1 @ sK0) = $false) | (spl6_1 | spl6_3)),
% 1.53/1.25 inference(superposition,[status(thm)],[f19,f64])).
% 1.53/1.25 thf(f64,plain,(
% 1.53/1.25 ( ! [X0 : $i] : (($false = (breln1 @ X0 @ sK2)) | ((in @ sK4 @ X0) = $false) | ((breln1 @ X0 @ sK0) = $false) | ((in @ sK3 @ X0) = $false)) ) | (spl6_1 | spl6_3)),
% 1.53/1.25 inference(subsumption_resolution,[status(thm)],[f63,f57])).
% 1.53/1.25 thf(f57,plain,(
% 1.53/1.25 ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true) | spl6_3),
% 1.53/1.25 inference(avatar_component_clause,[status(thm)],[f55])).
% 1.53/1.25 thf(f63,plain,(
% 1.53/1.25 ( ! [X0 : $i] : (($false = (breln1 @ X0 @ sK2)) | ((breln1 @ X0 @ sK0) = $false) | ((in @ (kpair @ sK3 @ sK4) @ sK0) = $true) | ((in @ sK3 @ X0) = $false) | ((in @ sK4 @ X0) = $false)) ) | spl6_1),
% 1.53/1.25 inference(subsumption_resolution,[status(thm)],[f62,f47])).
% 1.53/1.25 thf(f47,plain,(
% 1.53/1.25 ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true) | spl6_1),
% 1.53/1.25 inference(avatar_component_clause,[status(thm)],[f45])).
% 1.53/1.25 thf(f62,plain,(
% 1.53/1.25 ( ! [X0 : $i] : (((in @ sK3 @ X0) = $false) | ((in @ (kpair @ sK3 @ sK4) @ sK2) = $true) | ((in @ sK4 @ X0) = $false) | ($false = (breln1 @ X0 @ sK2)) | ((in @ (kpair @ sK3 @ sK4) @ sK0) = $true) | ((breln1 @ X0 @ sK0) = $false)) )),
% 1.53/1.25 inference(trivial_inequality_removal,[status(thm)],[f60])).
% 1.53/1.25 thf(f60,plain,(
% 1.53/1.25 ( ! [X0 : $i] : (((in @ sK3 @ X0) = $false) | ((in @ (kpair @ sK3 @ sK4) @ sK0) = $true) | ((breln1 @ X0 @ sK0) = $false) | ((in @ (kpair @ sK3 @ sK4) @ sK2) = $true) | ($true = $false) | ((in @ sK4 @ X0) = $false) | ($false = (breln1 @ X0 @ sK2))) )),
% 1.53/1.25 inference(superposition,[status(thm)],[f25,f43])).
% 1.53/1.25 thf(f43,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i,X5 : $i] : (((in @ (kpair @ X4 @ X5) @ (binunion @ X1 @ X3)) = $false) | ((breln1 @ X2 @ X3) = $false) | ((in @ (kpair @ X4 @ X5) @ X3) = $true) | ((in @ X5 @ X2) = $false) | ((breln1 @ X2 @ X1) = $false) | ((in @ X4 @ X2) = $false) | ((in @ (kpair @ X4 @ X5) @ X1) = $true)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f42])).
% 1.53/1.25 thf(f42,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i,X5 : $i] : (((breln1 @ X2 @ X1) = $false) | (((in @ (kpair @ X4 @ X5) @ X3) | (in @ (kpair @ X4 @ X5) @ X1)) = $true) | ((in @ (kpair @ X4 @ X5) @ (binunion @ X1 @ X3)) = $false) | ((breln1 @ X2 @ X3) = $false) | ((in @ X4 @ X2) = $false) | ((in @ X5 @ X2) = $false)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f41])).
% 1.53/1.25 thf(f41,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i,X5 : $i] : (((breln1 @ X2 @ X3) = $false) | ((breln1 @ X2 @ X1) = $false) | ((in @ X4 @ X2) = $false) | ((in @ X5 @ X2) = $false) | (((in @ (kpair @ X4 @ X5) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ X4 @ X5) @ X3) | (in @ (kpair @ X4 @ X5) @ X1))) = $true)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f40])).
% 1.53/1.25 thf(f40,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i,X5 : $i] : (((breln1 @ X2 @ X1) = $false) | ((breln1 @ X2 @ X3) = $false) | (((in @ X5 @ X2) => ((in @ (kpair @ X4 @ X5) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ X4 @ X5) @ X3) | (in @ (kpair @ X4 @ X5) @ X1)))) = $true) | ((in @ X4 @ X2) = $false)) )),
% 1.53/1.25 inference(beta_eta_normalization,[status(thm)],[f39])).
% 1.53/1.25 thf(f39,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i,X5 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ X2) => ((in @ (kpair @ X4 @ Y0) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ X4 @ Y0) @ X3) | (in @ (kpair @ X4 @ Y0) @ X1))))) @ X5) = $true) | ((breln1 @ X2 @ X1) = $false) | ((in @ X4 @ X2) = $false) | ((breln1 @ X2 @ X3) = $false)) )),
% 1.53/1.25 inference(pi_clausification,[status(thm)],[f38])).
% 1.53/1.25 thf(f38,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => ((in @ (kpair @ X4 @ Y0) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ X4 @ Y0) @ X3) | (in @ (kpair @ X4 @ Y0) @ X1)))))) = $true) | ((in @ X4 @ X2) = $false) | ((breln1 @ X2 @ X3) = $false) | ((breln1 @ X2 @ X1) = $false)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f37])).
% 1.53/1.25 thf(f37,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((breln1 @ X2 @ X3) = $false) | ((breln1 @ X2 @ X1) = $false) | (((in @ X4 @ X2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => ((in @ (kpair @ X4 @ Y0) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ X4 @ Y0) @ X3) | (in @ (kpair @ X4 @ Y0) @ X1))))))) = $true)) )),
% 1.53/1.25 inference(beta_eta_normalization,[status(thm)],[f36])).
% 1.53/1.25 thf(f36,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((breln1 @ X2 @ X3) = $false) | (((^[Y0 : $i]: ((in @ Y0 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((in @ (kpair @ Y0 @ Y1) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ Y0 @ Y1) @ X3) | (in @ (kpair @ Y0 @ Y1) @ X1)))))))) @ X4) = $true) | ((breln1 @ X2 @ X1) = $false)) )),
% 1.53/1.25 inference(pi_clausification,[status(thm)],[f35])).
% 1.53/1.25 thf(f35,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((breln1 @ X2 @ X3) = $false) | ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((in @ (kpair @ Y0 @ Y1) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ Y0 @ Y1) @ X3) | (in @ (kpair @ Y0 @ Y1) @ X1))))))))) = $true) | ((breln1 @ X2 @ X1) = $false)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f34])).
% 1.53/1.25 thf(f34,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((breln1 @ X2 @ X1) = $false) | (((breln1 @ X2 @ X3) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((in @ (kpair @ Y0 @ Y1) @ (binunion @ X1 @ X3)) => ((in @ (kpair @ Y0 @ Y1) @ X3) | (in @ (kpair @ Y0 @ Y1) @ X1)))))))))) = $true)) )),
% 1.53/1.25 inference(beta_eta_normalization,[status(thm)],[f33])).
% 1.53/1.25 thf(f33,plain,(
% 1.53/1.25 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((breln1 @ X2 @ X1) = $false) | (((^[Y0 : $i]: ((breln1 @ X2 @ Y0) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ (binunion @ X1 @ Y0)) => ((in @ (kpair @ Y1 @ Y2) @ Y0) | (in @ (kpair @ Y1 @ Y2) @ X1))))))))))) @ X3) = $true)) )),
% 1.53/1.25 inference(pi_clausification,[status(thm)],[f32])).
% 1.53/1.25 thf(f32,plain,(
% 1.53/1.25 ( ! [X2 : $i,X1 : $i] : (((breln1 @ X2 @ X1) = $false) | ((!! @ $i @ (^[Y0 : $i]: ((breln1 @ X2 @ Y0) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ (binunion @ X1 @ Y0)) => ((in @ (kpair @ Y1 @ Y2) @ Y0) | (in @ (kpair @ Y1 @ Y2) @ X1)))))))))))) = $true)) )),
% 1.53/1.25 inference(binary_proxy_clausification,[status(thm)],[f31])).
% 1.53/1.25 thf(f31,plain,(
% 1.53/1.25 ( ! [X2 : $i,X1 : $i] : ((((breln1 @ X2 @ X1) => (!! @ $i @ (^[Y0 : $i]: ((breln1 @ X2 @ Y0) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ (binunion @ X1 @ Y0)) => ((in @ (kpair @ Y1 @ Y2) @ Y0) | (in @ (kpair @ Y1 @ Y2) @ X1))))))))))))) = $true)) )),
% 1.53/1.25 inference(beta_eta_normalization,[status(thm)],[f30])).
% 1.53/1.25 thf(f30,plain,(
% 1.53/1.25 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: ((breln1 @ Y0 @ X1) => (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y0 @ Y1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((in @ (kpair @ Y2 @ Y3) @ (binunion @ X1 @ Y1)) => ((in @ (kpair @ Y2 @ Y3) @ Y1) | (in @ (kpair @ Y2 @ Y3) @ X1)))))))))))))) @ X2) = $true)) )),
% 1.53/1.25 inference(pi_clausification,[status(thm)],[f29])).
% 1.53/1.25 thf(f29,plain,(
% 1.53/1.25 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((breln1 @ Y0 @ X1) => (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y0 @ Y1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((in @ (kpair @ Y2 @ Y3) @ (binunion @ X1 @ Y1)) => ((in @ (kpair @ Y2 @ Y3) @ Y1) | (in @ (kpair @ Y2 @ Y3) @ X1))))))))))))))) = $true)) )),
% 1.53/1.25 inference(beta_eta_normalization,[status(thm)],[f28])).
% 1.53/1.25 thf(f28,plain,(
% 1.53/1.25 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((breln1 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ (binunion @ Y0 @ Y2)) => ((in @ (kpair @ Y3 @ Y4) @ Y2) | (in @ (kpair @ Y3 @ Y4) @ Y0)))))))))))))))) @ X1) = $true)) )),
% 1.53/1.25 inference(pi_clausification,[status(thm)],[f27])).
% 1.53/1.25 thf(f27,plain,(
% 1.53/1.25 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((breln1 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ (binunion @ Y0 @ Y2)) => ((in @ (kpair @ Y3 @ Y4) @ Y2) | (in @ (kpair @ Y3 @ Y4) @ Y0))))))))))))))))) = $true)),
% 1.53/1.25 inference(definition_unfolding,[status(thm)],[f26,f17])).
% 1.53/1.25 thf(f17,plain,(
% 1.53/1.25 (breln1unionE = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((breln1 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ (binunion @ Y0 @ Y2)) => ((in @ (kpair @ Y3 @ Y4) @ Y2) | (in @ (kpair @ Y3 @ Y4) @ Y0))))))))))))))))))),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f6])).
% 1.53/1.25 thf(f6,plain,(
% 1.53/1.25 (breln1unionE = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((breln1 @ Y1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((breln1 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ (binunion @ Y0 @ Y2)) => ((in @ (kpair @ Y3 @ Y4) @ Y2) | (in @ (kpair @ Y3 @ Y4) @ Y0))))))))))))))))))),
% 1.53/1.25 inference(fool_elimination,[status(thm)],[f5])).
% 1.53/1.25 thf(f5,plain,(
% 1.53/1.25 (breln1unionE = ! [X0 : $i,X1 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) => ((in @ (kpair @ X3 @ X4) @ X1) | (in @ (kpair @ X3 @ X4) @ X2))))))))),
% 1.53/1.25 inference(rectify,[status(thm)],[f1])).
% 1.53/1.25 thf(f1,axiom,(
% 1.53/1.25 (breln1unionE = ! [X0 : $i,X1 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) => ((in @ (kpair @ X3 @ X4) @ X1) | (in @ (kpair @ X3 @ X4) @ X2))))))))),
% 1.53/1.25 file('/export/starexec/sandbox/tmp/tmp.M5IsfpftDM/DTF2THF_22206.p',breln1unionE)).
% 1.53/1.25 thf(f26,plain,(
% 1.53/1.25 (breln1unionE = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f25,plain,(
% 1.53/1.25 ((in @ (kpair @ sK3 @ sK4) @ (binunion @ sK0 @ sK2)) = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f19,plain,(
% 1.53/1.25 ((breln1 @ sK1 @ sK2) = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f58,plain,(
% 1.53/1.25 ~spl6_3 | spl6_2),
% 1.53/1.25 inference(avatar_split_clause,[status(thm)],[f24,f49,f55])).
% 1.53/1.25 thf(f49,definition,(
% 1.53/1.25 spl6_2 <=> (sK5 = $true)),
% 1.53/1.25 introduced(definition,[new_symbols(naming,[spl6_2])],[avatar_definition])).
% 1.53/1.25 thf(f24,plain,(
% 1.53/1.25 ((in @ (kpair @ sK3 @ sK4) @ sK0) != $true) | (sK5 = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f53,plain,(
% 1.53/1.25 ~spl6_2),
% 1.53/1.25 inference(avatar_split_clause,[status(thm)],[f23,f49])).
% 1.53/1.25 thf(f23,plain,(
% 1.53/1.25 (sK5 != $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 thf(f52,plain,(
% 1.53/1.25 ~spl6_1 | spl6_2),
% 1.53/1.25 inference(avatar_split_clause,[status(thm)],[f22,f49,f45])).
% 1.53/1.25 thf(f22,plain,(
% 1.53/1.25 ((in @ (kpair @ sK3 @ sK4) @ sK2) != $true) | (sK5 = $true)),
% 1.53/1.25 inference(cnf_transformation,[status(thm)],[f16])).
% 1.53/1.25 % SZS output end Proof for DTF2THF_22206
% 1.53/1.25 % (22490)------------------------------
% 1.53/1.25 % (22490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.25 % (22490)Termination reason: Refutation
% 1.53/1.25
% 1.53/1.25 % (22490)Memory used [KB]: 5628
% 1.53/1.25 % (22490)Time elapsed: 0.007 s
% 1.53/1.25 % (22490)Instructions burned: 10 (million)
% 1.53/1.25 % (22490)------------------------------
% 1.53/1.25 % (22490)------------------------------
% 1.53/1.25 % (22489)Success in time 0.017 s
%------------------------------------------------------------------------------