%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU691^2 : TPTP v9.2.1. Released v3.7.0.
% 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 09:36:26 AM UTC 2026
% Result : Theorem 1.77s 1.55s
% Output : Refutation 1.77s
% Verified :
% SZS Type : Refutation
% Derivation depth : 52
% Number of leaves : 10
% Syntax : Number of formulae : 189 ( 119 unt; 0 typ; 2 def)
% Number of atoms : 2901 ( 381 equ; 0 cnn)
% Maximal formula atoms : 8 ( 15 avg)
% Number of connectives : 6380 ( 75 ~; 65 |; 140 &;4878 @)
% ( 2 <=>; 693 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 2 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 26 usr; 18 con; 0-4 aty)
% ( 457 !!; 70 ??; 0 @@+; 0 @@-)
% Number of variables : 915 ( 809 ^; 104 !; 2 ?; 915 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_4,type,
subset: $i > $i > $o ).
thf(func_def_5,type,
kpair: $i > $i > $i ).
thf(func_def_6,type,
cartprod: $i > $i > $i ).
thf(func_def_7,type,
singleton: $i > $o ).
thf(func_def_9,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_10,type,
breln: $i > $i > $i > $o ).
thf(func_def_11,type,
func: $i > $i > $i > $o ).
thf(func_def_12,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_33,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f415,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f205,f313,f414]) ).
thf(f414,plain,
~ spl0_1,
inference(avatar_contradiction_clause,[status(thm)],[f413]) ).
thf(f413,plain,
( $false
| ~ spl0_1 ),
inference(trivial_inequality_removal,[status(thm)],[f412]) ).
thf(f412,plain,
( ( $false = $true )
| ~ spl0_1 ),
inference(forward_demodulation,[status(thm)],[f406,f361]) ).
thf(f361,plain,
( ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 )
= $false )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f360]) ).
thf(f360,plain,
( ( ( $true
=> ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 ) )
= $false )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f357,f359]) ).
thf(f359,plain,
( ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
= $true )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f357]) ).
thf(f357,plain,
( ( $false
= ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 ) ) )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f356]) ).
thf(f356,plain,
( ( $false
= ( $true
=> ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 ) ) ) )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f351,f355]) ).
thf(f355,plain,
( ( $true
= ( in @ sK9 @ sK4 ) )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f351]) ).
thf(f351,plain,
( ( $false
= ( ( in @ sK9 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 ) ) ) )
| ~ spl0_1 ),
inference(beta_eta_normalization,[status(thm)],[f350]) ).
thf(f350,plain,
( ( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ Y0 ) @ sK5 ) ) )
@ sK9 )
= $false )
| ~ spl0_1 ),
inference(sigma_clausification,[status(thm)],[f328]) ).
thf(f328,plain,
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ Y0 ) @ sK5 ) ) ) ) )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f327]) ).
thf(f327,plain,
( ( $false
= ( ( in @ sK8 @ sK2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( in @ ( kpair @ sK8 @ Y0 ) @ sK5 ) ) ) ) ) )
| ~ spl0_1 ),
inference(beta_eta_normalization,[status(thm)],[f319]) ).
thf(f319,plain,
( ( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) )
@ sK8 )
= $false )
| ~ spl0_1 ),
inference(sigma_clausification,[status(thm)],[f201]) ).
thf(f201,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
= $false )
| ~ spl0_1 ),
inference(avatar_component_clause,[status(thm)],[f200]) ).
thf(f200,definition,
( spl0_1
<=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f406,plain,
( ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK5 )
= $true )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f366,f400]) ).
thf(f400,plain,
( ( sK9
= ( ap @ sK2 @ sK4 @ sK3 @ sK8 ) )
| ~ spl0_1 ),
inference(equality_proxy_clausification,[status(thm)],[f392]) ).
thf(f392,plain,
( ( ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= sK9 )
= $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f391]) ).
thf(f391,plain,
( ( ( $true
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= sK9 ) )
= $true )
| ~ spl0_1 ),
inference(forward_demodulation,[status(thm)],[f384,f359]) ).
thf(f384,plain,
( ( ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= sK9 ) )
= $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f382]) ).
thf(f382,plain,
( ( ( $true
=> ( ( in @ ( kpair @ sK8 @ sK9 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= sK9 ) ) )
= $true )
| ~ spl0_1 ),
inference(superposition,[status(thm)],[f377,f355]) ).
thf(f377,plain,
( ! [X1: $i] :
( ( ( in @ X1 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ X1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= X1 ) ) )
= $true )
| ~ spl0_1 ),
inference(beta_eta_normalization,[status(thm)],[f374]) ).
thf(f374,plain,
( ! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= Y0 ) ) )
@ X1 )
= $true )
| ~ spl0_1 ),
inference(pi_clausification,[status(thm)],[f342]) ).
thf(f342,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= Y0 ) ) ) )
= $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f338]) ).
thf(f338,plain,
( ( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK8 @ Y0 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK8 )
= Y0 ) ) ) ) )
= $true )
| ~ spl0_1 ),
inference(superposition,[status(thm)],[f162,f329]) ).
thf(f329,plain,
( ( ( in @ sK8 @ sK2 )
= $true )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f327]) ).
thf(f162,plain,
! [X1: $i] :
( ( ( in @ X1 @ sK2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ X1 @ Y0 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ X1 )
= Y0 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f159]) ).
thf(f159,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ Y0 )
= Y1 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f149]) ).
thf(f149,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ Y0 )
= Y1 ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f148]) ).
thf(f148,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f147,f83]) ).
thf(f83,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f82]) ).
thf(f82,plain,
( ( $true
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f73,f79]) ).
thf(f79,plain,
( ( subset @ sK3 @ ( cartprod @ sK2 @ sK4 ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f73]) ).
thf(f73,plain,
( ( ( subset @ sK3 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f69]) ).
thf(f69,plain,
( ( ( ( subset @ sK3 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) )
& ( in @ Y2
@ ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ Y0 @ Y1 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y1 ) ) ) )
=> ( sK3 = Y0 ) ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f68]) ).
thf(f68,plain,
( $false
= ( ^ [Y0: $i] :
( ( ( subset @ sK3 @ ( cartprod @ sK2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ sK3 ) ) )
& ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ sK3 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y1 @ ( cartprod @ sK2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ( ap @ sK2 @ Y0 @ Y1 @ Y2 )
= ( ap @ sK2 @ Y0 @ sK3 @ Y2 ) ) ) )
=> ( sK3 = Y1 ) ) ) ) )
@ sK4 ) ),
inference(sigma_clausification,[status(thm)],[f66]) ).
thf(f66,plain,
( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ( ( subset @ sK3 @ ( cartprod @ sK2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ sK3 ) ) )
& ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ sK3 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y1 @ ( cartprod @ sK2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ( ap @ sK2 @ Y0 @ Y1 @ Y2 )
= ( ap @ sK2 @ Y0 @ sK3 @ Y2 ) ) ) )
=> ( sK3 = Y1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f65]) ).
thf(f65,plain,
( $false
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y0 @ ( cartprod @ sK2 @ Y1 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ sK2 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ sK2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ sK2 )
=> ( ( ap @ sK2 @ Y1 @ Y2 @ Y3 )
= ( ap @ sK2 @ Y1 @ Y0 @ Y3 ) ) ) )
=> ( Y0 = Y2 ) ) ) ) ) )
@ sK3 ) ),
inference(sigma_clausification,[status(thm)],[f62]) ).
thf(f62,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y0 @ ( cartprod @ sK2 @ Y1 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ sK2 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ sK2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ sK2 )
=> ( ( ap @ sK2 @ Y1 @ Y2 @ Y3 )
= ( ap @ sK2 @ Y1 @ Y0 @ Y3 ) ) ) )
=> ( Y0 = Y2 ) ) ) ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f61]) ).
thf(f61,plain,
( $false
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) )
@ sK2 ) ),
inference(sigma_clausification,[status(thm)],[f58]) ).
thf(f58,plain,
( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f57]) ).
thf(f57,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) )
= $false ),
inference(backward_demodulation,[status(thm)],[f52,f56]) ).
thf(f56,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f52]) ).
thf(f52,plain,
( $false
= ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f51]) ).
thf(f51,plain,
( ( $true
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) )
= $false ),
inference(backward_demodulation,[status(thm)],[f48,f50]) ).
thf(f50,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f48]) ).
thf(f48,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) )
= $false ),
inference(boolean_simplification,[status(thm)],[f47]) ).
thf(f47,plain,
( ( $true
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) )
= $false ),
inference(backward_demodulation,[status(thm)],[f42,f46]) ).
thf(f46,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f42]) ).
thf(f42,plain,
( $false
= ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) ) ),
inference(not_proxy_clausification,[status(thm)],[f41]) ).
thf(f41,plain,
( ( ~ ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y2
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ( subset @ Y3 @ ( cartprod @ Y0 @ Y2 ) )
& ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ?? @ $i
@ ^ [Y5: $i] :
( ( ( setadjoin @ Y5 @ emptyset )
= ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( in @ Y5
@ ( dsetconstr @ Y2
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f40]) ).
thf(f40,plain,
( ( ~ ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] :
( ( ^ [Y6: $i,Y7: $i,Y8: $i] : ( subset @ Y8 @ ( cartprod @ Y6 @ Y7 ) )
@ Y3
@ Y4
@ Y5 )
& ( !! @ $i
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y3 )
=> ( ^ [Y7: $i,Y8: $i > $o] :
( ^ [Y9: $i] :
( ?? @ $i
@ ^ [Y10: $i] :
( ( ( setadjoin @ Y10 @ emptyset )
= Y9 )
& ( in @ Y10 @ Y9 ) ) )
@ ( dsetconstr @ Y7
@ ^ [Y9: $i] : ( Y8 @ Y9 ) ) )
@ Y4
@ ^ [Y7: $i] : ( in @ ( kpair @ Y6 @ Y7 ) @ Y5 ) ) ) ) )
@ Y2
@ Y0
@ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] :
( ( ^ [Y6: $i,Y7: $i,Y8: $i] : ( subset @ Y8 @ ( cartprod @ Y6 @ Y7 ) )
@ Y3
@ Y4
@ Y5 )
& ( !! @ $i
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y3 )
=> ( ^ [Y7: $i,Y8: $i > $o] :
( ^ [Y9: $i] :
( ?? @ $i
@ ^ [Y10: $i] :
( ( ( setadjoin @ Y10 @ emptyset )
= Y9 )
& ( in @ Y10 @ Y9 ) ) )
@ ( dsetconstr @ Y7
@ ^ [Y9: $i] : ( Y8 @ Y9 ) ) )
@ Y4
@ ^ [Y7: $i] : ( in @ ( kpair @ Y6 @ Y7 ) @ Y5 ) ) ) ) )
@ Y0
@ Y1
@ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] : ( subset @ Y5 @ ( cartprod @ Y3 @ Y4 ) )
@ Y2
@ Y1
@ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ^ [Y4: $i,Y5: $i,Y6: $i] : ( subset @ Y6 @ ( cartprod @ Y4 @ Y5 ) )
@ Y2
@ Y1
@ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] :
( ( ^ [Y6: $i,Y7: $i,Y8: $i] : ( subset @ Y8 @ ( cartprod @ Y6 @ Y7 ) )
@ Y3
@ Y4
@ Y5 )
& ( !! @ $i
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y3 )
=> ( ^ [Y7: $i,Y8: $i > $o] :
( ^ [Y9: $i] :
( ?? @ $i
@ ^ [Y10: $i] :
( ( ( setadjoin @ Y10 @ emptyset )
= Y9 )
& ( in @ Y10 @ Y9 ) ) )
@ ( dsetconstr @ Y7
@ ^ [Y9: $i] : ( Y8 @ Y9 ) ) )
@ Y4
@ ^ [Y7: $i] : ( in @ ( kpair @ Y6 @ Y7 ) @ Y5 ) ) ) ) )
@ Y0
@ Y2
@ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ^ [Y4: $i,Y5: $i,Y6: $i] :
( ( ^ [Y7: $i,Y8: $i,Y9: $i] : ( subset @ Y9 @ ( cartprod @ Y7 @ Y8 ) )
@ Y4
@ Y5
@ Y6 )
& ( !! @ $i
@ ^ [Y7: $i] :
( ( in @ Y7 @ Y4 )
=> ( ^ [Y8: $i,Y9: $i > $o] :
( ^ [Y10: $i] :
( ?? @ $i
@ ^ [Y11: $i] :
( ( ( setadjoin @ Y11 @ emptyset )
= Y10 )
& ( in @ Y11 @ Y10 ) ) )
@ ( dsetconstr @ Y8
@ ^ [Y10: $i] : ( Y9 @ Y10 ) ) )
@ Y5
@ ^ [Y8: $i] : ( in @ ( kpair @ Y7 @ Y8 ) @ Y6 ) ) ) ) )
@ Y0
@ Y2
@ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f31,f37,f38,f39,f36,f36]) ).
thf(f36,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] : ( subset @ Y5 @ ( cartprod @ Y3 @ Y4 ) )
@ Y0
@ Y1
@ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ^ [Y4: $i,Y5: $i > $o] :
( ^ [Y6: $i] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( ( setadjoin @ Y7 @ emptyset )
= Y6 )
& ( in @ Y7 @ Y6 ) ) )
@ ( dsetconstr @ Y4
@ ^ [Y6: $i] : ( Y5 @ Y6 ) ) )
@ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f33,f29,f35]) ).
thf(f35,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= Y2 )
& ( in @ Y3 @ Y2 ) ) )
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f28,f27]) ).
thf(f27,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= Y0 )
& ( in @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= Y0 )
& ( in @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ^ [X0: $i] :
? [X1: $i] :
( ( in @ X1 @ X0 )
& ( ( setadjoin @ X1 @ emptyset )
= X0 ) ) )
= singleton ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: $i] :
? [X1: $i] :
( ( in @ X1 @ X0 )
& ( ( setadjoin @ X1 @ emptyset )
= X0 ) ) )
= singleton ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',singleton) ).
thf(f28,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( singleton
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f26,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( singleton
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f25]) ).
thf(f25,plain,
( ex1
= ( ^ [X0: $i,X1: $i > $o] :
( singleton
@ ( dsetconstr @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ex1
= ( ^ [X0: $i,X2: $i > $o] :
( singleton
@ ( dsetconstr @ X0
@ ^ [X1: $i] : ( X2 @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',ex1) ).
thf(f29,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f21]) ).
thf(f21,plain,
( breln
= ( ^ [X0: $i,X1: $i,X2: $i] : ( subset @ X2 @ ( cartprod @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( breln
= ( ^ [X0: $i,X3: $i,X4: $i] : ( subset @ X4 @ ( cartprod @ X0 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',breln) ).
thf(f33,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( breln @ Y0 @ Y1 @ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ex1 @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( breln @ Y0 @ Y1 @ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ex1 @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f23]) ).
thf(f23,plain,
( func
= ( ^ [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ex1 @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) )
& ( breln @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( func
= ( ^ [X0: $i,X3: $i,X5: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ex1 @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X5 ) ) )
& ( breln @ X0 @ X3 @ X5 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',func) ).
thf(f39,plain,
( eqbreln
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] : ( subset @ Y5 @ ( cartprod @ Y3 @ Y4 ) )
@ Y2
@ Y1
@ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( ^ [Y4: $i,Y5: $i,Y6: $i] : ( subset @ Y6 @ ( cartprod @ Y4 @ Y5 ) )
@ Y2
@ Y1
@ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f32,f29,f29]) ).
thf(f32,plain,
( eqbreln
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( breln @ Y2 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( breln @ Y2 @ Y1 @ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f20,plain,
( eqbreln
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( breln @ Y2 @ Y1 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( breln @ Y2 @ Y1 @ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f19]) ).
thf(f19,plain,
( eqbreln
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( breln @ X0 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln @ X0 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ( ! [X6: $i] :
( ( in @ X6 @ X0 )
=> ! [X7: $i] :
( ( in @ X7 @ X1 )
=> ( ( in @ ( kpair @ X6 @ X7 ) @ X3 )
=> ( in @ ( kpair @ X6 @ X7 ) @ X2 ) ) ) )
=> ( X2 = X3 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
( eqbreln
= ( ! [X0: $i,X3: $i,X5: $i] :
( ( breln @ X0 @ X3 @ X5 )
=> ! [X8: $i] :
( ( breln @ X0 @ X3 @ X8 )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X5 )
=> ( in @ ( kpair @ X1 @ X6 ) @ X8 ) ) ) )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X8 )
=> ( in @ ( kpair @ X1 @ X6 ) @ X5 ) ) ) )
=> ( X5 = X8 ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',eqbreln) ).
thf(f38,plain,
( funcGraphProp2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] :
( ( ^ [Y6: $i,Y7: $i,Y8: $i] : ( subset @ Y8 @ ( cartprod @ Y6 @ Y7 ) )
@ Y3
@ Y4
@ Y5 )
& ( !! @ $i
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y3 )
=> ( ^ [Y7: $i,Y8: $i > $o] :
( ^ [Y9: $i] :
( ?? @ $i
@ ^ [Y10: $i] :
( ( ( setadjoin @ Y10 @ emptyset )
= Y9 )
& ( in @ Y10 @ Y9 ) ) )
@ ( dsetconstr @ Y7
@ ^ [Y9: $i] : ( Y8 @ Y9 ) ) )
@ Y4
@ ^ [Y7: $i] : ( in @ ( kpair @ Y6 @ Y7 ) @ Y5 ) ) ) ) )
@ Y0
@ Y1
@ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f34,f36]) ).
thf(f34,plain,
( funcGraphProp2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y0 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( funcGraphProp2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y0 @ Y1 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f17]) ).
thf(f17,plain,
( funcGraphProp2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
=> ( ( ap @ X2 @ X1 @ X0 @ X3 )
= X4 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( funcGraphProp2
= ( ! [X7: $i,X3: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ! [X6: $i] :
( ( in @ X6 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X6 ) @ X7 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= X6 ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcGraphProp2) ).
thf(f37,plain,
( funcGraphProp1
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] :
( ( ^ [Y6: $i,Y7: $i,Y8: $i] : ( subset @ Y8 @ ( cartprod @ Y6 @ Y7 ) )
@ Y3
@ Y4
@ Y5 )
& ( !! @ $i
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y3 )
=> ( ^ [Y7: $i,Y8: $i > $o] :
( ^ [Y9: $i] :
( ?? @ $i
@ ^ [Y10: $i] :
( ( ( setadjoin @ Y10 @ emptyset )
= Y9 )
& ( in @ Y10 @ Y9 ) ) )
@ ( dsetconstr @ Y7
@ ^ [Y9: $i] : ( Y8 @ Y9 ) ) )
@ Y4
@ ^ [Y7: $i] : ( in @ ( kpair @ Y6 @ Y7 ) @ Y5 ) ) ) ) )
@ Y2
@ Y0
@ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f30,f36]) ).
thf(f30,plain,
( funcGraphProp1
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y2 @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( funcGraphProp1
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y2 @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f15]) ).
thf(f15,plain,
( funcGraphProp1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X0 @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( kpair @ X3 @ ( ap @ X0 @ X2 @ X1 @ X3 ) ) @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
( funcGraphProp1
= ( ! [X0: $i,X7: $i,X3: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( kpair @ X1 @ ( ap @ X0 @ X3 @ X7 @ X1 ) ) @ X7 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcGraphProp1) ).
thf(f31,plain,
( ( ~ ( funcGraphProp1
=> ( funcGraphProp2
=> ( eqbreln
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y0 @ Y2 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( func @ Y0 @ Y2 @ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ( ~ ( funcGraphProp1
=> ( funcGraphProp2
=> ( eqbreln
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( func @ Y0 @ Y2 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( func @ Y0 @ Y2 @ Y3 )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( ap @ Y0 @ Y2 @ Y3 @ Y4 )
= ( ap @ Y0 @ Y2 @ Y1 @ Y4 ) ) ) )
=> ( Y1 = Y3 ) ) ) ) ) ) ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( funcGraphProp1
=> ( funcGraphProp2
=> ( eqbreln
=> ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X0 @ X1 )
=> ! [X3: $i] :
( ( func @ X2 @ X0 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ap @ X2 @ X0 @ X3 @ X4 )
= ( ap @ X2 @ X0 @ X1 @ X4 ) ) )
=> ( X1 = X3 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,negated_conjecture,
~ ( funcGraphProp1
=> ( funcGraphProp2
=> ( eqbreln
=> ! [X3: $i,X7: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X9: $i] :
( ( func @ X0 @ X3 @ X9 )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= ( ap @ X0 @ X3 @ X9 @ X1 ) ) )
=> ( X7 = X9 ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f8]) ).
thf(f8,conjecture,
( funcGraphProp1
=> ( funcGraphProp2
=> ( eqbreln
=> ! [X3: $i,X7: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X9: $i] :
( ( func @ X0 @ X3 @ X9 )
=> ( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ( ap @ X0 @ X3 @ X7 @ X1 )
= ( ap @ X0 @ X3 @ X9 @ X1 ) ) )
=> ( X7 = X9 ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcext) ).
thf(f147,plain,
( $true
= ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ Y0 )
= Y1 ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f143]) ).
thf(f143,plain,
( ( ( $true
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK3 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f134,f79]) ).
thf(f134,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ X3 ) ) )
& ( in @ Y1
@ ( dsetconstr @ X2
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ X3 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f133]) ).
thf(f133,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ^ [Y0: $i] :
( ( ( subset @ Y0 @ ( cartprod @ X1 @ X2 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) )
& ( in @ Y2
@ ( dsetconstr @ X2
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( ( ap @ X1 @ X2 @ Y0 @ Y1 )
= Y2 ) ) ) ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f126]) ).
thf(f126,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( ( subset @ Y0 @ ( cartprod @ X1 @ X2 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) )
& ( in @ Y2
@ ( dsetconstr @ X2
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( ( ap @ X1 @ X2 @ Y0 @ Y1 )
= Y2 ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f125]) ).
thf(f125,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y1 @ ( cartprod @ X1 @ Y0 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X1 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
=> ( ( ap @ X1 @ Y0 @ Y1 @ Y2 )
= Y3 ) ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f123]) ).
thf(f123,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y1 @ ( cartprod @ X1 @ Y0 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X1 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) )
& ( in @ Y3
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
=> ( ( ap @ X1 @ Y0 @ Y1 @ Y2 )
= Y3 ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f122]) ).
thf(f122,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( ( ap @ Y0 @ Y1 @ Y2 @ Y3 )
= Y4 ) ) ) ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f50]) ).
thf(f366,plain,
( ( ( in @ ( kpair @ sK8 @ ( ap @ sK2 @ sK4 @ sK3 @ sK8 ) ) @ sK5 )
= $true )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f347,f362]) ).
thf(f362,plain,
( ( ( ap @ sK2 @ sK4 @ sK5 @ sK8 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK8 ) )
| ~ spl0_1 ),
inference(equality_proxy_clausification,[status(thm)],[f343]) ).
thf(f343,plain,
( ( ( ( ap @ sK2 @ sK4 @ sK5 @ sK8 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK8 ) )
= $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f336]) ).
thf(f336,plain,
( ( ( $true
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK8 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK8 ) ) )
= $true )
| ~ spl0_1 ),
inference(superposition,[status(thm)],[f121,f329]) ).
thf(f121,plain,
! [X1: $i] :
( ( ( in @ X1 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ X1 )
= ( ap @ sK2 @ sK4 @ sK3 @ X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f120]) ).
thf(f120,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f109]) ).
thf(f109,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f102]) ).
thf(f102,plain,
( $false
= ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) ) )
=> ( sK3 = sK5 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f100]) ).
thf(f100,plain,
( $false
= ( ( ( subset @ sK5 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) ) )
=> ( sK3 = sK5 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f99]) ).
thf(f99,plain,
( $false
= ( ^ [Y0: $i] :
( ( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) )
& ( in @ Y2
@ ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ Y0 @ Y1 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y1 ) ) ) )
=> ( sK3 = Y0 ) ) )
@ sK5 ) ),
inference(sigma_clausification,[status(thm)],[f72]) ).
thf(f72,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) )
& ( in @ Y2
@ ( dsetconstr @ sK4
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ Y0 ) ) ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( ( ap @ sK2 @ sK4 @ Y0 @ Y1 )
= ( ap @ sK2 @ sK4 @ sK3 @ Y1 ) ) ) )
=> ( sK3 = Y0 ) ) ) )
= $false ),
inference(binary_proxy_clausification,[status(thm)],[f69]) ).
thf(f347,plain,
( ( ( in @ ( kpair @ sK8 @ ( ap @ sK2 @ sK4 @ sK5 @ sK8 ) ) @ sK5 )
= $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f337]) ).
thf(f337,plain,
( ( ( $true
=> ( in @ ( kpair @ sK8 @ ( ap @ sK2 @ sK4 @ sK5 @ sK8 ) ) @ sK5 ) )
= $true )
| ~ spl0_1 ),
inference(superposition,[status(thm)],[f157,f329]) ).
thf(f157,plain,
! [X1: $i] :
( ( ( in @ X1 @ sK2 )
=> ( in @ ( kpair @ X1 @ ( ap @ sK2 @ sK4 @ sK5 @ X1 ) ) @ sK5 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f154]) ).
thf(f154,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK5 @ Y0 ) ) @ sK5 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f130]) ).
thf(f130,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK5 @ Y0 ) ) @ sK5 ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f129]) ).
thf(f129,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK5 @ Y0 ) ) @ sK5 ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f64,f103]) ).
thf(f103,plain,
( ( ( subset @ sK5 @ ( cartprod @ sK2 @ sK4 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f100]) ).
thf(f64,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( ( subset @ X2 @ ( cartprod @ X3 @ X1 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ X1
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ X2 ) ) )
& ( in @ Y1
@ ( dsetconstr @ X1
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ X2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
=> ( in @ ( kpair @ Y0 @ ( ap @ X3 @ X1 @ X2 @ Y0 ) ) @ X2 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f63]) ).
thf(f63,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ^ [Y0: $i] :
( ( ( subset @ X2 @ ( cartprod @ Y0 @ X1 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ X1
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ X2 ) ) )
& ( in @ Y2
@ ( dsetconstr @ X1
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ X2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ ( kpair @ Y1 @ ( ap @ Y0 @ X1 @ X2 @ Y1 ) ) @ X2 ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f60]) ).
thf(f60,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( ( subset @ X2 @ ( cartprod @ Y0 @ X1 ) )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ X1
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ X2 ) ) )
& ( in @ Y2
@ ( dsetconstr @ X1
@ ^ [Y3: $i] : ( in @ ( kpair @ Y1 @ Y3 ) @ X2 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ ( kpair @ Y1 @ ( ap @ Y0 @ X1 @ X2 @ Y1 ) ) @ X2 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f59]) ).
thf(f59,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y0 @ ( cartprod @ Y1 @ X1 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ X1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) )
& ( in @ Y3
@ ( dsetconstr @ X1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ ( kpair @ Y2 @ ( ap @ Y1 @ X1 @ Y0 @ Y2 ) ) @ Y0 ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f54]) ).
thf(f54,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( ( subset @ Y0 @ ( cartprod @ Y1 @ X1 ) )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ X1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) )
& ( in @ Y3
@ ( dsetconstr @ X1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y2 @ Y4 ) @ Y0 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ ( kpair @ Y2 @ ( ap @ Y1 @ X1 @ Y0 @ Y2 ) ) @ Y0 ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f53]) ).
thf(f53,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( ( subset @ Y1 @ ( cartprod @ Y2 @ Y0 ) )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( setadjoin @ Y4 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) )
& ( in @ Y4
@ ( dsetconstr @ Y0
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( kpair @ Y3 @ ( ap @ Y2 @ Y0 @ Y1 @ Y3 ) ) @ Y1 ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f46]) ).
thf(f313,plain,
~ spl0_2,
inference(avatar_contradiction_clause,[status(thm)],[f312]) ).
thf(f312,plain,
( $false
| ~ spl0_2 ),
inference(trivial_inequality_removal,[status(thm)],[f311]) ).
thf(f311,plain,
( ( $false = $true )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f309,f265]) ).
thf(f265,plain,
( ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 )
= $false )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f264]) ).
thf(f264,plain,
( ( ( $true
=> ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 ) )
= $false )
| ~ spl0_2 ),
inference(backward_demodulation,[status(thm)],[f261,f263]) ).
thf(f263,plain,
( ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK5 )
= $true )
| ~ spl0_2 ),
inference(binary_proxy_clausification,[status(thm)],[f261]) ).
thf(f261,plain,
( ( ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 ) )
= $false )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f260]) ).
thf(f260,plain,
( ( ( $true
=> ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 ) ) )
= $false )
| ~ spl0_2 ),
inference(backward_demodulation,[status(thm)],[f255,f259]) ).
thf(f259,plain,
( ( ( in @ sK7 @ sK4 )
= $true )
| ~ spl0_2 ),
inference(binary_proxy_clausification,[status(thm)],[f255]) ).
thf(f255,plain,
( ( ( ( in @ sK7 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 ) ) )
= $false )
| ~ spl0_2 ),
inference(beta_eta_normalization,[status(thm)],[f254]) ).
thf(f254,plain,
( ( $false
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ Y0 ) @ sK3 ) ) )
@ sK7 ) )
| ~ spl0_2 ),
inference(sigma_clausification,[status(thm)],[f234]) ).
thf(f234,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ Y0 ) @ sK3 ) ) ) )
= $false )
| ~ spl0_2 ),
inference(binary_proxy_clausification,[status(thm)],[f228]) ).
thf(f228,plain,
( ( $false
= ( ( in @ sK6 @ sK2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( in @ ( kpair @ sK6 @ Y0 ) @ sK3 ) ) ) ) ) )
| ~ spl0_2 ),
inference(beta_eta_normalization,[status(thm)],[f225]) ).
thf(f225,plain,
( ( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) )
@ sK6 )
= $false )
| ~ spl0_2 ),
inference(sigma_clausification,[status(thm)],[f212]) ).
thf(f212,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) )
= $false )
| ~ spl0_2 ),
inference(not_proxy_clausification,[status(thm)],[f204]) ).
thf(f204,plain,
( ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) ) )
= $true )
| ~ spl0_2 ),
inference(avatar_component_clause,[status(thm)],[f203]) ).
thf(f203,definition,
( spl0_2
<=> ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition]) ).
thf(f309,plain,
( ( ( in @ ( kpair @ sK6 @ sK7 ) @ sK3 )
= $true )
| ~ spl0_2 ),
inference(backward_demodulation,[status(thm)],[f249,f300]) ).
thf(f300,plain,
( ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= sK7 )
| ~ spl0_2 ),
inference(equality_proxy_clausification,[status(thm)],[f295]) ).
thf(f295,plain,
( ( ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= sK7 )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f294]) ).
thf(f294,plain,
( ( ( $true
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= sK7 ) )
= $true )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f293,f259]) ).
thf(f293,plain,
( ( ( ( in @ sK7 @ sK4 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= sK7 ) )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f286]) ).
thf(f286,plain,
( ( ( ( in @ sK7 @ sK4 )
=> ( $true
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= sK7 ) ) )
= $true )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f284,f263]) ).
thf(f284,plain,
( ! [X1: $i] :
( $true
= ( ( in @ X1 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ X1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK3 @ sK6 )
= X1 ) ) ) )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f282,f268]) ).
thf(f268,plain,
( ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK6 ) )
| ~ spl0_2 ),
inference(equality_proxy_clausification,[status(thm)],[f252]) ).
thf(f252,plain,
( ( ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK6 ) )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f242]) ).
thf(f242,plain,
( ( ( $true
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= ( ap @ sK2 @ sK4 @ sK3 @ sK6 ) ) )
= $true )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f121,f235]) ).
thf(f235,plain,
( ( ( in @ sK6 @ sK2 )
= $true )
| ~ spl0_2 ),
inference(binary_proxy_clausification,[status(thm)],[f228]) ).
thf(f282,plain,
( ! [X1: $i] :
( ( ( in @ X1 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ X1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= X1 ) ) )
= $true )
| ~ spl0_2 ),
inference(beta_eta_normalization,[status(thm)],[f280]) ).
thf(f280,plain,
( ! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= Y0 ) ) )
@ X1 ) )
| ~ spl0_2 ),
inference(pi_clausification,[status(thm)],[f248]) ).
thf(f248,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= Y0 ) ) ) )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f245]) ).
thf(f245,plain,
( ( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ sK6 @ Y0 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ sK6 )
= Y0 ) ) ) ) )
= $true )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f166,f235]) ).
thf(f166,plain,
! [X1: $i] :
( ( ( in @ X1 @ sK2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK4 )
=> ( ( in @ ( kpair @ X1 @ Y0 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ X1 )
= Y0 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f164]) ).
thf(f164,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= Y1 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f151]) ).
thf(f151,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= Y1 ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f150]) ).
thf(f150,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f146,f127]) ).
thf(f127,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) ) ) ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f103]) ).
thf(f146,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f144]) ).
thf(f144,plain,
( ( ( $true
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK4
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK5 ) ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( ( ap @ sK2 @ sK4 @ sK5 @ Y0 )
= Y1 ) ) ) ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f134,f132]) ).
thf(f132,plain,
( ( subset @ sK5 @ ( cartprod @ sK2 @ sK4 ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f131]) ).
thf(f131,plain,
( ( ( subset @ sK5 @ ( cartprod @ sK2 @ sK4 ) )
& $true )
= $true ),
inference(backward_demodulation,[status(thm)],[f103,f127]) ).
thf(f249,plain,
( ( ( in @ ( kpair @ sK6 @ ( ap @ sK2 @ sK4 @ sK3 @ sK6 ) ) @ sK3 )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f241]) ).
thf(f241,plain,
( ( ( $true
=> ( in @ ( kpair @ sK6 @ ( ap @ sK2 @ sK4 @ sK3 @ sK6 ) ) @ sK3 ) )
= $true )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f97,f235]) ).
thf(f97,plain,
! [X1: $i] :
( $true
= ( ( in @ X1 @ sK2 )
=> ( in @ ( kpair @ X1 @ ( ap @ sK2 @ sK4 @ sK3 @ X1 ) ) @ sK3 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f95]) ).
thf(f95,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) @ sK3 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f81]) ).
thf(f81,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) @ sK3 ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f80]) ).
thf(f80,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( in @ ( kpair @ Y0 @ ( ap @ sK2 @ sK4 @ sK3 @ Y0 ) ) @ sK3 ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f64,f73]) ).
thf(f205,plain,
( spl0_1
| spl0_2 ),
inference(avatar_split_clause,[status(thm)],[f198,f203,f200]) ).
thf(f198,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
= $false )
| ( ( ~ ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f195]) ).
thf(f195,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
=> ~ ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f194]) ).
thf(f194,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) )
=> $false ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f193,f111]) ).
thf(f111,plain,
( ( sK3 = sK5 )
= $false ),
inference(boolean_simplification,[status(thm)],[f110]) ).
thf(f110,plain,
( ( $true
=> ( sK3 = sK5 ) )
= $false ),
inference(backward_demodulation,[status(thm)],[f102,f109]) ).
thf(f193,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = sK5 ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f187]) ).
thf(f187,plain,
( ( $true
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK5 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = sK5 ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f183,f132]) ).
thf(f183,plain,
! [X1: $i] :
( ( ( subset @ X1 @ ( cartprod @ sK2 @ sK4 ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ X1 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK4 )
=> ( ( in @ ( kpair @ Y0 @ Y1 ) @ X1 )
=> ( in @ ( kpair @ Y0 @ Y1 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = X1 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f180]) ).
thf(f180,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f119]) ).
thf(f119,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = Y0 ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f118]) ).
thf(f118,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( subset @ Y0 @ ( cartprod @ sK2 @ sK4 ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK4 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ sK3 ) ) ) ) ) )
=> ( sK3 = Y0 ) ) ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f116,f79]) ).
thf(f116,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( subset @ X1 @ ( cartprod @ X3 @ X2 ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( subset @ Y0 @ ( cartprod @ X3 @ X2 ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X3 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ X1 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X3 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
=> ( ( in @ ( kpair @ Y1 @ Y2 ) @ Y0 )
=> ( in @ ( kpair @ Y1 @ Y2 ) @ X1 ) ) ) ) ) )
=> ( X1 = Y0 ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f115]) ).
thf(f115,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ^ [Y0: $i] :
( ( subset @ X1 @ ( cartprod @ Y0 @ X2 ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( subset @ Y1 @ ( cartprod @ Y0 @ X2 ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ X2 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ X1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ X2 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ X1 ) ) ) ) ) )
=> ( X1 = Y1 ) ) ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f113]) ).
thf(f113,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( subset @ X1 @ ( cartprod @ Y0 @ X2 ) )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( subset @ Y1 @ ( cartprod @ Y0 @ X2 ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ X2 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ X1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ X2 )
=> ( ( in @ ( kpair @ Y2 @ Y3 ) @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ X1 ) ) ) ) ) )
=> ( X1 = Y1 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f112]) ).
thf(f112,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( subset @ X1 @ ( cartprod @ Y1 @ Y0 ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y2 @ ( cartprod @ Y1 @ Y0 ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ X1 )
=> ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( in @ ( kpair @ Y3 @ Y4 ) @ X1 ) ) ) ) ) )
=> ( X1 = Y2 ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f71]) ).
thf(f71,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( subset @ X1 @ ( cartprod @ Y1 @ Y0 ) )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y2 @ ( cartprod @ Y1 @ Y0 ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ X1 )
=> ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 )
=> ( in @ ( kpair @ Y3 @ Y4 ) @ X1 ) ) ) ) ) )
=> ( X1 = Y2 ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f70]) ).
thf(f70,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( subset @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( subset @ Y3 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 ) ) ) ) ) )
=> ( ( !! @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( !! @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ ( kpair @ Y4 @ Y5 ) @ Y3 )
=> ( in @ ( kpair @ Y4 @ Y5 ) @ Y0 ) ) ) ) ) )
=> ( Y0 = Y3 ) ) ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f56]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU691^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.36 % Computer : n019.cluster.edu
% 0.17/0.36 % Model : x86_64 x86_64
% 0.17/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.36 % Memory : 8042.1875MB
% 0.17/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.36 % CPULimit : 300
% 0.17/0.36 % WCLimit : 300
% 0.17/0.36 % DateTime : Tue Feb 24 12:31:09 EST 2026
% 0.17/0.36 % CPUTime :
% 0.17/0.36 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU691^2 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Functions - Extensionality and Beta Reduction
% 0.30/0.46 % Version : Especial > Reduced > Especial.
% 0.30/0.46 % English : (! A:i.! B:i.! f:i.func A B f -> (! g:i.func A B g ->
% 0.30/0.46 % (! x:i.in x A -> ap A B f x = ap A B g x) -> f = g))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC193l [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.50 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.43 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.80 v6.2.0, 0.71 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 23 ( 7 unt; 15 typ; 7 def)
% 0.30/0.46 % Number of atoms : 49 ( 12 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 8 ( 6 avg)
% 0.30/0.46 % Number of connectives : 123 ( 0 ~; 0 |; 2 &; 97 @)
% 0.30/0.46 % ( 0 <=>; 24 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 15 ( 3 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 16 ( 15 usr; 4 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 35 ( 11 ^; 23 !; 1 ?; 35 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=245
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpair_type,type,
% 0.30/0.46 kpair: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(cartprod_type,type,
% 0.30/0.46 cartprod: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(singleton_type,type,
% 0.30/0.46 singleton: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(singleton,definition,
% 0.30/0.46 ( singleton
% 0.30/0.46 = ( ^ [A: $i] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( A
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1_type,type,
% 0.30/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1,definition,
% 0.30/0.46 ( ex1
% 0.30/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln_type,type,
% 0.30/0.46 breln: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln,definition,
% 0.30/0.46 ( breln
% 0.30/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(func_type,type,
% 0.30/0.46 func: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(func,definition,
% 0.30/0.46 ( func
% 0.30/0.46 = ( ^ [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ex1 @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap_type,type,
% 0.30/0.46 ap: $i > $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1_type,type,
% 0.30/0.46 funcGraphProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1,definition,
% 0.30/0.46 ( funcGraphProp1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2_type,type,
% 0.30/0.46 funcGraphProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2,definition,
% 0.30/0.46 ( funcGraphProp2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln_type,type,
% 0.30/0.46 eqbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln,definition,
% 0.30/0.46 ( eqbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext,conjecture,
% 0.30/0.46 ( funcGraphProp1
% 0.30/0.46 => ( funcGraphProp2
% 0.30/0.46 => ( eqbreln
% 0.30/0.46 => ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 1.56/1.39 ---- Embedded in THF ---
% 1.56/1.39 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.56/1.39 %%% Generated on Tue Feb 24 12:31:10 EST 2026
% 1.56/1.39 %%% using '$$dhol' embedding, version 1.3.0.
% 1.56/1.39 %%% Logic specification used:
% 1.56/1.39 %%% thf(spec, logic, $$dhol).
% 1.56/1.39
% 1.56/1.39 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2DTF18641.p
% See solution above
% 1.56/1.39 ------------------------
% 1.77/1.40 ---- Cleaned THF ---
% 1.77/1.40 thf(in_type,type,
% 1.77/1.40 in: $i > $i > $o ).
% 1.77/1.40
% 1.77/1.40 thf(emptyset_type,type,
% 1.77/1.40 emptyset: $i ).
% 1.77/1.40
% 1.77/1.40 thf(setadjoin_type,type,
% 1.77/1.40 setadjoin: $i > $i > $i ).
% 1.77/1.40
% 1.77/1.40 thf(dsetconstr_type,type,
% 1.77/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.77/1.40
% 1.77/1.40 thf(subset_type,type,
% 1.77/1.40 subset: $i > $i > $o ).
% 1.77/1.40
% 1.77/1.40 thf(kpair_type,type,
% 1.77/1.40 kpair: $i > $i > $i ).
% 1.77/1.40
% 1.77/1.40 thf(cartprod_type,type,
% 1.77/1.40 cartprod: $i > $i > $i ).
% 1.77/1.40
% 1.77/1.40 thf(singleton_type,type,
% 1.77/1.40 singleton: $i > $o ).
% 1.77/1.40
% 1.77/1.40 thf(singleton,definition,
% 1.77/1.40 ( singleton
% 1.77/1.40 = ( ^ [A: $i] :
% 1.77/1.40 ? [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 & ( A
% 1.77/1.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(ex1_type,type,
% 1.77/1.40 ex1: $i > ( $i > $o ) > $o ).
% 1.77/1.40
% 1.77/1.40 thf(ex1,definition,
% 1.77/1.40 ( ex1
% 1.77/1.40 = ( ^ [A: $i,Xphi: $i > $o] :
% 1.77/1.40 ( singleton
% 1.77/1.40 @ ( dsetconstr @ A
% 1.77/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(breln_type,type,
% 1.77/1.40 breln: $i > $i > $i > $o ).
% 1.77/1.40
% 1.77/1.40 thf(breln,definition,
% 1.77/1.40 ( breln
% 1.77/1.40 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(func_type,type,
% 1.77/1.40 func: $i > $i > $i > $o ).
% 1.77/1.40
% 1.77/1.40 thf(func,definition,
% 1.77/1.40 ( func
% 1.77/1.40 = ( ^ [A: $i,B: $i,R: $i] :
% 1.77/1.40 ( ( breln @ A @ B @ R )
% 1.77/1.40 & ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ( ex1 @ B
% 1.77/1.40 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(ap_type,type,
% 1.77/1.40 ap: $i > $i > $i > $i > $i ).
% 1.77/1.40
% 1.77/1.40 thf(funcGraphProp1_type,type,
% 1.77/1.40 funcGraphProp1: $o ).
% 1.77/1.40
% 1.77/1.40 thf(funcGraphProp1,definition,
% 1.77/1.40 ( funcGraphProp1
% 1.77/1.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.77/1.40 ( ( func @ A @ B @ Xf )
% 1.77/1.40 => ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(funcGraphProp2_type,type,
% 1.77/1.40 funcGraphProp2: $o ).
% 1.77/1.40
% 1.77/1.40 thf(funcGraphProp2,definition,
% 1.77/1.40 ( funcGraphProp2
% 1.77/1.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.77/1.40 ( ( func @ A @ B @ Xf )
% 1.77/1.40 => ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ! [Xy: $i] :
% 1.77/1.40 ( ( in @ Xy @ B )
% 1.77/1.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 1.77/1.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 1.77/1.40 = Xy ) ) ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(eqbreln_type,type,
% 1.77/1.40 eqbreln: $o ).
% 1.77/1.40
% 1.77/1.40 thf(eqbreln,definition,
% 1.77/1.40 ( eqbreln
% 1.77/1.40 = ( ! [A: $i,B: $i,R: $i] :
% 1.77/1.40 ( ( breln @ A @ B @ R )
% 1.77/1.40 => ! [S: $i] :
% 1.77/1.40 ( ( breln @ A @ B @ S )
% 1.77/1.40 => ( ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ! [Xy: $i] :
% 1.77/1.40 ( ( in @ Xy @ B )
% 1.77/1.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 1.77/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 1.77/1.40 => ( ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ! [Xy: $i] :
% 1.77/1.40 ( ( in @ Xy @ B )
% 1.77/1.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 1.77/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 1.77/1.40 => ( R = S ) ) ) ) ) ) ) ).
% 1.77/1.40
% 1.77/1.40 thf(funcext,conjecture,
% 1.77/1.40 ( funcGraphProp1
% 1.77/1.40 => ( funcGraphProp2
% 1.77/1.40 => ( eqbreln
% 1.77/1.40 => ! [A: $i,B: $i,Xf: $i] :
% 1.77/1.40 ( ( func @ A @ B @ Xf )
% 1.77/1.40 => ! [Xg: $i] :
% 1.77/1.40 ( ( func @ A @ B @ Xg )
% 1.77/1.40 => ( ! [Xx: $i] :
% 1.77/1.40 ( ( in @ Xx @ A )
% 1.77/1.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 1.77/1.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 1.77/1.40 => ( Xf = Xg ) ) ) ) ) ) ) ).
% 1.77/1.40 ------------------------
% 1.77/1.42 % (18739)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/183Mi)
% 1.77/1.42 % (18740)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_18641 for (2999ds/4Mi)
% 1.77/1.42 % (18741)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_18641 for (2999ds/27Mi)
% 1.77/1.42 % (18742)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/2Mi)
% 1.77/1.42 % (18743)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_18641 for (2999ds/2Mi)
% 1.77/1.42 % (18745)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/18Mi)
% 1.77/1.42 % (18744)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_18641 for (2999ds/275Mi)
% 1.77/1.42 % (18742)Instruction limit reached!
% 1.77/1.42 % (18742)------------------------------
% 1.77/1.42 % (18742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.42 % (18742)Termination reason: Unknown
% 1.77/1.42 % (18743)Instruction limit reached!
% 1.77/1.42 % (18743)------------------------------
% 1.77/1.42 % (18743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.42 % (18743)Termination reason: Unknown
% 1.77/1.42 % (18743)Termination phase: Property scanning
% 1.77/1.42
% 1.77/1.42 % (18743)Memory used [KB]: 1023
% 1.77/1.42 % (18743)Time elapsed: 0.003 s
% 1.77/1.42 % (18743)Instructions burned: 3 (million)
% 1.77/1.42 % (18743)------------------------------
% 1.77/1.42 % (18743)------------------------------
% 1.77/1.42 % (18742)Termination phase: Property scanning
% 1.77/1.42
% 1.77/1.42 % (18742)Memory used [KB]: 1023
% 1.77/1.42 % (18742)Time elapsed: 0.003 s
% 1.77/1.42 % (18742)Instructions burned: 3 (million)
% 1.77/1.42 % (18742)------------------------------
% 1.77/1.42 % (18742)------------------------------
% 1.77/1.42 % (18740)Instruction limit reached!
% 1.77/1.42 % (18740)------------------------------
% 1.77/1.42 % (18740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.42 % (18740)Termination reason: Unknown
% 1.77/1.42 % (18740)Termination phase: Function definition elimination
% 1.77/1.42
% 1.77/1.42 % (18740)Memory used [KB]: 1023
% 1.77/1.42 % (18740)Time elapsed: 0.004 s
% 1.77/1.42 % (18740)Instructions burned: 5 (million)
% 1.77/1.42 % (18740)------------------------------
% 1.77/1.42 % (18740)------------------------------
% 1.77/1.43 % (18745)Instruction limit reached!
% 1.77/1.43 % (18745)------------------------------
% 1.77/1.43 % (18745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.43 % (18745)Termination reason: Unknown
% 1.77/1.43 % (18745)Termination phase: Saturation
% 1.77/1.43
% 1.77/1.43 % (18745)Memory used [KB]: 5628
% 1.77/1.43 % (18745)Time elapsed: 0.010 s
% 1.77/1.43 % (18745)Instructions burned: 18 (million)
% 1.77/1.43 % (18745)------------------------------
% 1.77/1.43 % (18745)------------------------------
% 1.77/1.43 % (18741)Instruction limit reached!
% 1.77/1.43 % (18741)------------------------------
% 1.77/1.43 % (18741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.43 % (18741)Termination reason: Unknown
% 1.77/1.43 % (18741)Termination phase: Saturation
% 1.77/1.43
% 1.77/1.43 % (18741)Memory used [KB]: 5756
% 1.77/1.43 % (18741)Time elapsed: 0.015 s
% 1.77/1.43 % (18741)Instructions burned: 28 (million)
% 1.77/1.43 % (18741)------------------------------
% 1.77/1.43 % (18741)------------------------------
% 1.77/1.43 % (18746)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/3Mi)
% 1.77/1.43 % (18747)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_18641 for (2999ds/37Mi)
% 1.77/1.44 % (18748)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_18641 for (2999ds/15Mi)
% 1.77/1.44 % (18746)Instruction limit reached!
% 1.77/1.44 % (18746)------------------------------
% 1.77/1.44 % (18746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.44 % (18746)Termination reason: Unknown
% 1.77/1.44 % (18746)Termination phase: Property scanning
% 1.77/1.44
% 1.77/1.44 % (18746)Memory used [KB]: 1023
% 1.77/1.44 % (18746)Time elapsed: 0.003 s
% 1.77/1.44 % (18746)Instructions burned: 4 (million)
% 1.77/1.44 % (18746)------------------------------
% 1.77/1.44 % (18746)------------------------------
% 1.77/1.44 % (18749)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_18641 for (2999ds/3Mi)
% 1.77/1.44 % (18748)Instruction limit reached!
% 1.77/1.44 % (18748)------------------------------
% 1.77/1.44 % (18748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.44 % (18748)Termination reason: Unknown
% 1.77/1.44 % (18748)Termination phase: Saturation
% 1.77/1.44
% 1.77/1.44 % (18748)Memory used [KB]: 5756
% 1.77/1.44 % (18748)Time elapsed: 0.009 s
% 1.77/1.44 % (18748)Instructions burned: 16 (million)
% 1.77/1.44 % (18748)------------------------------
% 1.77/1.44 % (18748)------------------------------
% 1.77/1.44 % (18749)Instruction limit reached!
% 1.77/1.44 % (18749)------------------------------
% 1.77/1.44 % (18749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.44 % (18749)Termination reason: Unknown
% 1.77/1.44 % (18749)Termination phase: Property scanning
% 1.77/1.44
% 1.77/1.44 % (18749)Memory used [KB]: 1023
% 1.77/1.44 % (18749)Time elapsed: 0.003 s
% 1.77/1.44 % (18749)Instructions burned: 4 (million)
% 1.77/1.44 % (18749)------------------------------
% 1.77/1.44 % (18749)------------------------------
% 1.77/1.45 % (18750)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/1041Mi)
% 1.77/1.45 % (18747)Instruction limit reached!
% 1.77/1.45 % (18747)------------------------------
% 1.77/1.45 % (18747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.45 % (18747)Termination reason: Unknown
% 1.77/1.45 % (18747)Termination phase: Saturation
% 1.77/1.45
% 1.77/1.45 % (18747)Memory used [KB]: 5756
% 1.77/1.45 % (18747)Time elapsed: 0.018 s
% 1.77/1.45 % (18747)Instructions burned: 37 (million)
% 1.77/1.45 % (18747)------------------------------
% 1.77/1.45 % (18747)------------------------------
% 1.77/1.45 % (18751)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/7Mi)
% 1.77/1.46 % (18751)Instruction limit reached!
% 1.77/1.46 % (18751)------------------------------
% 1.77/1.46 % (18751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.46 % (18751)Termination reason: Unknown
% 1.77/1.46 % (18751)Termination phase: Saturation
% 1.77/1.46
% 1.77/1.46 % (18751)Memory used [KB]: 1023
% 1.77/1.46 % (18751)Time elapsed: 0.005 s
% 1.77/1.46 % (18751)Instructions burned: 8 (million)
% 1.77/1.46 % (18751)------------------------------
% 1.77/1.46 % (18751)------------------------------
% 1.77/1.46 % (18752)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/16Mi)
% 1.77/1.46 % (18753)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/3Mi)
% 1.77/1.46 % (18753)Instruction limit reached!
% 1.77/1.46 % (18753)------------------------------
% 1.77/1.46 % (18753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.46 % (18753)Termination reason: Unknown
% 1.77/1.46 % (18753)Termination phase: Property scanning
% 1.77/1.46
% 1.77/1.46 % (18753)Memory used [KB]: 1023
% 1.77/1.46 % (18753)Time elapsed: 0.003 s
% 1.77/1.46 % (18753)Instructions burned: 4 (million)
% 1.77/1.46 % (18753)------------------------------
% 1.77/1.46 % (18753)------------------------------
% 1.77/1.47 % (18754)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/3Mi)
% 1.77/1.47 % (18752)Instruction limit reached!
% 1.77/1.47 % (18752)------------------------------
% 1.77/1.47 % (18752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.47 % (18752)Termination reason: Unknown
% 1.77/1.47 % (18752)Termination phase: Saturation
% 1.77/1.47
% 1.77/1.47 % (18752)Memory used [KB]: 6012
% 1.77/1.47 % (18752)Time elapsed: 0.010 s
% 1.77/1.47 % (18752)Instructions burned: 16 (million)
% 1.77/1.47 % (18752)------------------------------
% 1.77/1.47 % (18752)------------------------------
% 1.77/1.47 % (18754)Instruction limit reached!
% 1.77/1.47 % (18754)------------------------------
% 1.77/1.47 % (18754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.47 % (18754)Termination reason: Unknown
% 1.77/1.47 % (18754)Termination phase: Property scanning
% 1.77/1.47
% 1.77/1.47 % (18754)Memory used [KB]: 1023
% 1.77/1.47 % (18754)Time elapsed: 0.003 s
% 1.77/1.47 % (18754)Instructions burned: 4 (million)
% 1.77/1.47 % (18754)------------------------------
% 1.77/1.47 % (18754)------------------------------
% 1.77/1.47 % (18755)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/7Mi)
% 1.77/1.48 % (18755)Instruction limit reached!
% 1.77/1.48 % (18755)------------------------------
% 1.77/1.48 % (18755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.48 % (18755)Termination reason: Unknown
% 1.77/1.48 % (18755)Termination phase: Saturation
% 1.77/1.48
% 1.77/1.48 % (18755)Memory used [KB]: 5500
% 1.77/1.48 % (18755)Time elapsed: 0.005 s
% 1.77/1.48 % (18755)Instructions burned: 8 (million)
% 1.77/1.48 % (18755)------------------------------
% 1.77/1.48 % (18755)------------------------------
% 1.77/1.48 % (18756)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/3Mi)
% 1.77/1.48 % (18756)Instruction limit reached!
% 1.77/1.48 % (18756)------------------------------
% 1.77/1.48 % (18756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.48 % (18756)Termination reason: Unknown
% 1.77/1.48 % (18756)Termination phase: Property scanning
% 1.77/1.48
% 1.77/1.48 % (18756)Memory used [KB]: 1023
% 1.77/1.48 % (18756)Time elapsed: 0.003 s
% 1.77/1.48 % (18756)Instructions burned: 4 (million)
% 1.77/1.48 % (18756)------------------------------
% 1.77/1.48 % (18756)------------------------------
% 1.77/1.48 % (18757)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/4Mi)
% 1.77/1.48 % (18758)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/18Mi)
% 1.77/1.49 % (18757)Instruction limit reached!
% 1.77/1.49 % (18757)------------------------------
% 1.77/1.49 % (18757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.49 % (18757)Termination reason: Unknown
% 1.77/1.49 % (18757)Termination phase: shuffling
% 1.77/1.49
% 1.77/1.49 % (18757)Memory used [KB]: 1023
% 1.77/1.49 % (18757)Time elapsed: 0.003 s
% 1.77/1.49 % (18757)Instructions burned: 4 (million)
% 1.77/1.49 % (18757)------------------------------
% 1.77/1.49 % (18757)------------------------------
% 1.77/1.49 % (18759)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/710Mi)
% 1.77/1.49 % (18758)Instruction limit reached!
% 1.77/1.49 % (18758)------------------------------
% 1.77/1.49 % (18758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.49 % (18758)Termination reason: Unknown
% 1.77/1.49 % (18758)Termination phase: Saturation
% 1.77/1.49
% 1.77/1.49 % (18758)Memory used [KB]: 5628
% 1.77/1.49 % (18758)Time elapsed: 0.010 s
% 1.77/1.49 % (18758)Instructions burned: 19 (million)
% 1.77/1.49 % (18758)------------------------------
% 1.77/1.49 % (18758)------------------------------
% 1.77/1.49 % (18760)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/6Mi)
% 1.77/1.50 % (18760)Instruction limit reached!
% 1.77/1.50 % (18760)------------------------------
% 1.77/1.50 % (18760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.50 % (18760)Termination reason: Unknown
% 1.77/1.50 % (18760)Termination phase: Property scanning
% 1.77/1.50
% 1.77/1.50 % (18760)Memory used [KB]: 1023
% 1.77/1.50 % (18760)Time elapsed: 0.004 s
% 1.77/1.50 % (18760)Instructions burned: 6 (million)
% 1.77/1.50 % (18760)------------------------------
% 1.77/1.50 % (18760)------------------------------
% 1.77/1.50 % (18761)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/902Mi)
% 1.77/1.51 % (18739)Instruction limit reached!
% 1.77/1.51 % (18739)------------------------------
% 1.77/1.51 % (18739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.51 % (18739)Termination reason: Unknown
% 1.77/1.51 % (18739)Termination phase: Saturation
% 1.77/1.51
% 1.77/1.51 % (18739)Memory used [KB]: 6396
% 1.77/1.51 % (18739)Time elapsed: 0.091 s
% 1.77/1.51 % (18739)Instructions burned: 184 (million)
% 1.77/1.51 % (18739)------------------------------
% 1.77/1.51 % (18739)------------------------------
% 1.77/1.51 % (18762)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/21Mi)
% 1.77/1.51 % (18763)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/5Mi)
% 1.77/1.52 % (18763)Instruction limit reached!
% 1.77/1.52 % (18763)------------------------------
% 1.77/1.52 % (18763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.52 % (18763)Termination reason: Unknown
% 1.77/1.52 % (18763)Termination phase: Property scanning
% 1.77/1.52
% 1.77/1.52 % (18763)Memory used [KB]: 1023
% 1.77/1.52 % (18763)Time elapsed: 0.004 s
% 1.77/1.52 % (18763)Instructions burned: 6 (million)
% 1.77/1.52 % (18763)------------------------------
% 1.77/1.52 % (18763)------------------------------
% 1.77/1.52 % (18762)Instruction limit reached!
% 1.77/1.52 % (18762)------------------------------
% 1.77/1.52 % (18762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.52 % (18762)Termination reason: Unknown
% 1.77/1.52 % (18762)Termination phase: Saturation
% 1.77/1.52
% 1.77/1.52 % (18762)Memory used [KB]: 5756
% 1.77/1.52 % (18762)Time elapsed: 0.013 s
% 1.77/1.52 % (18762)Instructions burned: 22 (million)
% 1.77/1.52 % (18762)------------------------------
% 1.77/1.52 % (18762)------------------------------
% 1.77/1.52 % (18764)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_18641 for (2999ds/6Mi)
% 1.77/1.52 % (18764)Instruction limit reached!
% 1.77/1.52 % (18764)------------------------------
% 1.77/1.52 % (18764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.52 % (18764)Termination reason: Unknown
% 1.77/1.52 % (18764)Termination phase: Property scanning
% 1.77/1.52
% 1.77/1.53 % (18764)Memory used [KB]: 1023
% 1.77/1.53 % (18764)Time elapsed: 0.004 s
% 1.77/1.53 % (18764)Instructions burned: 6 (million)
% 1.77/1.53 % (18764)------------------------------
% 1.77/1.53 % (18764)------------------------------
% 1.77/1.53 % (18765)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18641 for (2998ds/377Mi)
% 1.77/1.53 % (18750)First to succeed.
% 1.77/1.54 % (18766)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_18641 for (2998ds/779Mi)
% 1.77/1.54 % (18767)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_18641 for (2998ds/19Mi)
% 1.77/1.54 % (18744)Instruction limit reached!
% 1.77/1.54 % (18744)------------------------------
% 1.77/1.54 % (18744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.54 % (18744)Termination reason: Unknown
% 1.77/1.54 % (18744)Termination phase: Saturation
% 1.77/1.54
% 1.77/1.54 % (18744)Memory used [KB]: 6140
% 1.77/1.54 % (18744)Time elapsed: 0.129 s
% 1.77/1.54 % (18744)Instructions burned: 276 (million)
% 1.77/1.54 % (18744)------------------------------
% 1.77/1.54 % (18744)------------------------------
% 1.77/1.55 % (18750)Refutation found. Thanks to Tanya!
% 1.77/1.55 % SZS status Theorem for DTF2THF_18641
% 1.77/1.55 % SZS output start Proof for DTF2THF_18641
% 1.77/1.55 thf(func_def_0, type, in: $i > $i > $o).
% 1.77/1.55 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.77/1.55 thf(func_def_3, type, dsetconstr: $i > ($i > $o) > $i).
% 1.77/1.55 thf(func_def_4, type, subset: $i > $i > $o).
% 1.77/1.55 thf(func_def_5, type, kpair: $i > $i > $i).
% 1.77/1.55 thf(func_def_6, type, cartprod: $i > $i > $i).
% 1.77/1.55 thf(func_def_7, type, singleton: $i > $o).
% 1.77/1.55 thf(func_def_9, type, ex1: $i > ($i > $o) > $o).
% 1.77/1.55 thf(func_def_10, type, breln: $i > $i > $i > $o).
% 1.77/1.55 thf(func_def_11, type, func: $i > $i > $i > $o).
% 1.77/1.55 thf(func_def_12, type, ap: $i > $i > $i > $i > $i).
% 1.77/1.55 thf(func_def_33, type, ph1: !>[X0: $tType]:(X0)).
% 1.77/1.55 thf(f415,plain,(
% 1.77/1.55 $false),
% 1.77/1.55 inference(avatar_sat_refutation,[status(thm)],[f205,f313,f414])).
% 1.77/1.55 thf(f414,plain,(
% 1.77/1.55 ~spl0_1),
% 1.77/1.55 inference(avatar_contradiction_clause,[status(thm)],[f413])).
% 1.77/1.55 thf(f413,plain,(
% 1.77/1.55 $false | ~spl0_1),
% 1.77/1.55 inference(trivial_inequality_removal,[status(thm)],[f412])).
% 1.77/1.55 thf(f412,plain,(
% 1.77/1.55 ($false = $true) | ~spl0_1),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f406,f361])).
% 1.77/1.55 thf(f361,plain,(
% 1.77/1.55 ((in @ (kpair @ sK8 @ sK9) @ sK5) = $false) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f360])).
% 1.77/1.55 thf(f360,plain,(
% 1.77/1.55 (($true => (in @ (kpair @ sK8 @ sK9) @ sK5)) = $false) | ~spl0_1),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f357,f359])).
% 1.77/1.55 thf(f359,plain,(
% 1.77/1.55 ((in @ (kpair @ sK8 @ sK9) @ sK3) = $true) | ~spl0_1),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f357])).
% 1.77/1.55 thf(f357,plain,(
% 1.77/1.55 ($false = ((in @ (kpair @ sK8 @ sK9) @ sK3) => (in @ (kpair @ sK8 @ sK9) @ sK5))) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f356])).
% 1.77/1.55 thf(f356,plain,(
% 1.77/1.55 ($false = ($true => ((in @ (kpair @ sK8 @ sK9) @ sK3) => (in @ (kpair @ sK8 @ sK9) @ sK5)))) | ~spl0_1),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f351,f355])).
% 1.77/1.55 thf(f355,plain,(
% 1.77/1.55 ($true = (in @ sK9 @ sK4)) | ~spl0_1),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f351])).
% 1.77/1.55 thf(f351,plain,(
% 1.77/1.55 ($false = ((in @ sK9 @ sK4) => ((in @ (kpair @ sK8 @ sK9) @ sK3) => (in @ (kpair @ sK8 @ sK9) @ sK5)))) | ~spl0_1),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f350])).
% 1.77/1.55 thf(f350,plain,(
% 1.77/1.55 (((^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => (in @ (kpair @ sK8 @ Y0) @ sK5)))) @ sK9) = $false) | ~spl0_1),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f328])).
% 1.77/1.55 thf(f328,plain,(
% 1.77/1.55 ($false = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => (in @ (kpair @ sK8 @ Y0) @ sK5)))))) | ~spl0_1),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f327])).
% 1.77/1.55 thf(f327,plain,(
% 1.77/1.55 ($false = ((in @ sK8 @ sK2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => (in @ (kpair @ sK8 @ Y0) @ sK5))))))) | ~spl0_1),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f319])).
% 1.77/1.55 thf(f319,plain,(
% 1.77/1.55 (((^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5))))))) @ sK8) = $false) | ~spl0_1),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f201])).
% 1.77/1.55 thf(f201,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) = $false) | ~spl0_1),
% 1.77/1.55 inference(avatar_component_clause,[status(thm)],[f200])).
% 1.77/1.55 thf(f200,definition,(
% 1.77/1.55 spl0_1 <=> ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) = $false)),
% 1.77/1.55 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 1.77/1.55 thf(f406,plain,(
% 1.77/1.55 ((in @ (kpair @ sK8 @ sK9) @ sK5) = $true) | ~spl0_1),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f366,f400])).
% 1.77/1.55 thf(f400,plain,(
% 1.77/1.55 (sK9 = (ap @ sK2 @ sK4 @ sK3 @ sK8)) | ~spl0_1),
% 1.77/1.55 inference(equality_proxy_clausification,[status(thm)],[f392])).
% 1.77/1.55 thf(f392,plain,(
% 1.77/1.55 (((ap @ sK2 @ sK4 @ sK3 @ sK8) = sK9) = $true) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f391])).
% 1.77/1.55 thf(f391,plain,(
% 1.77/1.55 (($true => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = sK9)) = $true) | ~spl0_1),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f384,f359])).
% 1.77/1.55 thf(f384,plain,(
% 1.77/1.55 (((in @ (kpair @ sK8 @ sK9) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = sK9)) = $true) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f382])).
% 1.77/1.55 thf(f382,plain,(
% 1.77/1.55 (($true => ((in @ (kpair @ sK8 @ sK9) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = sK9))) = $true) | ~spl0_1),
% 1.77/1.55 inference(superposition,[status(thm)],[f377,f355])).
% 1.77/1.55 thf(f377,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK4) => ((in @ (kpair @ sK8 @ X1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = X1))) = $true)) ) | ~spl0_1),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f374])).
% 1.77/1.55 thf(f374,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = Y0)))) @ X1) = $true)) ) | ~spl0_1),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f342])).
% 1.77/1.55 thf(f342,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = Y0))))) = $true) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f338])).
% 1.77/1.55 thf(f338,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK8 @ Y0) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ sK8) = Y0)))))) = $true) | ~spl0_1),
% 1.77/1.55 inference(superposition,[status(thm)],[f162,f329])).
% 1.77/1.55 thf(f329,plain,(
% 1.77/1.55 ((in @ sK8 @ sK2) = $true) | ~spl0_1),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f327])).
% 1.77/1.55 thf(f162,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ X1 @ Y0) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ X1) = Y0)))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f159])).
% 1.77/1.55 thf(f159,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ Y0) = Y1))))))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f149])).
% 1.77/1.55 thf(f149,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ Y0) = Y1)))))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f148])).
% 1.77/1.55 thf(f148,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ Y0) = Y1))))))))) = $true)),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f147,f83])).
% 1.77/1.55 thf(f83,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))))))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f82])).
% 1.77/1.55 thf(f82,plain,(
% 1.77/1.55 (($true & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3))))))))))) = $true)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f73,f79])).
% 1.77/1.55 thf(f79,plain,(
% 1.77/1.55 ((subset @ sK3 @ (cartprod @ sK2 @ sK4)) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f73])).
% 1.77/1.55 thf(f73,plain,(
% 1.77/1.55 (((subset @ sK3 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3))))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f69])).
% 1.77/1.55 thf(f69,plain,(
% 1.77/1.55 ((((subset @ sK3 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3))))))))))) => (!! @ $i @ (^[Y0 : $i]: (((subset @ Y0 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0)))) & (in @ Y2 @ (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0))))))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => ((ap @ sK2 @ sK4 @ Y0 @ Y1) = (ap @ sK2 @ sK4 @ sK3 @ Y1))))) => (sK3 = Y0)))))) = $false)),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f68])).
% 1.77/1.55 thf(f68,plain,(
% 1.77/1.55 ($false = ((^[Y0 : $i]: (((subset @ sK3 @ (cartprod @ sK2 @ Y0)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ Y0 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ sK3)))) & (in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ sK3))))))))))) => (!! @ $i @ (^[Y1 : $i]: (((subset @ Y1 @ (cartprod @ sK2 @ Y0)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1)))) & (in @ Y3 @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1))))))))))) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => ((ap @ sK2 @ Y0 @ Y1 @ Y2) = (ap @ sK2 @ Y0 @ sK3 @ Y2))))) => (sK3 = Y1))))))) @ sK4))),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f66])).
% 1.77/1.55 thf(f66,plain,(
% 1.77/1.55 ($false = (!! @ $i @ (^[Y0 : $i]: (((subset @ sK3 @ (cartprod @ sK2 @ Y0)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ Y0 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ sK3)))) & (in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ sK3))))))))))) => (!! @ $i @ (^[Y1 : $i]: (((subset @ Y1 @ (cartprod @ sK2 @ Y0)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1)))) & (in @ Y3 @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1))))))))))) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => ((ap @ sK2 @ Y0 @ Y1 @ Y2) = (ap @ sK2 @ Y0 @ sK3 @ Y2))))) => (sK3 = Y1)))))))))),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f65])).
% 1.77/1.55 thf(f65,plain,(
% 1.77/1.55 ($false = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y0 @ (cartprod @ sK2 @ Y1)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0)))) & (in @ Y3 @ (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0))))))))))) => (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ sK2 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ sK2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ sK2) => ((ap @ sK2 @ Y1 @ Y2 @ Y3) = (ap @ sK2 @ Y1 @ Y0 @ Y3))))) => (Y0 = Y2))))))))) @ sK3))),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f62])).
% 1.77/1.55 thf(f62,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y0 @ (cartprod @ sK2 @ Y1)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0)))) & (in @ Y3 @ (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0))))))))))) => (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ sK2 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ sK2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ sK2) => ((ap @ sK2 @ Y1 @ Y2 @ Y3) = (ap @ sK2 @ Y1 @ Y0 @ Y3))))) => (Y0 = Y2)))))))))) = $false)),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f61])).
% 1.77/1.55 thf(f61,plain,(
% 1.77/1.55 ($false = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3))))))))))) @ sK2))),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f58])).
% 1.77/1.55 thf(f58,plain,(
% 1.77/1.55 ($false = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f57])).
% 1.77/1.55 thf(f57,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3))))))))))))) = $false)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f52,f56])).
% 1.77/1.55 thf(f56,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f52])).
% 1.77/1.55 thf(f52,plain,(
% 1.77/1.55 ($false = ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3))))))))))))))),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f51])).
% 1.77/1.55 thf(f51,plain,(
% 1.77/1.55 (($true => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))) = $false)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f48,f50])).
% 1.77/1.55 thf(f50,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f48])).
% 1.77/1.55 thf(f48,plain,(
% 1.77/1.55 (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))) = $false)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f47])).
% 1.77/1.55 thf(f47,plain,(
% 1.77/1.55 (($true => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3))))))))))))))) = $false)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f42,f46])).
% 1.77/1.55 thf(f46,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y2 @ Y0)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f42])).
% 1.77/1.55 thf(f42,plain,(
% 1.77/1.55 ($false = ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y2 @ Y0)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3))))))))))))))))),
% 1.77/1.55 inference(not_proxy_clausification,[status(thm)],[f41])).
% 1.77/1.55 thf(f41,plain,(
% 1.77/1.55 ((~ ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y2 @ Y0)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y2 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: (((subset @ Y3 @ (cartprod @ Y0 @ Y2)) & (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => (?? @ $i @ (^[Y5 : $i]: (((setadjoin @ Y5 @ emptyset) = (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3)))) & (in @ Y5 @ (dsetconstr @ Y2 @ (^[Y6 : $i]: (in @ (kpair @ Y4 @ Y6) @ Y3))))))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))))) = $true)),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f40])).
% 1.77/1.55 thf(f40,plain,(
% 1.77/1.55 ((~ ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (((^[Y6 : $i]: ((^[Y7 : $i]: ((^[Y8 : $i]: (subset @ Y8 @ (cartprod @ Y6 @ Y7))))))) @ Y3 @ Y4 @ Y5) & (!! @ $i @ (^[Y6 : $i]: ((in @ Y6 @ Y3) => ((^[Y7 : $i]: ((^[Y8 : $i > $o]: ((^[Y9 : $i]: (?? @ $i @ (^[Y10 : $i]: (((setadjoin @ Y10 @ emptyset) = Y9) & (in @ Y10 @ Y9))))) @ (dsetconstr @ Y7 @ (^[Y9 : $i]: (Y8 @ Y9))))))) @ Y4 @ (^[Y7 : $i]: (in @ (kpair @ Y6 @ Y7) @ Y5)))))))))))) @ Y2 @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (((^[Y6 : $i]: ((^[Y7 : $i]: ((^[Y8 : $i]: (subset @ Y8 @ (cartprod @ Y6 @ Y7))))))) @ Y3 @ Y4 @ Y5) & (!! @ $i @ (^[Y6 : $i]: ((in @ Y6 @ Y3) => ((^[Y7 : $i]: ((^[Y8 : $i > $o]: ((^[Y9 : $i]: (?? @ $i @ (^[Y10 : $i]: (((setadjoin @ Y10 @ emptyset) = Y9) & (in @ Y10 @ Y9))))) @ (dsetconstr @ Y7 @ (^[Y9 : $i]: (Y8 @ Y9))))))) @ Y4 @ (^[Y7 : $i]: (in @ (kpair @ Y6 @ Y7) @ Y5)))))))))))) @ Y0 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))) => ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (subset @ Y5 @ (cartprod @ Y3 @ Y4))))))) @ Y2 @ Y1 @ Y0) => (!! @ $i @ (^[Y3 : $i]: (((^[Y4 : $i]: ((^[Y5 : $i]: ((^[Y6 : $i]: (subset @ Y6 @ (cartprod @ Y4 @ Y5))))))) @ Y2 @ Y1 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (((^[Y6 : $i]: ((^[Y7 : $i]: ((^[Y8 : $i]: (subset @ Y8 @ (cartprod @ Y6 @ Y7))))))) @ Y3 @ Y4 @ Y5) & (!! @ $i @ (^[Y6 : $i]: ((in @ Y6 @ Y3) => ((^[Y7 : $i]: ((^[Y8 : $i > $o]: ((^[Y9 : $i]: (?? @ $i @ (^[Y10 : $i]: (((setadjoin @ Y10 @ emptyset) = Y9) & (in @ Y10 @ Y9))))) @ (dsetconstr @ Y7 @ (^[Y9 : $i]: (Y8 @ Y9))))))) @ Y4 @ (^[Y7 : $i]: (in @ (kpair @ Y6 @ Y7) @ Y5)))))))))))) @ Y0 @ Y2 @ Y1) => (!! @ $i @ (^[Y3 : $i]: (((^[Y4 : $i]: ((^[Y5 : $i]: ((^[Y6 : $i]: (((^[Y7 : $i]: ((^[Y8 : $i]: ((^[Y9 : $i]: (subset @ Y9 @ (cartprod @ Y7 @ Y8))))))) @ Y4 @ Y5 @ Y6) & (!! @ $i @ (^[Y7 : $i]: ((in @ Y7 @ Y4) => ((^[Y8 : $i]: ((^[Y9 : $i > $o]: ((^[Y10 : $i]: (?? @ $i @ (^[Y11 : $i]: (((setadjoin @ Y11 @ emptyset) = Y10) & (in @ Y11 @ Y10))))) @ (dsetconstr @ Y8 @ (^[Y10 : $i]: (Y9 @ Y10))))))) @ Y5 @ (^[Y8 : $i]: (in @ (kpair @ Y7 @ Y8) @ Y6)))))))))))) @ Y0 @ Y2 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))))) = $true)),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f31,f37,f38,f39,f36,f36])).
% 1.77/1.55 thf(f36,plain,(
% 1.77/1.55 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (subset @ Y5 @ (cartprod @ Y3 @ Y4))))))) @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((^[Y4 : $i]: ((^[Y5 : $i > $o]: ((^[Y6 : $i]: (?? @ $i @ (^[Y7 : $i]: (((setadjoin @ Y7 @ emptyset) = Y6) & (in @ Y7 @ Y6))))) @ (dsetconstr @ Y4 @ (^[Y6 : $i]: (Y5 @ Y6))))))) @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f33,f29,f35])).
% 1.77/1.55 thf(f35,plain,(
% 1.77/1.55 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = Y2) & (in @ Y3 @ Y2))))) @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f28,f27])).
% 1.77/1.55 thf(f27,plain,(
% 1.77/1.55 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = Y0) & (in @ Y1 @ Y0))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f14])).
% 1.77/1.55 thf(f14,plain,(
% 1.77/1.55 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = Y0) & (in @ Y1 @ Y0))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f13])).
% 1.77/1.55 thf(f13,plain,(
% 1.77/1.55 ((^[X0 : $i] : (? [X1 : $i] : ((in @ X1 @ X0) & ((setadjoin @ X1 @ emptyset) = X0)))) = singleton)),
% 1.77/1.55 inference(rectify,[status(thm)],[f1])).
% 1.77/1.55 thf(f1,axiom,(
% 1.77/1.55 ((^[X0 : $i] : (? [X1 : $i] : ((in @ X1 @ X0) & ((setadjoin @ X1 @ emptyset) = X0)))) = singleton)),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',singleton)).
% 1.77/1.55 thf(f28,plain,(
% 1.77/1.55 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: (singleton @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f26])).
% 1.77/1.55 thf(f26,plain,(
% 1.77/1.55 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: (singleton @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f25])).
% 1.77/1.55 thf(f25,plain,(
% 1.77/1.55 (ex1 = (^[X0 : $i, X1 : $i > $o] : (singleton @ (dsetconstr @ X0 @ (^[X2 : $i] : (X1 @ X2))))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f2])).
% 1.77/1.55 thf(f2,axiom,(
% 1.77/1.55 (ex1 = (^[X0 : $i, X2 : $i > $o] : (singleton @ (dsetconstr @ X0 @ (^[X1 : $i] : (X2 @ X1))))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',ex1)).
% 1.77/1.55 thf(f29,plain,(
% 1.77/1.55 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f22])).
% 1.77/1.55 thf(f22,plain,(
% 1.77/1.55 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f21])).
% 1.77/1.55 thf(f21,plain,(
% 1.77/1.55 (breln = (^[X0 : $i, X1 : $i, X2 : $i] : (subset @ X2 @ (cartprod @ X0 @ X1))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f3])).
% 1.77/1.55 thf(f3,axiom,(
% 1.77/1.55 (breln = (^[X0 : $i, X3 : $i, X4 : $i] : (subset @ X4 @ (cartprod @ X0 @ X3))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',breln)).
% 1.77/1.55 thf(f33,plain,(
% 1.77/1.55 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((breln @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (ex1 @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f24])).
% 1.77/1.55 thf(f24,plain,(
% 1.77/1.55 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((breln @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (ex1 @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f23])).
% 1.77/1.55 thf(f23,plain,(
% 1.77/1.55 (func = (^[X0 : $i, X1 : $i, X2 : $i] : (! [X3 : $i] : ((in @ X3 @ X0) => (ex1 @ X1 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X2)))) & (breln @ X0 @ X1 @ X2))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f4])).
% 1.77/1.55 thf(f4,axiom,(
% 1.77/1.55 (func = (^[X0 : $i, X3 : $i, X5 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) => (ex1 @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X5)))) & (breln @ X0 @ X3 @ X5))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',func)).
% 1.77/1.55 thf(f39,plain,(
% 1.77/1.55 (eqbreln = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (subset @ Y5 @ (cartprod @ Y3 @ Y4))))))) @ Y2 @ Y1 @ Y0) => (!! @ $i @ (^[Y3 : $i]: (((^[Y4 : $i]: ((^[Y5 : $i]: ((^[Y6 : $i]: (subset @ Y6 @ (cartprod @ Y4 @ Y5))))))) @ Y2 @ Y1 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))))),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f32,f29,f29])).
% 1.77/1.55 thf(f32,plain,(
% 1.77/1.55 (eqbreln = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((breln @ Y2 @ Y1 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((breln @ Y2 @ Y1 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f20])).
% 1.77/1.55 thf(f20,plain,(
% 1.77/1.55 (eqbreln = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((breln @ Y2 @ Y1 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((breln @ Y2 @ Y1 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3))))))))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f19])).
% 1.77/1.55 thf(f19,plain,(
% 1.77/1.55 (eqbreln = ! [X0 : $i,X1 : $i,X2 : $i] : ((breln @ X0 @ X1 @ X2) => ! [X3 : $i] : ((breln @ X0 @ X1 @ X3) => (! [X4 : $i] : ((in @ X4 @ X0) => ! [X5 : $i] : ((in @ X5 @ X1) => ((in @ (kpair @ X4 @ X5) @ X2) => (in @ (kpair @ X4 @ X5) @ X3)))) => (! [X6 : $i] : ((in @ X6 @ X0) => ! [X7 : $i] : ((in @ X7 @ X1) => ((in @ (kpair @ X6 @ X7) @ X3) => (in @ (kpair @ X6 @ X7) @ X2)))) => (X2 = X3))))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f7])).
% 1.77/1.55 thf(f7,axiom,(
% 1.77/1.55 (eqbreln = ! [X0 : $i,X3 : $i,X5 : $i] : ((breln @ X0 @ X3 @ X5) => ! [X8 : $i] : ((breln @ X0 @ X3 @ X8) => (! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X5) => (in @ (kpair @ X1 @ X6) @ X8)))) => (! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X8) => (in @ (kpair @ X1 @ X6) @ X5)))) => (X5 = X8))))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',eqbreln)).
% 1.77/1.55 thf(f38,plain,(
% 1.77/1.55 (funcGraphProp2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (((^[Y6 : $i]: ((^[Y7 : $i]: ((^[Y8 : $i]: (subset @ Y8 @ (cartprod @ Y6 @ Y7))))))) @ Y3 @ Y4 @ Y5) & (!! @ $i @ (^[Y6 : $i]: ((in @ Y6 @ Y3) => ((^[Y7 : $i]: ((^[Y8 : $i > $o]: ((^[Y9 : $i]: (?? @ $i @ (^[Y10 : $i]: (((setadjoin @ Y10 @ emptyset) = Y9) & (in @ Y10 @ Y9))))) @ (dsetconstr @ Y7 @ (^[Y9 : $i]: (Y8 @ Y9))))))) @ Y4 @ (^[Y7 : $i]: (in @ (kpair @ Y6 @ Y7) @ Y5)))))))))))) @ Y0 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))))),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f34,f36])).
% 1.77/1.55 thf(f34,plain,(
% 1.77/1.55 (funcGraphProp2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y0 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f18])).
% 1.77/1.55 thf(f18,plain,(
% 1.77/1.55 (funcGraphProp2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y0 @ Y1 @ Y2) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4))))))))))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f17])).
% 1.77/1.55 thf(f17,plain,(
% 1.77/1.55 (funcGraphProp2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X2) => ! [X4 : $i] : ((in @ X4 @ X1) => ((in @ (kpair @ X3 @ X4) @ X0) => ((ap @ X2 @ X1 @ X0 @ X3) = X4))))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f6])).
% 1.77/1.55 thf(f6,axiom,(
% 1.77/1.55 (funcGraphProp2 = ! [X7 : $i,X3 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => ! [X6 : $i] : ((in @ X6 @ X3) => ((in @ (kpair @ X1 @ X6) @ X7) => ((ap @ X0 @ X3 @ X7 @ X1) = X6))))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcGraphProp2)).
% 1.77/1.55 thf(f37,plain,(
% 1.77/1.55 (funcGraphProp1 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (((^[Y6 : $i]: ((^[Y7 : $i]: ((^[Y8 : $i]: (subset @ Y8 @ (cartprod @ Y6 @ Y7))))))) @ Y3 @ Y4 @ Y5) & (!! @ $i @ (^[Y6 : $i]: ((in @ Y6 @ Y3) => ((^[Y7 : $i]: ((^[Y8 : $i > $o]: ((^[Y9 : $i]: (?? @ $i @ (^[Y10 : $i]: (((setadjoin @ Y10 @ emptyset) = Y9) & (in @ Y10 @ Y9))))) @ (dsetconstr @ Y7 @ (^[Y9 : $i]: (Y8 @ Y9))))))) @ Y4 @ (^[Y7 : $i]: (in @ (kpair @ Y6 @ Y7) @ Y5)))))))))))) @ Y2 @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))))),
% 1.77/1.55 inference(definition_unfolding,[status(thm)],[f30,f36])).
% 1.77/1.55 thf(f30,plain,(
% 1.77/1.55 (funcGraphProp1 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y2 @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))))),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f16])).
% 1.77/1.55 thf(f16,plain,(
% 1.77/1.55 (funcGraphProp1 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y2 @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1))))))))))))),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f15])).
% 1.77/1.55 thf(f15,plain,(
% 1.77/1.55 (funcGraphProp1 = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (kpair @ X3 @ (ap @ X0 @ X2 @ X1 @ X3)) @ X1))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f5])).
% 1.77/1.55 thf(f5,axiom,(
% 1.77/1.55 (funcGraphProp1 = ! [X0 : $i,X7 : $i,X3 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (kpair @ X1 @ (ap @ X0 @ X3 @ X7 @ X1)) @ X7))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcGraphProp1)).
% 1.77/1.55 thf(f31,plain,(
% 1.77/1.55 ((~ (funcGraphProp1 => (funcGraphProp2 => (eqbreln => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y0 @ Y2 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((func @ Y0 @ Y2 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))))) = $true)),
% 1.77/1.55 inference(cnf_transformation,[status(thm)],[f12])).
% 1.77/1.55 thf(f12,plain,(
% 1.77/1.55 ((~ (funcGraphProp1 => (funcGraphProp2 => (eqbreln => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((func @ Y0 @ Y2 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((func @ Y0 @ Y2 @ Y3) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((ap @ Y0 @ Y2 @ Y3 @ Y4) = (ap @ Y0 @ Y2 @ Y1 @ Y4))))) => (Y1 = Y3)))))))))))))))) = $true)),
% 1.77/1.55 inference(fool_elimination,[status(thm)],[f11])).
% 1.77/1.55 thf(f11,plain,(
% 1.77/1.55 ~(funcGraphProp1 => (funcGraphProp2 => (eqbreln => ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X0 @ X1) => ! [X3 : $i] : ((func @ X2 @ X0 @ X3) => (! [X4 : $i] : ((in @ X4 @ X2) => ((ap @ X2 @ X0 @ X3 @ X4) = (ap @ X2 @ X0 @ X1 @ X4))) => (X1 = X3)))))))),
% 1.77/1.55 inference(rectify,[status(thm)],[f9])).
% 1.77/1.55 thf(f9,negated_conjecture,(
% 1.77/1.55 ~(funcGraphProp1 => (funcGraphProp2 => (eqbreln => ! [X3 : $i,X7 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X9 : $i] : ((func @ X0 @ X3 @ X9) => (! [X1 : $i] : ((in @ X1 @ X0) => ((ap @ X0 @ X3 @ X7 @ X1) = (ap @ X0 @ X3 @ X9 @ X1))) => (X7 = X9)))))))),
% 1.77/1.55 inference(negated_conjecture,[status(cth)],[f8])).
% 1.77/1.55 thf(f8,conjecture,(
% 1.77/1.55 funcGraphProp1 => (funcGraphProp2 => (eqbreln => ! [X3 : $i,X7 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X9 : $i] : ((func @ X0 @ X3 @ X9) => (! [X1 : $i] : ((in @ X1 @ X0) => ((ap @ X0 @ X3 @ X7 @ X1) = (ap @ X0 @ X3 @ X9 @ X1))) => (X7 = X9))))))),
% 1.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.cfa2aMtMY7/DTF2THF_18641.p',funcext)).
% 1.77/1.55 thf(f147,plain,(
% 1.77/1.55 ($true = ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ Y0) = Y1))))))))))),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f143])).
% 1.77/1.55 thf(f143,plain,(
% 1.77/1.55 ((($true & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK3))))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => ((ap @ sK2 @ sK4 @ sK3 @ Y0) = Y1))))))))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f134,f79])).
% 1.77/1.55 thf(f134,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((((subset @ X3 @ (cartprod @ X1 @ X2)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X1) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ X2 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ X3)))) & (in @ Y1 @ (dsetconstr @ X2 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ X3))))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X1) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((in @ (kpair @ Y0 @ Y1) @ X3) => ((ap @ X1 @ X2 @ X3 @ Y0) = Y1))))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f133])).
% 1.77/1.55 thf(f133,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: (((subset @ Y0 @ (cartprod @ X1 @ X2)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ X2 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0)))) & (in @ Y2 @ (dsetconstr @ X2 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0))))))))))) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => ((ap @ X1 @ X2 @ Y0 @ Y1) = Y2)))))))))) @ X3) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f126])).
% 1.77/1.55 thf(f126,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (((subset @ Y0 @ (cartprod @ X1 @ X2)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ X2 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0)))) & (in @ Y2 @ (dsetconstr @ X2 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0))))))))))) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => ((ap @ X1 @ X2 @ Y0 @ Y1) = Y2))))))))))))) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f125])).
% 1.77/1.55 thf(f125,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y1 @ (cartprod @ X1 @ Y0)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X1) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1)))) & (in @ Y3 @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1))))))))))) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((in @ (kpair @ Y2 @ Y3) @ Y1) => ((ap @ X1 @ Y0 @ Y1 @ Y2) = Y3)))))))))))) @ X2) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f123])).
% 1.77/1.55 thf(f123,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y1 @ (cartprod @ X1 @ Y0)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X1) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1)))) & (in @ Y3 @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y1))))))))))) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((in @ (kpair @ Y2 @ Y3) @ Y1) => ((ap @ X1 @ Y0 @ Y1 @ Y2) = Y3))))))))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f122])).
% 1.77/1.55 thf(f122,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y2 @ (cartprod @ Y0 @ Y1)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2)))) & (in @ Y4 @ (dsetconstr @ Y1 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y2))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => ((ap @ Y0 @ Y1 @ Y2 @ Y3) = Y4)))))))))))))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f50])).
% 1.77/1.55 thf(f366,plain,(
% 1.77/1.55 ((in @ (kpair @ sK8 @ (ap @ sK2 @ sK4 @ sK3 @ sK8)) @ sK5) = $true) | ~spl0_1),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f347,f362])).
% 1.77/1.55 thf(f362,plain,(
% 1.77/1.55 ((ap @ sK2 @ sK4 @ sK5 @ sK8) = (ap @ sK2 @ sK4 @ sK3 @ sK8)) | ~spl0_1),
% 1.77/1.55 inference(equality_proxy_clausification,[status(thm)],[f343])).
% 1.77/1.55 thf(f343,plain,(
% 1.77/1.55 (((ap @ sK2 @ sK4 @ sK5 @ sK8) = (ap @ sK2 @ sK4 @ sK3 @ sK8)) = $true) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f336])).
% 1.77/1.55 thf(f336,plain,(
% 1.77/1.55 (($true => ((ap @ sK2 @ sK4 @ sK5 @ sK8) = (ap @ sK2 @ sK4 @ sK3 @ sK8))) = $true) | ~spl0_1),
% 1.77/1.55 inference(superposition,[status(thm)],[f121,f329])).
% 1.77/1.55 thf(f121,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK2) => ((ap @ sK2 @ sK4 @ sK5 @ X1) = (ap @ sK2 @ sK4 @ sK3 @ X1))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f120])).
% 1.77/1.55 thf(f120,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ sK2) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = (ap @ sK2 @ sK4 @ sK3 @ Y0)))) @ X1))) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f109])).
% 1.77/1.55 thf(f109,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = (ap @ sK2 @ sK4 @ sK3 @ Y0))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f102])).
% 1.77/1.55 thf(f102,plain,(
% 1.77/1.55 ($false = ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = (ap @ sK2 @ sK4 @ sK3 @ Y0))))) => (sK3 = sK5)))),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f100])).
% 1.77/1.55 thf(f100,plain,(
% 1.77/1.55 ($false = (((subset @ sK5 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5))))))))))) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = (ap @ sK2 @ sK4 @ sK3 @ Y0))))) => (sK3 = sK5))))),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f99])).
% 1.77/1.55 thf(f99,plain,(
% 1.77/1.55 ($false = ((^[Y0 : $i]: (((subset @ Y0 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0)))) & (in @ Y2 @ (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0))))))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => ((ap @ sK2 @ sK4 @ Y0 @ Y1) = (ap @ sK2 @ sK4 @ sK3 @ Y1))))) => (sK3 = Y0)))) @ sK5))),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f72])).
% 1.77/1.55 thf(f72,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: (((subset @ Y0 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0)))) & (in @ Y2 @ (dsetconstr @ sK4 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ Y0))))))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => ((ap @ sK2 @ sK4 @ Y0 @ Y1) = (ap @ sK2 @ sK4 @ sK3 @ Y1))))) => (sK3 = Y0))))) = $false)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f69])).
% 1.77/1.55 thf(f347,plain,(
% 1.77/1.55 ((in @ (kpair @ sK8 @ (ap @ sK2 @ sK4 @ sK5 @ sK8)) @ sK5) = $true) | ~spl0_1),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f337])).
% 1.77/1.55 thf(f337,plain,(
% 1.77/1.55 (($true => (in @ (kpair @ sK8 @ (ap @ sK2 @ sK4 @ sK5 @ sK8)) @ sK5)) = $true) | ~spl0_1),
% 1.77/1.55 inference(superposition,[status(thm)],[f157,f329])).
% 1.77/1.55 thf(f157,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK2) => (in @ (kpair @ X1 @ (ap @ sK2 @ sK4 @ sK5 @ X1)) @ sK5)) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f154])).
% 1.77/1.55 thf(f154,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK5 @ Y0)) @ sK5))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f130])).
% 1.77/1.55 thf(f130,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK5 @ Y0)) @ sK5)))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f129])).
% 1.77/1.55 thf(f129,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK5 @ Y0)) @ sK5))))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f64,f103])).
% 1.77/1.55 thf(f103,plain,(
% 1.77/1.55 (((subset @ sK5 @ (cartprod @ sK2 @ sK4)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5))))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f100])).
% 1.77/1.55 thf(f64,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((((subset @ X2 @ (cartprod @ X3 @ X1)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X3) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ X1 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ X2)))) & (in @ Y1 @ (dsetconstr @ X1 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ X2))))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X3) => (in @ (kpair @ Y0 @ (ap @ X3 @ X1 @ X2 @ Y0)) @ X2))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f63])).
% 1.77/1.55 thf(f63,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: (((subset @ X2 @ (cartprod @ Y0 @ X1)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ X1 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ X2)))) & (in @ Y2 @ (dsetconstr @ X1 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ X2))))))))))) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (in @ (kpair @ Y1 @ (ap @ Y0 @ X1 @ X2 @ Y1)) @ X2)))))) @ X3) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f60])).
% 1.77/1.55 thf(f60,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (((subset @ X2 @ (cartprod @ Y0 @ X1)) & (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (?? @ $i @ (^[Y2 : $i]: (((setadjoin @ Y2 @ emptyset) = (dsetconstr @ X1 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ X2)))) & (in @ Y2 @ (dsetconstr @ X1 @ (^[Y3 : $i]: (in @ (kpair @ Y1 @ Y3) @ X2))))))))))) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (in @ (kpair @ Y1 @ (ap @ Y0 @ X1 @ X2 @ Y1)) @ X2))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f59])).
% 1.77/1.55 thf(f59,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y0 @ (cartprod @ Y1 @ X1)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ X1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0)))) & (in @ Y3 @ (dsetconstr @ X1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0))))))))))) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (in @ (kpair @ Y2 @ (ap @ Y1 @ X1 @ Y0 @ Y2)) @ Y0)))))))) @ X2) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f54])).
% 1.77/1.55 thf(f54,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (((subset @ Y0 @ (cartprod @ Y1 @ X1)) & (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = (dsetconstr @ X1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0)))) & (in @ Y3 @ (dsetconstr @ X1 @ (^[Y4 : $i]: (in @ (kpair @ Y2 @ Y4) @ Y0))))))))))) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (in @ (kpair @ Y2 @ (ap @ Y1 @ X1 @ Y0 @ Y2)) @ Y0))))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f53])).
% 1.77/1.55 thf(f53,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (((subset @ Y1 @ (cartprod @ Y2 @ Y0)) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (?? @ $i @ (^[Y4 : $i]: (((setadjoin @ Y4 @ emptyset) = (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1)))) & (in @ Y4 @ (dsetconstr @ Y0 @ (^[Y5 : $i]: (in @ (kpair @ Y3 @ Y5) @ Y1))))))))))) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (kpair @ Y3 @ (ap @ Y2 @ Y0 @ Y1 @ Y3)) @ Y1)))))))))) @ X1))) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f46])).
% 1.77/1.55 thf(f313,plain,(
% 1.77/1.55 ~spl0_2),
% 1.77/1.55 inference(avatar_contradiction_clause,[status(thm)],[f312])).
% 1.77/1.55 thf(f312,plain,(
% 1.77/1.55 $false | ~spl0_2),
% 1.77/1.55 inference(trivial_inequality_removal,[status(thm)],[f311])).
% 1.77/1.55 thf(f311,plain,(
% 1.77/1.55 ($false = $true) | ~spl0_2),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f309,f265])).
% 1.77/1.55 thf(f265,plain,(
% 1.77/1.55 ((in @ (kpair @ sK6 @ sK7) @ sK3) = $false) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f264])).
% 1.77/1.55 thf(f264,plain,(
% 1.77/1.55 (($true => (in @ (kpair @ sK6 @ sK7) @ sK3)) = $false) | ~spl0_2),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f261,f263])).
% 1.77/1.55 thf(f263,plain,(
% 1.77/1.55 ((in @ (kpair @ sK6 @ sK7) @ sK5) = $true) | ~spl0_2),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f261])).
% 1.77/1.55 thf(f261,plain,(
% 1.77/1.55 (((in @ (kpair @ sK6 @ sK7) @ sK5) => (in @ (kpair @ sK6 @ sK7) @ sK3)) = $false) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f260])).
% 1.77/1.55 thf(f260,plain,(
% 1.77/1.55 (($true => ((in @ (kpair @ sK6 @ sK7) @ sK5) => (in @ (kpair @ sK6 @ sK7) @ sK3))) = $false) | ~spl0_2),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f255,f259])).
% 1.77/1.55 thf(f259,plain,(
% 1.77/1.55 ((in @ sK7 @ sK4) = $true) | ~spl0_2),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f255])).
% 1.77/1.55 thf(f255,plain,(
% 1.77/1.55 (((in @ sK7 @ sK4) => ((in @ (kpair @ sK6 @ sK7) @ sK5) => (in @ (kpair @ sK6 @ sK7) @ sK3))) = $false) | ~spl0_2),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f254])).
% 1.77/1.55 thf(f254,plain,(
% 1.77/1.55 ($false = ((^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => (in @ (kpair @ sK6 @ Y0) @ sK3)))) @ sK7)) | ~spl0_2),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f234])).
% 1.77/1.55 thf(f234,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => (in @ (kpair @ sK6 @ Y0) @ sK3))))) = $false) | ~spl0_2),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f228])).
% 1.77/1.55 thf(f228,plain,(
% 1.77/1.55 ($false = ((in @ sK6 @ sK2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => (in @ (kpair @ sK6 @ Y0) @ sK3))))))) | ~spl0_2),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f225])).
% 1.77/1.55 thf(f225,plain,(
% 1.77/1.55 (((^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3))))))) @ sK6) = $false) | ~spl0_2),
% 1.77/1.55 inference(sigma_clausification,[status(thm)],[f212])).
% 1.77/1.55 thf(f212,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))) = $false) | ~spl0_2),
% 1.77/1.55 inference(not_proxy_clausification,[status(thm)],[f204])).
% 1.77/1.55 thf(f204,plain,(
% 1.77/1.55 ((~ (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3))))))))) = $true) | ~spl0_2),
% 1.77/1.55 inference(avatar_component_clause,[status(thm)],[f203])).
% 1.77/1.55 thf(f203,definition,(
% 1.77/1.55 spl0_2 <=> ((~ (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3))))))))) = $true)),
% 1.77/1.55 introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition])).
% 1.77/1.55 thf(f309,plain,(
% 1.77/1.55 ((in @ (kpair @ sK6 @ sK7) @ sK3) = $true) | ~spl0_2),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f249,f300])).
% 1.77/1.55 thf(f300,plain,(
% 1.77/1.55 ((ap @ sK2 @ sK4 @ sK3 @ sK6) = sK7) | ~spl0_2),
% 1.77/1.55 inference(equality_proxy_clausification,[status(thm)],[f295])).
% 1.77/1.55 thf(f295,plain,(
% 1.77/1.55 (((ap @ sK2 @ sK4 @ sK3 @ sK6) = sK7) = $true) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f294])).
% 1.77/1.55 thf(f294,plain,(
% 1.77/1.55 (($true => ((ap @ sK2 @ sK4 @ sK3 @ sK6) = sK7)) = $true) | ~spl0_2),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f293,f259])).
% 1.77/1.55 thf(f293,plain,(
% 1.77/1.55 (((in @ sK7 @ sK4) => ((ap @ sK2 @ sK4 @ sK3 @ sK6) = sK7)) = $true) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f286])).
% 1.77/1.55 thf(f286,plain,(
% 1.77/1.55 (((in @ sK7 @ sK4) => ($true => ((ap @ sK2 @ sK4 @ sK3 @ sK6) = sK7))) = $true) | ~spl0_2),
% 1.77/1.55 inference(superposition,[status(thm)],[f284,f263])).
% 1.77/1.55 thf(f284,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (($true = ((in @ X1 @ sK4) => ((in @ (kpair @ sK6 @ X1) @ sK5) => ((ap @ sK2 @ sK4 @ sK3 @ sK6) = X1))))) ) | ~spl0_2),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f282,f268])).
% 1.77/1.55 thf(f268,plain,(
% 1.77/1.55 ((ap @ sK2 @ sK4 @ sK5 @ sK6) = (ap @ sK2 @ sK4 @ sK3 @ sK6)) | ~spl0_2),
% 1.77/1.55 inference(equality_proxy_clausification,[status(thm)],[f252])).
% 1.77/1.55 thf(f252,plain,(
% 1.77/1.55 (((ap @ sK2 @ sK4 @ sK5 @ sK6) = (ap @ sK2 @ sK4 @ sK3 @ sK6)) = $true) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f242])).
% 1.77/1.55 thf(f242,plain,(
% 1.77/1.55 (($true => ((ap @ sK2 @ sK4 @ sK5 @ sK6) = (ap @ sK2 @ sK4 @ sK3 @ sK6))) = $true) | ~spl0_2),
% 1.77/1.55 inference(superposition,[status(thm)],[f121,f235])).
% 1.77/1.55 thf(f235,plain,(
% 1.77/1.55 ((in @ sK6 @ sK2) = $true) | ~spl0_2),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f228])).
% 1.77/1.55 thf(f282,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK4) => ((in @ (kpair @ sK6 @ X1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ sK6) = X1))) = $true)) ) | ~spl0_2),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f280])).
% 1.77/1.55 thf(f280,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ sK6) = Y0)))) @ X1))) ) | ~spl0_2),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f248])).
% 1.77/1.55 thf(f248,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ sK6) = Y0))))) = $true) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f245])).
% 1.77/1.55 thf(f245,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ sK6 @ Y0) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ sK6) = Y0)))))) = $true) | ~spl0_2),
% 1.77/1.55 inference(superposition,[status(thm)],[f166,f235])).
% 1.77/1.55 thf(f166,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((in @ X1 @ sK2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK4) => ((in @ (kpair @ X1 @ Y0) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ X1) = Y0)))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f164])).
% 1.77/1.55 thf(f164,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = Y1))))))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f151])).
% 1.77/1.55 thf(f151,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = Y1)))))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f150])).
% 1.77/1.55 thf(f150,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = Y1))))))))) = $true)),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f146,f127])).
% 1.77/1.55 thf(f127,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f103])).
% 1.77/1.55 thf(f146,plain,(
% 1.77/1.55 (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = Y1))))))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f144])).
% 1.77/1.55 thf(f144,plain,(
% 1.77/1.55 ((($true & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5)))) & (in @ Y1 @ (dsetconstr @ sK4 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK5))))))))))) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => ((ap @ sK2 @ sK4 @ sK5 @ Y0) = Y1))))))))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f134,f132])).
% 1.77/1.55 thf(f132,plain,(
% 1.77/1.55 ((subset @ sK5 @ (cartprod @ sK2 @ sK4)) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f131])).
% 1.77/1.55 thf(f131,plain,(
% 1.77/1.55 (((subset @ sK5 @ (cartprod @ sK2 @ sK4)) & $true) = $true)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f103,f127])).
% 1.77/1.55 thf(f249,plain,(
% 1.77/1.55 ((in @ (kpair @ sK6 @ (ap @ sK2 @ sK4 @ sK3 @ sK6)) @ sK3) = $true) | ~spl0_2),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f241])).
% 1.77/1.55 thf(f241,plain,(
% 1.77/1.55 (($true => (in @ (kpair @ sK6 @ (ap @ sK2 @ sK4 @ sK3 @ sK6)) @ sK3)) = $true) | ~spl0_2),
% 1.77/1.55 inference(superposition,[status(thm)],[f97,f235])).
% 1.77/1.55 thf(f97,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (($true = ((in @ X1 @ sK2) => (in @ (kpair @ X1 @ (ap @ sK2 @ sK4 @ sK3 @ X1)) @ sK3)))) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f95])).
% 1.77/1.55 thf(f95,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK3 @ Y0)) @ sK3))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f81])).
% 1.77/1.55 thf(f81,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK3 @ Y0)) @ sK3)))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f80])).
% 1.77/1.55 thf(f80,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (in @ (kpair @ Y0 @ (ap @ sK2 @ sK4 @ sK3 @ Y0)) @ sK3))))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f64,f73])).
% 1.77/1.55 thf(f205,plain,(
% 1.77/1.55 spl0_1 | spl0_2),
% 1.77/1.55 inference(avatar_split_clause,[status(thm)],[f198,f203,f200])).
% 1.77/1.55 thf(f198,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) = $false) | ((~ (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3))))))))) = $true)),
% 1.77/1.55 inference(binary_proxy_clausification,[status(thm)],[f195])).
% 1.77/1.55 thf(f195,plain,(
% 1.77/1.55 (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) => (~ (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f194])).
% 1.77/1.55 thf(f194,plain,(
% 1.77/1.55 (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))) => $false)) = $true)),
% 1.77/1.55 inference(forward_demodulation,[status(thm)],[f193,f111])).
% 1.77/1.55 thf(f111,plain,(
% 1.77/1.55 ((sK3 = sK5) = $false)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f110])).
% 1.77/1.55 thf(f110,plain,(
% 1.77/1.55 (($true => (sK3 = sK5)) = $false)),
% 1.77/1.55 inference(backward_demodulation,[status(thm)],[f102,f109])).
% 1.77/1.55 thf(f193,plain,(
% 1.77/1.55 (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))) => (sK3 = sK5))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f187])).
% 1.77/1.55 thf(f187,plain,(
% 1.77/1.55 (($true => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ sK5)))))))) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK5) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))) => (sK3 = sK5)))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f183,f132])).
% 1.77/1.55 thf(f183,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((subset @ X1 @ (cartprod @ sK2 @ sK4)) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ sK3) => (in @ (kpair @ Y0 @ Y1) @ X1)))))))) => ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK4) => ((in @ (kpair @ Y0 @ Y1) @ X1) => (in @ (kpair @ Y0 @ Y1) @ sK3)))))))) => (sK3 = X1)))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f180])).
% 1.77/1.55 thf(f180,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((subset @ Y0 @ (cartprod @ sK2 @ sK4)) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ sK3) => (in @ (kpair @ Y1 @ Y2) @ Y0)))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => (in @ (kpair @ Y1 @ Y2) @ sK3)))))))) => (sK3 = Y0))))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f119])).
% 1.77/1.55 thf(f119,plain,(
% 1.77/1.55 ((!! @ $i @ (^[Y0 : $i]: ((subset @ Y0 @ (cartprod @ sK2 @ sK4)) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ sK3) => (in @ (kpair @ Y1 @ Y2) @ Y0)))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => (in @ (kpair @ Y1 @ Y2) @ sK3)))))))) => (sK3 = Y0)))))) = $true)),
% 1.77/1.55 inference(boolean_simplification,[status(thm)],[f118])).
% 1.77/1.55 thf(f118,plain,(
% 1.77/1.55 (($true => (!! @ $i @ (^[Y0 : $i]: ((subset @ Y0 @ (cartprod @ sK2 @ sK4)) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ sK3) => (in @ (kpair @ Y1 @ Y2) @ Y0)))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK4) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => (in @ (kpair @ Y1 @ Y2) @ sK3)))))))) => (sK3 = Y0))))))) = $true)),
% 1.77/1.55 inference(superposition,[status(thm)],[f116,f79])).
% 1.77/1.55 thf(f116,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((subset @ X1 @ (cartprod @ X3 @ X2)) => (!! @ $i @ (^[Y0 : $i]: ((subset @ Y0 @ (cartprod @ X3 @ X2)) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X3) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ X1) => (in @ (kpair @ Y1 @ Y2) @ Y0)))))))) => ((!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X3) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ X2) => ((in @ (kpair @ Y1 @ Y2) @ Y0) => (in @ (kpair @ Y1 @ Y2) @ X1)))))))) => (X1 = Y0))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f115])).
% 1.77/1.55 thf(f115,plain,(
% 1.77/1.55 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: ((subset @ X1 @ (cartprod @ Y0 @ X2)) => (!! @ $i @ (^[Y1 : $i]: ((subset @ Y1 @ (cartprod @ Y0 @ X2)) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ X2) => ((in @ (kpair @ Y2 @ Y3) @ X1) => (in @ (kpair @ Y2 @ Y3) @ Y1)))))))) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ X2) => ((in @ (kpair @ Y2 @ Y3) @ Y1) => (in @ (kpair @ Y2 @ Y3) @ X1)))))))) => (X1 = Y1)))))))) @ X3) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f113])).
% 1.77/1.55 thf(f113,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((subset @ X1 @ (cartprod @ Y0 @ X2)) => (!! @ $i @ (^[Y1 : $i]: ((subset @ Y1 @ (cartprod @ Y0 @ X2)) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ X2) => ((in @ (kpair @ Y2 @ Y3) @ X1) => (in @ (kpair @ Y2 @ Y3) @ Y1)))))))) => ((!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ X2) => ((in @ (kpair @ Y2 @ Y3) @ Y1) => (in @ (kpair @ Y2 @ Y3) @ X1)))))))) => (X1 = Y1))))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f112])).
% 1.77/1.55 thf(f112,plain,(
% 1.77/1.55 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((subset @ X1 @ (cartprod @ Y1 @ Y0)) => (!! @ $i @ (^[Y2 : $i]: ((subset @ Y2 @ (cartprod @ Y1 @ Y0)) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((in @ (kpair @ Y3 @ Y4) @ X1) => (in @ (kpair @ Y3 @ Y4) @ Y2)))))))) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => (in @ (kpair @ Y3 @ Y4) @ X1)))))))) => (X1 = Y2)))))))))) @ X2) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f71])).
% 1.77/1.55 thf(f71,plain,(
% 1.77/1.55 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((subset @ X1 @ (cartprod @ Y1 @ Y0)) => (!! @ $i @ (^[Y2 : $i]: ((subset @ Y2 @ (cartprod @ Y1 @ Y0)) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((in @ (kpair @ Y3 @ Y4) @ X1) => (in @ (kpair @ Y3 @ Y4) @ Y2)))))))) => ((!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) => ((in @ (kpair @ Y3 @ Y4) @ Y2) => (in @ (kpair @ Y3 @ Y4) @ X1)))))))) => (X1 = Y2))))))))))) = $true)) )),
% 1.77/1.55 inference(beta_eta_normalization,[status(thm)],[f70])).
% 1.77/1.55 thf(f70,plain,(
% 1.77/1.55 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((subset @ Y0 @ (cartprod @ Y2 @ Y1)) => (!! @ $i @ (^[Y3 : $i]: ((subset @ Y3 @ (cartprod @ Y2 @ Y1)) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y0) => (in @ (kpair @ Y4 @ Y5) @ Y3)))))))) => ((!! @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y2) => (!! @ $i @ (^[Y5 : $i]: ((in @ Y5 @ Y1) => ((in @ (kpair @ Y4 @ Y5) @ Y3) => (in @ (kpair @ Y4 @ Y5) @ Y0)))))))) => (Y0 = Y3)))))))))))) @ X1) = $true)) )),
% 1.77/1.55 inference(pi_clausification,[status(thm)],[f56])).
% 1.77/1.55 % SZS output end Proof for DTF2THF_18641
% 1.77/1.55 % (18750)------------------------------
% 1.77/1.55 % (18750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.77/1.55 % (18750)Termination reason: Refutation
% 1.77/1.55
% 1.77/1.55 % (18750)Memory used [KB]: 6652
% 1.77/1.55 % (18750)Time elapsed: 0.100 s
% 1.77/1.55 % (18750)Instructions burned: 179 (million)
% 1.77/1.55 % (18750)------------------------------
% 1.77/1.55 % (18750)------------------------------
% 1.77/1.55 % (18738)Success in time 0.134 s
%------------------------------------------------------------------------------