%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV303^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n001.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 09:40:47 AM UTC 2026
% Result : Theorem 1.37s 1.16s
% Output : Refutation 1.37s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 8
% Syntax : Number of formulae : 106 ( 39 unt; 0 typ; 4 def)
% Number of atoms : 1324 ( 273 equ; 0 cnn)
% Maximal formula atoms : 14 ( 12 avg)
% Number of connectives : 4483 ( 428 ~; 86 |; 598 &;2430 @)
% ( 4 <=>; 355 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 1732 (1732 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 14 usr; 8 con; 0-2 aty)
% ( 259 !!; 323 ??; 0 @@+; 0 @@-)
% Number of variables : 1022 ( 978 ^; 40 !; 4 ?;1022 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
cNAT: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_1,type,
cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
thf(func_def_2,type,
cZERO: ( $i > $o ) > $o ).
thf(func_def_18,type,
ph1:
!>[X0: $tType] : X0 ).
thf(func_def_19,type,
sK2: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_20,type,
sK3: ( $i > $o ) > $o ).
thf(func_def_22,type,
sK4: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_23,type,
sK5: ( $i > $o ) > $o ).
thf(func_def_25,type,
sK6: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_26,type,
sK7: ( ( $i > $o ) > $o ) > $o ).
thf(f232,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f79,f91,f137,f189,f231]) ).
thf(f231,plain,
( ~ spl0_4
| ~ spl0_8 ),
inference(avatar_contradiction_clause,[status(thm)],[f230]) ).
thf(f230,plain,
( $false
| ~ spl0_4
| ~ spl0_8 ),
inference(trivial_inequality_removal,[status(thm)],[f229]) ).
thf(f229,plain,
( ( $true = $false )
| ~ spl0_4
| ~ spl0_8 ),
inference(boolean_simplification,[status(thm)],[f228]) ).
thf(f228,plain,
( ( $true = ( ~ $true ) )
| ~ spl0_4
| ~ spl0_8 ),
inference(backward_demodulation,[status(thm)],[f227,f226]) ).
thf(f226,plain,
( ( $true
= ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
| ~ spl0_8 ),
inference(boolean_simplification,[status(thm)],[f225]) ).
thf(f225,plain,
( ( $true
= ( $true
& ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ) )
| ~ spl0_8 ),
inference(backward_demodulation,[status(thm)],[f215,f223]) ).
thf(f223,plain,
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK7 @ Y0 )
=> ( sK7
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) )
= $true )
| ~ spl0_8 ),
inference(binary_proxy_clausification,[status(thm)],[f215]) ).
thf(f215,plain,
( ( $true
= ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK7 @ Y0 )
=> ( sK7
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) )
& ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ) )
| ~ spl0_8 ),
inference(binary_proxy_clausification,[status(thm)],[f195]) ).
thf(f195,plain,
( ( $false
= ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK7 @ Y0 )
=> ( sK7
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) )
& ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
=> ( sK7 @ sK5 ) ) )
| ~ spl0_8 ),
inference(beta_eta_normalization,[status(thm)],[f192]) ).
thf(f192,plain,
( ( ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) )
@ sK7 )
= $false )
| ~ spl0_8 ),
inference(sigma_clausification,[status(thm)],[f136]) ).
thf(f136,plain,
( ( $false
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) ) ) )
| ~ spl0_8 ),
inference(avatar_component_clause,[status(thm)],[f135]) ).
thf(f135,definition,
( spl0_8
<=> ( $false
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_8])],[avatar_definition]) ).
thf(f227,plain,
( ( $true
= ( ~ ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ) )
| ~ spl0_4
| ~ spl0_8 ),
inference(boolean_simplification,[status(thm)],[f224]) ).
thf(f224,plain,
( ( $true
= ( ~ ( ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& $true ) ) )
| ~ spl0_4
| ~ spl0_8 ),
inference(backward_demodulation,[status(thm)],[f221,f223]) ).
thf(f221,plain,
( ( $true
= ( ~ ( ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK7 @ Y0 )
=> ( sK7
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_8 ),
inference(boolean_simplification,[status(thm)],[f220]) ).
thf(f220,plain,
( ( $true
= ( ( ( sK7
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK7 @ Y0 )
=> ( sK7
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> $false ) )
| ~ spl0_4
| ~ spl0_8 ),
inference(superposition,[status(thm)],[f120,f214]) ).
thf(f214,plain,
( ( ( sK7 @ sK5 )
= $false )
| ~ spl0_8 ),
inference(binary_proxy_clausification,[status(thm)],[f195]) ).
thf(f120,plain,
( ! [X1: ( ( $i > $o ) > $o ) > $o] :
( $true
= ( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( X1 @ sK5 ) ) )
| ~ spl0_4 ),
inference(beta_eta_normalization,[status(thm)],[f119]) ).
thf(f119,plain,
( ! [X1: ( ( $i > $o ) > $o ) > $o] :
( $true
= ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) )
@ X1 ) )
| ~ spl0_4 ),
inference(pi_clausification,[status(thm)],[f118]) ).
thf(f118,plain,
( ( $true
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) ) )
| ~ spl0_4 ),
inference(boolean_simplification,[status(thm)],[f117]) ).
thf(f117,plain,
( ( $true
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
& $true ) )
| ~ spl0_4 ),
inference(backward_demodulation,[status(thm)],[f110,f115]) ).
thf(f115,plain,
( ( $true
= ( sK2 @ sK5 ) )
| ~ spl0_4 ),
inference(binary_proxy_clausification,[status(thm)],[f110]) ).
thf(f110,plain,
( ( $true
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
& ( sK2 @ sK5 ) ) )
| ~ spl0_4 ),
inference(binary_proxy_clausification,[status(thm)],[f105]) ).
thf(f105,plain,
( ( $false
= ( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
& ( sK2 @ sK5 ) )
=> ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( sK5
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) ) )
| ~ spl0_4 ),
inference(beta_eta_normalization,[status(thm)],[f104]) ).
thf(f104,plain,
( ( $false
= ( ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) )
@ sK5 ) )
| ~ spl0_4 ),
inference(sigma_clausification,[status(thm)],[f78]) ).
thf(f78,plain,
( ( $false
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
| ~ spl0_4 ),
inference(avatar_component_clause,[status(thm)],[f77]) ).
thf(f77,definition,
( spl0_4
<=> ( $false
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_4])],[avatar_definition]) ).
thf(f189,plain,
( ~ spl0_4
| ~ spl0_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f188]) ).
thf(f188,plain,
( $false
| ~ spl0_4
| ~ spl0_7 ),
inference(trivial_inequality_removal,[status(thm)],[f187]) ).
thf(f187,plain,
( ( $true = $false )
| ~ spl0_4
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f186]) ).
thf(f186,plain,
( ( $true = ( ~ $true ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(forward_demodulation,[status(thm)],[f177,f157]) ).
thf(f157,plain,
( ( $true
= ( sK6 @ sK5 ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f156]) ).
thf(f156,plain,
( ( $true
= ( $true
=> ( sK6 @ sK5 ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(superposition,[status(thm)],[f120,f151]) ).
thf(f151,plain,
( ( $true
= ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(binary_proxy_clausification,[status(thm)],[f143]) ).
thf(f143,plain,
( ( ( ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) )
= $false )
| ~ spl0_4
| ~ spl0_7 ),
inference(beta_eta_normalization,[status(thm)],[f142]) ).
thf(f142,plain,
( ( $false
= ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( sK5
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) )
@ sK6 ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(sigma_clausification,[status(thm)],[f140]) ).
thf(f140,plain,
( ( $false
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( sK5
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f139]) ).
thf(f139,plain,
( ( $false
= ( $true
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( sK5
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(backward_demodulation,[status(thm)],[f109,f133]) ).
thf(f133,plain,
( ( $true
= ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) )
| ~ spl0_7 ),
inference(avatar_component_clause,[status(thm)],[f132]) ).
thf(f132,definition,
( spl0_7
<=> ( $true
= ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_7])],[avatar_definition]) ).
thf(f109,plain,
( ( ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( sK5
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
= $false )
| ~ spl0_4 ),
inference(binary_proxy_clausification,[status(thm)],[f105]) ).
thf(f177,plain,
( ( $true
= ( ~ ( sK6 @ sK5 ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f173]) ).
thf(f173,plain,
( ( $true
= ( ( sK6 @ sK5 )
=> $false ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(superposition,[status(thm)],[f169,f153]) ).
thf(f153,plain,
( ( ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
= $false )
| ~ spl0_4
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f152]) ).
thf(f152,plain,
( ( ( $true
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) )
= $false )
| ~ spl0_4
| ~ spl0_7 ),
inference(backward_demodulation,[status(thm)],[f143,f151]) ).
thf(f169,plain,
( ! [X1: ( $i > $o ) > $o] :
( $true
= ( ( sK6 @ X1 )
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( X1
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(beta_eta_normalization,[status(thm)],[f166]) ).
thf(f166,plain,
( ! [X1: ( $i > $o ) > $o] :
( $true
= ( ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) )
@ X1 ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(pi_clausification,[status(thm)],[f154]) ).
thf(f154,plain,
( ( $true
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
| ~ spl0_4
| ~ spl0_7 ),
inference(binary_proxy_clausification,[status(thm)],[f151]) ).
thf(f137,plain,
( spl0_7
| spl0_8
| ~ spl0_4 ),
inference(avatar_split_clause,[status(thm)],[f130,f77,f135,f132]) ).
thf(f130,plain,
( ( $true
= ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) )
| ( $false
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) ) ) )
| ~ spl0_4 ),
inference(binary_proxy_clausification,[status(thm)],[f122]) ).
thf(f122,plain,
( ( $true
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) ) )
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) )
| ~ spl0_4 ),
inference(boolean_simplification,[status(thm)],[f121]) ).
thf(f121,plain,
( ( $true
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK5 ) ) )
=> ( $true
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK5
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) ) )
| ~ spl0_4 ),
inference(superposition,[status(thm)],[f38,f115]) ).
thf(f38,plain,
! [X1: ( $i > $o ) > $o] :
( $true
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ X1 ) ) )
=> ( ( sK2 @ X1 )
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( X1
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( Y1 != Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f37]) ).
thf(f37,plain,
! [X1: ( $i > $o ) > $o] :
( $true
= ( ^ [Y0: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f36]) ).
thf(f36,plain,
( $true
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f35]) ).
thf(f35,plain,
( $true
= ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
& $true ) ),
inference(backward_demodulation,[status(thm)],[f28,f33]) ).
thf(f33,plain,
( $true
= ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f28]) ).
thf(f28,plain,
( $true
= ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f26]) ).
thf(f26,plain,
( ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( sK2 @ Y1 ) )
=> ( ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) ) ) )
=> ( sK2 @ Y0 ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f25]) ).
thf(f25,plain,
( $false
= ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) ) ) )
=> ( Y0 @ Y1 ) ) ) )
@ sK2 ) ),
inference(sigma_clausification,[status(thm)],[f22]) ).
thf(f22,plain,
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) ) ) )
=> ( Y0 @ Y1 ) ) ) ) )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f21]) ).
thf(f21,plain,
( $true
= ( ~ ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i @ Y4 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ^ [Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) )
& ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) ) )
=> ( Y2 @ Y1 ) ) ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f20]) ).
thf(f20,plain,
( $true
= ( ~ ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ^ [Y2: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ( ^ [Y5: ( $i > $o ) > $o,Y6: $i > $o] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( Y5
@ ^ [Y8: $i] :
( ( Y6 @ Y8 )
& ( Y7 != Y8 ) ) )
& ( Y6 @ Y7 ) ) )
@ Y4 ) ) ) )
& ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i
@ ^ [Y5: $i] : ( Y4 @ Y5 ) ) ) )
=> ( Y3 @ Y2 ) ) )
@ Y1 )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ( ^ [Y2: ( $i > $o ) > $o,Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) )
@ Y1 ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i
@ ^ [Y4: $i] : ( Y3 @ Y4 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ( ^ [Y4: ( $i > $o ) > $o,Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) )
@ Y3 ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i
@ ^ [Y4: $i] : ( Y3 @ Y4 ) ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i
@ ^ [Y5: $i] : ( Y4 @ Y5 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ( ^ [Y5: ( $i > $o ) > $o,Y6: $i > $o] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( Y5
@ ^ [Y8: $i] :
( ( Y6 @ Y8 )
& ( Y7 != Y8 ) ) )
& ( Y6 @ Y7 ) ) )
@ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) )
@ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i
@ ^ [Y5: $i] : ( Y4 @ Y5 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ( ^ [Y5: ( $i > $o ) > $o,Y6: $i > $o] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( Y5
@ ^ [Y8: $i] :
( ( Y6 @ Y8 )
& ( Y7 != Y8 ) ) )
& ( Y6 @ Y7 ) ) )
@ Y4 ) ) ) ) )
=> ( Y3
@ ( ^ [Y4: ( $i > $o ) > $o,Y5: $i > $o] :
( ?? @ $i
@ ^ [Y6: $i] :
( ( Y4
@ ^ [Y7: $i] :
( ( Y5 @ Y7 )
& ( Y6 != Y7 ) ) )
& ( Y5 @ Y6 ) ) )
@ Y2 ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( ^ [Y2: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3
@ ( ^ [Y5: ( $i > $o ) > $o,Y6: $i > $o] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( Y5
@ ^ [Y8: $i] :
( ( Y6 @ Y8 )
& ( Y7 != Y8 ) ) )
& ( Y6 @ Y7 ) ) )
@ Y4 ) ) ) )
& ( Y3
@ ^ [Y4: $i > $o] :
~ ( ?? @ $i
@ ^ [Y5: $i] : ( Y4 @ Y5 ) ) ) )
=> ( Y3 @ Y2 ) ) )
@ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f17,f19,f18,f16,f16,f16,f18,f16,f16,f18,f18,f16,f18,f18,f19]) ).
thf(f16,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) )
= cZERO ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) )
= cZERO ),
file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cZERO_def) ).
thf(f18,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
( cSUCC
= ( ^ [X0: ( $i > $o ) > $o,X1: $i > $o] :
? [X2: $i] :
( ( X1 @ X2 )
& ( X0
@ ^ [X3: $i] :
( ( X2 != X3 )
& ( X1 @ X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( cSUCC
= ( ^ [X2: ( $i > $o ) > $o,X0: $i > $o] :
? [X1: $i] :
( ( X0 @ X1 )
& ( X2
@ ^ [X3: $i] :
( ( X1 != X3 )
& ( X0 @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cSUCC_def) ).
thf(f19,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) )
@ Y2 ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f15,f18,f16]) ).
thf(f15,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
( ( ^ [X0: ( $i > $o ) > $o] :
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1 @ cZERO )
& ! [X2: ( $i > $o ) > $o] :
( ( X1 @ X2 )
=> ( X1 @ ( cSUCC @ X2 ) ) ) )
=> ( X1 @ X0 ) ) )
= cNAT ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ^ [X2: ( $i > $o ) > $o] :
! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ( X0 @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) ) )
=> ( X0 @ X2 ) ) )
= cNAT ),
file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cNAT_def) ).
thf(f17,plain,
( $true
= ( ~ ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( cNAT @ Y1 )
=> ( ( Y0 @ Y1 )
=> ( Y0 @ ( cSUCC @ Y1 ) ) ) ) )
& ( Y0 @ cZERO ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0 @ cZERO )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2 @ ( cSUCC @ Y3 ) ) ) ) )
=> ( Y2 @ cZERO ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3 @ ( cSUCC @ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0 @ ( cSUCC @ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3 @ ( cSUCC @ Y4 ) ) ) ) )
=> ( Y3 @ ( cSUCC @ Y2 ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( cNAT @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( $true
= ( ~ ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( cNAT @ Y1 )
=> ( ( Y0 @ Y1 )
=> ( Y0 @ ( cSUCC @ Y1 ) ) ) ) )
& ( Y0 @ cZERO ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0 @ cZERO )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2 @ ( cSUCC @ Y3 ) ) ) ) )
=> ( Y2 @ cZERO ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3 @ ( cSUCC @ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
& ( Y0 @ Y2 ) )
=> ( ( Y0 @ ( cSUCC @ Y2 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y3: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y3 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y4: ( $i > $o ) > $o] :
( ( Y3 @ Y4 )
=> ( Y3 @ ( cSUCC @ Y4 ) ) ) ) )
=> ( Y3 @ ( cSUCC @ Y2 ) ) ) ) ) ) ) )
=> ( ( Y0 @ Y1 )
& ( cNAT @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ( X0 @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( cNAT @ X1 )
=> ( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) ) ) )
=> ! [X2: ( $i > $o ) > $o] :
( ( ( ! [X3: ( $i > $o ) > $o] :
( ( ( X0 @ X3 )
& ! [X4: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X5: ( $i > $o ) > $o] :
( ( X4 @ X5 )
=> ( X4 @ ( cSUCC @ X5 ) ) )
& ( X4 @ cZERO ) )
=> ( X4 @ X3 ) ) )
=> ( ! [X6: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X7: ( $i > $o ) > $o] :
( ( X6 @ X7 )
=> ( X6 @ ( cSUCC @ X7 ) ) )
& ( X6 @ cZERO ) )
=> ( X6 @ ( cSUCC @ X3 ) ) )
& ( X0 @ ( cSUCC @ X3 ) ) ) )
& ! [X8: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X9: ( $i > $o ) > $o] :
( ( X8 @ X9 )
=> ( X8 @ ( cSUCC @ X9 ) ) )
& ( X8 @ cZERO ) )
=> ( X8 @ cZERO ) )
& ( X0 @ cZERO ) )
=> ( ( cNAT @ X2 )
& ( X0 @ X2 ) ) )
=> ( X0 @ X2 ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ( X0 @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( cNAT @ X1 )
=> ( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) ) ) )
=> ! [X4: ( $i > $o ) > $o] :
( ( ( ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
& ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ X1 ) ) )
=> ( ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ ( cSUCC @ X1 ) ) )
& ( X0 @ ( cSUCC @ X1 ) ) ) )
& ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( X5 @ X1 )
=> ( X5 @ ( cSUCC @ X1 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ cZERO ) )
& ( X0 @ cZERO ) )
=> ( ( cNAT @ X4 )
& ( X0 @ X4 ) ) )
=> ( X0 @ X4 ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ( X0 @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( cNAT @ X1 )
=> ( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) ) ) )
=> ! [X4: ( $i > $o ) > $o] :
( ( ( ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
& ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ X1 ) ) )
=> ( ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ ( cSUCC @ X1 ) ) )
& ( X0 @ ( cSUCC @ X1 ) ) ) )
& ! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( X5 @ X1 )
=> ( X5 @ ( cSUCC @ X1 ) ) )
& ( X5 @ cZERO ) )
=> ( X5 @ cZERO ) )
& ( X0 @ cZERO ) )
=> ( ( cNAT @ X4 )
& ( X0 @ X4 ) ) )
=> ( X0 @ X4 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cX6102_C) ).
thf(f91,plain,
~ spl0_3,
inference(avatar_contradiction_clause,[status(thm)],[f90]) ).
thf(f90,plain,
( $false
| ~ spl0_3 ),
inference(trivial_inequality_removal,[status(thm)],[f89]) ).
thf(f89,plain,
( ( $true = $false )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f88]) ).
thf(f88,plain,
( ( $true
= ( $false
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_3 ),
inference(backward_demodulation,[status(thm)],[f85,f87]) ).
thf(f87,plain,
( ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f86]) ).
thf(f86,plain,
( ( $false
= ( $true
=> ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ) )
| ~ spl0_3 ),
inference(backward_demodulation,[status(thm)],[f83,f85]) ).
thf(f83,plain,
( ( ( ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
= $false )
| ~ spl0_3 ),
inference(beta_eta_normalization,[status(thm)],[f82]) ).
thf(f82,plain,
( ( ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
@ sK4 )
= $false )
| ~ spl0_3 ),
inference(sigma_clausification,[status(thm)],[f75]) ).
thf(f75,plain,
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
= $false )
| ~ spl0_3 ),
inference(avatar_component_clause,[status(thm)],[f74]) ).
thf(f74,definition,
( spl0_3
<=> ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_3])],[avatar_definition]) ).
thf(f85,plain,
( ( $true
= ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_3 ),
inference(binary_proxy_clausification,[status(thm)],[f83]) ).
thf(f79,plain,
( spl0_3
| spl0_4 ),
inference(avatar_split_clause,[status(thm)],[f72,f77,f74]) ).
thf(f72,plain,
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
= $false )
| ( $false
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f66]) ).
thf(f66,plain,
( $false
= ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f65]) ).
thf(f65,plain,
( $false
= ( ~ ~ ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f64]) ).
thf(f64,plain,
( ( ~ ( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> $false ) )
= $false ),
inference(boolean_simplification,[status(thm)],[f63]) ).
thf(f63,plain,
( $false
= ( ~ ( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> ( $false
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK3 ) ) ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f62]) ).
thf(f62,plain,
( ( ( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> ( $false
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK3 ) ) ) ) )
=> $false )
= $false ),
inference(backward_demodulation,[status(thm)],[f56,f60]) ).
thf(f60,plain,
( $false
= ( sK2 @ sK3 ) ),
inference(binary_proxy_clausification,[status(thm)],[f56]) ).
thf(f56,plain,
( ( ( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ sK3 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK3 ) ) ) ) )
=> ( sK2 @ sK3 ) )
= $false ),
inference(boolean_simplification,[status(thm)],[f55]) ).
thf(f55,plain,
( $false
= ( ( ( $true
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ sK3 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK3 ) ) ) ) )
=> ( sK2 @ sK3 ) ) ),
inference(forward_demodulation,[status(thm)],[f54,f33]) ).
thf(f54,plain,
( $false
= ( ( ( ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) )
=> ( ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
& ( Y2 != Y3 ) ) )
& ( Y1 @ Y2 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y0
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ sK3 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( Y0 @ sK3 ) ) ) ) )
=> ( sK2 @ sK3 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f53]) ).
thf(f53,plain,
( $false
= ( ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( sK2 @ Y1 ) )
=> ( ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) ) ) )
=> ( sK2 @ Y0 ) )
@ sK3 ) ),
inference(sigma_clausification,[status(thm)],[f27]) ).
thf(f27,plain,
( $false
= ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( sK2 @ Y1 ) )
=> ( ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y2 @ Y4 )
& ( Y3 != Y4 ) ) )
& ( Y2 @ Y3 ) ) ) )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y2: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y4 @ Y6 )
& ( Y5 != Y6 ) ) )
& ( Y4 @ Y5 ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y1
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y2
@ ^ [Y5: $i] :
( ( Y3 @ Y5 )
& ( Y4 != Y5 ) ) )
& ( Y3 @ Y4 ) ) ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) ) )
=> ( Y1 @ Y0 ) ) ) ) )
=> ( sK2 @ Y0 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f26]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEV303^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.32 % Computer : n001.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Tue Feb 24 12:44:35 EST 2026
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.39 ---- Original DTF file ---
% 0.19/0.39 thf(spec,logic,$$dhol).
% 0.19/0.39 %------------------------------------------------------------------------------
% 0.19/0.39 % File : SEV303^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.19/0.39 % Domain : Set Theory
% 0.19/0.39 % Problem : TPS problem from TTTP-NATS-THMS
% 0.19/0.39 % Version : Especial.
% 0.19/0.39 % English :
% 0.19/0.39
% 0.19/0.39 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.19/0.39 % Source : [Bro09]
% 0.19/0.39 % Names : tps_1084 [Bro09]
% 0.19/0.39
% 0.19/0.39 % Status : Theorem
% 0.19/0.39 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.11 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0
% 0.19/0.39 % Syntax : Number of formulae : 7 ( 3 unt; 3 typ; 3 def)
% 0.19/0.39 % Number of atoms : 23 ( 4 equ; 0 cnn)
% 0.19/0.39 % Maximal formula atoms : 14 ( 5 avg)
% 0.19/0.39 % Number of connectives : 65 ( 2 ~; 0 |; 12 &; 37 @)
% 0.19/0.39 % ( 0 <=>; 14 =>; 0 <=; 0 <~>)
% 0.19/0.39 % Maximal formula depth : 18 ( 5 avg)
% 0.19/0.39 % Number of types : 2 ( 0 usr)
% 0.19/0.39 % Number of type conns : 45 ( 45 >; 0 *; 0 +; 0 <<)
% 0.19/0.39 % Number of symbols : 4 ( 3 usr; 0 con; 1-2 aty)
% 0.19/0.39 % Number of variables : 19 ( 5 ^; 12 !; 2 ?; 19 :)
% 0.19/0.39 % SPC : TH0_THM_EQU_NAR
% 0.19/0.39
% 0.19/0.39 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.19/0.39 % project in the Department of Mathematical Sciences at Carnegie
% 0.19/0.39 % Mellon University. Distributed under the Creative Commons copyleft
% 0.19/0.39 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.19/0.39 % Bugfixes : v5.2.0 - Added missing type declarations.
% 0.19/0.39 % : v6.2.0 - Reordered definitions.
% 0.19/0.39 %------------------------------------------------------------------------------
% 0.19/0.39 thf(cNAT_type,type,
% 0.19/0.39 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 0.19/0.39
% 0.19/0.39 thf(cSUCC_type,type,
% 0.19/0.39 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 0.19/0.39
% 0.19/0.39 thf(cZERO_type,type,
% 0.19/0.39 cZERO: ( $i > $o ) > $o ).
% 0.19/0.39
% 0.19/0.39 thf(cZERO_def,definition,
% 0.19/0.39 ( cZERO
% 0.19/0.39 = ( ^ [Xp: $i > $o] :
% 0.19/0.39 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cSUCC_def,definition,
% 0.19/0.39 ( cSUCC
% 0.19/0.39 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 0.19/0.39 ? [Xx: $i] :
% 0.19/0.39 ( ( Xp @ Xx )
% 0.19/0.39 & ( Xn
% 0.19/0.39 @ ^ [Xt: $i] :
% 0.19/0.39 ( ( Xt != Xx )
% 0.19/0.39 & ( Xp @ Xt ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cNAT_def,definition,
% 0.19/0.39 ( cNAT
% 0.19/0.39 = ( ^ [Xn: ( $i > $o ) > $o] :
% 0.19/0.39 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.39 ( ( ( Xp @ cZERO )
% 0.19/0.39 & ! [Xx: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( Xp @ Xx )
% 0.19/0.39 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 0.19/0.39 => ( Xp @ Xn ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cX6102_C,conjecture,
% 0.19/0.39 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.39 ( ( ( Xp @ cZERO )
% 0.19/0.39 & ! [Xx: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( cNAT @ Xx )
% 0.19/0.39 => ( ( Xp @ Xx )
% 0.19/0.39 => ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 0.19/0.39 => ! [Xm: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.39 ( ( ( Xp0 @ cZERO )
% 0.19/0.39 & ! [Xx: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( Xp0 @ Xx )
% 0.19/0.39 => ( Xp0 @ ( cSUCC @ Xx ) ) ) )
% 0.19/0.39 => ( Xp0 @ cZERO ) )
% 0.19/0.39 & ( Xp @ cZERO )
% 0.19/0.39 & ! [Xx: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.39 ( ( ( Xp0 @ cZERO )
% 0.19/0.39 & ! [Xx0: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( Xp0 @ Xx0 )
% 0.19/0.39 => ( Xp0 @ ( cSUCC @ Xx0 ) ) ) )
% 0.19/0.39 => ( Xp0 @ Xx ) )
% 0.19/0.39 & ( Xp @ Xx ) )
% 0.19/0.39 => ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.39 ( ( ( Xp0 @ cZERO )
% 0.19/0.39 & ! [Xx0: ( $i > $o ) > $o] :
% 0.19/0.39 ( ( Xp0 @ Xx0 )
% 0.19/0.39 => ( Xp0 @ ( cSUCC @ Xx0 ) ) ) )
% 0.19/0.39 => ( Xp0 @ ( cSUCC @ Xx ) ) )
% 0.19/0.39 & ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 0.19/0.39 => ( ( cNAT @ Xm )
% 0.19/0.39 & ( Xp @ Xm ) ) )
% 0.19/0.39 => ( Xp @ Xm ) ) ) ).
% 0.19/0.39
% 0.19/0.39 %------------------------------------------------------------------------------
% 0.19/0.39 ------------------------
% 1.22/1.09 ---- Embedded in THF ---
% 1.22/1.09 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.22/1.09 %%% Generated on Tue Feb 24 12:44:36 EST 2026
% 1.22/1.09 %%% using '$$dhol' embedding, version 1.3.0.
% 1.22/1.09 %%% Logic specification used:
% 1.22/1.09 %%% thf(spec, logic, $$dhol).
% 1.22/1.09
% 1.22/1.09 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2DTF16140.p
% See solution above
% 1.22/1.09 ------------------------
% 1.22/1.09 ---- Cleaned THF ---
% 1.22/1.10 thf(cNAT_type,type,
% 1.22/1.10 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 1.22/1.10
% 1.22/1.10 thf(cSUCC_type,type,
% 1.22/1.10 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 1.22/1.10
% 1.22/1.10 thf(cZERO_type,type,
% 1.22/1.10 cZERO: ( $i > $o ) > $o ).
% 1.22/1.10
% 1.22/1.10 thf(cZERO_def,definition,
% 1.22/1.10 ( cZERO
% 1.22/1.10 = ( ^ [Xp: $i > $o] :
% 1.22/1.10 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 1.22/1.10
% 1.22/1.10 thf(cSUCC_def,definition,
% 1.22/1.10 ( cSUCC
% 1.22/1.10 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 1.22/1.10 ? [Xx: $i] :
% 1.22/1.10 ( ( Xp @ Xx )
% 1.22/1.10 & ( Xn
% 1.22/1.10 @ ^ [Xt: $i] :
% 1.22/1.10 ( ( Xt != Xx )
% 1.22/1.10 & ( Xp @ Xt ) ) ) ) ) ) ).
% 1.22/1.10
% 1.22/1.10 thf(cNAT_def,definition,
% 1.22/1.10 ( cNAT
% 1.22/1.10 = ( ^ [Xn: ( $i > $o ) > $o] :
% 1.22/1.10 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 1.22/1.10 ( ( ( Xp @ cZERO )
% 1.22/1.10 & ! [Xx: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( Xp @ Xx )
% 1.22/1.10 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 1.22/1.10 => ( Xp @ Xn ) ) ) ) ).
% 1.22/1.10
% 1.22/1.10 thf(cX6102_C,conjecture,
% 1.22/1.10 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 1.22/1.10 ( ( ( Xp @ cZERO )
% 1.22/1.10 & ! [Xx: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( cNAT @ Xx )
% 1.22/1.10 => ( ( Xp @ Xx )
% 1.22/1.10 => ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 1.22/1.10 => ! [Xm: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 1.22/1.10 ( ( ( Xp0 @ cZERO )
% 1.22/1.10 & ! [Xx: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( Xp0 @ Xx )
% 1.22/1.10 => ( Xp0 @ ( cSUCC @ Xx ) ) ) )
% 1.22/1.10 => ( Xp0 @ cZERO ) )
% 1.22/1.10 & ( Xp @ cZERO )
% 1.22/1.10 & ! [Xx: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 1.22/1.10 ( ( ( Xp0 @ cZERO )
% 1.22/1.10 & ! [Xx0: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( Xp0 @ Xx0 )
% 1.22/1.10 => ( Xp0 @ ( cSUCC @ Xx0 ) ) ) )
% 1.22/1.10 => ( Xp0 @ Xx ) )
% 1.22/1.10 & ( Xp @ Xx ) )
% 1.22/1.10 => ( ! [Xp0: ( ( $i > $o ) > $o ) > $o] :
% 1.22/1.10 ( ( ( Xp0 @ cZERO )
% 1.22/1.10 & ! [Xx0: ( $i > $o ) > $o] :
% 1.22/1.10 ( ( Xp0 @ Xx0 )
% 1.22/1.10 => ( Xp0 @ ( cSUCC @ Xx0 ) ) ) )
% 1.22/1.10 => ( Xp0 @ ( cSUCC @ Xx ) ) )
% 1.22/1.10 & ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 1.22/1.10 => ( ( cNAT @ Xm )
% 1.22/1.10 & ( Xp @ Xm ) ) )
% 1.22/1.10 => ( Xp @ Xm ) ) ) ).
% 1.22/1.10 ------------------------
% 1.22/1.11 % (16267)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16140 for (3000ds/2Mi)
% 1.22/1.11 % (16268)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_16140 for (3000ds/2Mi)
% 1.22/1.11 % (16264)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16140 for (3000ds/183Mi)
% 1.22/1.11 % (16266)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_16140 for (3000ds/27Mi)
% 1.22/1.11 % (16265)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_16140 for (3000ds/4Mi)
% 1.22/1.11 % (16270)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16140 for (3000ds/18Mi)
% 1.22/1.11 % (16267)Instruction limit reached!
% 1.22/1.11 % (16267)------------------------------
% 1.22/1.11 % (16267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.11 % (16267)Termination reason: Unknown
% 1.22/1.11 % (16267)Termination phase: Property scanning
% 1.22/1.11
% 1.22/1.11 % (16267)Memory used [KB]: 1023
% 1.22/1.11 % (16267)Time elapsed: 0.002 s
% 1.22/1.11 % (16267)Instructions burned: 2 (million)
% 1.22/1.11 % (16267)------------------------------
% 1.22/1.11 % (16267)------------------------------
% 1.22/1.11 % (16268)Instruction limit reached!
% 1.22/1.11 % (16268)------------------------------
% 1.22/1.11 % (16268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.11 % (16268)Termination reason: Unknown
% 1.22/1.11 % (16268)Termination phase: Property scanning
% 1.22/1.11
% 1.22/1.11 % (16268)Memory used [KB]: 1023
% 1.22/1.11 % (16268)Time elapsed: 0.002 s
% 1.22/1.11 % (16268)Instructions burned: 2 (million)
% 1.22/1.11 % (16268)------------------------------
% 1.22/1.11 % (16268)------------------------------
% 1.22/1.11 % (16265)Instruction limit reached!
% 1.22/1.11 % (16265)------------------------------
% 1.22/1.11 % (16265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.11 % (16265)Termination reason: Unknown
% 1.22/1.11 % (16265)Termination phase: Property scanning
% 1.22/1.11
% 1.22/1.11 % (16265)Memory used [KB]: 1023
% 1.22/1.11 % (16265)Time elapsed: 0.003 s
% 1.22/1.11 % (16265)Instructions burned: 5 (million)
% 1.22/1.11 % (16265)------------------------------
% 1.22/1.11 % (16265)------------------------------
% 1.22/1.11 % (16270)Instruction limit reached!
% 1.22/1.11 % (16270)------------------------------
% 1.22/1.11 % (16270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.11 % (16270)Termination reason: Unknown
% 1.22/1.11 % (16270)Termination phase: Saturation
% 1.22/1.11
% 1.22/1.11 % (16270)Memory used [KB]: 5628
% 1.22/1.11 % (16270)Time elapsed: 0.007 s
% 1.22/1.11 % (16270)Instructions burned: 19 (million)
% 1.22/1.11 % (16270)------------------------------
% 1.22/1.11 % (16270)------------------------------
% 1.22/1.12 % (16266)Instruction limit reached!
% 1.22/1.12 % (16266)------------------------------
% 1.22/1.12 % (16266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.12 % (16266)Termination reason: Unknown
% 1.22/1.12 % (16266)Termination phase: Saturation
% 1.22/1.12
% 1.22/1.12 % (16266)Memory used [KB]: 5500
% 1.22/1.12 % (16266)Time elapsed: 0.009 s
% 1.22/1.12 % (16266)Instructions burned: 28 (million)
% 1.22/1.12 % (16266)------------------------------
% 1.22/1.12 % (16266)------------------------------
% 1.22/1.12 % (16269)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_16140 for (3000ds/275Mi)
% 1.22/1.12 % (16275)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/3Mi)
% 1.22/1.12 % (16276)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_16140 for (2999ds/37Mi)
% 1.22/1.12 % (16278)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_16140 for (2999ds/15Mi)
% 1.22/1.12 % (16275)Instruction limit reached!
% 1.22/1.12 % (16275)------------------------------
% 1.22/1.12 % (16275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.12 % (16275)Termination reason: Unknown
% 1.22/1.12 % (16275)Termination phase: Property scanning
% 1.22/1.12
% 1.22/1.12 % (16275)Memory used [KB]: 1023
% 1.22/1.12 % (16275)Time elapsed: 0.002 s
% 1.22/1.12 % (16275)Instructions burned: 4 (million)
% 1.22/1.12 % (16275)------------------------------
% 1.22/1.12 % (16275)------------------------------
% 1.22/1.12 % (16269)Refutation not found, incomplete strategy
% 1.22/1.12 % (16269)------------------------------
% 1.22/1.12 % (16269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.22/1.12 % (16269)Termination reason: Refutation not found, incomplete strategy
% 1.22/1.12
% 1.22/1.12
% 1.22/1.12 % (16269)Memory used [KB]: 5500
% 1.22/1.12 % (16269)Time elapsed: 0.007 s
% 1.22/1.12 % (16269)Instructions burned: 7 (million)
% 1.22/1.12 % (16269)------------------------------
% 1.22/1.12 % (16269)------------------------------
% 1.37/1.12 % (16281)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_16140 for (2999ds/3Mi)
% 1.37/1.12 % (16278)Instruction limit reached!
% 1.37/1.12 % (16278)------------------------------
% 1.37/1.12 % (16278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.12 % (16278)Termination reason: Unknown
% 1.37/1.12 % (16278)Termination phase: Saturation
% 1.37/1.12
% 1.37/1.12 % (16278)Memory used [KB]: 5756
% 1.37/1.12 % (16278)Time elapsed: 0.006 s
% 1.37/1.12 % (16278)Instructions burned: 15 (million)
% 1.37/1.12 % (16278)------------------------------
% 1.37/1.12 % (16278)------------------------------
% 1.37/1.12 % (16281)Instruction limit reached!
% 1.37/1.12 % (16281)------------------------------
% 1.37/1.12 % (16281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.12 % (16281)Termination reason: Unknown
% 1.37/1.12 % (16281)Termination phase: Preprocessing 3
% 1.37/1.12
% 1.37/1.12 % (16281)Memory used [KB]: 1023
% 1.37/1.12 % (16281)Time elapsed: 0.002 s
% 1.37/1.12 % (16281)Instructions burned: 3 (million)
% 1.37/1.12 % (16281)------------------------------
% 1.37/1.12 % (16281)------------------------------
% 1.37/1.13 % (16282)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_16140 for (2999ds/1041Mi)
% 1.37/1.13 % (16285)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/7Mi)
% 1.37/1.13 % (16276)Instruction limit reached!
% 1.37/1.13 % (16276)------------------------------
% 1.37/1.13 % (16276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.13 % (16276)Termination reason: Unknown
% 1.37/1.13 % (16276)Termination phase: Saturation
% 1.37/1.13
% 1.37/1.13 % (16276)Memory used [KB]: 5628
% 1.37/1.13 % (16276)Time elapsed: 0.012 s
% 1.37/1.13 % (16276)Instructions burned: 39 (million)
% 1.37/1.13 % (16276)------------------------------
% 1.37/1.13 % (16276)------------------------------
% 1.37/1.13 % (16285)Instruction limit reached!
% 1.37/1.13 % (16285)------------------------------
% 1.37/1.13 % (16285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.13 % (16285)Termination reason: Unknown
% 1.37/1.13 % (16285)Termination phase: Saturation
% 1.37/1.13
% 1.37/1.13 % (16285)Memory used [KB]: 1023
% 1.37/1.13 % (16285)Time elapsed: 0.003 s
% 1.37/1.13 % (16285)Instructions burned: 7 (million)
% 1.37/1.13 % (16285)------------------------------
% 1.37/1.13 % (16285)------------------------------
% 1.37/1.13 % (16290)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_16140 for (2999ds/3Mi)
% 1.37/1.13 % (16291)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_16140 for (2999ds/3Mi)
% 1.37/1.13 % (16290)Instruction limit reached!
% 1.37/1.13 % (16290)------------------------------
% 1.37/1.13 % (16290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.13 % (16290)Termination reason: Unknown
% 1.37/1.13 % (16290)Termination phase: Property scanning
% 1.37/1.13
% 1.37/1.13 % (16290)Memory used [KB]: 1023
% 1.37/1.13 % (16290)Time elapsed: 0.002 s
% 1.37/1.13 % (16290)Instructions burned: 3 (million)
% 1.37/1.13 % (16290)------------------------------
% 1.37/1.13 % (16290)------------------------------
% 1.37/1.13 % (16291)Instruction limit reached!
% 1.37/1.13 % (16291)------------------------------
% 1.37/1.13 % (16291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.13 % (16291)Termination reason: Unknown
% 1.37/1.13 % (16291)Termination phase: Preprocessing 3
% 1.37/1.13
% 1.37/1.13 % (16291)Memory used [KB]: 1023
% 1.37/1.13 % (16291)Time elapsed: 0.002 s
% 1.37/1.13 % (16291)Instructions burned: 3 (million)
% 1.37/1.13 % (16291)------------------------------
% 1.37/1.13 % (16291)------------------------------
% 1.37/1.14 % (16296)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/7Mi)
% 1.37/1.14 % (16297)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/3Mi)
% 1.37/1.14 % (16287)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/16Mi)
% 1.37/1.14 % (16296)Instruction limit reached!
% 1.37/1.14 % (16296)------------------------------
% 1.37/1.14 % (16296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.14 % (16296)Termination reason: Unknown
% 1.37/1.14 % (16296)Termination phase: Saturation
% 1.37/1.14
% 1.37/1.14 % (16296)Memory used [KB]: 5500
% 1.37/1.14 % (16296)Time elapsed: 0.003 s
% 1.37/1.14 % (16296)Instructions burned: 7 (million)
% 1.37/1.14 % (16296)------------------------------
% 1.37/1.14 % (16296)------------------------------
% 1.37/1.14 % (16297)Instruction limit reached!
% 1.37/1.14 % (16297)------------------------------
% 1.37/1.14 % (16297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.14 % (16297)Termination reason: Unknown
% 1.37/1.14 % (16297)Termination phase: Property scanning
% 1.37/1.14
% 1.37/1.14 % (16297)Memory used [KB]: 1023
% 1.37/1.14 % (16297)Time elapsed: 0.002 s
% 1.37/1.14 % (16297)Instructions burned: 4 (million)
% 1.37/1.14 % (16297)------------------------------
% 1.37/1.14 % (16297)------------------------------
% 1.37/1.14 % (16300)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/4Mi)
% 1.37/1.14 % (16301)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/18Mi)
% 1.37/1.14 % (16300)Instruction limit reached!
% 1.37/1.14 % (16300)------------------------------
% 1.37/1.14 % (16300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.14 % (16300)Termination reason: Unknown
% 1.37/1.14 % (16300)Termination phase: Function definition elimination
% 1.37/1.14
% 1.37/1.14 % (16300)Memory used [KB]: 1023
% 1.37/1.14 % (16300)Time elapsed: 0.002 s
% 1.37/1.14 % (16300)Instructions burned: 4 (million)
% 1.37/1.14 % (16300)------------------------------
% 1.37/1.14 % (16300)------------------------------
% 1.37/1.15 % (16287)Instruction limit reached!
% 1.37/1.15 % (16287)------------------------------
% 1.37/1.15 % (16287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.15 % (16287)Termination reason: Unknown
% 1.37/1.15 % (16287)Termination phase: Saturation
% 1.37/1.15
% 1.37/1.15 % (16287)Memory used [KB]: 5756
% 1.37/1.15 % (16287)Time elapsed: 0.012 s
% 1.37/1.15 % (16287)Instructions burned: 16 (million)
% 1.37/1.15 % (16287)------------------------------
% 1.37/1.15 % (16287)------------------------------
% 1.37/1.15 % (16301)Instruction limit reached!
% 1.37/1.15 % (16301)------------------------------
% 1.37/1.15 % (16301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.15 % (16301)Termination reason: Unknown
% 1.37/1.15 % (16301)Termination phase: Saturation
% 1.37/1.15
% 1.37/1.15 % (16301)Memory used [KB]: 5628
% 1.37/1.15 % (16301)Time elapsed: 0.007 s
% 1.37/1.15 % (16301)Instructions burned: 18 (million)
% 1.37/1.15 % (16301)------------------------------
% 1.37/1.15 % (16301)------------------------------
% 1.37/1.15 % (16304)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/710Mi)
% 1.37/1.15 % (16305)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16140 for (2999ds/6Mi)
% 1.37/1.15 % (16282)First to succeed.
% 1.37/1.15 % (16305)Instruction limit reached!
% 1.37/1.15 % (16305)------------------------------
% 1.37/1.15 % (16305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.15 % (16305)Termination reason: Unknown
% 1.37/1.15 % (16305)Termination phase: Saturation
% 1.37/1.15
% 1.37/1.15 % (16305)Memory used [KB]: 5500
% 1.37/1.15 % (16305)Time elapsed: 0.003 s
% 1.37/1.15 % (16305)Instructions burned: 7 (million)
% 1.37/1.15 % (16305)------------------------------
% 1.37/1.15 % (16305)------------------------------
% 1.37/1.15 % (16307)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_16140 for (2999ds/902Mi)
% 1.37/1.16 % (16307)Refutation not found, incomplete strategy
% 1.37/1.16 % (16307)------------------------------
% 1.37/1.16 % (16307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.16 % (16307)Termination reason: Refutation not found, incomplete strategy
% 1.37/1.16
% 1.37/1.16
% 1.37/1.16 % (16307)Memory used [KB]: 5500
% 1.37/1.16 % (16307)Time elapsed: 0.003 s
% 1.37/1.16 % (16307)Instructions burned: 6 (million)
% 1.37/1.16 % (16307)------------------------------
% 1.37/1.16 % (16307)------------------------------
% 1.37/1.16 % (16282)Refutation found. Thanks to Tanya!
% 1.37/1.16 % SZS status Theorem for DTF2THF_16140
% 1.37/1.16 % SZS output start Proof for DTF2THF_16140
% 1.37/1.16 thf(func_def_0, type, cNAT: (($i > $o) > $o) > $o).
% 1.37/1.16 thf(func_def_1, type, cSUCC: (($i > $o) > $o) > ($i > $o) > $o).
% 1.37/1.16 thf(func_def_2, type, cZERO: ($i > $o) > $o).
% 1.37/1.16 thf(func_def_18, type, ph1: !>[X0: $tType]:(X0)).
% 1.37/1.16 thf(func_def_19, type, sK2: (($i > $o) > $o) > $o).
% 1.37/1.16 thf(func_def_20, type, sK3: ($i > $o) > $o).
% 1.37/1.16 thf(func_def_22, type, sK4: (($i > $o) > $o) > $o).
% 1.37/1.16 thf(func_def_23, type, sK5: ($i > $o) > $o).
% 1.37/1.16 thf(func_def_25, type, sK6: (($i > $o) > $o) > $o).
% 1.37/1.16 thf(func_def_26, type, sK7: (($i > $o) > $o) > $o).
% 1.37/1.16 thf(f232,plain,(
% 1.37/1.16 $false),
% 1.37/1.16 inference(avatar_sat_refutation,[status(thm)],[f79,f91,f137,f189,f231])).
% 1.37/1.16 thf(f231,plain,(
% 1.37/1.16 ~spl0_4 | ~spl0_8),
% 1.37/1.16 inference(avatar_contradiction_clause,[status(thm)],[f230])).
% 1.37/1.16 thf(f230,plain,(
% 1.37/1.16 $false | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(trivial_inequality_removal,[status(thm)],[f229])).
% 1.37/1.16 thf(f229,plain,(
% 1.37/1.16 ($true = $false) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f228])).
% 1.37/1.16 thf(f228,plain,(
% 1.37/1.16 ($true = (~ $true)) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f227,f226])).
% 1.37/1.16 thf(f226,plain,(
% 1.37/1.16 ($true = (sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) | ~spl0_8),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f225])).
% 1.37/1.16 thf(f225,plain,(
% 1.37/1.16 ($true = ($true & (sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))) | ~spl0_8),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f215,f223])).
% 1.37/1.16 thf(f223,plain,(
% 1.37/1.16 ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK7 @ Y0) => (sK7 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))) = $true) | ~spl0_8),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f215])).
% 1.37/1.16 thf(f215,plain,(
% 1.37/1.16 ($true = ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK7 @ Y0) => (sK7 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))) & (sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))) | ~spl0_8),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f195])).
% 1.37/1.16 thf(f195,plain,(
% 1.37/1.16 ($false = (((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK7 @ Y0) => (sK7 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))) & (sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) => (sK7 @ sK5))) | ~spl0_8),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f192])).
% 1.37/1.16 thf(f192,plain,(
% 1.37/1.16 (((^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5))) @ sK7) = $false) | ~spl0_8),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f136])).
% 1.37/1.16 thf(f136,plain,(
% 1.37/1.16 ($false = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5))))) | ~spl0_8),
% 1.37/1.16 inference(avatar_component_clause,[status(thm)],[f135])).
% 1.37/1.16 thf(f135,definition,(
% 1.37/1.16 spl0_8 <=> ($false = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5)))))),
% 1.37/1.16 introduced(definition,[new_symbols(naming,[spl0_8])],[avatar_definition])).
% 1.37/1.16 thf(f227,plain,(
% 1.37/1.16 ($true = (~ (sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f224])).
% 1.37/1.16 thf(f224,plain,(
% 1.37/1.16 ($true = (~ ((sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & $true))) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f221,f223])).
% 1.37/1.16 thf(f221,plain,(
% 1.37/1.16 ($true = (~ ((sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK7 @ Y0) => (sK7 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))))) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f220])).
% 1.37/1.16 thf(f220,plain,(
% 1.37/1.16 ($true = (((sK7 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK7 @ Y0) => (sK7 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) => $false)) | (~spl0_4 | ~spl0_8)),
% 1.37/1.16 inference(superposition,[status(thm)],[f120,f214])).
% 1.37/1.16 thf(f214,plain,(
% 1.37/1.16 ((sK7 @ sK5) = $false) | ~spl0_8),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f195])).
% 1.37/1.16 thf(f120,plain,(
% 1.37/1.16 ( ! [X1 : (($i > $o) > $o) > $o] : (($true = (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) => (X1 @ sK5)))) ) | ~spl0_4),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f119])).
% 1.37/1.16 thf(f119,plain,(
% 1.37/1.16 ( ! [X1 : (($i > $o) > $o) > $o] : (($true = ((^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ sK5))) @ X1))) ) | ~spl0_4),
% 1.37/1.16 inference(pi_clausification,[status(thm)],[f118])).
% 1.37/1.16 thf(f118,plain,(
% 1.37/1.16 ($true = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ sK5))))) | ~spl0_4),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f117])).
% 1.37/1.16 thf(f117,plain,(
% 1.37/1.16 ($true = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ sK5)))) & $true)) | ~spl0_4),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f110,f115])).
% 1.37/1.16 thf(f115,plain,(
% 1.37/1.16 ($true = (sK2 @ sK5)) | ~spl0_4),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f110])).
% 1.37/1.16 thf(f110,plain,(
% 1.37/1.16 ($true = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ sK5)))) & (sK2 @ sK5))) | ~spl0_4),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f105])).
% 1.37/1.16 thf(f105,plain,(
% 1.37/1.16 ($false = (((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ sK5)))) & (sK2 @ sK5)) => ((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((sK5 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))))) | ~spl0_4),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f104])).
% 1.37/1.16 thf(f104,plain,(
% 1.37/1.16 ($false = ((^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))) @ sK5)) | ~spl0_4),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f78])).
% 1.37/1.16 thf(f78,plain,(
% 1.37/1.16 ($false = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) | ~spl0_4),
% 1.37/1.16 inference(avatar_component_clause,[status(thm)],[f77])).
% 1.37/1.16 thf(f77,definition,(
% 1.37/1.16 spl0_4 <=> ($false = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))))))))),
% 1.37/1.16 introduced(definition,[new_symbols(naming,[spl0_4])],[avatar_definition])).
% 1.37/1.16 thf(f189,plain,(
% 1.37/1.16 ~spl0_4 | ~spl0_7),
% 1.37/1.16 inference(avatar_contradiction_clause,[status(thm)],[f188])).
% 1.37/1.16 thf(f188,plain,(
% 1.37/1.16 $false | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(trivial_inequality_removal,[status(thm)],[f187])).
% 1.37/1.16 thf(f187,plain,(
% 1.37/1.16 ($true = $false) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f186])).
% 1.37/1.16 thf(f186,plain,(
% 1.37/1.16 ($true = (~ $true)) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(forward_demodulation,[status(thm)],[f177,f157])).
% 1.37/1.16 thf(f157,plain,(
% 1.37/1.16 ($true = (sK6 @ sK5)) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f156])).
% 1.37/1.16 thf(f156,plain,(
% 1.37/1.16 ($true = ($true => (sK6 @ sK5))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(superposition,[status(thm)],[f120,f151])).
% 1.37/1.16 thf(f151,plain,(
% 1.37/1.16 ($true = ((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f143])).
% 1.37/1.16 thf(f143,plain,(
% 1.37/1.16 ((((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))) = $false) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f142])).
% 1.37/1.16 thf(f142,plain,(
% 1.37/1.16 ($false = ((^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((sK5 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))) @ sK6)) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f140])).
% 1.37/1.16 thf(f140,plain,(
% 1.37/1.16 ($false = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((sK5 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f139])).
% 1.37/1.16 thf(f139,plain,(
% 1.37/1.16 ($false = ($true & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((sK5 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f109,f133])).
% 1.37/1.16 thf(f133,plain,(
% 1.37/1.16 ($true = (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))) | ~spl0_7),
% 1.37/1.16 inference(avatar_component_clause,[status(thm)],[f132])).
% 1.37/1.16 thf(f132,definition,(
% 1.37/1.16 spl0_7 <=> ($true = (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))))),
% 1.37/1.16 introduced(definition,[new_symbols(naming,[spl0_7])],[avatar_definition])).
% 1.37/1.16 thf(f109,plain,(
% 1.37/1.16 (((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((sK5 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) = $false) | ~spl0_4),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f105])).
% 1.37/1.16 thf(f177,plain,(
% 1.37/1.16 ($true = (~ (sK6 @ sK5))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f173])).
% 1.37/1.16 thf(f173,plain,(
% 1.37/1.16 ($true = ((sK6 @ sK5) => $false)) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(superposition,[status(thm)],[f169,f153])).
% 1.37/1.16 thf(f153,plain,(
% 1.37/1.16 ((sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))) = $false) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f152])).
% 1.37/1.16 thf(f152,plain,(
% 1.37/1.16 (($true => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))) = $false) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f143,f151])).
% 1.37/1.16 thf(f169,plain,(
% 1.37/1.16 ( ! [X1 : ($i > $o) > $o] : (($true = ((sK6 @ X1) => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((X1 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))))) ) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f166])).
% 1.37/1.16 thf(f166,plain,(
% 1.37/1.16 ( ! [X1 : ($i > $o) > $o] : (($true = ((^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))) @ X1))) ) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(pi_clausification,[status(thm)],[f154])).
% 1.37/1.16 thf(f154,plain,(
% 1.37/1.16 ($true = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) | (~spl0_4 | ~spl0_7)),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f151])).
% 1.37/1.16 thf(f137,plain,(
% 1.37/1.16 spl0_7 | spl0_8 | ~spl0_4),
% 1.37/1.16 inference(avatar_split_clause,[status(thm)],[f130,f77,f135,f132])).
% 1.37/1.16 thf(f130,plain,(
% 1.37/1.16 ($true = (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))) | ($false = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5))))) | ~spl0_4),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f122])).
% 1.37/1.16 thf(f122,plain,(
% 1.37/1.16 ($true = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5)))) => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))))) | ~spl0_4),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f121])).
% 1.37/1.16 thf(f121,plain,(
% 1.37/1.16 ($true = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK5)))) => ($true => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK5 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1))))))))) | ~spl0_4),
% 1.37/1.16 inference(superposition,[status(thm)],[f38,f115])).
% 1.37/1.16 thf(f38,plain,(
% 1.37/1.16 ( ! [X1 : ($i > $o) > $o] : (($true = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ X1)))) => ((sK2 @ X1) => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((X1 @ (^[Y2 : $i]: ((Y0 @ Y2) & (~ (Y1 = Y2))))) & (Y0 @ Y1)))))))))) )),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f37])).
% 1.37/1.16 thf(f37,plain,(
% 1.37/1.16 ( ! [X1 : ($i > $o) > $o] : (($true = ((^[Y0 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))) @ X1))) )),
% 1.37/1.16 inference(pi_clausification,[status(thm)],[f36])).
% 1.37/1.16 thf(f36,plain,(
% 1.37/1.16 ($true = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))))),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f35])).
% 1.37/1.16 thf(f35,plain,(
% 1.37/1.16 ($true = ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) & $true))),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f28,f33])).
% 1.37/1.16 thf(f33,plain,(
% 1.37/1.16 ($true = (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f28])).
% 1.37/1.16 thf(f28,plain,(
% 1.37/1.16 ($true = ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) & (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))))),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f26])).
% 1.37/1.16 thf(f26,plain,(
% 1.37/1.16 ((((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) & (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) => (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ Y1)))) & (sK2 @ Y1)) => ((sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y1 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))))))) => ((sK2 @ Y0) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))))) => (sK2 @ Y0))))) = $false)),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f25])).
% 1.37/1.16 thf(f25,plain,(
% 1.37/1.16 ($false = ((^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y2 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))))))) => ((Y0 @ Y1) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))))) => (Y0 @ Y1)))))) @ sK2))),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f22])).
% 1.37/1.16 thf(f22,plain,(
% 1.37/1.16 ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y2 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))))))) => ((Y0 @ Y1) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))))) => (Y0 @ Y1))))))) = $false)),
% 1.37/1.16 inference(not_proxy_clausification,[status(thm)],[f21])).
% 1.37/1.16 thf(f21,plain,(
% 1.37/1.16 ($true = (~ (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ Y4)))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6)))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y2 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))))))) => ((Y0 @ Y1) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))))) & (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3))))) => (Y2 @ Y1)))))) => (Y0 @ Y1)))))))))),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f20])).
% 1.37/1.16 thf(f20,plain,(
% 1.37/1.16 ($true = (~ (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((^[Y2 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ ((^[Y5 : ($i > $o) > $o]: ((^[Y6 : $i > $o]: (?? @ $i @ (^[Y7 : $i]: ((Y5 @ (^[Y8 : $i]: ((Y6 @ Y8) & (~ (Y7 = Y8))))) & (Y6 @ Y7))))))) @ Y4))))) & (Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ (^[Y5 : $i]: (Y4 @ Y5))))))) => (Y3 @ Y2))))) @ Y1) => ((Y0 @ Y1) => (Y0 @ ((^[Y2 : ($i > $o) > $o]: ((^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))) @ Y1)))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ (^[Y2 : $i]: (Y1 @ Y2))))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ (^[Y4 : $i]: (Y3 @ Y4)))))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ ((^[Y4 : ($i > $o) > $o]: ((^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6))))))) @ Y3)))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ (^[Y4 : $i]: (Y3 @ Y4)))))))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ (^[Y5 : $i]: (Y4 @ Y5)))))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ ((^[Y5 : ($i > $o) > $o]: ((^[Y6 : $i > $o]: (?? @ $i @ (^[Y7 : $i]: ((Y5 @ (^[Y8 : $i]: ((Y6 @ Y8) & (~ (Y7 = Y8))))) & (Y6 @ Y7))))))) @ Y4)))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))) @ Y2)) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ (^[Y5 : $i]: (Y4 @ Y5)))))) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ ((^[Y5 : ($i > $o) > $o]: ((^[Y6 : $i > $o]: (?? @ $i @ (^[Y7 : $i]: ((Y5 @ (^[Y8 : $i]: ((Y6 @ Y8) & (~ (Y7 = Y8))))) & (Y6 @ Y7))))))) @ Y4)))))) => (Y3 @ ((^[Y4 : ($i > $o) > $o]: ((^[Y5 : $i > $o]: (?? @ $i @ (^[Y6 : $i]: ((Y4 @ (^[Y7 : $i]: ((Y5 @ Y7) & (~ (Y6 = Y7))))) & (Y5 @ Y6))))))) @ Y2)))))))))) => ((Y0 @ Y1) & ((^[Y2 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ ((^[Y5 : ($i > $o) > $o]: ((^[Y6 : $i > $o]: (?? @ $i @ (^[Y7 : $i]: ((Y5 @ (^[Y8 : $i]: ((Y6 @ Y8) & (~ (Y7 = Y8))))) & (Y6 @ Y7))))))) @ Y4))))) & (Y3 @ (^[Y4 : $i > $o]: (~ (?? @ $i @ (^[Y5 : $i]: (Y4 @ Y5))))))) => (Y3 @ Y2))))) @ Y1))) => (Y0 @ Y1)))))))))),
% 1.37/1.16 inference(definition_unfolding,[status(thm)],[f17,f19,f18,f16,f16,f16,f18,f16,f16,f18,f18,f16,f18,f18,f19])).
% 1.37/1.16 thf(f16,plain,(
% 1.37/1.16 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.37/1.16 inference(cnf_transformation,[status(thm)],[f14])).
% 1.37/1.16 thf(f14,plain,(
% 1.37/1.16 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.37/1.16 inference(fool_elimination,[status(thm)],[f13])).
% 1.37/1.16 thf(f13,plain,(
% 1.37/1.16 ((^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))) = cZERO)),
% 1.37/1.16 inference(rectify,[status(thm)],[f1])).
% 1.37/1.16 thf(f1,axiom,(
% 1.37/1.16 ((^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))) = cZERO)),
% 1.37/1.16 file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cZERO_def)).
% 1.37/1.16 thf(f18,plain,(
% 1.37/1.16 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))),
% 1.37/1.16 inference(cnf_transformation,[status(thm)],[f10])).
% 1.37/1.16 thf(f10,plain,(
% 1.37/1.16 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))),
% 1.37/1.16 inference(fool_elimination,[status(thm)],[f9])).
% 1.37/1.16 thf(f9,plain,(
% 1.37/1.16 (cSUCC = (^[X0 : ($i > $o) > $o, X1 : $i > $o] : (? [X2 : $i] : ((X1 @ X2) & (X0 @ (^[X3 : $i] : ((X2 != X3) & (X1 @ X3))))))))),
% 1.37/1.16 inference(rectify,[status(thm)],[f2])).
% 1.37/1.16 thf(f2,axiom,(
% 1.37/1.16 (cSUCC = (^[X2 : ($i > $o) > $o, X0 : $i > $o] : (? [X1 : $i] : ((X0 @ X1) & (X2 @ (^[X3 : $i] : ((X1 != X3) & (X0 @ X3))))))))),
% 1.37/1.16 file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cSUCC_def)).
% 1.37/1.16 thf(f19,plain,(
% 1.37/1.16 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5))))))) @ Y2))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3))))))) => (Y1 @ Y0))))))),
% 1.37/1.16 inference(definition_unfolding,[status(thm)],[f15,f18,f16])).
% 1.37/1.16 thf(f15,plain,(
% 1.37/1.16 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))))),
% 1.37/1.16 inference(cnf_transformation,[status(thm)],[f12])).
% 1.37/1.16 thf(f12,plain,(
% 1.37/1.16 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))))),
% 1.37/1.16 inference(fool_elimination,[status(thm)],[f11])).
% 1.37/1.16 thf(f11,plain,(
% 1.37/1.16 ((^[X0 : ($i > $o) > $o] : (! [X1 : (($i > $o) > $o) > $o] : (((X1 @ cZERO) & ! [X2 : ($i > $o) > $o] : ((X1 @ X2) => (X1 @ (cSUCC @ X2)))) => (X1 @ X0)))) = cNAT)),
% 1.37/1.16 inference(rectify,[status(thm)],[f3])).
% 1.37/1.16 thf(f3,axiom,(
% 1.37/1.16 ((^[X2 : ($i > $o) > $o] : (! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((X0 @ X1) => (X0 @ (cSUCC @ X1)))) => (X0 @ X2)))) = cNAT)),
% 1.37/1.16 file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cNAT_def)).
% 1.37/1.16 thf(f17,plain,(
% 1.37/1.16 ($true = (~ (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((cNAT @ Y1) => ((Y0 @ Y1) => (Y0 @ (cSUCC @ Y1)))))) & (Y0 @ cZERO)) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ cZERO) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (cSUCC @ Y3)))))) => (Y2 @ cZERO))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (cSUCC @ Y4)))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ (cSUCC @ Y2)) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (cSUCC @ Y4)))))) => (Y3 @ (cSUCC @ Y2)))))))))) => ((Y0 @ Y1) & (cNAT @ Y1))) => (Y0 @ Y1)))))))))),
% 1.37/1.16 inference(cnf_transformation,[status(thm)],[f8])).
% 1.37/1.16 thf(f8,plain,(
% 1.37/1.16 ($true = (~ (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((cNAT @ Y1) => ((Y0 @ Y1) => (Y0 @ (cSUCC @ Y1)))))) & (Y0 @ cZERO)) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ cZERO) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (cSUCC @ Y3)))))) => (Y2 @ cZERO))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (cSUCC @ Y4)))))) => (Y3 @ Y2)))) & (Y0 @ Y2)) => ((Y0 @ (cSUCC @ Y2)) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y3 : (($i > $o) > $o) > $o]: (((Y3 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y4 : ($i > $o) > $o]: ((Y3 @ Y4) => (Y3 @ (cSUCC @ Y4)))))) => (Y3 @ (cSUCC @ Y2)))))))))) => ((Y0 @ Y1) & (cNAT @ Y1))) => (Y0 @ Y1)))))))))),
% 1.37/1.16 inference(fool_elimination,[status(thm)],[f7])).
% 1.37/1.16 thf(f7,plain,(
% 1.37/1.16 ~! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X2 : ($i > $o) > $o] : (((! [X3 : ($i > $o) > $o] : (((X0 @ X3) & ! [X4 : (($i > $o) > $o) > $o] : ((! [X5 : ($i > $o) > $o] : ((X4 @ X5) => (X4 @ (cSUCC @ X5))) & (X4 @ cZERO)) => (X4 @ X3))) => (! [X6 : (($i > $o) > $o) > $o] : ((! [X7 : ($i > $o) > $o] : ((X6 @ X7) => (X6 @ (cSUCC @ X7))) & (X6 @ cZERO)) => (X6 @ (cSUCC @ X3))) & (X0 @ (cSUCC @ X3)))) & ! [X8 : (($i > $o) > $o) > $o] : ((! [X9 : ($i > $o) > $o] : ((X8 @ X9) => (X8 @ (cSUCC @ X9))) & (X8 @ cZERO)) => (X8 @ cZERO)) & (X0 @ cZERO)) => ((cNAT @ X2) & (X0 @ X2))) => (X0 @ X2)))),
% 1.37/1.16 inference(rectify,[status(thm)],[f5])).
% 1.37/1.16 thf(f5,negated_conjecture,(
% 1.37/1.16 ~! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X4 : ($i > $o) > $o] : (((! [X1 : ($i > $o) > $o] : (((X0 @ X1) & ! [X5 : (($i > $o) > $o) > $o] : ((! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6))) & (X5 @ cZERO)) => (X5 @ X1))) => (! [X5 : (($i > $o) > $o) > $o] : ((! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6))) & (X5 @ cZERO)) => (X5 @ (cSUCC @ X1))) & (X0 @ (cSUCC @ X1)))) & ! [X5 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : ((X5 @ X1) => (X5 @ (cSUCC @ X1))) & (X5 @ cZERO)) => (X5 @ cZERO)) & (X0 @ cZERO)) => ((cNAT @ X4) & (X0 @ X4))) => (X0 @ X4)))),
% 1.37/1.16 inference(negated_conjecture,[status(cth)],[f4])).
% 1.37/1.16 thf(f4,conjecture,(
% 1.37/1.16 ! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X4 : ($i > $o) > $o] : (((! [X1 : ($i > $o) > $o] : (((X0 @ X1) & ! [X5 : (($i > $o) > $o) > $o] : ((! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6))) & (X5 @ cZERO)) => (X5 @ X1))) => (! [X5 : (($i > $o) > $o) > $o] : ((! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6))) & (X5 @ cZERO)) => (X5 @ (cSUCC @ X1))) & (X0 @ (cSUCC @ X1)))) & ! [X5 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : ((X5 @ X1) => (X5 @ (cSUCC @ X1))) & (X5 @ cZERO)) => (X5 @ cZERO)) & (X0 @ cZERO)) => ((cNAT @ X4) & (X0 @ X4))) => (X0 @ X4)))),
% 1.37/1.16 file('/export/starexec/sandbox2/tmp/tmp.nBbtWJALOu/DTF2THF_16140.p',cX6102_C)).
% 1.37/1.16 thf(f91,plain,(
% 1.37/1.16 ~spl0_3),
% 1.37/1.16 inference(avatar_contradiction_clause,[status(thm)],[f90])).
% 1.37/1.16 thf(f90,plain,(
% 1.37/1.16 $false | ~spl0_3),
% 1.37/1.16 inference(trivial_inequality_removal,[status(thm)],[f89])).
% 1.37/1.16 thf(f89,plain,(
% 1.37/1.16 ($true = $false) | ~spl0_3),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f88])).
% 1.37/1.16 thf(f88,plain,(
% 1.37/1.16 ($true = ($false & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))))) | ~spl0_3),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f85,f87])).
% 1.37/1.16 thf(f87,plain,(
% 1.37/1.16 ((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ~spl0_3),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f86])).
% 1.37/1.16 thf(f86,plain,(
% 1.37/1.16 ($false = ($true => (sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))) | ~spl0_3),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f83,f85])).
% 1.37/1.16 thf(f83,plain,(
% 1.37/1.16 ((((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))))))) => (sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) = $false) | ~spl0_3),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f82])).
% 1.37/1.16 thf(f82,plain,(
% 1.37/1.16 (((^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))) @ sK4) = $false) | ~spl0_3),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f75])).
% 1.37/1.16 thf(f75,plain,(
% 1.37/1.16 ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) = $false) | ~spl0_3),
% 1.37/1.16 inference(avatar_component_clause,[status(thm)],[f74])).
% 1.37/1.16 thf(f74,definition,(
% 1.37/1.16 spl0_3 <=> ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) = $false)),
% 1.37/1.16 introduced(definition,[new_symbols(naming,[spl0_3])],[avatar_definition])).
% 1.37/1.16 thf(f85,plain,(
% 1.37/1.16 ($true = ((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2))))))))))) | ~spl0_3),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f83])).
% 1.37/1.16 thf(f79,plain,(
% 1.37/1.16 spl0_3 | spl0_4),
% 1.37/1.16 inference(avatar_split_clause,[status(thm)],[f72,f77,f74])).
% 1.37/1.16 thf(f72,plain,(
% 1.37/1.16 ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) = $false) | ($false = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))))))))),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f66])).
% 1.37/1.16 thf(f66,plain,(
% 1.37/1.16 ($false = ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))))),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f65])).
% 1.37/1.16 thf(f65,plain,(
% 1.37/1.16 ($false = (~ (~ ((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))))))),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f64])).
% 1.37/1.16 thf(f64,plain,(
% 1.37/1.16 ((~ (((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => $false)) = $false)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f63])).
% 1.37/1.16 thf(f63,plain,(
% 1.37/1.16 ($false = (~ (((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => ($false & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK3))))))))),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f62])).
% 1.37/1.16 thf(f62,plain,(
% 1.37/1.16 (((((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => ($false & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK3)))))) => $false) = $false)),
% 1.37/1.16 inference(backward_demodulation,[status(thm)],[f56,f60])).
% 1.37/1.16 thf(f60,plain,(
% 1.37/1.16 ($false = (sK2 @ sK3))),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f56])).
% 1.37/1.16 thf(f56,plain,(
% 1.37/1.16 (((((!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => ((sK2 @ sK3) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK3)))))) => (sK2 @ sK3)) = $false)),
% 1.37/1.16 inference(boolean_simplification,[status(thm)],[f55])).
% 1.37/1.16 thf(f55,plain,(
% 1.37/1.16 ($false = (((($true & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => ((sK2 @ sK3) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK3)))))) => (sK2 @ sK3)))),
% 1.37/1.16 inference(forward_demodulation,[status(thm)],[f54,f33])).
% 1.37/1.16 thf(f54,plain,(
% 1.37/1.16 ($false = (((((sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0)) => ((sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((Y1 @ Y3) & (~ (Y2 = Y3))))) & (Y1 @ Y2)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y0 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))))))))))) => ((sK2 @ sK3) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (Y0 @ sK3)))))) => (sK2 @ sK3)))),
% 1.37/1.16 inference(beta_eta_normalization,[status(thm)],[f53])).
% 1.37/1.16 thf(f53,plain,(
% 1.37/1.16 ($false = ((^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ Y1)))) & (sK2 @ Y1)) => ((sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y1 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))))))) => ((sK2 @ Y0) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))))) => (sK2 @ Y0))) @ sK3))),
% 1.37/1.16 inference(sigma_clausification,[status(thm)],[f27])).
% 1.37/1.16 thf(f27,plain,(
% 1.37/1.16 ($false = (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ Y1)))) & (sK2 @ Y1)) => ((sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((Y2 @ Y4) & (~ (Y3 = Y4))))) & (Y2 @ Y3)))))) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y2 : (($i > $o) > $o) > $o]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((Y4 @ Y6) & (~ (Y5 = Y6))))) & (Y4 @ Y5)))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y1 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4)))))))))))))) => ((sK2 @ Y0) & (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y2 @ (^[Y5 : $i]: ((Y3 @ Y5) & (~ (Y4 = Y5))))) & (Y3 @ Y4))))))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2))))) => (Y1 @ Y0)))))) => (sK2 @ Y0)))))),
% 1.37/1.16 inference(binary_proxy_clausification,[status(thm)],[f26])).
% 1.37/1.16 % SZS output end Proof for DTF2THF_16140
% 1.37/1.16 % (16282)------------------------------
% 1.37/1.16 % (16282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.37/1.16 % (16282)Termination reason: Refutation
% 1.37/1.16
% 1.37/1.16 % (16282)Memory used [KB]: 6140
% 1.37/1.16 % (16282)Time elapsed: 0.033 s
% 1.37/1.16 % (16282)Instructions burned: 85 (million)
% 1.37/1.16 % (16282)------------------------------
% 1.37/1.16 % (16282)------------------------------
% 1.37/1.16 % (16262)Success in time 0.048 s
%------------------------------------------------------------------------------