%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV302^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n021.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 1.69s 1.47s
% Output : Refutation 1.69s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 8
% Syntax : Number of formulae : 100 ( 40 unt; 0 typ; 4 def)
% Number of atoms : 1234 ( 265 equ; 0 cnn)
% Maximal formula atoms : 11 ( 12 avg)
% Number of connectives : 4180 ( 389 ~; 64 |; 572 &;2284 @)
% ( 4 <=>; 316 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 1580 (1580 >; 0 *; 0 +; 0 <<)
% Number of symbols : 18 ( 13 usr; 8 con; 0-2 aty)
% ( 238 !!; 313 ??; 0 @@+; 0 @@-)
% Number of variables : 971 ( 945 ^; 22 !; 4 ?; 971 :)
% 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_24,type,
sK6: ( ( $i > $o ) > $o ) > $o ).
thf(f198,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f74,f92,f122,f131,f197]) ).
thf(f197,plain,
( ~ spl0_3
| ~ spl0_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f196]) ).
thf(f196,plain,
( $false
| ~ spl0_3
| ~ spl0_7 ),
inference(trivial_inequality_removal,[status(thm)],[f195]) ).
thf(f195,plain,
( ( $false = $true )
| ~ spl0_3
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f194]) ).
thf(f194,plain,
( ( ( ~ $true ) = $true )
| ~ spl0_3
| ~ spl0_7 ),
inference(forward_demodulation,[status(thm)],[f185,f173]) ).
thf(f173,plain,
( ( ( sK6 @ sK5 )
= $true )
| ~ spl0_3
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f172]) ).
thf(f172,plain,
( ( ( $true
=> ( sK6 @ sK5 ) )
= $true )
| ~ spl0_3
| ~ spl0_7 ),
inference(forward_demodulation,[status(thm)],[f171,f151]) ).
thf(f151,plain,
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
= $true )
| ~ spl0_7 ),
inference(binary_proxy_clausification,[status(thm)],[f148]) ).
thf(f148,plain,
( ( ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $true )
| ~ spl0_7 ),
inference(binary_proxy_clausification,[status(thm)],[f138]) ).
thf(f138,plain,
( ( ( ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) )
= $false )
| ~ spl0_7 ),
inference(beta_eta_normalization,[status(thm)],[f134]) ).
thf(f134,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) )
@ sK6 ) )
| ~ spl0_7 ),
inference(sigma_clausification,[status(thm)],[f121]) ).
thf(f121,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
= $false )
| ~ spl0_7 ),
inference(avatar_component_clause,[status(thm)],[f120]) ).
thf(f120,definition,
( spl0_7
<=> ( ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_7])],[avatar_definition]) ).
thf(f171,plain,
( ( $true
= ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
=> ( sK6 @ sK5 ) ) )
| ~ spl0_3
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f167]) ).
thf(f167,plain,
( ( ( ( $true
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
=> ( sK6 @ sK5 ) )
= $true )
| ~ spl0_3
| ~ spl0_7 ),
inference(superposition,[status(thm)],[f158,f154]) ).
thf(f154,plain,
( ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $true )
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f153]) ).
thf(f153,plain,
( ( ( ( sK6
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& $true )
= $true )
| ~ spl0_7 ),
inference(backward_demodulation,[status(thm)],[f148,f151]) ).
thf(f158,plain,
( ! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
=> ( X1 @ sK5 ) )
= $true )
| ~ spl0_3 ),
inference(beta_eta_normalization,[status(thm)],[f155]) ).
thf(f155,plain,
( ! [X1: ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) )
@ X1 )
= $true )
| ~ spl0_3 ),
inference(pi_clausification,[status(thm)],[f114]) ).
thf(f114,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
= $true )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f113]) ).
thf(f113,plain,
( ( $true
= ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) ) ) )
| ~ spl0_3 ),
inference(backward_demodulation,[status(thm)],[f108,f112]) ).
thf(f112,plain,
( ( ( sK2 @ sK5 )
= $true )
| ~ spl0_3 ),
inference(binary_proxy_clausification,[status(thm)],[f108]) ).
thf(f108,plain,
( ( $true
= ( ( sK2 @ sK5 )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) ) ) )
| ~ spl0_3 ),
inference(binary_proxy_clausification,[status(thm)],[f96]) ).
thf(f96,plain,
( ( ( ( ( sK2 @ sK5 )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
= $false )
| ~ spl0_3 ),
inference(beta_eta_normalization,[status(thm)],[f95]) ).
thf(f95,plain,
( ( ( ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
@ sK5 )
= $false )
| ~ spl0_3 ),
inference(sigma_clausification,[status(thm)],[f73]) ).
thf(f73,plain,
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $false )
| ~ spl0_3 ),
inference(avatar_component_clause,[status(thm)],[f72]) ).
thf(f72,definition,
( spl0_3
<=> ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_3])],[avatar_definition]) ).
thf(f185,plain,
( ( ( ~ ( sK6 @ sK5 ) )
= $true )
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f181]) ).
thf(f181,plain,
( ( ( ( sK6 @ sK5 )
=> $false )
= $true )
| ~ spl0_7 ),
inference(superposition,[status(thm)],[f162,f150]) ).
thf(f150,plain,
( ( ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) )
= $false )
| ~ spl0_7 ),
inference(boolean_simplification,[status(thm)],[f149]) ).
thf(f149,plain,
( ( ( $true
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) )
= $false )
| ~ spl0_7 ),
inference(backward_demodulation,[status(thm)],[f138,f148]) ).
thf(f162,plain,
( ! [X1: ( $i > $o ) > $o] :
( $true
= ( ( sK6 @ X1 )
=> ( sK6
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( X1
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_7 ),
inference(beta_eta_normalization,[status(thm)],[f161]) ).
thf(f161,plain,
( ! [X1: ( $i > $o ) > $o] :
( ( ^ [Y0: ( $i > $o ) > $o] :
( ( sK6 @ Y0 )
=> ( sK6
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) )
@ X1 )
= $true )
| ~ spl0_7 ),
inference(pi_clausification,[status(thm)],[f151]) ).
thf(f131,plain,
( ~ spl0_3
| ~ spl0_6 ),
inference(avatar_contradiction_clause,[status(thm)],[f130]) ).
thf(f130,plain,
( $false
| ~ spl0_3
| ~ spl0_6 ),
inference(trivial_inequality_removal,[status(thm)],[f129]) ).
thf(f129,plain,
( ( $false = $true )
| ~ spl0_3
| ~ spl0_6 ),
inference(forward_demodulation,[status(thm)],[f128,f118]) ).
thf(f118,plain,
( ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) )
= $false )
| ~ spl0_6 ),
inference(avatar_component_clause,[status(thm)],[f117]) ).
thf(f117,definition,
( spl0_6
<=> ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_6])],[avatar_definition]) ).
thf(f128,plain,
( ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) )
= $true )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f127]) ).
thf(f127,plain,
( ( $true
= ( $true
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_3 ),
inference(forward_demodulation,[status(thm)],[f126,f114]) ).
thf(f126,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f125]) ).
thf(f125,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK5 ) ) )
=> ( $true
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
= $true )
| ~ spl0_3 ),
inference(superposition,[status(thm)],[f38,f112]) ).
thf(f38,plain,
! [X1: ( $i > $o ) > $o] :
( ( ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ X1 ) ) )
=> ( ( sK2 @ X1 )
=> ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( X1
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f37]) ).
thf(f37,plain,
! [X1: ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f34]) ).
thf(f34,plain,
( $true
= ( !! @ ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ( !! @ ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f26]) ).
thf(f26,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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
=> ( ( sK2 @ Y0 )
=> ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( sK2 @ Y1 )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y1
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) ) )
=> ( sK2 @ Y0 ) ) ) ) ),
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] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y2
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) )
@ sK2 ) ),
inference(sigma_clausification,[status(thm)],[f22]) ).
thf(f22,plain,
( $false
= ( !! @ ( ( ( $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] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y2
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) ),
inference(not_proxy_clausification,[status(thm)],[f21]) ).
thf(f21,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] :
( ( ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ Y3 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y3: ( $i > $o ) > $o] :
( ( Y2 @ Y3 )
=> ( Y2
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
=> ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( ( Y0
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ Y2 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3 @ Y2 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y5 @ Y6 )
& ( Y4
@ ^ [Y7: $i] :
( ( Y6 != Y7 )
& ( Y5 @ Y7 ) ) ) ) ) ) ) ) )
=> ( Y3
@ ^ [Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y2
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) )
& ( Y0
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ~ ( !! @ ( ( ( $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] :
( ( ( 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] :
( ( Y6 @ Y7 )
& ( Y5
@ ^ [Y8: $i] :
( ( Y7 != Y8 )
& ( Y6 @ Y8 ) ) ) ) )
@ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
@ Y1 )
=> ( ( Y0 @ Y1 )
=> ( Y0
@ ( ^ [Y2: ( $i > $o ) > $o,Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) )
@ 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 )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( ^ [Y3: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y4: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y4
@ ^ [Y5: $i > $o] :
~ ( ?? @ $i
@ ^ [Y6: $i] : ( Y5 @ Y6 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y5: ( $i > $o ) > $o] :
( ( Y4 @ Y5 )
=> ( Y4
@ ( ^ [Y6: ( $i > $o ) > $o,Y7: $i > $o] :
( ?? @ $i
@ ^ [Y8: $i] :
( ( Y7 @ Y8 )
& ( Y6
@ ^ [Y9: $i] :
( ( Y8 != Y9 )
& ( Y7 @ Y9 ) ) ) ) )
@ Y5 ) ) ) ) )
=> ( Y4 @ Y3 ) ) )
@ Y2 ) )
=> ( ( ^ [Y3: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y4: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y4
@ ^ [Y5: $i > $o] :
~ ( ?? @ $i
@ ^ [Y6: $i] : ( Y5 @ Y6 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y5: ( $i > $o ) > $o] :
( ( Y4 @ Y5 )
=> ( Y4
@ ( ^ [Y6: ( $i > $o ) > $o,Y7: $i > $o] :
( ?? @ $i
@ ^ [Y8: $i] :
( ( Y7 @ Y8 )
& ( Y6
@ ^ [Y9: $i] :
( ( Y8 != Y9 )
& ( Y7 @ Y9 ) ) ) ) )
@ Y5 ) ) ) ) )
=> ( Y4 @ Y3 ) ) )
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) )
@ Y2 ) )
& ( Y0
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) )
@ Y2 ) ) ) ) )
& ( ^ [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] :
( ( Y6 @ Y7 )
& ( Y5
@ ^ [Y8: $i] :
( ( Y7 != Y8 )
& ( Y6 @ Y8 ) ) ) ) )
@ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) )
=> ( ( ^ [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] :
( ( Y6 @ Y7 )
& ( Y5
@ ^ [Y8: $i] :
( ( Y7 != Y8 )
& ( Y6 @ Y8 ) ) ) ) )
@ Y4 ) ) ) ) )
=> ( Y3 @ Y2 ) ) )
@ Y1 )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f15,f19,f18,f16,f16,f19,f19,f18,f18,f19,f16,f19]) ).
thf(f16,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
( cZERO
= ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( cZERO
= ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cZERO_def) ).
thf(f18,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
( cSUCC
= ( ^ [X0: ( $i > $o ) > $o,X1: $i > $o] :
? [X2: $i] :
( ( X0
@ ^ [X3: $i] :
( ( X1 @ X3 )
& ( X2 != X3 ) ) )
& ( X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( cSUCC
= ( ^ [X2: ( $i > $o ) > $o,X0: $i > $o] :
? [X1: $i] :
( ( X2
@ ^ [X3: $i] :
( ( X0 @ X3 )
& ( X1 != X3 ) ) )
& ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cSUCC_def) ).
thf(f19,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) )
@ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f17,f16,f18]) ).
thf(f17,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ^ [X0: ( $i > $o ) > $o] :
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X2: ( $i > $o ) > $o] :
( ( X1 @ X2 )
=> ( X1 @ ( cSUCC @ X2 ) ) )
& ( X1 @ cZERO ) )
=> ( X1 @ X0 ) ) )
= cNAT ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ^ [X2: ( $i > $o ) > $o] :
! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) )
& ( X0 @ cZERO ) )
=> ( X0 @ X2 ) ) )
= cNAT ),
file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cNAT_def) ).
thf(f15,plain,
( ( ~ ( !! @ ( ( ( $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 )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( cNAT @ Y2 ) )
=> ( ( cNAT @ ( cSUCC @ Y2 ) )
& ( Y0 @ ( cSUCC @ Y2 ) ) ) ) )
& ( cNAT @ cZERO ) )
=> ( ( cNAT @ Y1 )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ( ~ ( !! @ ( ( ( $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 )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( ( Y0 @ Y2 )
& ( cNAT @ Y2 ) )
=> ( ( cNAT @ ( cSUCC @ Y2 ) )
& ( Y0 @ ( cSUCC @ Y2 ) ) ) ) )
& ( cNAT @ cZERO ) )
=> ( ( cNAT @ Y1 )
& ( Y0 @ Y1 ) ) )
=> ( Y0 @ Y1 ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
~ ! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ( X0 @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( cNAT @ X1 )
=> ( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) ) ) )
=> ! [X2: ( $i > $o ) > $o] :
( ( ( ( cNAT @ cZERO )
& ! [X3: ( $i > $o ) > $o] :
( ( ( cNAT @ X3 )
& ( X0 @ X3 ) )
=> ( ( X0 @ ( cSUCC @ X3 ) )
& ( cNAT @ ( cSUCC @ X3 ) ) ) )
& ( X0 @ cZERO ) )
=> ( ( X0 @ X2 )
& ( cNAT @ 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] :
( ( ( ( cNAT @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( ( cNAT @ X1 )
& ( X0 @ X1 ) )
=> ( ( X0 @ ( cSUCC @ X1 ) )
& ( cNAT @ ( cSUCC @ X1 ) ) ) )
& ( X0 @ cZERO ) )
=> ( ( X0 @ X4 )
& ( cNAT @ 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] :
( ( ( ( cNAT @ cZERO )
& ! [X1: ( $i > $o ) > $o] :
( ( ( cNAT @ X1 )
& ( X0 @ X1 ) )
=> ( ( X0 @ ( cSUCC @ X1 ) )
& ( cNAT @ ( cSUCC @ X1 ) ) ) )
& ( X0 @ cZERO ) )
=> ( ( X0 @ X4 )
& ( cNAT @ X4 ) ) )
=> ( X0 @ X4 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cX6102_B) ).
thf(f122,plain,
( spl0_6
| spl0_7
| ~ spl0_3 ),
inference(avatar_split_clause,[status(thm)],[f115,f72,f120,f117]) ).
thf(f115,plain,
( ( ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) )
= $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
= $false )
| ~ spl0_3 ),
inference(binary_proxy_clausification,[status(thm)],[f110]) ).
thf(f110,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
| ~ spl0_3 ),
inference(boolean_simplification,[status(thm)],[f109]) ).
thf(f109,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( sK5
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( sK5
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) ) ) ) ) ) )
= $false )
| ~ spl0_3 ),
inference(backward_demodulation,[status(thm)],[f96,f108]) ).
thf(f92,plain,
~ spl0_1,
inference(avatar_contradiction_clause,[status(thm)],[f91]) ).
thf(f91,plain,
( $false
| ~ spl0_1 ),
inference(trivial_inequality_removal,[status(thm)],[f90]) ).
thf(f90,plain,
( ( $false = $true )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f89]) ).
thf(f89,plain,
( ( ( $false
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $true )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f86,f88]) ).
thf(f88,plain,
( ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f87]) ).
thf(f87,plain,
( ( ( $true
=> ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
= $false )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f84,f86]) ).
thf(f84,plain,
( ( ( ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
=> ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
= $false )
| ~ spl0_1 ),
inference(beta_eta_normalization,[status(thm)],[f80]) ).
thf(f80,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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) )
@ sK4 ) )
| ~ spl0_1 ),
inference(sigma_clausification,[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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) )
| ~ spl0_1 ),
inference(avatar_component_clause,[status(thm)],[f62]) ).
thf(f62,definition,
( spl0_1
<=> ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f86,plain,
( ( ( ( sK4
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( sK4 @ Y0 )
=> ( sK4
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $true )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f84]) ).
thf(f74,plain,
( spl0_3
| spl0_1 ),
inference(avatar_split_clause,[status(thm)],[f70,f62,f72]) ).
thf(f70,plain,
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
= $false )
| ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f57]) ).
thf(f57,plain,
( $false
= ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) ) ),
inference(not_proxy_clausification,[status(thm)],[f53]) ).
thf(f53,plain,
( ( ~ ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f52]) ).
thf(f52,plain,
( ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) ) ) ) )
=> $false )
= $true ),
inference(boolean_simplification,[status(thm)],[f51]) ).
thf(f51,plain,
( ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK3 ) ) )
& $false ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f48,f50]) ).
thf(f50,plain,
( ( sK2 @ sK3 )
= $false ),
inference(boolean_simplification,[status(thm)],[f49]) ).
thf(f49,plain,
( $false
= ( $true
=> ( sK2 @ sK3 ) ) ),
inference(backward_demodulation,[status(thm)],[f42,f48]) ).
thf(f42,plain,
( $false
= ( ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK3 ) ) )
& ( sK2 @ sK3 ) ) )
=> ( sK2 @ sK3 ) ) ),
inference(boolean_simplification,[status(thm)],[f41]) ).
thf(f41,plain,
( $false
= ( ( ( $true
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK3 ) ) )
& ( sK2 @ sK3 ) ) )
=> ( sK2 @ sK3 ) ) ),
inference(forward_demodulation,[status(thm)],[f40,f36]) ).
thf(f36,plain,
( $true
= ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) ),
inference(boolean_simplification,[status(thm)],[f35]) ).
thf(f35,plain,
( ( $true
& ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f28,f34]) ).
thf(f40,plain,
( ( ( ( ( sK2
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK3 ) ) )
& ( sK2 @ sK3 ) ) )
=> ( sK2 @ sK3 ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f39]) ).
thf(f39,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( sK2 @ Y1 )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y1
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) ) )
=> ( sK2 @ Y0 ) )
@ sK3 )
= $false ),
inference(sigma_clausification,[status(thm)],[f27]) ).
thf(f27,plain,
( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( ( sK2
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( ( sK2 @ Y1 )
& ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2 @ Y1 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) ) ) ) ) )
=> ( Y2
@ ^ [Y3: $i > $o] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( Y3 @ Y4 )
& ( Y1
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
& ( sK2 @ Y0 ) ) )
=> ( sK2 @ Y0 ) ) )
= $false ),
inference(binary_proxy_clausification,[status(thm)],[f26]) ).
thf(f48,plain,
( ( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( ( sK2 @ Y0 )
& ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
=> ( ( !! @ ( ( ( $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] :
( ( Y3 @ Y4 )
& ( Y2
@ ^ [Y5: $i] :
( ( Y4 != Y5 )
& ( Y3 @ Y5 ) ) ) ) ) ) ) ) )
=> ( Y1
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y2 @ Y3 )
& ( Y0
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) )
& ( sK2
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ) ) )
& ( !! @ ( ( ( $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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ 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] :
( ( Y2 @ Y3 )
& ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) ) ) ) ) ) ) )
=> ( Y0 @ sK3 ) ) )
& ( sK2 @ sK3 ) ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f42]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEV302^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.12/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n021.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:43:24 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.48 ---- Original DTF file ---
% 0.20/0.48 thf(spec,logic,$$dhol).
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.20/0.48 % File : SEV302^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.20/0.48 % Domain : Set Theory
% 0.20/0.48 % Problem : TPS problem from TTTP-NATS-THMS
% 0.20/0.48 % Version : Especial.
% 0.20/0.48 % English :
% 0.20/0.48
% 0.20/0.48 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.20/0.48 % Source : [Bro09]
% 0.20/0.48 % Names : tps_0931 [Bro09]
% 0.20/0.48
% 0.20/0.48 % Status : Theorem
% 0.20/0.48 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 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
% 0.20/0.48 % Syntax : Number of formulae : 7 ( 3 unt; 3 typ; 3 def)
% 0.20/0.48 % Number of atoms : 20 ( 4 equ; 0 cnn)
% 0.20/0.48 % Maximal formula atoms : 11 ( 5 avg)
% 0.20/0.48 % Number of connectives : 44 ( 2 ~; 0 |; 9 &; 25 @)
% 0.20/0.48 % ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% 0.20/0.48 % Maximal formula depth : 13 ( 4 avg)
% 0.20/0.48 % Number of types : 2 ( 0 usr)
% 0.20/0.48 % Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% 0.20/0.48 % Number of symbols : 4 ( 3 usr; 0 con; 1-2 aty)
% 0.20/0.48 % Number of variables : 13 ( 5 ^; 6 !; 2 ?; 13 :)
% 0.20/0.48 % SPC : TH0_THM_EQU_NAR
% 0.20/0.48
% 0.20/0.48 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.20/0.48 % project in the Department of Mathematical Sciences at Carnegie
% 0.20/0.48 % Mellon University. Distributed under the Creative Commons copyleft
% 0.20/0.48 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.20/0.48 % Bugfixes : v5.2.0 - Added missing type declarations.
% 0.20/0.48 % : v6.2.0 - Reordered definitions.
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.20/0.48 thf(cNAT_type,type,
% 0.20/0.48 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(cSUCC_type,type,
% 0.20/0.48 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(cZERO_type,type,
% 0.20/0.48 cZERO: ( $i > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(cZERO_def,definition,
% 0.20/0.48 ( cZERO
% 0.20/0.48 = ( ^ [Xp: $i > $o] :
% 0.20/0.48 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 0.20/0.48
% 0.20/0.48 thf(cSUCC_def,definition,
% 0.20/0.48 ( cSUCC
% 0.20/0.48 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 0.20/0.48 ? [Xx: $i] :
% 0.20/0.48 ( ( Xp @ Xx )
% 0.20/0.48 & ( Xn
% 0.20/0.48 @ ^ [Xt: $i] :
% 0.20/0.48 ( ( Xt != Xx )
% 0.20/0.48 & ( Xp @ Xt ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 thf(cNAT_def,definition,
% 0.20/0.48 ( cNAT
% 0.20/0.48 = ( ^ [Xn: ( $i > $o ) > $o] :
% 0.20/0.48 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 0.20/0.48 ( ( ( Xp @ cZERO )
% 0.20/0.48 & ! [Xx: ( $i > $o ) > $o] :
% 0.20/0.48 ( ( Xp @ Xx )
% 0.20/0.48 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 0.20/0.48 => ( Xp @ Xn ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 thf(cX6102_B,conjecture,
% 0.20/0.48 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 0.20/0.48 ( ( ( Xp @ cZERO )
% 0.20/0.48 & ! [Xx: ( $i > $o ) > $o] :
% 0.20/0.48 ( ( cNAT @ Xx )
% 0.20/0.48 => ( ( Xp @ Xx )
% 0.20/0.48 => ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 0.20/0.48 => ! [Xm: ( $i > $o ) > $o] :
% 0.20/0.48 ( ( ( ( cNAT @ cZERO )
% 0.20/0.48 & ( Xp @ cZERO )
% 0.20/0.48 & ! [Xx: ( $i > $o ) > $o] :
% 0.20/0.48 ( ( ( cNAT @ Xx )
% 0.20/0.48 & ( Xp @ Xx ) )
% 0.20/0.48 => ( ( cNAT @ ( cSUCC @ Xx ) )
% 0.20/0.48 & ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 0.20/0.48 => ( ( cNAT @ Xm )
% 0.20/0.48 & ( Xp @ Xm ) ) )
% 0.20/0.48 => ( Xp @ Xm ) ) ) ).
% 0.20/0.48
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 1.53/1.37 ---- Embedded in THF ---
% 1.53/1.37 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.53/1.37 %%% Generated on Tue Feb 24 12:43:24 EST 2026
% 1.53/1.37 %%% using '$$dhol' embedding, version 1.3.0.
% 1.53/1.37 %%% Logic specification used:
% 1.53/1.37 %%% thf(spec, logic, $$dhol).
% 1.53/1.37
% 1.53/1.37 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2DTF3555.p
% See solution above
% 1.53/1.37 ------------------------
% 1.53/1.38 ---- Cleaned THF ---
% 1.53/1.38 thf(cNAT_type,type,
% 1.53/1.38 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 1.53/1.38
% 1.53/1.38 thf(cSUCC_type,type,
% 1.53/1.38 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 1.53/1.38
% 1.53/1.38 thf(cZERO_type,type,
% 1.53/1.38 cZERO: ( $i > $o ) > $o ).
% 1.53/1.38
% 1.53/1.38 thf(cZERO_def,definition,
% 1.53/1.38 ( cZERO
% 1.53/1.38 = ( ^ [Xp: $i > $o] :
% 1.53/1.38 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 1.53/1.38
% 1.53/1.38 thf(cSUCC_def,definition,
% 1.53/1.38 ( cSUCC
% 1.53/1.38 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 1.53/1.38 ? [Xx: $i] :
% 1.53/1.38 ( ( Xp @ Xx )
% 1.53/1.38 & ( Xn
% 1.53/1.38 @ ^ [Xt: $i] :
% 1.53/1.38 ( ( Xt != Xx )
% 1.53/1.38 & ( Xp @ Xt ) ) ) ) ) ) ).
% 1.53/1.38
% 1.53/1.38 thf(cNAT_def,definition,
% 1.53/1.38 ( cNAT
% 1.53/1.38 = ( ^ [Xn: ( $i > $o ) > $o] :
% 1.53/1.38 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 1.53/1.38 ( ( ( Xp @ cZERO )
% 1.53/1.38 & ! [Xx: ( $i > $o ) > $o] :
% 1.53/1.38 ( ( Xp @ Xx )
% 1.53/1.38 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 1.53/1.38 => ( Xp @ Xn ) ) ) ) ).
% 1.53/1.38
% 1.53/1.38 thf(cX6102_B,conjecture,
% 1.53/1.38 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 1.53/1.38 ( ( ( Xp @ cZERO )
% 1.53/1.38 & ! [Xx: ( $i > $o ) > $o] :
% 1.53/1.38 ( ( cNAT @ Xx )
% 1.53/1.38 => ( ( Xp @ Xx )
% 1.53/1.38 => ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 1.53/1.38 => ! [Xm: ( $i > $o ) > $o] :
% 1.53/1.38 ( ( ( ( cNAT @ cZERO )
% 1.53/1.38 & ( Xp @ cZERO )
% 1.53/1.38 & ! [Xx: ( $i > $o ) > $o] :
% 1.53/1.38 ( ( ( cNAT @ Xx )
% 1.53/1.38 & ( Xp @ Xx ) )
% 1.53/1.38 => ( ( cNAT @ ( cSUCC @ Xx ) )
% 1.53/1.38 & ( Xp @ ( cSUCC @ Xx ) ) ) ) )
% 1.53/1.38 => ( ( cNAT @ Xm )
% 1.53/1.38 & ( Xp @ Xm ) ) )
% 1.53/1.38 => ( Xp @ Xm ) ) ) ).
% 1.53/1.38 ------------------------
% 1.53/1.39 % (3706)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3555 for (3000ds/183Mi)
% 1.69/1.40 % (3709)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3555 for (3000ds/2Mi)
% 1.69/1.40 % (3708)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_3555 for (3000ds/27Mi)
% 1.69/1.40 % (3707)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_3555 for (3000ds/4Mi)
% 1.69/1.40 % (3712)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3555 for (3000ds/18Mi)
% 1.69/1.40 % (3710)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_3555 for (3000ds/2Mi)
% 1.69/1.40 % (3711)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_3555 for (3000ds/275Mi)
% 1.69/1.40 % (3709)Instruction limit reached!
% 1.69/1.40 % (3709)------------------------------
% 1.69/1.40 % (3709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.40 % (3709)Termination reason: Unknown
% 1.69/1.40 % (3710)Instruction limit reached!
% 1.69/1.40 % (3710)------------------------------
% 1.69/1.40 % (3710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.40 % (3710)Termination reason: Unknown
% 1.69/1.40 % (3710)Termination phase: Property scanning
% 1.69/1.40
% 1.69/1.40 % (3710)Memory used [KB]: 1023
% 1.69/1.40 % (3710)Time elapsed: 0.003 s
% 1.69/1.40 % (3710)Instructions burned: 3 (million)
% 1.69/1.40 % (3710)------------------------------
% 1.69/1.40 % (3710)------------------------------
% 1.69/1.40 % (3709)Termination phase: Function definition elimination
% 1.69/1.40
% 1.69/1.40 % (3709)Memory used [KB]: 1023
% 1.69/1.40 % (3709)Time elapsed: 0.003 s
% 1.69/1.40 % (3709)Instructions burned: 4 (million)
% 1.69/1.40 % (3709)------------------------------
% 1.69/1.40 % (3709)------------------------------
% 1.69/1.40 % (3707)Instruction limit reached!
% 1.69/1.40 % (3707)------------------------------
% 1.69/1.40 % (3707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.40 % (3707)Termination reason: Unknown
% 1.69/1.40 % (3707)Termination phase: Property scanning
% 1.69/1.40
% 1.69/1.40 % (3707)Memory used [KB]: 1023
% 1.69/1.40 % (3707)Time elapsed: 0.004 s
% 1.69/1.40 % (3707)Instructions burned: 5 (million)
% 1.69/1.40 % (3707)------------------------------
% 1.69/1.40 % (3707)------------------------------
% 1.69/1.40 % (3711)Refutation not found, incomplete strategy
% 1.69/1.40 % (3711)------------------------------
% 1.69/1.40 % (3711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.40 % (3711)Termination reason: Refutation not found, incomplete strategy
% 1.69/1.40
% 1.69/1.40
% 1.69/1.40 % (3711)Memory used [KB]: 5500
% 1.69/1.40 % (3711)Time elapsed: 0.004 s
% 1.69/1.40 % (3711)Instructions burned: 4 (million)
% 1.69/1.40 % (3711)------------------------------
% 1.69/1.40 % (3711)------------------------------
% 1.69/1.41 % (3712)Instruction limit reached!
% 1.69/1.41 % (3712)------------------------------
% 1.69/1.41 % (3712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.41 % (3712)Termination reason: Unknown
% 1.69/1.41 % (3712)Termination phase: Saturation
% 1.69/1.41
% 1.69/1.41 % (3712)Memory used [KB]: 5500
% 1.69/1.41 % (3712)Time elapsed: 0.010 s
% 1.69/1.41 % (3712)Instructions burned: 19 (million)
% 1.69/1.41 % (3712)------------------------------
% 1.69/1.41 % (3712)------------------------------
% 1.69/1.41 % (3708)Instruction limit reached!
% 1.69/1.41 % (3708)------------------------------
% 1.69/1.41 % (3708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.41 % (3708)Termination reason: Unknown
% 1.69/1.41 % (3708)Termination phase: Saturation
% 1.69/1.41
% 1.69/1.41 % (3708)Memory used [KB]: 5500
% 1.69/1.41 % (3708)Time elapsed: 0.014 s
% 1.69/1.41 % (3708)Instructions burned: 28 (million)
% 1.69/1.41 % (3708)------------------------------
% 1.69/1.41 % (3708)------------------------------
% 1.69/1.41 % (3713)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/3Mi)
% 1.69/1.41 % (3714)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_3555 for (2999ds/37Mi)
% 1.69/1.42 % (3715)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_3555 for (2999ds/15Mi)
% 1.69/1.42 % (3713)Instruction limit reached!
% 1.69/1.42 % (3713)------------------------------
% 1.69/1.42 % (3713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.42 % (3713)Termination reason: Unknown
% 1.69/1.42 % (3713)Termination phase: Function definition elimination
% 1.69/1.42
% 1.69/1.42 % (3713)Memory used [KB]: 1023
% 1.69/1.42 % (3713)Time elapsed: 0.003 s
% 1.69/1.42 % (3713)Instructions burned: 4 (million)
% 1.69/1.42 % (3713)------------------------------
% 1.69/1.42 % (3713)------------------------------
% 1.69/1.42 % (3716)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_3555 for (2999ds/3Mi)
% 1.69/1.42 % (3716)Instruction limit reached!
% 1.69/1.42 % (3716)------------------------------
% 1.69/1.42 % (3716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.42 % (3716)Termination reason: Unknown
% 1.69/1.42 % (3716)Termination phase: Property scanning
% 1.69/1.42
% 1.69/1.42 % (3716)Memory used [KB]: 1023
% 1.69/1.42 % (3716)Time elapsed: 0.003 s
% 1.69/1.42 % (3716)Instructions burned: 4 (million)
% 1.69/1.42 % (3716)------------------------------
% 1.69/1.42 % (3716)------------------------------
% 1.69/1.42 % (3717)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_3555 for (2999ds/1041Mi)
% 1.69/1.42 % (3715)Instruction limit reached!
% 1.69/1.42 % (3715)------------------------------
% 1.69/1.42 % (3715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.42 % (3715)Termination reason: Unknown
% 1.69/1.42 % (3715)Termination phase: Saturation
% 1.69/1.42
% 1.69/1.42 % (3715)Memory used [KB]: 5500
% 1.69/1.42 % (3715)Time elapsed: 0.009 s
% 1.69/1.42 % (3715)Instructions burned: 16 (million)
% 1.69/1.42 % (3715)------------------------------
% 1.69/1.42 % (3715)------------------------------
% 1.69/1.43 % (3718)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/7Mi)
% 1.69/1.43 % (3718)Instruction limit reached!
% 1.69/1.43 % (3718)------------------------------
% 1.69/1.43 % (3718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.43 % (3718)Termination reason: Unknown
% 1.69/1.43 % (3718)Termination phase: Saturation
% 1.69/1.43
% 1.69/1.43 % (3718)Memory used [KB]: 1023
% 1.69/1.43 % (3718)Time elapsed: 0.005 s
% 1.69/1.43 % (3718)Instructions burned: 8 (million)
% 1.69/1.43 % (3718)------------------------------
% 1.69/1.43 % (3718)------------------------------
% 1.69/1.43 % (3714)Instruction limit reached!
% 1.69/1.43 % (3714)------------------------------
% 1.69/1.43 % (3714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.43 % (3714)Termination reason: Unknown
% 1.69/1.43 % (3714)Termination phase: Saturation
% 1.69/1.43
% 1.69/1.43 % (3714)Memory used [KB]: 5500
% 1.69/1.43 % (3714)Time elapsed: 0.018 s
% 1.69/1.43 % (3714)Instructions burned: 37 (million)
% 1.69/1.43 % (3714)------------------------------
% 1.69/1.43 % (3714)------------------------------
% 1.69/1.43 % (3720)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_3555 for (2999ds/3Mi)
% 1.69/1.43 % (3719)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/16Mi)
% 1.69/1.44 % (3720)Instruction limit reached!
% 1.69/1.44 % (3720)------------------------------
% 1.69/1.44 % (3720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.44 % (3720)Termination reason: Unknown
% 1.69/1.44 % (3720)Termination phase: Function definition elimination
% 1.69/1.44
% 1.69/1.44 % (3720)Memory used [KB]: 1023
% 1.69/1.44 % (3720)Time elapsed: 0.003 s
% 1.69/1.44 % (3720)Instructions burned: 4 (million)
% 1.69/1.44 % (3720)------------------------------
% 1.69/1.44 % (3720)------------------------------
% 1.69/1.44 % (3721)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_3555 for (2999ds/3Mi)
% 1.69/1.44 % (3721)Instruction limit reached!
% 1.69/1.44 % (3721)------------------------------
% 1.69/1.44 % (3721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.44 % (3721)Termination reason: Unknown
% 1.69/1.44 % (3721)Termination phase: Function definition elimination
% 1.69/1.44
% 1.69/1.44 % (3721)Memory used [KB]: 1023
% 1.69/1.44 % (3721)Time elapsed: 0.003 s
% 1.69/1.44 % (3721)Instructions burned: 4 (million)
% 1.69/1.44 % (3721)------------------------------
% 1.69/1.44 % (3721)------------------------------
% 1.69/1.44 % (3719)Instruction limit reached!
% 1.69/1.44 % (3719)------------------------------
% 1.69/1.44 % (3719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.44 % (3719)Termination reason: Unknown
% 1.69/1.44 % (3719)Termination phase: Saturation
% 1.69/1.44
% 1.69/1.44 % (3719)Memory used [KB]: 5756
% 1.69/1.44 % (3719)Time elapsed: 0.010 s
% 1.69/1.44 % (3719)Instructions burned: 16 (million)
% 1.69/1.44 % (3719)------------------------------
% 1.69/1.44 % (3719)------------------------------
% 1.69/1.44 % (3706)Instruction limit reached!
% 1.69/1.44 % (3706)------------------------------
% 1.69/1.44 % (3706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.44 % (3706)Termination reason: Unknown
% 1.69/1.44 % (3706)Termination phase: Saturation
% 1.69/1.44
% 1.69/1.44 % (3706)Memory used [KB]: 6652
% 1.69/1.44 % (3706)Time elapsed: 0.054 s
% 1.69/1.44 % (3706)Instructions burned: 184 (million)
% 1.69/1.44 % (3706)------------------------------
% 1.69/1.44 % (3706)------------------------------
% 1.69/1.45 % (3722)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/7Mi)
% 1.69/1.45 % (3723)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/3Mi)
% 1.69/1.45 % (3723)Instruction limit reached!
% 1.69/1.45 % (3723)------------------------------
% 1.69/1.45 % (3723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.45 % (3723)Termination reason: Unknown
% 1.69/1.45 % (3723)Termination phase: Property scanning
% 1.69/1.45
% 1.69/1.45 % (3723)Memory used [KB]: 1023
% 1.69/1.45 % (3723)Time elapsed: 0.003 s
% 1.69/1.45 % (3723)Instructions burned: 4 (million)
% 1.69/1.45 % (3723)------------------------------
% 1.69/1.45 % (3723)------------------------------
% 1.69/1.45 % (3722)Instruction limit reached!
% 1.69/1.45 % (3722)------------------------------
% 1.69/1.45 % (3722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.45 % (3722)Termination reason: Unknown
% 1.69/1.45 % (3722)Termination phase: Saturation
% 1.69/1.45
% 1.69/1.45 % (3722)Memory used [KB]: 5500
% 1.69/1.45 % (3722)Time elapsed: 0.005 s
% 1.69/1.45 % (3722)Instructions burned: 8 (million)
% 1.69/1.45 % (3722)------------------------------
% 1.69/1.45 % (3722)------------------------------
% 1.69/1.45 % (3724)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/4Mi)
% 1.69/1.45 % (3724)Instruction limit reached!
% 1.69/1.45 % (3724)------------------------------
% 1.69/1.45 % (3724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.45 % (3724)Termination reason: Unknown
% 1.69/1.45 % (3724)Termination phase: Property scanning
% 1.69/1.45
% 1.69/1.45 % (3724)Memory used [KB]: 1023
% 1.69/1.45 % (3724)Time elapsed: 0.003 s
% 1.69/1.45 % (3724)Instructions burned: 4 (million)
% 1.69/1.45 % (3724)------------------------------
% 1.69/1.45 % (3724)------------------------------
% 1.69/1.45 % (3727)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/6Mi)
% 1.69/1.45 % (3727)Instruction limit reached!
% 1.69/1.45 % (3727)------------------------------
% 1.69/1.45 % (3727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.45 % (3727)Termination reason: Unknown
% 1.69/1.45 % (3727)Termination phase: Saturation
% 1.69/1.45
% 1.69/1.45 % (3727)Memory used [KB]: 5500
% 1.69/1.45 % (3727)Time elapsed: 0.003 s
% 1.69/1.45 % (3727)Instructions burned: 8 (million)
% 1.69/1.45 % (3727)------------------------------
% 1.69/1.45 % (3727)------------------------------
% 1.69/1.46 % (3725)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/18Mi)
% 1.69/1.46 % (3717)First to succeed.
% 1.69/1.46 % (3726)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/710Mi)
% 1.69/1.46 % (3730)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/5Mi)
% 1.69/1.46 % (3728)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_3555 for (2999ds/902Mi)
% 1.69/1.46 % (3729)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_3555 for (2999ds/21Mi)
% 1.69/1.46 % (3730)Instruction limit reached!
% 1.69/1.46 % (3730)------------------------------
% 1.69/1.46 % (3730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.46 % (3730)Termination reason: Unknown
% 1.69/1.47 % (3730)Termination phase: Saturation
% 1.69/1.47
% 1.69/1.47 % (3730)Memory used [KB]: 1023
% 1.69/1.47 % (3730)Time elapsed: 0.002 s
% 1.69/1.47 % (3730)Instructions burned: 5 (million)
% 1.69/1.47 % (3730)------------------------------
% 1.69/1.47 % (3730)------------------------------
% 1.69/1.47 % (3725)Instruction limit reached!
% 1.69/1.47 % (3725)------------------------------
% 1.69/1.47 % (3725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (3725)Termination reason: Unknown
% 1.69/1.47 % (3725)Termination phase: Saturation
% 1.69/1.47
% 1.69/1.47 % (3725)Memory used [KB]: 5628
% 1.69/1.47 % (3725)Time elapsed: 0.011 s
% 1.69/1.47 % (3725)Instructions burned: 18 (million)
% 1.69/1.47 % (3725)------------------------------
% 1.69/1.47 % (3725)------------------------------
% 1.69/1.47 % (3728)Refutation not found, incomplete strategy
% 1.69/1.47 % (3728)------------------------------
% 1.69/1.47 % (3728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (3728)Termination reason: Refutation not found, incomplete strategy
% 1.69/1.47
% 1.69/1.47
% 1.69/1.47 % (3728)Memory used [KB]: 5500
% 1.69/1.47 % (3728)Time elapsed: 0.003 s
% 1.69/1.47 % (3728)Instructions burned: 3 (million)
% 1.69/1.47 % (3728)------------------------------
% 1.69/1.47 % (3728)------------------------------
% 1.69/1.47 % (3717)Refutation found. Thanks to Tanya!
% 1.69/1.47 % SZS status Theorem for DTF2THF_3555
% 1.69/1.47 % SZS output start Proof for DTF2THF_3555
% 1.69/1.47 thf(func_def_0, type, cNAT: (($i > $o) > $o) > $o).
% 1.69/1.47 thf(func_def_1, type, cSUCC: (($i > $o) > $o) > ($i > $o) > $o).
% 1.69/1.47 thf(func_def_2, type, cZERO: ($i > $o) > $o).
% 1.69/1.47 thf(func_def_18, type, ph1: !>[X0: $tType]:(X0)).
% 1.69/1.47 thf(func_def_19, type, sK2: (($i > $o) > $o) > $o).
% 1.69/1.47 thf(func_def_20, type, sK3: ($i > $o) > $o).
% 1.69/1.47 thf(func_def_22, type, sK4: (($i > $o) > $o) > $o).
% 1.69/1.47 thf(func_def_23, type, sK5: ($i > $o) > $o).
% 1.69/1.47 thf(func_def_24, type, sK6: (($i > $o) > $o) > $o).
% 1.69/1.47 thf(f198,plain,(
% 1.69/1.47 $false),
% 1.69/1.47 inference(avatar_sat_refutation,[status(thm)],[f74,f92,f122,f131,f197])).
% 1.69/1.47 thf(f197,plain,(
% 1.69/1.47 ~spl0_3 | ~spl0_7),
% 1.69/1.47 inference(avatar_contradiction_clause,[status(thm)],[f196])).
% 1.69/1.47 thf(f196,plain,(
% 1.69/1.47 $false | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f195])).
% 1.69/1.47 thf(f195,plain,(
% 1.69/1.47 ($false = $true) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f194])).
% 1.69/1.47 thf(f194,plain,(
% 1.69/1.47 ((~ $true) = $true) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f185,f173])).
% 1.69/1.47 thf(f173,plain,(
% 1.69/1.47 ((sK6 @ sK5) = $true) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f172])).
% 1.69/1.47 thf(f172,plain,(
% 1.69/1.47 (($true => (sK6 @ sK5)) = $true) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f171,f151])).
% 1.69/1.47 thf(f151,plain,(
% 1.69/1.47 ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) = $true) | ~spl0_7),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f148])).
% 1.69/1.47 thf(f148,plain,(
% 1.69/1.47 (((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $true) | ~spl0_7),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f138])).
% 1.69/1.47 thf(f138,plain,(
% 1.69/1.47 ((((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))))))))) = $false) | ~spl0_7),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f134])).
% 1.69/1.47 thf(f134,plain,(
% 1.69/1.47 ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))) @ sK6)) | ~spl0_7),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f121])).
% 1.69/1.47 thf(f121,plain,(
% 1.69/1.47 ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) = $false) | ~spl0_7),
% 1.69/1.47 inference(avatar_component_clause,[status(thm)],[f120])).
% 1.69/1.47 thf(f120,definition,(
% 1.69/1.47 spl0_7 <=> ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) = $false)),
% 1.69/1.47 introduced(definition,[new_symbols(naming,[spl0_7])],[avatar_definition])).
% 1.69/1.47 thf(f171,plain,(
% 1.69/1.47 ($true = ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) => (sK6 @ sK5))) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f167])).
% 1.69/1.47 thf(f167,plain,(
% 1.69/1.47 ((($true & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) => (sK6 @ sK5)) = $true) | (~spl0_3 | ~spl0_7)),
% 1.69/1.47 inference(superposition,[status(thm)],[f158,f154])).
% 1.69/1.47 thf(f154,plain,(
% 1.69/1.47 ((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $true) | ~spl0_7),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f153])).
% 1.69/1.47 thf(f153,plain,(
% 1.69/1.47 (((sK6 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & $true) = $true) | ~spl0_7),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f148,f151])).
% 1.69/1.47 thf(f158,plain,(
% 1.69/1.47 ( ! [X1 : (($i > $o) > $o) > $o] : (((((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) => (X1 @ sK5)) = $true)) ) | ~spl0_3),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f155])).
% 1.69/1.47 thf(f155,plain,(
% 1.69/1.47 ( ! [X1 : (($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5))) @ X1) = $true)) ) | ~spl0_3),
% 1.69/1.47 inference(pi_clausification,[status(thm)],[f114])).
% 1.69/1.47 thf(f114,plain,(
% 1.69/1.47 ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5)))) = $true) | ~spl0_3),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f113])).
% 1.69/1.47 thf(f113,plain,(
% 1.69/1.47 ($true = ($true & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5)))))) | ~spl0_3),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f108,f112])).
% 1.69/1.47 thf(f112,plain,(
% 1.69/1.47 ((sK2 @ sK5) = $true) | ~spl0_3),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f108])).
% 1.69/1.47 thf(f108,plain,(
% 1.69/1.47 ($true = ((sK2 @ sK5) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5)))))) | ~spl0_3),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f96])).
% 1.69/1.47 thf(f96,plain,(
% 1.69/1.47 ((((sK2 @ sK5) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) & (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) = $false) | ~spl0_3),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f95])).
% 1.69/1.47 thf(f95,plain,(
% 1.69/1.47 (((^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) @ sK5) = $false) | ~spl0_3),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f73])).
% 1.69/1.47 thf(f73,plain,(
% 1.69/1.47 ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $false) | ~spl0_3),
% 1.69/1.47 inference(avatar_component_clause,[status(thm)],[f72])).
% 1.69/1.47 thf(f72,definition,(
% 1.69/1.47 spl0_3 <=> ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $false)),
% 1.69/1.47 introduced(definition,[new_symbols(naming,[spl0_3])],[avatar_definition])).
% 1.69/1.47 thf(f185,plain,(
% 1.69/1.47 ((~ (sK6 @ sK5)) = $true) | ~spl0_7),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f181])).
% 1.69/1.47 thf(f181,plain,(
% 1.69/1.47 (((sK6 @ sK5) => $false) = $true) | ~spl0_7),
% 1.69/1.47 inference(superposition,[status(thm)],[f162,f150])).
% 1.69/1.47 thf(f150,plain,(
% 1.69/1.47 ((sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))) = $false) | ~spl0_7),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f149])).
% 1.69/1.47 thf(f149,plain,(
% 1.69/1.47 (($true => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))))))))) = $false) | ~spl0_7),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f138,f148])).
% 1.69/1.47 thf(f162,plain,(
% 1.69/1.47 ( ! [X1 : ($i > $o) > $o] : (($true = ((sK6 @ X1) => (sK6 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (X1 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))))))))))) ) | ~spl0_7),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f161])).
% 1.69/1.47 thf(f161,plain,(
% 1.69/1.47 ( ! [X1 : ($i > $o) > $o] : ((((^[Y0 : ($i > $o) > $o]: ((sK6 @ Y0) => (sK6 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))) @ X1) = $true)) ) | ~spl0_7),
% 1.69/1.47 inference(pi_clausification,[status(thm)],[f151])).
% 1.69/1.47 thf(f131,plain,(
% 1.69/1.47 ~spl0_3 | ~spl0_6),
% 1.69/1.47 inference(avatar_contradiction_clause,[status(thm)],[f130])).
% 1.69/1.47 thf(f130,plain,(
% 1.69/1.47 $false | (~spl0_3 | ~spl0_6)),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f129])).
% 1.69/1.47 thf(f129,plain,(
% 1.69/1.47 ($false = $true) | (~spl0_3 | ~spl0_6)),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f128,f118])).
% 1.69/1.47 thf(f118,plain,(
% 1.69/1.47 ((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))) = $false) | ~spl0_6),
% 1.69/1.47 inference(avatar_component_clause,[status(thm)],[f117])).
% 1.69/1.47 thf(f117,definition,(
% 1.69/1.47 spl0_6 <=> ((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))) = $false)),
% 1.69/1.47 introduced(definition,[new_symbols(naming,[spl0_6])],[avatar_definition])).
% 1.69/1.47 thf(f128,plain,(
% 1.69/1.47 ((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))) = $true) | ~spl0_3),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f127])).
% 1.69/1.47 thf(f127,plain,(
% 1.69/1.47 ($true = ($true => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) | ~spl0_3),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f126,f114])).
% 1.69/1.47 thf(f126,plain,(
% 1.69/1.47 ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5)))) => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) | ~spl0_3),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f125])).
% 1.69/1.47 thf(f125,plain,(
% 1.69/1.47 (((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK5)))) => ($true => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) = $true) | ~spl0_3),
% 1.69/1.47 inference(superposition,[status(thm)],[f38,f112])).
% 1.69/1.47 thf(f38,plain,(
% 1.69/1.47 ( ! [X1 : ($i > $o) > $o] : ((((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ X1)))) => ((sK2 @ X1) => (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (X1 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) = $true)) )),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f37])).
% 1.69/1.47 thf(f37,plain,(
% 1.69/1.47 ( ! [X1 : ($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) @ X1) = $true)) )),
% 1.69/1.47 inference(pi_clausification,[status(thm)],[f34])).
% 1.69/1.47 thf(f34,plain,(
% 1.69/1.47 ($true = (!! @ (($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))))),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f28])).
% 1.69/1.47 thf(f28,plain,(
% 1.69/1.47 (((!! @ (($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) = $true)),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f26])).
% 1.69/1.47 thf(f26,plain,(
% 1.69/1.47 ($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) => ((sK2 @ Y0) => (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) => (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((sK2 @ Y1) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y1 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))) & (sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))))) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0))) => (sK2 @ Y0))))))),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f25])).
% 1.69/1.47 thf(f25,plain,(
% 1.69/1.47 ($false = ((^[Y0 : (($i > $o) > $o) > $o]: (((!! @ (($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & (!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ Y2))))) => ((!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y2 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))))) & (Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))))) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) & (Y0 @ Y1))) => (Y0 @ Y1)))))) @ sK2))),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f22])).
% 1.69/1.47 thf(f22,plain,(
% 1.69/1.47 ($false = (!! @ ((($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]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & (!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ Y2))))) => ((!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y2 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))))) & (Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))))) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) & (Y0 @ Y1))) => (Y0 @ Y1))))))))),
% 1.69/1.47 inference(not_proxy_clausification,[status(thm)],[f21])).
% 1.69/1.47 thf(f21,plain,(
% 1.69/1.47 ((~ (!! @ ((($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]: (((Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))) & (!! @ (($i > $o) > $o) @ (^[Y3 : ($i > $o) > $o]: ((Y2 @ Y3) => (Y2 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) => ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) & (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))) => (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((((Y0 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & (!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ Y2))))) => ((!! @ ((($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]: ((Y5 @ Y6) & (Y4 @ (^[Y7 : $i]: ((~ (Y6 = Y7)) & (Y5 @ Y7))))))))))))) => (Y3 @ (^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y2 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))))) & (Y0 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))))) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (~ (?? @ $i @ Y3)))))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1)))) & (Y0 @ Y1))) => (Y0 @ Y1)))))))) = $true)),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f20])).
% 1.69/1.47 thf(f20,plain,(
% 1.69/1.47 ((~ (!! @ ((($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]: (((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]: ((Y6 @ Y7) & (Y5 @ (^[Y8 : $i]: ((~ (Y7 = Y8)) & (Y6 @ Y8)))))))))) @ Y4)))))) => (Y3 @ Y2))))) @ Y1) => ((Y0 @ Y1) => (Y0 @ ((^[Y2 : ($i > $o) > $o]: ((^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))) @ 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) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & ((^[Y3 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y4 : (($i > $o) > $o) > $o]: (((Y4 @ (^[Y5 : $i > $o]: (~ (?? @ $i @ (^[Y6 : $i]: (Y5 @ Y6)))))) & (!! @ (($i > $o) > $o) @ (^[Y5 : ($i > $o) > $o]: ((Y4 @ Y5) => (Y4 @ ((^[Y6 : ($i > $o) > $o]: ((^[Y7 : $i > $o]: (?? @ $i @ (^[Y8 : $i]: ((Y7 @ Y8) & (Y6 @ (^[Y9 : $i]: ((~ (Y8 = Y9)) & (Y7 @ Y9)))))))))) @ Y5)))))) => (Y4 @ Y3))))) @ Y2)) => (((^[Y3 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y4 : (($i > $o) > $o) > $o]: (((Y4 @ (^[Y5 : $i > $o]: (~ (?? @ $i @ (^[Y6 : $i]: (Y5 @ Y6)))))) & (!! @ (($i > $o) > $o) @ (^[Y5 : ($i > $o) > $o]: ((Y4 @ Y5) => (Y4 @ ((^[Y6 : ($i > $o) > $o]: ((^[Y7 : $i > $o]: (?? @ $i @ (^[Y8 : $i]: ((Y7 @ Y8) & (Y6 @ (^[Y9 : $i]: ((~ (Y8 = Y9)) & (Y7 @ Y9)))))))))) @ Y5)))))) => (Y4 @ Y3))))) @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))) @ Y2)) & (Y0 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))) @ Y2))))))) & ((^[Y2 : ($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]: ((Y6 @ Y7) & (Y5 @ (^[Y8 : $i]: ((~ (Y7 = Y8)) & (Y6 @ Y8)))))))))) @ Y4)))))) => (Y3 @ Y2))))) @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3))))))) => (((^[Y2 : ($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]: ((Y6 @ Y7) & (Y5 @ (^[Y8 : $i]: ((~ (Y7 = Y8)) & (Y6 @ Y8)))))))))) @ Y4)))))) => (Y3 @ Y2))))) @ Y1) & (Y0 @ Y1))) => (Y0 @ Y1)))))))) = $true)),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f15,f19,f18,f16,f16,f19,f19,f18,f18,f19,f16,f19])).
% 1.69/1.47 thf(f16,plain,(
% 1.69/1.47 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f10])).
% 1.69/1.47 thf(f10,plain,(
% 1.69/1.47 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f9])).
% 1.69/1.47 thf(f9,plain,(
% 1.69/1.47 (cZERO = (^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))))),
% 1.69/1.47 inference(rectify,[status(thm)],[f1])).
% 1.69/1.47 thf(f1,axiom,(
% 1.69/1.47 (cZERO = (^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cZERO_def)).
% 1.69/1.47 thf(f18,plain,(
% 1.69/1.47 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f8])).
% 1.69/1.47 thf(f8,plain,(
% 1.69/1.47 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f7])).
% 1.69/1.47 thf(f7,plain,(
% 1.69/1.47 (cSUCC = (^[X0 : ($i > $o) > $o, X1 : $i > $o] : (? [X2 : $i] : ((X0 @ (^[X3 : $i] : ((X1 @ X3) & (X2 != X3)))) & (X1 @ X2)))))),
% 1.69/1.47 inference(rectify,[status(thm)],[f2])).
% 1.69/1.47 thf(f2,axiom,(
% 1.69/1.47 (cSUCC = (^[X2 : ($i > $o) > $o, X0 : $i > $o] : (? [X1 : $i] : ((X2 @ (^[X3 : $i] : ((X0 @ X3) & (X1 != X3)))) & (X0 @ X1)))))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cSUCC_def)).
% 1.69/1.47 thf(f19,plain,(
% 1.69/1.47 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3)))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))) @ Y2)))))) => (Y1 @ Y0))))))),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f17,f16,f18])).
% 1.69/1.47 thf(f17,plain,(
% 1.69/1.47 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2)))))) => (Y1 @ Y0))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f14])).
% 1.69/1.47 thf(f14,plain,(
% 1.69/1.47 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2)))))) => (Y1 @ Y0))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f13])).
% 1.69/1.47 thf(f13,plain,(
% 1.69/1.47 ((^[X0 : ($i > $o) > $o] : (! [X1 : (($i > $o) > $o) > $o] : ((! [X2 : ($i > $o) > $o] : ((X1 @ X2) => (X1 @ (cSUCC @ X2))) & (X1 @ cZERO)) => (X1 @ X0)))) = cNAT)),
% 1.69/1.47 inference(rectify,[status(thm)],[f3])).
% 1.69/1.47 thf(f3,axiom,(
% 1.69/1.47 ((^[X2 : ($i > $o) > $o] : (! [X0 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : ((X0 @ X1) => (X0 @ (cSUCC @ X1))) & (X0 @ cZERO)) => (X0 @ X2)))) = cNAT)),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cNAT_def)).
% 1.69/1.47 thf(f15,plain,(
% 1.69/1.47 ((~ (!! @ ((($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) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & (cNAT @ Y2)) => ((cNAT @ (cSUCC @ Y2)) & (Y0 @ (cSUCC @ Y2))))))) & (cNAT @ cZERO)) => ((cNAT @ Y1) & (Y0 @ Y1))) => (Y0 @ Y1)))))))) = $true)),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f12])).
% 1.69/1.47 thf(f12,plain,(
% 1.69/1.47 ((~ (!! @ ((($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) > $o) @ (^[Y2 : ($i > $o) > $o]: (((Y0 @ Y2) & (cNAT @ Y2)) => ((cNAT @ (cSUCC @ Y2)) & (Y0 @ (cSUCC @ Y2))))))) & (cNAT @ cZERO)) => ((cNAT @ Y1) & (Y0 @ Y1))) => (Y0 @ Y1)))))))) = $true)),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f11])).
% 1.69/1.47 thf(f11,plain,(
% 1.69/1.47 ~! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X2 : ($i > $o) > $o] : ((((cNAT @ cZERO) & ! [X3 : ($i > $o) > $o] : (((cNAT @ X3) & (X0 @ X3)) => ((X0 @ (cSUCC @ X3)) & (cNAT @ (cSUCC @ X3)))) & (X0 @ cZERO)) => ((X0 @ X2) & (cNAT @ X2))) => (X0 @ X2)))),
% 1.69/1.47 inference(rectify,[status(thm)],[f5])).
% 1.69/1.47 thf(f5,negated_conjecture,(
% 1.69/1.47 ~! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X4 : ($i > $o) > $o] : ((((cNAT @ cZERO) & ! [X1 : ($i > $o) > $o] : (((cNAT @ X1) & (X0 @ X1)) => ((X0 @ (cSUCC @ X1)) & (cNAT @ (cSUCC @ X1)))) & (X0 @ cZERO)) => ((X0 @ X4) & (cNAT @ X4))) => (X0 @ X4)))),
% 1.69/1.47 inference(negated_conjecture,[status(cth)],[f4])).
% 1.69/1.47 thf(f4,conjecture,(
% 1.69/1.47 ! [X0 : (($i > $o) > $o) > $o] : (((X0 @ cZERO) & ! [X1 : ($i > $o) > $o] : ((cNAT @ X1) => ((X0 @ X1) => (X0 @ (cSUCC @ X1))))) => ! [X4 : ($i > $o) > $o] : ((((cNAT @ cZERO) & ! [X1 : ($i > $o) > $o] : (((cNAT @ X1) & (X0 @ X1)) => ((X0 @ (cSUCC @ X1)) & (cNAT @ (cSUCC @ X1)))) & (X0 @ cZERO)) => ((X0 @ X4) & (cNAT @ X4))) => (X0 @ X4)))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.1ZcplIdUSW/DTF2THF_3555.p',cX6102_B)).
% 1.69/1.47 thf(f122,plain,(
% 1.69/1.47 spl0_6 | spl0_7 | ~spl0_3),
% 1.69/1.47 inference(avatar_split_clause,[status(thm)],[f115,f72,f120,f117])).
% 1.69/1.47 thf(f115,plain,(
% 1.69/1.47 ((sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))) = $false) | ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) = $false) | ~spl0_3),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f110])).
% 1.69/1.47 thf(f110,plain,(
% 1.69/1.47 ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) & (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) | ~spl0_3),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f109])).
% 1.69/1.47 thf(f109,plain,(
% 1.69/1.47 (($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (sK5 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))) & (sK2 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (sK5 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2))))))))))) = $false) | ~spl0_3),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f96,f108])).
% 1.69/1.47 thf(f92,plain,(
% 1.69/1.47 ~spl0_1),
% 1.69/1.47 inference(avatar_contradiction_clause,[status(thm)],[f91])).
% 1.69/1.47 thf(f91,plain,(
% 1.69/1.47 $false | ~spl0_1),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f90])).
% 1.69/1.47 thf(f90,plain,(
% 1.69/1.47 ($false = $true) | ~spl0_1),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f89])).
% 1.69/1.47 thf(f89,plain,(
% 1.69/1.47 (($false & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $true) | ~spl0_1),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f86,f88])).
% 1.69/1.47 thf(f88,plain,(
% 1.69/1.47 ((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ~spl0_1),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f87])).
% 1.69/1.47 thf(f87,plain,(
% 1.69/1.47 (($true => (sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) = $false) | ~spl0_1),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f84,f86])).
% 1.69/1.47 thf(f84,plain,(
% 1.69/1.47 ((((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) => (sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) = $false) | ~spl0_1),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f80])).
% 1.69/1.47 thf(f80,plain,(
% 1.69/1.47 ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))) @ sK4)) | ~spl0_1),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f63])).
% 1.69/1.47 thf(f63,plain,(
% 1.69/1.47 ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) | ~spl0_1),
% 1.69/1.47 inference(avatar_component_clause,[status(thm)],[f62])).
% 1.69/1.47 thf(f62,definition,(
% 1.69/1.47 spl0_1 <=> ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))))),
% 1.69/1.47 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 1.69/1.47 thf(f86,plain,(
% 1.69/1.47 (((sK4 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((sK4 @ Y0) => (sK4 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $true) | ~spl0_1),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f84])).
% 1.69/1.47 thf(f74,plain,(
% 1.69/1.47 spl0_3 | spl0_1),
% 1.69/1.47 inference(avatar_split_clause,[status(thm)],[f70,f62,f72])).
% 1.69/1.47 thf(f70,plain,(
% 1.69/1.47 ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) = $false) | ($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))))),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f57])).
% 1.69/1.47 thf(f57,plain,(
% 1.69/1.47 ($false = ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))))),
% 1.69/1.47 inference(not_proxy_clausification,[status(thm)],[f53])).
% 1.69/1.47 thf(f53,plain,(
% 1.69/1.47 ((~ ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1))))))))) = $true)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f52])).
% 1.69/1.47 thf(f52,plain,(
% 1.69/1.47 ((((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => $false) = $true)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f51])).
% 1.69/1.47 thf(f51,plain,(
% 1.69/1.47 ((((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK3)))) & $false)) = $true)),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f48,f50])).
% 1.69/1.47 thf(f50,plain,(
% 1.69/1.47 ((sK2 @ sK3) = $false)),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f49])).
% 1.69/1.47 thf(f49,plain,(
% 1.69/1.47 ($false = ($true => (sK2 @ sK3)))),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f42,f48])).
% 1.69/1.47 thf(f42,plain,(
% 1.69/1.47 ($false = ((((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK3)))) & (sK2 @ sK3))) => (sK2 @ sK3)))),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f41])).
% 1.69/1.47 thf(f41,plain,(
% 1.69/1.47 ($false = (((($true & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK3)))) & (sK2 @ sK3))) => (sK2 @ sK3)))),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f40,f36])).
% 1.69/1.47 thf(f36,plain,(
% 1.69/1.47 ($true = (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))))),
% 1.69/1.47 inference(boolean_simplification,[status(thm)],[f35])).
% 1.69/1.47 thf(f35,plain,(
% 1.69/1.47 (($true & (sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0))))) = $true)),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f28,f34])).
% 1.69/1.47 thf(f40,plain,(
% 1.69/1.47 ((((((sK2 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK3)))) & (sK2 @ sK3))) => (sK2 @ sK3)) = $false)),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f39])).
% 1.69/1.47 thf(f39,plain,(
% 1.69/1.47 (((^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((sK2 @ Y1) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y1 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))) & (sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))))) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0))) => (sK2 @ Y0))) @ sK3) = $false)),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f27])).
% 1.69/1.47 thf(f27,plain,(
% 1.69/1.47 ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((((sK2 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: (((sK2 @ Y1) & (!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ Y1))))) => ((!! @ ((($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]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6))))))))))))) => (Y2 @ (^[Y3 : $i > $o]: (?? @ $i @ (^[Y4 : $i]: ((Y3 @ Y4) & (Y1 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5)))))))))))) & (sK2 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))))) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ Y2)))))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0)))) & (sK2 @ Y0))) => (sK2 @ Y0)))) = $false)),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f26])).
% 1.69/1.47 thf(f48,plain,(
% 1.69/1.47 ((((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: (((sK2 @ Y0) & (!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ Y0))))) => ((!! @ ((($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]: ((Y3 @ Y4) & (Y2 @ (^[Y5 : $i]: ((~ (Y4 = Y5)) & (Y3 @ Y5))))))))))))) => (Y1 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y2 @ Y3) & (Y0 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))))))))))) & (sK2 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3))))))))))))) & (!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))))))) => ((!! @ ((($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]: ((Y2 @ Y3) & (Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4))))))))))))) => (Y0 @ sK3)))) & (sK2 @ sK3))) = $true)),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f42])).
% 1.69/1.47 % SZS output end Proof for DTF2THF_3555
% 1.69/1.47 % (3717)------------------------------
% 1.69/1.47 % (3717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (3717)Termination reason: Refutation
% 1.69/1.47
% 1.69/1.47 % (3717)Memory used [KB]: 6140
% 1.69/1.47 % (3717)Time elapsed: 0.046 s
% 1.69/1.47 % (3717)Instructions burned: 75 (million)
% 1.69/1.47 % (3717)------------------------------
% 1.69/1.47 % (3717)------------------------------
% 1.69/1.47 % (3705)Success in time 0.076 s
%------------------------------------------------------------------------------