%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SYN041^4 : TPTP v9.2.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n009.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 : Thu Oct 9 03:16:09 PM UTC 2025
% Result : Theorem 0.18s 0.44s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 87
% Syntax : Number of formulae : 100 ( 37 unt; 0 typ; 21 def)
% Number of atoms : 300 ( 21 equ; 3 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 473 ( 69 ~; 34 |; 1 &; 247 @)
% ( 29 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 57 usr; 57 con; 0-2 aty)
% Number of variables : 100 ( 35 ^; 63 !; 2 ?; 100 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_q,type,
q: $i > $o ).
thf(ty_eigen__4,type,
eigen__4: $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_p,type,
p: $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_irel,type,
irel: $i > $i > $o ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__3,definition,
( eigen__3
= ( eps__0
@ ^ [X1: $i] :
~ ( ( irel @ eigen__2 @ X1 )
=> ( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( q @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__4,definition,
( eigen__4
= ( eps__0
@ ^ [X1: $i] :
~ ( ( irel @ eigen__3 @ X1 )
=> ( q @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( irel @ X1 @ X2 )
=> ~ ( irel @ X2 @ X3 ) )
=> ( irel @ X1 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i] :
( ( irel @ eigen__0 @ X1 )
=> ~ ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( p @ X3 ) )
=> ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( q @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( irel @ eigen__1 @ X1 )
=> ( q @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] :
( ~ ( ( irel @ eigen__1 @ eigen__3 )
=> ~ ( irel @ eigen__3 @ X1 ) )
=> ( irel @ eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( irel @ eigen__0 @ eigen__1 )
=> ~ ( irel @ eigen__1 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( irel @ eigen__1 @ eigen__2 )
=> ~ ( irel @ eigen__2 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i] :
( ( irel @ eigen__3 @ X1 )
=> ( q @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: $i] :
( ~ ( ( irel @ eigen__0 @ eigen__1 )
=> ~ ( irel @ eigen__1 @ X1 ) )
=> ( irel @ eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( q @ eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ~ sP5
=> ( irel @ eigen__0 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( irel @ eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( ~ ( ( irel @ eigen__1 @ eigen__3 )
=> ~ ( irel @ eigen__3 @ eigen__4 ) )
=> ( irel @ eigen__1 @ eigen__4 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ! [X1: $i] :
( ( irel @ eigen__3 @ X1 )
=> ( p @ X1 ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( irel @ eigen__1 @ eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( irel @ eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( irel @ eigen__3 @ eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( ( irel @ eigen__1 @ eigen__3 )
=> ~ sP16 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $i] :
( ~ ( ( irel @ eigen__1 @ eigen__2 )
=> ~ ( irel @ eigen__2 @ X1 ) )
=> ( irel @ eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ( irel @ eigen__2 @ eigen__3 )
=> sP13 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP16
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i,X2: $i] :
( ~ ( ( irel @ eigen__1 @ X1 )
=> ~ ( irel @ X1 @ X2 ) )
=> ( irel @ eigen__1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( irel @ eigen__1 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( sP15
=> ~ ! [X1: $i] :
( ( irel @ eigen__2 @ X1 )
=> ( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( q @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( sP14
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: $i] :
( ( irel @ eigen__2 @ X1 )
=> ( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( q @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ! [X1: $i,X2: $i] :
( ~ ( ( irel @ eigen__0 @ X1 )
=> ~ ( irel @ X1 @ X2 ) )
=> ( irel @ eigen__0 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ~ sP6
=> sP22 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( irel @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( irel @ eigen__2 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(def_mnot,definition,
( mnot
= ( ^ [X1: $i > $o,X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_mor,definition,
( mor
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
| ( X2 @ X3 ) ) ) ) ).
thf(def_mand,definition,
( mand
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
& ( X2 @ X3 ) ) ) ) ).
thf(def_mimplies,definition,
( mimplies
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mnot @ X1 ) @ X2 ) ) ) ).
thf(def_mbox_s4,definition,
( mbox_s4
= ( ^ [X1: $i > $o,X2: $i] :
! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( irel @ X2 @ X3 )
@ ( X1 @ X3 ) ) ) ) ).
thf(def_iatom,definition,
( iatom
= ( ^ [X1: $i > $o] : X1 ) ) ).
thf(def_inot,definition,
( inot
= ( ^ [X1: $i > $o] : ( mnot @ ( mbox_s4 @ X1 ) ) ) ) ).
thf(def_itrue,definition,
( itrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_ifalse,definition,
( ifalse
= ( inot @ itrue ) ) ).
thf(def_iand,definition,
( iand
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ X1 @ X2 ) ) ) ).
thf(def_ior,definition,
( ior
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mbox_s4 @ X1 ) @ ( mbox_s4 @ X2 ) ) ) ) ).
thf(def_iimplies,definition,
( iimplies
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mimplies @ ( mbox_s4 @ X1 ) @ ( mbox_s4 @ X2 ) ) ) ) ).
thf(def_iimplied,definition,
( iimplied
= ( ^ [X1: $i > $o,X2: $i > $o] : ( iimplies @ X2 @ X1 ) ) ) ).
thf(def_iequiv,definition,
( iequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( iand @ ( iimplies @ X1 @ X2 ) @ ( iimplies @ X2 @ X1 ) ) ) ) ).
thf(def_ixor,definition,
( ixor
= ( ^ [X1: $i > $o,X2: $i > $o] : ( inot @ ( iequiv @ X1 @ X2 ) ) ) ) ).
thf(def_ivalid,definition,
( ivalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_isatisfiable,definition,
( isatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_icountersatisfiable,definition,
( icountersatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_iinvalid,definition,
( iinvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(pel3,conjecture,
! [X1: $i] :
( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( q @ X4 ) ) ) ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( q @ X3 ) )
=> ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i] :
( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( q @ X4 ) ) ) ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( q @ X3 ) )
=> ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ),
inference(assume_negation,[status(cth)],[pel3]) ).
thf(h2,assumption,
~ ( sP2
=> ! [X1: $i] :
( ( irel @ eigen__0 @ X1 )
=> ( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( q @ X2 ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
sP2,
introduced(assumption,[]) ).
thf(h4,assumption,
~ ! [X1: $i] :
( ( irel @ eigen__0 @ X1 )
=> ( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( q @ X2 ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( p @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
~ ( sP11
=> ( sP3
=> ! [X1: $i] :
( ( irel @ eigen__1 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
sP11,
introduced(assumption,[]) ).
thf(h7,assumption,
~ ( sP3
=> ! [X1: $i] :
( ( irel @ eigen__1 @ X1 )
=> ( p @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h8,assumption,
sP3,
introduced(assumption,[]) ).
thf(h9,assumption,
~ ! [X1: $i] :
( ( irel @ eigen__1 @ X1 )
=> ( p @ X1 ) ),
introduced(assumption,[]) ).
thf(h10,assumption,
~ ( sP28
=> ( p @ eigen__2 ) ),
introduced(assumption,[]) ).
thf(h11,assumption,
sP28,
introduced(assumption,[]) ).
thf(h12,assumption,
~ ( p @ eigen__2 ),
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP17
| ~ sP22
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP12
| sP17
| sP14 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP24
| ~ sP14
| sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP3
| sP24 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP4
| sP12 ),
inference(all_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP6
| ~ sP28
| ~ sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP27
| sP6
| sP22 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP21
| sP4 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP18
| sP27 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP5
| ~ sP11
| ~ sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP10
| sP5
| sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP8
| sP10 ),
inference(all_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP26
| sP8 ),
inference(all_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP1
| sP26 ),
inference(all_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP21
| sP18 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP1
| sP21 ),
inference(all_rule,[status(thm)],[]) ).
thf(17,plain,
( sP20
| ~ sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( sP20
| sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(19,plain,
( sP7
| ~ sP20 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(20,plain,
( sP13
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(21,plain,
( sP19
| ~ sP13 ),
inference(prop_rule,[status(thm)],[]) ).
thf(22,plain,
( sP19
| sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( sP25
| ~ sP19 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(24,plain,
( ~ sP23
| ~ sP15
| ~ sP25 ),
inference(prop_rule,[status(thm)],[]) ).
thf(25,plain,
( ~ sP2
| sP23 ),
inference(all_rule,[status(thm)],[]) ).
thf(trans_axiom,axiom,
sP1 ).
thf(26,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h11,h12,h10,h8,h9,h6,h7,h5,h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,h3,h6,h8,h11,trans_axiom]) ).
thf(27,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h10,h8,h9,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h11,h12])],[h10,26,h11,h12]) ).
thf(28,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h8,h9,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h10]),tab_negall(eigenvar,eigen__2)],[h9,27,h10]) ).
thf(29,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,28,h8,h9]) ).
thf(30,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,29,h6,h7]) ).
thf(31,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__1)],[h4,30,h5]) ).
thf(32,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,31,h3,h4]) ).
thf(33,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,32,h2]) ).
thf(34,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[33,h0]) ).
thf(0,theorem,
! [X1: $i] :
( ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( irel @ X3 @ X4 )
=> ( q @ X4 ) ) ) ) )
=> ! [X2: $i] :
( ( irel @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( q @ X3 ) )
=> ! [X3: $i] :
( ( irel @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[33,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SYN041^4 : TPTP v9.2.0. Released v4.0.0.
% 0.06/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Oct 8 15:21:08 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.18/0.44 % SZS status Theorem
% 0.18/0.44 % Mode: cade22grackle2xfee4
% 0.18/0.44 % Steps: 439
% 0.18/0.44 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------