%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : PLA033^7 : TPTP v9.2.0. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n011.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 02:14:00 PM UTC 2025
% Result : Theorem 0.19s 0.56s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 142
% Syntax : Number of formulae : 153 ( 50 unt; 0 typ; 35 def)
% Number of atoms : 559 ( 39 equ; 5 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 1226 ( 186 ~; 66 |; 8 &; 687 @)
% ( 52 <=>; 227 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 5 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 61 ( 61 >; 0 *; 0 +; 0 <<)
% Number of symbols : 103 ( 99 usr; 96 con; 0-3 aty)
% Number of variables : 220 ( 70 ^; 144 !; 6 ?; 220 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_d,type,
d: mu ).
thf(ty_h,type,
h: mu > $i > $o ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_o,type,
o: mu ).
thf(ty_closed,type,
closed: mu > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__4,type,
eigen__4: mu ).
thf(ty_rel_s4,type,
rel_s4: $i > $i > $o ).
thf(ty_open,type,
open: mu > $i > $o ).
thf(ty_combo,type,
combo: mu > mu > $i > $o ).
thf(ty_exists_in_world,type,
exists_in_world: mu > $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__5,type,
eigen__5: mu ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__1 @ X1 )
=> ( open @ d @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(h1,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__4,definition,
( eigen__4
= ( eps__1
@ ^ [X1: mu] :
~ ( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( combo @ d @ X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(eigendef_eigen__5,definition,
( eigen__5
= ( eps__1
@ ^ [X1: mu] :
~ ( ( exists_in_world @ X1 @ eigen__1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ~ ( ( closed @ d @ X2 )
=> ( ( combo @ d @ d @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( open @ d @ X3 ) ) ) )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ~ ( ( closed @ d @ X2 )
=> ( ~ ( combo @ d @ d @ X2 )
=> ~ ( h @ o @ X2 ) ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( closed @ d @ X3 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__5])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i] : ( rel_s4 @ X1 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ( ! [X3: $i] :
( ( rel_s4 @ eigen__1 @ X3 )
=> ( ~ ( ( closed @ d @ X3 )
=> ( ( combo @ d @ X1 @ X3 )
=> ~ ( h @ X2 @ X3 ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( open @ d @ X4 ) ) ) )
=> ~ ! [X3: $i] :
( ( rel_s4 @ eigen__1 @ X3 )
=> ( ~ ( ( closed @ d @ X3 )
=> ( ~ ( combo @ d @ X1 @ X3 )
=> ~ ( h @ o @ X3 ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( closed @ d @ X4 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ~ ( ( closed @ d @ X1 )
=> ( ( combo @ d @ d @ X1 )
=> ~ ( h @ eigen__5 @ X1 ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( open @ d @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] : ( exists_in_world @ d @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( rel_s4 @ eigen__2 @ eigen__2 )
=> ( closed @ d @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ( open @ d @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( exists_in_world @ d @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( ( combo @ d @ eigen__4 @ eigen__2 )
=> ~ ( h @ eigen__5 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( rel_s4 @ eigen__0 @ eigen__1 )
=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ eigen__1 )
=> ( ! [X4: $i] :
( ( rel_s4 @ eigen__1 @ X4 )
=> ( ~ ( ( closed @ X1 @ X4 )
=> ( ( combo @ X1 @ X2 @ X4 )
=> ~ ( h @ X3 @ X4 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( open @ X1 @ X5 ) ) ) )
=> ~ ! [X4: $i] :
( ( rel_s4 @ eigen__1 @ X4 )
=> ( ~ ( ( closed @ X1 @ X4 )
=> ( ~ ( combo @ X1 @ X2 @ X4 )
=> ~ ( h @ o @ X4 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( closed @ X1 @ X5 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( exists_in_world @ eigen__4 @ eigen__1 )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( combo @ d @ eigen__4 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( combo @ d @ d @ eigen__2 )
=> ~ ( h @ d @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ( open @ d @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( exists_in_world @ eigen__5 @ eigen__1 )
=> ( sP3
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ~ ( ( closed @ d @ X1 )
=> ( ~ ( combo @ d @ d @ X1 )
=> ~ ( h @ o @ X1 ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( closed @ d @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ~ sP8 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( sP7
=> ~ ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( ( combo @ d @ d @ X1 )
=> ~ ( h @ d @ X1 ) ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( open @ d @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( sP7
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ~ ( ( closed @ d @ X2 )
=> ( ( combo @ d @ d @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( open @ d @ X3 ) ) ) )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ~ ( ( closed @ d @ X2 )
=> ( ~ ( combo @ d @ d @ X2 )
=> ~ ( h @ o @ X2 ) ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( closed @ d @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( sP3
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ~ ( ( closed @ d @ X1 )
=> ( ~ ( combo @ d @ d @ X1 )
=> ~ ( h @ o @ X1 ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( closed @ d @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ( rel_s4 @ eigen__2 @ eigen__2 )
=> ( open @ d @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ( ~ ( ( closed @ d @ eigen__2 )
=> ( ( combo @ d @ d @ eigen__2 )
=> ~ ( h @ eigen__5 @ eigen__2 ) ) )
=> sP6 ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( sP7
=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ~ ( ( combo @ d @ d @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( open @ d @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( ( combo @ d @ d @ X1 )
=> ~ ( h @ d @ X1 ) ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( open @ d @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( rel_s4 @ eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ~ sP11 ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( ( combo @ d @ eigen__4 @ X1 )
=> ~ ( h @ eigen__5 @ X1 ) ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( open @ d @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( closed @ d @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ( exists_in_world @ eigen__4 @ eigen__1 )
=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ~ ( ( combo @ d @ eigen__4 @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( open @ d @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: $i] :
~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( combo @ d @ X2 @ X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( ( exists_in_world @ eigen__5 @ eigen__1 )
=> ~ sP25 ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ eigen__1 )
=> ( ! [X4: $i] :
( ( rel_s4 @ eigen__1 @ X4 )
=> ( ~ ( ( closed @ X1 @ X4 )
=> ( ( combo @ X1 @ X2 @ X4 )
=> ~ ( h @ X3 @ X4 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( open @ X1 @ X5 ) ) ) )
=> ~ ! [X4: $i] :
( ( rel_s4 @ eigen__1 @ X4 )
=> ( ~ ( ( closed @ X1 @ X4 )
=> ( ~ ( combo @ X1 @ X2 @ X4 )
=> ~ ( h @ o @ X4 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( closed @ X1 @ X5 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( closed @ d @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( exists_in_world @ eigen__5 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( ( combo @ d @ eigen__4 @ X1 )
=> ~ ( h @ eigen__5 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( open @ d @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( open @ d @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( combo @ d @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( rel_s4 @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( sP31
=> ( ( combo @ d @ d @ eigen__2 )
=> ~ ( h @ eigen__5 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ~ ( ( combo @ d @ d @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> sP34 ) ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( ( combo @ d @ d @ X1 )
=> ~ ( h @ d @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ~ ( ( combo @ d @ eigen__4 @ X2 )
=> ~ ( h @ X1 @ X2 ) ) )
=> sP34 ) ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ! [X5: mu] :
( ( exists_in_world @ X5 @ X2 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X2 @ X6 )
=> ( ~ ( ( closed @ X3 @ X6 )
=> ( ( combo @ X3 @ X4 @ X6 )
=> ~ ( h @ X5 @ X6 ) ) )
=> ! [X7: $i] :
( ( rel_s4 @ X6 @ X7 )
=> ( open @ X3 @ X7 ) ) ) )
=> ~ ! [X6: $i] :
( ( rel_s4 @ X2 @ X6 )
=> ( ~ ( ( closed @ X3 @ X6 )
=> ( ~ ( combo @ X3 @ X4 @ X6 )
=> ~ ( h @ o @ X6 ) ) )
=> ! [X7: $i] :
( ( rel_s4 @ X6 @ X7 )
=> ( closed @ X3 @ X7 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ eigen__1 )
=> ~ ( ! [X3: $i] :
( ( rel_s4 @ eigen__1 @ X3 )
=> ~ ( ( combo @ d @ X1 @ X3 )
=> ~ ( h @ X2 @ X3 ) ) )
=> sP34 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( h @ eigen__5 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( combo @ d @ d @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ( closed @ d @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( ~ sP38
=> sP6 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( rel_s4 @ eigen__2 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( sP45
=> ~ sP44 ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( exists_in_world @ eigen__4 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X1 )
=> ~ ! [X4: mu] :
( ( exists_in_world @ X4 @ X1 )
=> ( ! [X5: $i] :
( ( rel_s4 @ X1 @ X5 )
=> ( ~ ( ( closed @ X2 @ X5 )
=> ( ( combo @ X2 @ X3 @ X5 )
=> ~ ( h @ X4 @ X5 ) ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( open @ X2 @ X6 ) ) ) )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X1 @ X5 )
=> ( ~ ( ( closed @ X2 @ X5 )
=> ( ~ ( combo @ X2 @ X3 @ X5 )
=> ~ ( h @ o @ X5 ) ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( closed @ X2 @ X6 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ( sP7
=> ~ sP17 ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(def_meq_prop,definition,
( meq_prop
= ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
( ( X1 @ X3 )
= ( X2 @ X3 ) ) ) ) ).
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_mbox,definition,
( mbox
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( (~) @ ( X1 @ X3 @ X4 ) )
| ( X2 @ X4 ) ) ) ) ).
thf(def_mforall_prop,definition,
( mforall_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mtrue,definition,
( mtrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_mfalse,definition,
( mfalse
= ( mnot @ mtrue ) ) ).
thf(def_mand,definition,
( mand
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mnot @ ( mor @ ( mnot @ X1 ) @ ( mnot @ X2 ) ) ) ) ) ).
thf(def_mimplies,definition,
( mimplies
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mnot @ X1 ) @ X2 ) ) ) ).
thf(def_mimplied,definition,
( mimplied
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mor @ ( mnot @ X2 ) @ X1 ) ) ) ).
thf(def_mequiv,definition,
( mequiv
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mand @ ( mimplies @ X1 @ X2 ) @ ( mimplies @ X2 @ X1 ) ) ) ) ).
thf(def_mxor,definition,
( mxor
= ( ^ [X1: $i > $o,X2: $i > $o] : ( mnot @ ( mequiv @ X1 @ X2 ) ) ) ) ).
thf(def_mdia,definition,
( mdia
= ( ^ [X1: $i > $i > $o,X2: $i > $o] : ( mnot @ ( mbox @ X1 @ ( mnot @ X2 ) ) ) ) ) ).
thf(def_mforall_ind,definition,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( exists_in_world @ X3 @ X2 )
@ ( X1 @ X3 @ X2 ) ) ) ) ).
thf(def_mexists_ind,definition,
( mexists_ind
= ( ^ [X1: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [X2: mu] : ( mnot @ ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_mexists_prop,definition,
( mexists_prop
= ( ^ [X1: ( $i > $o ) > $i > $o] :
( mnot
@ ( mforall_prop
@ ^ [X2: $i > $o] : ( mnot @ ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_mreflexive,definition,
( mreflexive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) ).
thf(def_msymmetric,definition,
( msymmetric
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 @ X3 )
@ ( X1 @ X3 @ X2 ) ) ) ) ).
thf(def_mserial,definition,
( mserial
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
? [X3: $i] : ( X1 @ X2 @ X3 ) ) ) ).
thf(def_mtransitive,definition,
( mtransitive
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X3 @ X4 ) )
@ ( X1 @ X2 @ X4 ) ) ) ) ).
thf(def_meuclidean,definition,
( meuclidean
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( X1 @ X3 @ X4 ) ) ) ) ).
thf(def_mpartially_functional,definition,
( mpartially_functional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( X3 = X4 ) ) ) ) ).
thf(def_mfunctional,definition,
( mfunctional
= ( ^ [X1: $i > $i > $o] :
! [X2: $i] :
? [X3: $i] :
( ( X1 @ X2 @ X3 )
& ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X1 @ X2 @ X4 )
@ ( X3 = X4 ) ) ) ) ) ).
thf(def_mweakly_dense,definition,
( mweakly_dense
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X1 @ X2 @ X3 )
@ ? [X5: $i] :
( ( X1 @ X2 @ X5 )
& ( X1 @ X5 @ X3 ) ) ) ) ) ).
thf(def_mweakly_connected,definition,
( mweakly_connected
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ( ( X1 @ X3 @ X4 )
| ( X3 = X4 )
| ( X1 @ X4 @ X3 ) ) ) ) ) ).
thf(def_mweakly_directed,definition,
( mweakly_directed
= ( ^ [X1: $i > $i > $o] :
! [X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( X1 @ X2 @ X3 )
& ( X1 @ X2 @ X4 ) )
@ ? [X5: $i] :
( ( X1 @ X3 @ X5 )
& ( X1 @ X4 @ X5 ) ) ) ) ) ).
thf(def_mvalid,definition,
( mvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_msatisfiable,definition,
( msatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( X1 @ X2 ) ) ) ).
thf(def_mcountersatisfiable,definition,
( mcountersatisfiable
= ( ^ [X1: $i > $o] :
? [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_minvalid,definition,
( minvalid
= ( ^ [X1: $i > $o] :
! [X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).
thf(def_mbox_s4,definition,
( mbox_s4
= ( ^ [X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( (~) @ ( rel_s4 @ X2 @ X3 ) )
| ( X1 @ X3 ) ) ) ) ).
thf(def_mdia_s4,definition,
( mdia_s4
= ( ^ [X1: $i > $o] : ( mnot @ ( mbox_s4 @ ( mnot @ X1 ) ) ) ) ) ).
thf(con,conjecture,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ~ ( ( combo @ d @ X3 @ X5 )
=> ~ ( h @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( open @ d @ X5 ) ) ) ) ) ) ).
thf(h2,negated_conjecture,
~ ! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ~ ( ( combo @ d @ X3 @ X5 )
=> ~ ( h @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( open @ d @ X5 ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[con]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ~ ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X1 )
=> ~ ( ! [X4: $i] :
( ( rel_s4 @ X1 @ X4 )
=> ~ ( ( combo @ d @ X2 @ X4 )
=> ~ ( h @ X3 @ X4 ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X1 @ X4 )
=> ( open @ d @ X4 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( sP23
=> ~ sP43 ),
introduced(assumption,[]) ).
thf(h5,assumption,
sP23,
introduced(assumption,[]) ).
thf(h6,assumption,
sP43,
introduced(assumption,[]) ).
thf(1,plain,
( sP8
| sP44 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP14
| ~ sP37
| ~ sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP33
| sP14 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( sP25
| sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP29
| ~ sP32
| ~ sP25 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP41
| sP29 ),
inference(all_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP27
| ~ sP50
| sP41 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP43
| sP27 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP49
| ~ sP45
| ~ sP44 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP38
| ~ sP31
| sP49 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP47
| sP38
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP19
| ~ sP48
| sP35 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP6
| sP19 ),
inference(all_rule,[status(thm)],[]) ).
thf(14,plain,
( sP11
| sP45 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP24
| ~ sP37
| ~ sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP20
| ~ sP37
| sP47 ),
inference(prop_rule,[status(thm)],[]) ).
thf(17,plain,
( ~ sP5
| ~ sP48
| sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( ~ sP46
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(19,plain,
( ~ sP1
| sP48 ),
inference(all_rule,[status(thm)],[]) ).
thf(20,plain,
( ~ sP26
| sP46 ),
inference(all_rule,[status(thm)],[]) ).
thf(21,plain,
( ~ sP40
| sP24 ),
inference(all_rule,[status(thm)],[]) ).
thf(22,plain,
( ~ sP3
| sP20 ),
inference(all_rule,[status(thm)],[]) ).
thf(23,plain,
( ~ sP9
| ~ sP23
| sP30 ),
inference(prop_rule,[status(thm)],[]) ).
thf(24,plain,
( ~ sP51
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(25,plain,
( ~ sP42
| sP51 ),
inference(all_rule,[status(thm)],[]) ).
thf(26,plain,
( sP18
| sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(27,plain,
( sP13
| ~ sP18 ),
inference(prop_rule,[status(thm)],[]) ).
thf(28,plain,
( sP13
| sP32 ),
inference(prop_rule,[status(thm)],[]) ).
thf(29,plain,
( sP17
| ~ sP13 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__5]) ).
thf(30,plain,
( ~ sP52
| ~ sP7
| ~ sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(31,plain,
( ~ sP2
| sP52 ),
inference(all_rule,[status(thm)],[]) ).
thf(32,plain,
( ~ sP16
| ~ sP7
| sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(33,plain,
( ~ sP30
| sP16 ),
inference(all_rule,[status(thm)],[]) ).
thf(34,plain,
( sP10
| sP50 ),
inference(prop_rule,[status(thm)],[]) ).
thf(35,plain,
( sP36
| ~ sP10 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__4]) ).
thf(36,plain,
( ~ sP4
| sP7 ),
inference(all_rule,[status(thm)],[]) ).
thf(37,plain,
( ~ sP28
| ~ sP36 ),
inference(all_rule,[status(thm)],[]) ).
thf(38,plain,
( sP22
| ~ sP34 ),
inference(prop_rule,[status(thm)],[]) ).
thf(39,plain,
( sP22
| sP40 ),
inference(prop_rule,[status(thm)],[]) ).
thf(40,plain,
( ~ sP15
| ~ sP7
| ~ sP22 ),
inference(prop_rule,[status(thm)],[]) ).
thf(41,plain,
( ~ sP39
| sP15 ),
inference(all_rule,[status(thm)],[]) ).
thf(42,plain,
( ~ sP21
| ~ sP7
| sP39 ),
inference(prop_rule,[status(thm)],[]) ).
thf(43,plain,
( ~ sP43
| sP21 ),
inference(all_rule,[status(thm)],[]) ).
thf(44,plain,
( sP12
| ~ sP35 ),
inference(prop_rule,[status(thm)],[]) ).
thf(45,plain,
( sP12
| sP37 ),
inference(prop_rule,[status(thm)],[]) ).
thf(46,plain,
( sP34
| ~ sP12 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(ax5,axiom,
sP28 ).
thf(ax2,axiom,
sP26 ).
thf(ax1,axiom,
sP42 ).
thf(existence_of_d_ax,axiom,
sP4 ).
thf(a1,axiom,
sP1 ).
thf(47,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,h5,h6,ax5,ax2,ax1,existence_of_d_ax,a1]) ).
thf(48,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,47,h5,h6]) ).
thf(49,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,48,h4]) ).
thf(50,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,49,h3]) ).
thf(51,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[50,h1]) ).
thf(52,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[51,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ~ ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X2 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ~ ( ( combo @ d @ X3 @ X5 )
=> ~ ( h @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( rel_s4 @ X2 @ X5 )
=> ( open @ d @ X5 ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h2])],[50,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : PLA033^7 : TPTP v9.2.0. Released v5.5.0.
% 0.07/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n011.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 12:21:08 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.56 % SZS status Theorem
% 0.19/0.56 % Mode: cade22grackle2xfee4
% 0.19/0.56 % Steps: 2996
% 0.19/0.56 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------