TSTP Solution File: SYN052^7 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SYN052^7 : TPTP v8.1.2. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n015.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 : Fri Sep 1 03:19:39 EDT 2023
% Result : Theorem 0.90s 1.15s
% Output : Proof 0.90s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 215
% Syntax : Number of formulae : 231 ( 55 unt; 14 typ; 39 def)
% Number of atoms : 798 ( 43 equ; 5 cnn)
% Maximal formula atoms : 27 ( 3 avg)
% Number of connectives : 1539 ( 176 ~; 103 |; 8 &; 843 @)
% ( 81 <=>; 328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 68 ( 68 >; 0 *; 0 +; 0 <<)
% Number of symbols : 132 ( 128 usr; 127 con; 0-2 aty)
% Number of variables : 331 ( 74 ^; 251 !; 6 ?; 331 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_mu,type,
mu: $tType ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__7,type,
eigen__7: $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_p,type,
p: $i > $o ).
thf(ty_eigen__5,type,
eigen__5: $i ).
thf(ty_eigen__9,type,
eigen__9: mu ).
thf(ty_rel_s4,type,
rel_s4: $i > $i > $o ).
thf(ty_big_f,type,
big_f: mu > $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_exists_in_world,type,
exists_in_world: mu > $i > $o ).
thf(ty_eigen__6,type,
eigen__6: mu ).
thf(ty_eigen__17,type,
eigen__17: $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] :
~ ( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__17,definition,
( eigen__17
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__7 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__17])]) ).
thf(h1,assumption,
! [X1: mu > $o,X2: mu] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__1 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__6,definition,
( eigen__6
= ( eps__1
@ ^ [X1: mu] :
~ ( ( exists_in_world @ X1 @ eigen__5 )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__5 @ X2 )
=> ( big_f @ X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__6])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(eigendef_eigen__9,definition,
( eigen__9
= ( eps__1
@ ^ [X1: mu] :
~ ~ ( exists_in_world @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[eigen__9])]) ).
thf(eigendef_eigen__7,definition,
( eigen__7
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__5 @ X1 )
=> ( big_f @ eigen__6 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__7])]) ).
thf(eigendef_eigen__5,definition,
( eigen__5
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s4 @ eigen__2 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ 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
<=> ( ( rel_s4 @ eigen__5 @ eigen__7 )
=> ( big_f @ eigen__6 @ eigen__7 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i,X2: $i] :
( ~ ( ( rel_s4 @ eigen__1 @ X1 )
=> ~ ( rel_s4 @ X1 @ X2 ) )
=> ( rel_s4 @ eigen__1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: $i] :
( ~ ( ( exists_in_world @ eigen__9 @ X1 )
=> ~ ( rel_s4 @ X1 @ X2 ) )
=> ( exists_in_world @ eigen__9 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( rel_s4 @ eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( rel_s4 @ eigen__7 @ eigen__17 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( rel_s4 @ eigen__1 @ eigen__5 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__1 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_f @ X1 @ X3 ) ) ) )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_f @ X1 @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( exists_in_world @ eigen__9 @ eigen__1 )
=> ~ ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__9 @ X2 ) ) ) )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__9 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( exists_in_world @ eigen__6 @ eigen__5 )
=> ~ ( ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__6 @ X2 ) ) ) )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__6 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ( p @ X1 ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__7 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( rel_s4 @ eigen__7 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__6 @ X2 ) ) ) )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__6 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ( rel_s4 @ eigen__5 @ eigen__7 )
=> ( sP12
=> ! [X1: $i] :
( ( rel_s4 @ eigen__7 @ X1 )
=> ( big_f @ eigen__6 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ~ ( ( rel_s4 @ eigen__2 @ eigen__5 )
=> ~ ( rel_s4 @ eigen__5 @ eigen__7 ) )
=> ( rel_s4 @ eigen__2 @ eigen__7 ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( rel_s4 @ eigen__1 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( exists_in_world @ eigen__9 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i] :
( ~ ( ( exists_in_world @ eigen__9 @ eigen__1 )
=> ~ ( rel_s4 @ eigen__1 @ X1 ) )
=> ( exists_in_world @ eigen__9 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) ) ) )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( big_f @ eigen__6 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( sP13
=> ( big_f @ eigen__6 @ eigen__7 ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_f @ eigen__9 @ X1 ) )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__5 )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ eigen__5 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_f @ X1 @ X3 ) ) ) )
=> ~ ! [X2: $i] :
( ( rel_s4 @ eigen__5 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( big_f @ X1 @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ( exists_in_world @ eigen__9 @ eigen__3 )
=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_f @ eigen__9 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( exists_in_world @ eigen__6 @ eigen__5 ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ( rel_s4 @ eigen__2 @ eigen__5 )
=> ~ ( rel_s4 @ eigen__5 @ eigen__7 ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__3 )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__3 @ X2 )
=> ( big_f @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( ( rel_s4 @ eigen__2 @ eigen__17 )
=> ( p @ eigen__17 ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ~ ( ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_f @ X2 @ X4 ) ) ) )
=> ~ ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( big_f @ X2 @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__9 @ X2 ) ) ) )
=> ~ ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__9 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( ( rel_s4 @ eigen__3 @ eigen__3 )
=> sP28 ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( exists_in_world @ eigen__9 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( big_f @ eigen__9 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ! [X1: $i] :
( ~ ( ( rel_s4 @ eigen__2 @ eigen__7 )
=> ~ ( rel_s4 @ eigen__7 @ X1 ) )
=> ( rel_s4 @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( sP17
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__5 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__6 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( sP6
=> ( p @ eigen__17 ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ( sP12
=> ! [X1: $i] :
( ( rel_s4 @ eigen__7 @ X1 )
=> ( big_f @ eigen__6 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( ( rel_s4 @ eigen__2 @ eigen__7 )
=> ~ sP6 ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ( p @ X1 ) )
=> sP36 ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( ( rel_s4 @ eigen__1 @ eigen__3 )
=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ X3 ) ) ) )
=> sP38 ) ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( ~ ( sP33
=> ~ ( rel_s4 @ eigen__1 @ eigen__3 ) )
=> ( exists_in_world @ eigen__9 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( sP26
=> sP21 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( ( rel_s4 @ eigen__1 @ eigen__3 )
=> sP23 ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ! [X1: mu] :
~ ( exists_in_world @ X1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( sP33
=> ~ ( rel_s4 @ eigen__1 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( rel_s4 @ eigen__2 @ eigen__5 ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__2 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( rel_s4 @ X1 @ X2 )
=> ~ ( rel_s4 @ X2 @ X3 ) )
=> ( rel_s4 @ X1 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ! [X1: $i] :
( ~ ( sP51
=> ~ ( rel_s4 @ eigen__5 @ X1 ) )
=> ( rel_s4 @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( p @ eigen__17 ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(sP56,plain,
( sP56
<=> ( rel_s4 @ eigen__3 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP56])]) ).
thf(sP57,plain,
( sP57
<=> ( sP51
=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__5 )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__5 @ X2 )
=> ( big_f @ X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP57])]) ).
thf(sP58,plain,
( sP58
<=> ! [X1: $i] :
~ ! [X2: mu] :
~ ( exists_in_world @ X2 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP58])]) ).
thf(sP59,plain,
( sP59
<=> ( ~ ( sP18
=> ~ sP5 )
=> sP33 ) ),
introduced(definition,[new_symbols(definition,[sP59])]) ).
thf(sP60,plain,
( sP60
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( big_f @ eigen__9 @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP60])]) ).
thf(sP61,plain,
( sP61
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__7 @ X1 )
=> ( big_f @ eigen__6 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP61])]) ).
thf(sP62,plain,
( sP62
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__1 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( p @ X2 ) )
=> ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( big_f @ X3 @ X4 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP62])]) ).
thf(sP63,plain,
( sP63
<=> ( ~ ( ( rel_s4 @ eigen__1 @ eigen__2 )
=> ~ sP51 )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP63])]) ).
thf(sP64,plain,
( sP64
<=> ( sP7
=> sP24 ) ),
introduced(definition,[new_symbols(definition,[sP64])]) ).
thf(sP65,plain,
( sP65
<=> ( rel_s4 @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP65])]) ).
thf(sP66,plain,
( sP66
<=> ( big_f @ eigen__6 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP66])]) ).
thf(sP67,plain,
( sP67
<=> ( sP18
=> ~ sP5 ) ),
introduced(definition,[new_symbols(definition,[sP67])]) ).
thf(sP68,plain,
( sP68
<=> ! [X1: $i] :
( ~ ( sP18
=> ~ ( rel_s4 @ eigen__0 @ X1 ) )
=> ( exists_in_world @ eigen__9 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP68])]) ).
thf(sP69,plain,
( sP69
<=> ( ~ sP42
=> ( rel_s4 @ eigen__2 @ eigen__17 ) ) ),
introduced(definition,[new_symbols(definition,[sP69])]) ).
thf(sP70,plain,
( sP70
<=> ( ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ X3 ) ) ) )
=> sP38 ) ),
introduced(definition,[new_symbols(definition,[sP70])]) ).
thf(sP71,plain,
( sP71
<=> ( rel_s4 @ eigen__1 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP71])]) ).
thf(sP72,plain,
( sP72
<=> ( exists_in_world @ eigen__9 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP72])]) ).
thf(sP73,plain,
( sP73
<=> ( sP65
=> ~ sP51 ) ),
introduced(definition,[new_symbols(definition,[sP73])]) ).
thf(sP74,plain,
( sP74
<=> ( rel_s4 @ eigen__5 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP74])]) ).
thf(sP75,plain,
( sP75
<=> ( rel_s4 @ eigen__2 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP75])]) ).
thf(sP76,plain,
( sP76
<=> ! [X1: mu,X2: $i,X3: $i] :
( ~ ( ( exists_in_world @ X1 @ X2 )
=> ~ ( rel_s4 @ X2 @ X3 ) )
=> ( exists_in_world @ X1 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP76])]) ).
thf(sP77,plain,
( sP77
<=> ( rel_s4 @ eigen__2 @ eigen__17 ) ),
introduced(definition,[new_symbols(definition,[sP77])]) ).
thf(sP78,plain,
( sP78
<=> ! [X1: mu] :
( ( exists_in_world @ X1 @ eigen__5 )
=> ! [X2: $i] :
( ( rel_s4 @ eigen__5 @ X2 )
=> ( big_f @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP78])]) ).
thf(sP79,plain,
( sP79
<=> ! [X1: $i] :
( ~ ( sP65
=> ~ ( rel_s4 @ eigen__2 @ X1 ) )
=> ( rel_s4 @ eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP79])]) ).
thf(sP80,plain,
( sP80
<=> ! [X1: $i,X2: $i] :
( ~ ( ( rel_s4 @ eigen__2 @ X1 )
=> ~ ( rel_s4 @ X1 @ X2 ) )
=> ( rel_s4 @ eigen__2 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP80])]) ).
thf(sP81,plain,
( sP81
<=> ! [X1: $i] :
( ( rel_s4 @ eigen__3 @ X1 )
=> ! [X2: mu] :
( ( exists_in_world @ X2 @ X1 )
=> ! [X3: $i] :
( ( rel_s4 @ X1 @ X3 )
=> ( big_f @ X2 @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP81])]) ).
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(pel22,conjecture,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X3 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) ) ) )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) ) ) )
=> ~ ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ) ) ).
thf(h2,negated_conjecture,
~ ! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X3 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) ) ) )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) ) ) )
=> ~ ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[pel22]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( rel_s4 @ eigen__0 @ X1 )
=> ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ! [X3: mu] :
( ( exists_in_world @ X3 @ X2 )
=> ~ ( ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( p @ X5 ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( big_f @ X3 @ X5 ) ) ) )
=> ~ ! [X4: $i] :
( ( rel_s4 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( big_f @ X3 @ X5 ) )
=> ! [X5: $i] :
( ( rel_s4 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X3 )
=> ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( big_f @ X4 @ X5 ) ) ) ) ) )
=> ~ ! [X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X3 )
=> ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( big_f @ X4 @ X5 ) ) ) )
=> ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( sP5
=> ( sP30
=> ~ sP20 ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
sP5,
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP30
=> ~ sP20 ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP30,
introduced(assumption,[]) ).
thf(h8,assumption,
sP20,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP42
| ~ sP75
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP69
| sP42
| sP77 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP29
| ~ sP77
| sP55 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP52
| sP29 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP35
| sP69 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP15
| ~ sP74
| sP41 ),
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP39
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( sP14
| sP39 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP41
| ~ sP12
| sP61 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP10
| ~ sP26
| ~ sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP22
| ~ sP13
| sP66 ),
inference(prop_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP61
| sP22 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP24
| sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP48
| ~ sP71
| sP23 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( ~ sP60
| sP48 ),
inference(all_rule,[status(thm)],]) ).
thf(16,plain,
( sP31
| sP60 ),
inference(prop_rule,[status(thm)],]) ).
thf(17,plain,
( ~ sP23
| ~ sP34
| sP38 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP9
| ~ sP33
| ~ sP31 ),
inference(prop_rule,[status(thm)],]) ).
thf(19,plain,
( ~ sP50
| ~ sP33
| ~ sP71 ),
inference(prop_rule,[status(thm)],]) ).
thf(20,plain,
( ~ sP46
| sP50
| sP72 ),
inference(prop_rule,[status(thm)],]) ).
thf(21,plain,
( ~ sP67
| ~ sP18
| ~ sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(22,plain,
( ~ sP59
| sP67
| sP33 ),
inference(prop_rule,[status(thm)],]) ).
thf(23,plain,
( ~ sP19
| sP46 ),
inference(all_rule,[status(thm)],]) ).
thf(24,plain,
( ~ sP68
| sP59 ),
inference(all_rule,[status(thm)],]) ).
thf(25,plain,
( ~ sP4
| sP19 ),
inference(all_rule,[status(thm)],]) ).
thf(26,plain,
( ~ sP4
| sP68 ),
inference(all_rule,[status(thm)],]) ).
thf(27,plain,
( ~ sP25
| ~ sP72
| sP34 ),
inference(prop_rule,[status(thm)],]) ).
thf(28,plain,
( ~ sP76
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(29,plain,
( ~ sP8
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(30,plain,
( ~ sP28
| sP25 ),
inference(all_rule,[status(thm)],]) ).
thf(31,plain,
( sP40
| ~ sP55 ),
inference(prop_rule,[status(thm)],]) ).
thf(32,plain,
( sP40
| sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(33,plain,
( sP12
| ~ sP40 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__17]) ).
thf(34,plain,
( ~ sP32
| ~ sP56
| sP28 ),
inference(prop_rule,[status(thm)],]) ).
thf(35,plain,
( ~ sP1
| sP56 ),
inference(all_rule,[status(thm)],]) ).
thf(36,plain,
( ~ sP81
| sP32 ),
inference(all_rule,[status(thm)],]) ).
thf(37,plain,
( ~ sP27
| ~ sP51
| ~ sP74 ),
inference(prop_rule,[status(thm)],]) ).
thf(38,plain,
( ~ sP16
| sP27
| sP75 ),
inference(prop_rule,[status(thm)],]) ).
thf(39,plain,
( ~ sP1
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(40,plain,
( ~ sP80
| sP35 ),
inference(all_rule,[status(thm)],]) ).
thf(41,plain,
( ~ sP54
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(42,plain,
( ~ sP64
| ~ sP7
| sP24 ),
inference(prop_rule,[status(thm)],]) ).
thf(43,plain,
( ~ sP73
| ~ sP65
| ~ sP51 ),
inference(prop_rule,[status(thm)],]) ).
thf(44,plain,
( ~ sP63
| sP73
| sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(45,plain,
( ~ sP30
| sP64 ),
inference(all_rule,[status(thm)],]) ).
thf(46,plain,
( ~ sP79
| sP63 ),
inference(all_rule,[status(thm)],]) ).
thf(47,plain,
( ~ sP80
| sP54 ),
inference(all_rule,[status(thm)],]) ).
thf(48,plain,
( ~ sP53
| sP80 ),
inference(all_rule,[status(thm)],]) ).
thf(49,plain,
( ~ sP3
| sP79 ),
inference(all_rule,[status(thm)],]) ).
thf(50,plain,
( sP49
| sP18 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__9]) ).
thf(51,plain,
( ~ sP58
| ~ sP49 ),
inference(all_rule,[status(thm)],]) ).
thf(52,plain,
( ~ sP37
| ~ sP17
| sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(53,plain,
( ~ sP1
| sP17 ),
inference(all_rule,[status(thm)],]) ).
thf(54,plain,
( ~ sP53
| sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(55,plain,
( ~ sP30
| sP37 ),
inference(all_rule,[status(thm)],]) ).
thf(56,plain,
( sP2
| ~ sP66 ),
inference(prop_rule,[status(thm)],]) ).
thf(57,plain,
( sP2
| sP74 ),
inference(prop_rule,[status(thm)],]) ).
thf(58,plain,
( sP21
| ~ sP2 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__7]) ).
thf(59,plain,
( sP47
| ~ sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(60,plain,
( sP47
| sP26 ),
inference(prop_rule,[status(thm)],]) ).
thf(61,plain,
( sP78
| ~ sP47 ),
inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__6]) ).
thf(62,plain,
( sP57
| ~ sP78 ),
inference(prop_rule,[status(thm)],]) ).
thf(63,plain,
( sP57
| sP51 ),
inference(prop_rule,[status(thm)],]) ).
thf(64,plain,
( sP36
| ~ sP57 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__5]) ).
thf(65,plain,
( sP70
| ~ sP38 ),
inference(prop_rule,[status(thm)],]) ).
thf(66,plain,
( sP70
| sP81 ),
inference(prop_rule,[status(thm)],]) ).
thf(67,plain,
( sP43
| ~ sP36 ),
inference(prop_rule,[status(thm)],]) ).
thf(68,plain,
( sP43
| sP52 ),
inference(prop_rule,[status(thm)],]) ).
thf(69,plain,
( sP45
| ~ sP70 ),
inference(prop_rule,[status(thm)],]) ).
thf(70,plain,
( sP45
| sP71 ),
inference(prop_rule,[status(thm)],]) ).
thf(71,plain,
( sP11
| ~ sP43 ),
inference(prop_rule,[status(thm)],]) ).
thf(72,plain,
( sP11
| sP65 ),
inference(prop_rule,[status(thm)],]) ).
thf(73,plain,
( sP44
| ~ sP45 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(74,plain,
( sP62
| ~ sP11 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(75,plain,
( ~ sP20
| ~ sP62
| ~ sP44 ),
inference(prop_rule,[status(thm)],]) ).
thf(cumulative_ax,axiom,
sP76 ).
thf(a2,axiom,
sP53 ).
thf(a1,axiom,
sP1 ).
thf(nonempty_ax,axiom,
sP58 ).
thf(76,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,h5,h7,h8,cumulative_ax,a2,a1,nonempty_ax]) ).
thf(77,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,76,h7,h8]) ).
thf(78,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,77,h5,h6]) ).
thf(79,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,78,h4]) ).
thf(80,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,79,h3]) ).
thf(81,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[80,h1]) ).
thf(82,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[81,h0]) ).
thf(0,theorem,
! [X1: $i,X2: $i] :
( ( rel_s4 @ X1 @ X2 )
=> ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ! [X4: mu] :
( ( exists_in_world @ X4 @ X3 )
=> ~ ( ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) ) ) )
=> ~ ! [X5: $i] :
( ( rel_s4 @ X3 @ X5 )
=> ( ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( big_f @ X4 @ X6 ) )
=> ! [X6: $i] :
( ( rel_s4 @ X5 @ X6 )
=> ( p @ X6 ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) ) ) )
=> ~ ! [X3: $i] :
( ( rel_s4 @ X2 @ X3 )
=> ( ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ! [X5: mu] :
( ( exists_in_world @ X5 @ X4 )
=> ! [X6: $i] :
( ( rel_s4 @ X4 @ X6 )
=> ( big_f @ X5 @ X6 ) ) ) )
=> ! [X4: $i] :
( ( rel_s4 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h2])],[80,h2]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SYN052^7 : TPTP v8.1.2. Released v5.5.0.
% 0.00/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n015.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Aug 26 19:12:38 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.90/1.15 % SZS status Theorem
% 0.90/1.15 % Mode: cade22grackle2xfee4
% 0.90/1.15 % Steps: 10900
% 0.90/1.15 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------