TSTP Solution File: SYO442^1 by Lash---1.13
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- Process Solution
%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SYO442^1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n002.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 04:46:45 EDT 2023
% Result : Theorem 0.20s 0.70s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 4
% Number of leaves : 156
% Syntax : Number of formulae : 166 ( 46 unt; 8 typ; 38 def)
% Number of atoms : 490 ( 43 equ; 5 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 832 ( 128 ~; 74 |; 8 &; 432 @)
% ( 55 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 63 ( 63 >; 0 *; 0 +; 0 <<)
% Number of symbols : 102 ( 98 usr; 99 con; 0-2 aty)
% Number of variables : 203 ( 73 ^; 124 !; 6 ?; 203 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__6,type,
eigen__6: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_eigen__5,type,
eigen__5: $i ).
thf(ty_rel_s5,type,
rel_s5: $i > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_p,type,
p: $i > $o ).
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_s5 @ eigen__0 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__6,definition,
( eigen__6
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s5 @ eigen__5 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__6])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s5 @ eigen__1 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(eigendef_eigen__5,definition,
( eigen__5
= ( eps__0
@ ^ [X1: $i] :
~ ( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__5])]) ).
thf(sP1,plain,
( sP1
<=> ( ( rel_s5 @ eigen__0 @ eigen__0 )
=> ~ ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( rel_s5 @ X1 @ X2 )
=> ~ ( rel_s5 @ X2 @ X3 ) )
=> ( rel_s5 @ X1 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__6 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ( p @ X4 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__5 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( rel_s5 @ eigen__3 @ eigen__3 )
=> ! [X1: $i] :
( ( rel_s5 @ eigen__3 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__3 @ X1 )
=> ~ ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( ( rel_s5 @ eigen__6 @ eigen__0 )
=> ~ ( rel_s5 @ eigen__0 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( rel_s5 @ eigen__5 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( rel_s5 @ eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( rel_s5 @ eigen__5 @ eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( sP11
=> ~ sP3 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( rel_s5 @ eigen__6 @ eigen__2 )
=> ( p @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( sP4
=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ( rel_s5 @ eigen__0 @ eigen__3 )
=> ( rel_s5 @ eigen__3 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( rel_s5 @ eigen__3 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( sP10
=> ~ ( rel_s5 @ eigen__1 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ( rel_s5 @ X1 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ( rel_s5 @ eigen__0 @ eigen__5 )
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ! [X1: $i] :
( ~ ( ( rel_s5 @ eigen__6 @ eigen__5 )
=> ~ ( rel_s5 @ eigen__5 @ X1 ) )
=> ( rel_s5 @ eigen__6 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( rel_s5 @ eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ! [X1: $i] :
( ~ ( ( rel_s5 @ eigen__6 @ eigen__0 )
=> ~ ( rel_s5 @ eigen__0 @ X1 ) )
=> ( rel_s5 @ eigen__6 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( sP11
=> ( rel_s5 @ eigen__6 @ eigen__5 ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( rel_s5 @ eigen__0 @ eigen__5 ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) )
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__5 @ X1 )
=> ( rel_s5 @ X1 @ eigen__5 ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__3 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( ( rel_s5 @ eigen__0 @ eigen__0 )
=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( rel_s5 @ eigen__1 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ! [X1: $i] :
( ~ ( sP10
=> ~ ( rel_s5 @ eigen__1 @ X1 ) )
=> ( rel_s5 @ eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( sP10
=> ! [X1: $i] :
( ( rel_s5 @ eigen__1 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__1 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( sP16
=> ~ ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( ~ sP17
=> sP21 ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ( ~ ( ( rel_s5 @ eigen__6 @ eigen__5 )
=> ~ sP9 )
=> ( rel_s5 @ eigen__6 @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( sP25
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ! [X1: $i,X2: $i] :
( ~ ( ( rel_s5 @ eigen__0 @ X1 )
=> ~ ( rel_s5 @ X1 @ X2 ) )
=> ( rel_s5 @ eigen__0 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( rel_s5 @ eigen__0 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ( p @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ! [X1: $i,X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( rel_s5 @ X2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( rel_s5 @ eigen__3 @ eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( rel_s5 @ eigen__0 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ! [X1: $i] : ( rel_s5 @ X1 @ X1 ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( rel_s5 @ eigen__6 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ! [X1: $i] :
( ( rel_s5 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ( p @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( sP30
=> ( p @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ! [X1: $i,X2: $i] :
( ~ ( ( rel_s5 @ eigen__6 @ X1 )
=> ~ ( rel_s5 @ X1 @ X2 ) )
=> ( rel_s5 @ eigen__6 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( ( rel_s5 @ eigen__6 @ eigen__5 )
=> ~ sP9 ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( ~ sP8
=> ( rel_s5 @ eigen__6 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( sP39
=> ~ sP28 ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ( p @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ( rel_s5 @ eigen__6 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ( sP14
=> ~ sP26 ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( rel_s5 @ eigen__6 @ eigen__5 ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(def_meq_ind,definition,
( meq_ind
= ( ^ [X1: mu,X2: mu,X3: $i] : ( X1 = X2 ) ) ) ).
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_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_mforall_ind,definition,
( mforall_ind
= ( ^ [X1: mu > $i > $o,X2: $i] :
! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ).
thf(def_mforall_prop,definition,
( mforall_prop
= ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
! [X3: $i > $o] : ( 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_mtrue,definition,
( mtrue
= ( ^ [X1: $i] : $true ) ) ).
thf(def_mfalse,definition,
( mfalse
= ( mnot @ mtrue ) ) ).
thf(def_mbox,definition,
( mbox
= ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
! [X4: $i] :
( ( (~) @ ( X1 @ X3 @ X4 ) )
| ( X2 @ X4 ) ) ) ) ).
thf(def_mdia,definition,
( mdia
= ( ^ [X1: $i > $i > $o,X2: $i > $o] : ( mnot @ ( mbox @ X1 @ ( mnot @ 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_minvalid,definition,
( minvalid
= ( ^ [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_mbox_s5,definition,
( mbox_s5
= ( ^ [X1: $i > $o,X2: $i] :
! [X3: $i] :
( ( (~) @ ( rel_s5 @ X2 @ X3 ) )
| ( X1 @ X3 ) ) ) ) ).
thf(def_mdia_s5,definition,
( mdia_s5
= ( ^ [X1: $i > $o] : ( mnot @ ( mbox_s5 @ ( mnot @ X1 ) ) ) ) ) ).
thf(prove,conjecture,
! [X1: $i] :
~ ( ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) )
=> ~ ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [X1: $i] :
~ ( ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) )
=> ~ ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[prove]) ).
thf(h2,assumption,
sP54,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP8
| ~ sP45
| ~ sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP50
| sP8
| sP53 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP49
| ~ sP55
| ~ sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP36
| sP49
| sP45 ),
inference(prop_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP23
| sP50 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP20
| sP36 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP48
| sP23 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP48
| sP20 ),
inference(all_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP24
| ~ sP11
| sP55 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP2
| sP48 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP27
| sP24 ),
inference(all_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP19
| ~ sP25
| sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP41
| sP27 ),
inference(all_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP18
| sP19 ),
inference(all_rule,[status(thm)],]) ).
thf(15,plain,
( ~ sP17
| ~ sP10
| ~ sP30 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
( ~ sP35
| sP17
| sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(17,plain,
( ~ sP13
| ~ sP53
| sP52 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP3
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(19,plain,
( ~ sP31
| sP35 ),
inference(all_rule,[status(thm)],]) ).
thf(20,plain,
( ~ sP38
| sP31 ),
inference(all_rule,[status(thm)],]) ).
thf(21,plain,
( ~ sP34
| ~ sP16
| ~ sP40 ),
inference(prop_rule,[status(thm)],]) ).
thf(22,plain,
( ~ sP7
| sP34 ),
inference(all_rule,[status(thm)],]) ).
thf(23,plain,
( ~ sP6
| ~ sP42
| sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(24,plain,
( ~ sP15
| ~ sP39
| sP16 ),
inference(prop_rule,[status(thm)],]) ).
thf(25,plain,
( ~ sP44
| sP42 ),
inference(all_rule,[status(thm)],]) ).
thf(26,plain,
( ~ sP28
| sP6 ),
inference(all_rule,[status(thm)],]) ).
thf(27,plain,
( ~ sP18
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(28,plain,
( sP12
| sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(29,plain,
( sP12
| sP11 ),
inference(prop_rule,[status(thm)],]) ).
thf(30,plain,
( sP5
| ~ sP12 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__6]) ).
thf(31,plain,
( sP37
| ~ sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(32,plain,
( sP37
| sP25 ),
inference(prop_rule,[status(thm)],]) ).
thf(33,plain,
( sP22
| ~ sP37 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__5]) ).
thf(34,plain,
( ~ sP1
| ~ sP43
| ~ sP22 ),
inference(prop_rule,[status(thm)],]) ).
thf(35,plain,
( ~ sP29
| ~ sP43
| sP40 ),
inference(prop_rule,[status(thm)],]) ).
thf(36,plain,
( ~ sP44
| sP43 ),
inference(all_rule,[status(thm)],]) ).
thf(37,plain,
( ~ sP41
| sP18 ),
inference(all_rule,[status(thm)],]) ).
thf(38,plain,
( ~ sP2
| sP38 ),
inference(all_rule,[status(thm)],]) ).
thf(39,plain,
( ~ sP4
| sP1 ),
inference(all_rule,[status(thm)],]) ).
thf(40,plain,
( ~ sP46
| sP29 ),
inference(all_rule,[status(thm)],]) ).
thf(41,plain,
( sP51
| sP28 ),
inference(prop_rule,[status(thm)],]) ).
thf(42,plain,
( sP51
| sP39 ),
inference(prop_rule,[status(thm)],]) ).
thf(43,plain,
( sP4
| ~ sP51 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(44,plain,
( sP26
| ~ sP4 ),
inference(prop_rule,[status(thm)],]) ).
thf(45,plain,
( sP26
| sP46 ),
inference(prop_rule,[status(thm)],]) ).
thf(46,plain,
( sP47
| ~ sP52 ),
inference(prop_rule,[status(thm)],]) ).
thf(47,plain,
( sP47
| sP30 ),
inference(prop_rule,[status(thm)],]) ).
thf(48,plain,
( sP33
| ~ sP47 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(49,plain,
( sP32
| ~ sP33 ),
inference(prop_rule,[status(thm)],]) ).
thf(50,plain,
( sP32
| sP10 ),
inference(prop_rule,[status(thm)],]) ).
thf(51,plain,
( sP46
| ~ sP32 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(52,plain,
( sP14
| ~ sP46 ),
inference(prop_rule,[status(thm)],]) ).
thf(53,plain,
( sP14
| sP4 ),
inference(prop_rule,[status(thm)],]) ).
thf(54,plain,
( ~ sP54
| ~ sP14
| ~ sP26 ),
inference(prop_rule,[status(thm)],]) ).
thf(a3,axiom,
sP41 ).
thf(a2,axiom,
sP2 ).
thf(a1,axiom,
sP44 ).
thf(55,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([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,h2,a3,a2,a1]) ).
thf(56,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,55,h2]) ).
thf(57,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[56,h0]) ).
thf(0,theorem,
! [X1: $i] :
~ ( ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) ) )
=> ~ ( ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ( p @ X3 ) ) )
=> ! [X2: $i] :
( ( rel_s5 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( rel_s5 @ X2 @ X3 )
=> ! [X4: $i] :
( ( rel_s5 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( rel_s5 @ X4 @ X5 )
=> ( p @ X5 ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[56,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SYO442^1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.33 % Computer : n002.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sat Aug 26 07:21:03 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.70 % SZS status Theorem
% 0.20/0.70 % Mode: cade22grackle2xfee4
% 0.20/0.70 % Steps: 3408
% 0.20/0.70 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------