TSTP Solution File: SET575^7 by Lash---1.13

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : SET575^7 : 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 : 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 Aug 31 15:17:32 EDT 2023

% Result   : Theorem 0.19s 0.46s
% Output   : Proof 0.19s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_mu,type,
    mu: $tType ).

thf(ty_eigen__4,type,
    eigen__4: mu ).

thf(ty_eigen__0,type,
    eigen__0: $i ).

thf(ty_eigen__3,type,
    eigen__3: $i ).

thf(ty_eigen__2,type,
    eigen__2: mu ).

thf(ty_member,type,
    member: mu > mu > $i > $o ).

thf(ty_eigen__5,type,
    eigen__5: $i ).

thf(ty_rel_s4,type,
    rel_s4: $i > $i > $o ).

thf(ty_intersect,type,
    intersect: mu > mu > $i > $o ).

thf(ty_eigen__1,type,
    eigen__1: $i ).

thf(ty_exists_in_world,type,
    exists_in_world: mu > $i > $o ).

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__5 )
       => ( ! [X2: $i] :
              ( ( rel_s4 @ eigen__5 @ X2 )
             => ( member @ X1 @ eigen__2 @ X2 ) )
         => ~ ! [X2: $i] :
                ( ( rel_s4 @ eigen__5 @ X2 )
               => ( member @ X1 @ eigen__4 @ X2 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ( ( rel_s4 @ eigen__3 @ eigen__3 )
     => ! [X1: mu] :
          ( ( exists_in_world @ X1 @ eigen__3 )
         => ~ ( ! [X2: $i] :
                  ( ( rel_s4 @ eigen__3 @ X2 )
                 => ( ! [X3: $i] :
                        ( ( rel_s4 @ X2 @ X3 )
                       => ( intersect @ eigen__2 @ X1 @ X3 ) )
                   => ~ ! [X3: mu] :
                          ( ( exists_in_world @ X3 @ X2 )
                         => ( ! [X4: $i] :
                                ( ( rel_s4 @ X2 @ X4 )
                               => ( member @ X3 @ eigen__2 @ X4 ) )
                           => ~ ! [X4: $i] :
                                  ( ( rel_s4 @ X2 @ X4 )
                                 => ( member @ X3 @ X1 @ X4 ) ) ) ) ) )
             => ~ ! [X2: $i] :
                    ( ( rel_s4 @ eigen__3 @ X2 )
                   => ( ~ ! [X3: mu] :
                            ( ( exists_in_world @ X3 @ X2 )
                           => ( ! [X4: $i] :
                                  ( ( rel_s4 @ X2 @ X4 )
                                 => ( member @ X3 @ eigen__2 @ X4 ) )
                             => ~ ! [X4: $i] :
                                    ( ( rel_s4 @ X2 @ X4 )
                                   => ( member @ X3 @ X1 @ X4 ) ) ) )
                     => ! [X3: $i] :
                          ( ( rel_s4 @ X2 @ X3 )
                         => ( intersect @ eigen__2 @ X1 @ X3 ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( exists_in_world @ eigen__4 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ( exists_in_world @ eigen__2 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ( rel_s4 @ eigen__3 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: $i,X2: $i] :
        ( ( rel_s4 @ X1 @ X2 )
       => ! [X3: mu] :
            ( ( exists_in_world @ X3 @ X2 )
           => ! [X4: $i] :
                ( ( rel_s4 @ X2 @ X4 )
               => ! [X5: mu] :
                    ( ( exists_in_world @ X5 @ X4 )
                   => ~ ( ! [X6: $i] :
                            ( ( rel_s4 @ X4 @ X6 )
                           => ( ! [X7: $i] :
                                  ( ( rel_s4 @ X6 @ X7 )
                                 => ( intersect @ X3 @ X5 @ X7 ) )
                             => ~ ! [X7: mu] :
                                    ( ( exists_in_world @ X7 @ X6 )
                                   => ( ! [X8: $i] :
                                          ( ( rel_s4 @ X6 @ X8 )
                                         => ( member @ X7 @ X3 @ X8 ) )
                                     => ~ ! [X8: $i] :
                                            ( ( rel_s4 @ X6 @ X8 )
                                           => ( member @ X7 @ X5 @ X8 ) ) ) ) ) )
                       => ~ ! [X6: $i] :
                              ( ( rel_s4 @ X4 @ X6 )
                             => ( ~ ! [X7: mu] :
                                      ( ( exists_in_world @ X7 @ X6 )
                                     => ( ! [X8: $i] :
                                            ( ( rel_s4 @ X6 @ X8 )
                                           => ( member @ X7 @ X3 @ X8 ) )
                                       => ~ ! [X8: $i] :
                                              ( ( rel_s4 @ X6 @ X8 )
                                             => ( member @ X7 @ X5 @ X8 ) ) ) )
                               => ! [X7: $i] :
                                    ( ( rel_s4 @ X6 @ X7 )
                                   => ( intersect @ X3 @ X5 @ X7 ) ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( sP5
     => ! [X1: $i] :
          ( ( rel_s4 @ eigen__3 @ X1 )
         => ! [X2: mu] :
              ( ( exists_in_world @ X2 @ X1 )
             => ~ ( ! [X3: $i] :
                      ( ( rel_s4 @ X1 @ X3 )
                     => ( ! [X4: $i] :
                            ( ( rel_s4 @ X3 @ X4 )
                           => ( intersect @ eigen__2 @ X2 @ X4 ) )
                       => ~ ! [X4: mu] :
                              ( ( exists_in_world @ X4 @ X3 )
                             => ( ! [X5: $i] :
                                    ( ( rel_s4 @ X3 @ X5 )
                                   => ( member @ X4 @ eigen__2 @ X5 ) )
                               => ~ ! [X5: $i] :
                                      ( ( rel_s4 @ X3 @ X5 )
                                     => ( member @ X4 @ X2 @ X5 ) ) ) ) ) )
                 => ~ ! [X3: $i] :
                        ( ( rel_s4 @ X1 @ X3 )
                       => ( ~ ! [X4: mu] :
                                ( ( exists_in_world @ X4 @ X3 )
                               => ( ! [X5: $i] :
                                      ( ( rel_s4 @ X3 @ X5 )
                                     => ( member @ X4 @ eigen__2 @ X5 ) )
                                 => ~ ! [X5: $i] :
                                        ( ( rel_s4 @ X3 @ X5 )
                                       => ( member @ X4 @ X2 @ X5 ) ) ) )
                         => ! [X4: $i] :
                              ( ( rel_s4 @ X3 @ X4 )
                             => ( intersect @ eigen__2 @ X2 @ X4 ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ( ! [X1: $i] :
          ( ( rel_s4 @ eigen__5 @ X1 )
         => ( intersect @ eigen__2 @ eigen__4 @ X1 ) )
     => ~ sP2 ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ( ( exists_in_world @ eigen__2 @ eigen__1 )
     => ~ ( rel_s4 @ eigen__1 @ eigen__3 ) ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(sP11,plain,
    ( sP11
  <=> ! [X1: $i] :
        ( ( rel_s4 @ eigen__3 @ X1 )
       => ( ! [X2: $i] :
              ( ( rel_s4 @ X1 @ X2 )
             => ( intersect @ eigen__2 @ eigen__4 @ X2 ) )
         => ~ ! [X2: mu] :
                ( ( exists_in_world @ X2 @ X1 )
               => ( ! [X3: $i] :
                      ( ( rel_s4 @ X1 @ X3 )
                     => ( member @ X2 @ eigen__2 @ X3 ) )
                 => ~ ! [X3: $i] :
                        ( ( rel_s4 @ X1 @ X3 )
                       => ( member @ X2 @ eigen__4 @ X3 ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP11])]) ).

thf(sP12,plain,
    ( sP12
  <=> ( sP11
     => ~ ! [X1: $i] :
            ( ( rel_s4 @ eigen__3 @ X1 )
           => ( ~ ! [X2: mu] :
                    ( ( exists_in_world @ X2 @ X1 )
                   => ( ! [X3: $i] :
                          ( ( rel_s4 @ X1 @ X3 )
                         => ( member @ X2 @ eigen__2 @ X3 ) )
                     => ~ ! [X3: $i] :
                            ( ( rel_s4 @ X1 @ X3 )
                           => ( member @ X2 @ eigen__4 @ X3 ) ) ) )
             => ! [X2: $i] :
                  ( ( rel_s4 @ X1 @ X2 )
                 => ( intersect @ eigen__2 @ eigen__4 @ X2 ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP12])]) ).

thf(sP13,plain,
    ( sP13
  <=> ! [X1: mu] :
        ( ( exists_in_world @ X1 @ eigen__3 )
       => ! [X2: $i] :
            ( ( rel_s4 @ eigen__3 @ X2 )
           => ! [X3: mu] :
                ( ( exists_in_world @ X3 @ X2 )
               => ~ ( ! [X4: $i] :
                        ( ( rel_s4 @ X2 @ X4 )
                       => ( ! [X5: $i] :
                              ( ( rel_s4 @ X4 @ X5 )
                             => ( intersect @ X1 @ X3 @ X5 ) )
                         => ~ ! [X5: mu] :
                                ( ( exists_in_world @ X5 @ X4 )
                               => ( ! [X6: $i] :
                                      ( ( rel_s4 @ X4 @ X6 )
                                     => ( member @ X5 @ X1 @ X6 ) )
                                 => ~ ! [X6: $i] :
                                        ( ( rel_s4 @ X4 @ X6 )
                                       => ( member @ X5 @ X3 @ X6 ) ) ) ) ) )
                   => ~ ! [X4: $i] :
                          ( ( rel_s4 @ X2 @ X4 )
                         => ( ~ ! [X5: mu] :
                                  ( ( exists_in_world @ X5 @ X4 )
                                 => ( ! [X6: $i] :
                                        ( ( rel_s4 @ X4 @ X6 )
                                       => ( member @ X5 @ X1 @ X6 ) )
                                   => ~ ! [X6: $i] :
                                          ( ( rel_s4 @ X4 @ X6 )
                                         => ( member @ X5 @ X3 @ X6 ) ) ) )
                           => ! [X5: $i] :
                                ( ( rel_s4 @ X4 @ X5 )
                               => ( intersect @ X1 @ X3 @ X5 ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP13])]) ).

thf(sP14,plain,
    ( sP14
  <=> ! [X1: $i] :
        ( ( rel_s4 @ eigen__3 @ X1 )
       => ! [X2: mu] :
            ( ( exists_in_world @ X2 @ X1 )
           => ! [X3: $i] :
                ( ( rel_s4 @ X1 @ X3 )
               => ! [X4: mu] :
                    ( ( exists_in_world @ X4 @ X3 )
                   => ~ ( ! [X5: $i] :
                            ( ( rel_s4 @ X3 @ X5 )
                           => ( ! [X6: $i] :
                                  ( ( rel_s4 @ X5 @ X6 )
                                 => ( intersect @ X2 @ X4 @ X6 ) )
                             => ~ ! [X6: mu] :
                                    ( ( exists_in_world @ X6 @ X5 )
                                   => ( ! [X7: $i] :
                                          ( ( rel_s4 @ X5 @ X7 )
                                         => ( member @ X6 @ X2 @ X7 ) )
                                     => ~ ! [X7: $i] :
                                            ( ( rel_s4 @ X5 @ X7 )
                                           => ( member @ X6 @ X4 @ X7 ) ) ) ) ) )
                       => ~ ! [X5: $i] :
                              ( ( rel_s4 @ X3 @ X5 )
                             => ( ~ ! [X6: mu] :
                                      ( ( exists_in_world @ X6 @ X5 )
                                     => ( ! [X7: $i] :
                                            ( ( rel_s4 @ X5 @ X7 )
                                           => ( member @ X6 @ X2 @ X7 ) )
                                       => ~ ! [X7: $i] :
                                              ( ( rel_s4 @ X5 @ X7 )
                                             => ( member @ X6 @ X4 @ X7 ) ) ) )
                               => ! [X6: $i] :
                                    ( ( rel_s4 @ X5 @ X6 )
                                   => ( intersect @ X2 @ X4 @ X6 ) ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP14])]) ).

thf(sP15,plain,
    ( sP15
  <=> ! [X1: mu] :
        ( ( exists_in_world @ X1 @ eigen__3 )
       => ~ ( ! [X2: $i] :
                ( ( rel_s4 @ eigen__3 @ X2 )
               => ( ! [X3: $i] :
                      ( ( rel_s4 @ X2 @ X3 )
                     => ( intersect @ eigen__2 @ X1 @ X3 ) )
                 => ~ ! [X3: mu] :
                        ( ( exists_in_world @ X3 @ X2 )
                       => ( ! [X4: $i] :
                              ( ( rel_s4 @ X2 @ X4 )
                             => ( member @ X3 @ eigen__2 @ X4 ) )
                         => ~ ! [X4: $i] :
                                ( ( rel_s4 @ X2 @ X4 )
                               => ( member @ X3 @ X1 @ X4 ) ) ) ) ) )
           => ~ ! [X2: $i] :
                  ( ( rel_s4 @ eigen__3 @ X2 )
                 => ( ~ ! [X3: mu] :
                          ( ( exists_in_world @ X3 @ X2 )
                         => ( ! [X4: $i] :
                                ( ( rel_s4 @ X2 @ X4 )
                               => ( member @ X3 @ eigen__2 @ X4 ) )
                           => ~ ! [X4: $i] :
                                  ( ( rel_s4 @ X2 @ X4 )
                                 => ( member @ X3 @ X1 @ X4 ) ) ) )
                   => ! [X3: $i] :
                        ( ( rel_s4 @ X2 @ X3 )
                       => ( intersect @ eigen__2 @ X1 @ X3 ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP15])]) ).

thf(sP16,plain,
    ( sP16
  <=> ( rel_s4 @ eigen__3 @ eigen__5 ) ),
    introduced(definition,[new_symbols(definition,[sP16])]) ).

thf(sP17,plain,
    ( sP17
  <=> ( rel_s4 @ eigen__1 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP17])]) ).

thf(sP18,plain,
    ( sP18
  <=> ! [X1: $i,X2: $i] :
        ( ~ ( ( exists_in_world @ eigen__2 @ X1 )
           => ~ ( rel_s4 @ X1 @ X2 ) )
       => ( exists_in_world @ eigen__2 @ X2 ) ) ),
    introduced(definition,[new_symbols(definition,[sP18])]) ).

thf(sP19,plain,
    ( sP19
  <=> ( ~ sP10
     => sP5 ) ),
    introduced(definition,[new_symbols(definition,[sP19])]) ).

thf(sP20,plain,
    ( sP20
  <=> ! [X1: $i] :
        ( ~ ( ( exists_in_world @ eigen__2 @ eigen__1 )
           => ~ ( rel_s4 @ eigen__1 @ X1 ) )
       => ( exists_in_world @ eigen__2 @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP20])]) ).

thf(sP21,plain,
    ( sP21
  <=> ( sP16
     => sP9 ) ),
    introduced(definition,[new_symbols(definition,[sP21])]) ).

thf(sP22,plain,
    ( sP22
  <=> ! [X1: $i] :
        ( ( rel_s4 @ eigen__5 @ X1 )
       => ( intersect @ eigen__2 @ eigen__4 @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP22])]) ).

thf(sP23,plain,
    ( sP23
  <=> ( exists_in_world @ eigen__2 @ eigen__1 ) ),
    introduced(definition,[new_symbols(definition,[sP23])]) ).

thf(sP24,plain,
    ( sP24
  <=> ( sP4
     => ~ sP12 ) ),
    introduced(definition,[new_symbols(definition,[sP24])]) ).

thf(sP25,plain,
    ( sP25
  <=> ( sP6
     => sP13 ) ),
    introduced(definition,[new_symbols(definition,[sP25])]) ).

thf(sP26,plain,
    ( sP26
  <=> ! [X1: $i] :
        ( ( rel_s4 @ eigen__3 @ X1 )
       => ! [X2: mu] :
            ( ( exists_in_world @ X2 @ X1 )
           => ~ ( ! [X3: $i] :
                    ( ( rel_s4 @ X1 @ X3 )
                   => ( ! [X4: $i] :
                          ( ( rel_s4 @ X3 @ X4 )
                         => ( intersect @ eigen__2 @ X2 @ X4 ) )
                     => ~ ! [X4: mu] :
                            ( ( exists_in_world @ X4 @ X3 )
                           => ( ! [X5: $i] :
                                  ( ( rel_s4 @ X3 @ X5 )
                                 => ( member @ X4 @ eigen__2 @ X5 ) )
                             => ~ ! [X5: $i] :
                                    ( ( rel_s4 @ X3 @ X5 )
                                   => ( member @ X4 @ X2 @ X5 ) ) ) ) ) )
               => ~ ! [X3: $i] :
                      ( ( rel_s4 @ X1 @ X3 )
                     => ( ~ ! [X4: mu] :
                              ( ( exists_in_world @ X4 @ X3 )
                             => ( ! [X5: $i] :
                                    ( ( rel_s4 @ X3 @ X5 )
                                   => ( member @ X4 @ eigen__2 @ X5 ) )
                               => ~ ! [X5: $i] :
                                      ( ( rel_s4 @ X3 @ X5 )
                                     => ( member @ X4 @ X2 @ X5 ) ) ) )
                       => ! [X4: $i] :
                            ( ( rel_s4 @ X3 @ X4 )
                           => ( intersect @ eigen__2 @ X2 @ X4 ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP26])]) ).

thf(sP27,plain,
    ( sP27
  <=> ! [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,[sP27])]) ).

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(prove_th15,conjecture,
    ! [X1: $i,X2: $i] :
      ( ( rel_s4 @ X1 @ X2 )
     => ! [X3: mu] :
          ( ( exists_in_world @ X3 @ X2 )
         => ! [X4: $i] :
              ( ( rel_s4 @ X2 @ X4 )
             => ! [X5: mu] :
                  ( ( exists_in_world @ X5 @ X4 )
                 => ! [X6: $i] :
                      ( ( rel_s4 @ X4 @ X6 )
                     => ( ! [X7: $i] :
                            ( ( rel_s4 @ X6 @ X7 )
                           => ( intersect @ X3 @ X5 @ X7 ) )
                       => ~ ! [X7: mu] :
                              ( ( exists_in_world @ X7 @ X6 )
                             => ( ! [X8: $i] :
                                    ( ( rel_s4 @ X6 @ X8 )
                                   => ( member @ X7 @ X3 @ X8 ) )
                               => ~ ! [X8: $i] :
                                      ( ( rel_s4 @ X6 @ X8 )
                                     => ( member @ X7 @ X5 @ X8 ) ) ) ) ) ) ) ) ) ) ).

thf(h0,negated_conjecture,
    ~ ! [X1: $i,X2: $i] :
        ( ( rel_s4 @ X1 @ X2 )
       => ! [X3: mu] :
            ( ( exists_in_world @ X3 @ X2 )
           => ! [X4: $i] :
                ( ( rel_s4 @ X2 @ X4 )
               => ! [X5: mu] :
                    ( ( exists_in_world @ X5 @ X4 )
                   => ! [X6: $i] :
                        ( ( rel_s4 @ X4 @ X6 )
                       => ( ! [X7: $i] :
                              ( ( rel_s4 @ X6 @ X7 )
                             => ( intersect @ X3 @ X5 @ X7 ) )
                         => ~ ! [X7: mu] :
                                ( ( exists_in_world @ X7 @ X6 )
                               => ( ! [X8: $i] :
                                      ( ( rel_s4 @ X6 @ X8 )
                                     => ( member @ X7 @ X3 @ X8 ) )
                                 => ~ ! [X8: $i] :
                                        ( ( rel_s4 @ X6 @ X8 )
                                       => ( member @ X7 @ X5 @ X8 ) ) ) ) ) ) ) ) ) ),
    inference(assume_negation,[status(cth)],[prove_th15]) ).

thf(h1,assumption,
    ~ ! [X1: $i] :
        ( ( rel_s4 @ eigen__0 @ X1 )
       => ! [X2: mu] :
            ( ( exists_in_world @ X2 @ X1 )
           => ! [X3: $i] :
                ( ( rel_s4 @ X1 @ X3 )
               => ! [X4: mu] :
                    ( ( exists_in_world @ X4 @ X3 )
                   => ! [X5: $i] :
                        ( ( rel_s4 @ X3 @ X5 )
                       => ( ! [X6: $i] :
                              ( ( rel_s4 @ X5 @ X6 )
                             => ( intersect @ X2 @ X4 @ X6 ) )
                         => ~ ! [X6: mu] :
                                ( ( exists_in_world @ X6 @ X5 )
                               => ( ! [X7: $i] :
                                      ( ( rel_s4 @ X5 @ X7 )
                                     => ( member @ X6 @ X2 @ X7 ) )
                                 => ~ ! [X7: $i] :
                                        ( ( rel_s4 @ X5 @ X7 )
                                       => ( member @ X6 @ X4 @ X7 ) ) ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h2,assumption,
    ~ ( ( rel_s4 @ eigen__0 @ eigen__1 )
     => ! [X1: mu] :
          ( ( exists_in_world @ X1 @ eigen__1 )
         => ! [X2: $i] :
              ( ( rel_s4 @ eigen__1 @ X2 )
             => ! [X3: mu] :
                  ( ( exists_in_world @ X3 @ X2 )
                 => ! [X4: $i] :
                      ( ( rel_s4 @ X2 @ X4 )
                     => ( ! [X5: $i] :
                            ( ( rel_s4 @ X4 @ X5 )
                           => ( intersect @ X1 @ X3 @ X5 ) )
                       => ~ ! [X5: mu] :
                              ( ( exists_in_world @ X5 @ X4 )
                             => ( ! [X6: $i] :
                                    ( ( rel_s4 @ X4 @ X6 )
                                   => ( member @ X5 @ X1 @ X6 ) )
                               => ~ ! [X6: $i] :
                                      ( ( rel_s4 @ X4 @ X6 )
                                     => ( member @ X5 @ X3 @ X6 ) ) ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h3,assumption,
    rel_s4 @ eigen__0 @ eigen__1,
    introduced(assumption,[]) ).

thf(h4,assumption,
    ~ ! [X1: mu] :
        ( ( exists_in_world @ X1 @ eigen__1 )
       => ! [X2: $i] :
            ( ( rel_s4 @ eigen__1 @ X2 )
           => ! [X3: mu] :
                ( ( exists_in_world @ X3 @ X2 )
               => ! [X4: $i] :
                    ( ( rel_s4 @ X2 @ X4 )
                   => ( ! [X5: $i] :
                          ( ( rel_s4 @ X4 @ X5 )
                         => ( intersect @ X1 @ X3 @ X5 ) )
                     => ~ ! [X5: mu] :
                            ( ( exists_in_world @ X5 @ X4 )
                           => ( ! [X6: $i] :
                                  ( ( rel_s4 @ X4 @ X6 )
                                 => ( member @ X5 @ X1 @ X6 ) )
                             => ~ ! [X6: $i] :
                                    ( ( rel_s4 @ X4 @ X6 )
                                   => ( member @ X5 @ X3 @ X6 ) ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h5,assumption,
    ~ ( sP23
     => ! [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 )
                       => ( intersect @ eigen__2 @ X2 @ X4 ) )
                   => ~ ! [X4: mu] :
                          ( ( exists_in_world @ X4 @ X3 )
                         => ( ! [X5: $i] :
                                ( ( rel_s4 @ X3 @ X5 )
                               => ( member @ X4 @ eigen__2 @ X5 ) )
                           => ~ ! [X5: $i] :
                                  ( ( rel_s4 @ X3 @ X5 )
                                 => ( member @ X4 @ X2 @ X5 ) ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h6,assumption,
    sP23,
    introduced(assumption,[]) ).

thf(h7,assumption,
    ~ ! [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 )
                     => ( intersect @ eigen__2 @ X2 @ X4 ) )
                 => ~ ! [X4: mu] :
                        ( ( exists_in_world @ X4 @ X3 )
                       => ( ! [X5: $i] :
                              ( ( rel_s4 @ X3 @ X5 )
                             => ( member @ X4 @ eigen__2 @ X5 ) )
                         => ~ ! [X5: $i] :
                                ( ( rel_s4 @ X3 @ X5 )
                               => ( member @ X4 @ X2 @ X5 ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h8,assumption,
    ~ ( sP17
     => ! [X1: mu] :
          ( ( exists_in_world @ X1 @ eigen__3 )
         => ! [X2: $i] :
              ( ( rel_s4 @ eigen__3 @ X2 )
             => ( ! [X3: $i] :
                    ( ( rel_s4 @ X2 @ X3 )
                   => ( intersect @ eigen__2 @ X1 @ X3 ) )
               => ~ ! [X3: mu] :
                      ( ( exists_in_world @ X3 @ X2 )
                     => ( ! [X4: $i] :
                            ( ( rel_s4 @ X2 @ X4 )
                           => ( member @ X3 @ eigen__2 @ X4 ) )
                       => ~ ! [X4: $i] :
                              ( ( rel_s4 @ X2 @ X4 )
                             => ( member @ X3 @ X1 @ X4 ) ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h9,assumption,
    sP17,
    introduced(assumption,[]) ).

thf(h10,assumption,
    ~ ! [X1: mu] :
        ( ( exists_in_world @ X1 @ eigen__3 )
       => ! [X2: $i] :
            ( ( rel_s4 @ eigen__3 @ X2 )
           => ( ! [X3: $i] :
                  ( ( rel_s4 @ X2 @ X3 )
                 => ( intersect @ eigen__2 @ X1 @ X3 ) )
             => ~ ! [X3: mu] :
                    ( ( exists_in_world @ X3 @ X2 )
                   => ( ! [X4: $i] :
                          ( ( rel_s4 @ X2 @ X4 )
                         => ( member @ X3 @ eigen__2 @ X4 ) )
                     => ~ ! [X4: $i] :
                            ( ( rel_s4 @ X2 @ X4 )
                           => ( member @ X3 @ X1 @ X4 ) ) ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h11,assumption,
    ~ ( sP4
     => sP11 ),
    introduced(assumption,[]) ).

thf(h12,assumption,
    sP4,
    introduced(assumption,[]) ).

thf(h13,assumption,
    ~ sP11,
    introduced(assumption,[]) ).

thf(h14,assumption,
    ~ sP21,
    introduced(assumption,[]) ).

thf(h15,assumption,
    sP16,
    introduced(assumption,[]) ).

thf(h16,assumption,
    ~ sP9,
    introduced(assumption,[]) ).

thf(h17,assumption,
    sP22,
    introduced(assumption,[]) ).

thf(h18,assumption,
    sP2,
    introduced(assumption,[]) ).

thf(1,plain,
    ( ~ sP10
    | ~ sP23
    | ~ sP17 ),
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP19
    | sP10
    | sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( ~ sP20
    | sP19 ),
    inference(all_rule,[status(thm)],]) ).

thf(4,plain,
    ( ~ sP18
    | sP20 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ( ~ sP3
    | ~ sP6
    | sP15 ),
    inference(prop_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP26
    | sP3 ),
    inference(all_rule,[status(thm)],]) ).

thf(7,plain,
    ( ~ sP8
    | ~ sP5
    | sP26 ),
    inference(prop_rule,[status(thm)],]) ).

thf(8,plain,
    ( ~ sP25
    | ~ sP6
    | sP13 ),
    inference(prop_rule,[status(thm)],]) ).

thf(9,plain,
    ( ~ sP13
    | sP8 ),
    inference(all_rule,[status(thm)],]) ).

thf(10,plain,
    ( ~ sP14
    | sP25 ),
    inference(all_rule,[status(thm)],]) ).

thf(11,plain,
    ( ~ sP21
    | ~ sP16
    | sP9 ),
    inference(prop_rule,[status(thm)],]) ).

thf(12,plain,
    ( ~ sP11
    | sP21 ),
    inference(all_rule,[status(thm)],]) ).

thf(13,plain,
    ( sP12
    | sP11 ),
    inference(prop_rule,[status(thm)],]) ).

thf(14,plain,
    ( ~ sP24
    | ~ sP4
    | ~ sP12 ),
    inference(prop_rule,[status(thm)],]) ).

thf(15,plain,
    ( ~ sP15
    | sP24 ),
    inference(all_rule,[status(thm)],]) ).

thf(16,plain,
    ( ~ sP1
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(17,plain,
    ( ~ sP7
    | sP14 ),
    inference(all_rule,[status(thm)],]) ).

thf(18,plain,
    ( ~ sP9
    | ~ sP22
    | ~ sP2 ),
    inference(prop_rule,[status(thm)],]) ).

thf(19,plain,
    ( ~ sP27
    | sP18 ),
    inference(all_rule,[status(thm)],]) ).

thf(intersect_defn,axiom,
    sP7 ).

thf(cumulative_ax,axiom,
    sP27 ).

thf(a1,axiom,
    sP1 ).

thf(20,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h17,h18,h15,h16,h14,h12,h13,h11,h9,h10,h8,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,h6,h9,h12,h15,h17,h18,intersect_defn,cumulative_ax,a1]) ).

thf(21,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h15,h16,h14,h12,h13,h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h17,h18])],[h16,20,h17,h18]) ).

thf(22,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h14,h12,h13,h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h15,h16])],[h14,21,h15,h16]) ).

thf(23,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h12,h13,h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h14]),tab_negall(eigenvar,eigen__5)],[h13,22,h14]) ).

thf(24,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h12,h13])],[h11,23,h12,h13]) ).

thf(25,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h11]),tab_negall(eigenvar,eigen__4)],[h10,24,h11]) ).

thf(26,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h9,h10])],[h8,25,h9,h10]) ).

thf(27,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h8]),tab_negall(eigenvar,eigen__3)],[h7,26,h8]) ).

thf(28,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,27,h6,h7]) ).

thf(29,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__2)],[h4,28,h5]) ).

thf(30,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,29,h3,h4]) ).

thf(31,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__1)],[h1,30,h2]) ).

thf(32,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,31,h1]) ).

thf(0,theorem,
    ! [X1: $i,X2: $i] :
      ( ( rel_s4 @ X1 @ X2 )
     => ! [X3: mu] :
          ( ( exists_in_world @ X3 @ X2 )
         => ! [X4: $i] :
              ( ( rel_s4 @ X2 @ X4 )
             => ! [X5: mu] :
                  ( ( exists_in_world @ X5 @ X4 )
                 => ! [X6: $i] :
                      ( ( rel_s4 @ X4 @ X6 )
                     => ( ! [X7: $i] :
                            ( ( rel_s4 @ X6 @ X7 )
                           => ( intersect @ X3 @ X5 @ X7 ) )
                       => ~ ! [X7: mu] :
                              ( ( exists_in_world @ X7 @ X6 )
                             => ( ! [X8: $i] :
                                    ( ( rel_s4 @ X6 @ X8 )
                                   => ( member @ X7 @ X3 @ X8 ) )
                               => ~ ! [X8: $i] :
                                      ( ( rel_s4 @ X6 @ X8 )
                                     => ( member @ X7 @ X5 @ X8 ) ) ) ) ) ) ) ) ) ),
    inference(contra,[status(thm),contra(discharge,[h0])],[32,h0]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13  % Problem  : SET575^7 : TPTP v8.1.2. Released v5.5.0.
% 0.11/0.14  % Command  : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34  % Computer : n011.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Sat Aug 26 08:40:54 EDT 2023
% 0.13/0.35  % CPUTime  : 
% 0.19/0.46  % SZS status Theorem
% 0.19/0.46  % Mode: cade22grackle2xfee4
% 0.19/0.46  % Steps: 775
% 0.19/0.46  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------