TSTP Solution File: PHI005^2 by Lash---1.13

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : PHI005^2 : TPTP v8.1.2. Released v6.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : lash -P picomus -M modes -p tstp -t %d %s

% Computer : n028.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 12:56:23 EDT 2023

% Result   : Theorem 23.70s 24.07s
% Output   : Proof 23.70s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    7
%            Number of leaves      :   84
% Syntax   : Number of formulae    :   94 (  41 unt;   8 typ;  27 def)
%            Number of atoms       :  210 (  28 equ;   5 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :  397 (  63   ~;  26   |;   4   &; 229   @)
%                                         (  23 <=>;  52  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   4 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :  110 ( 110   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   60 (  56 usr;  56 con; 0-3 aty)
%            Number of variables   :  149 (  71   ^;  74   !;   4   ?; 149   :)

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

thf(ty_rel,type,
    rel: $i > $i > $o ).

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

thf(ty_essence,type,
    essence: ( mu > $i > $o ) > mu > $i > $o ).

thf(ty_positive,type,
    positive: ( mu > $i > $o ) > $i > $o ).

thf(ty_eigen__6,type,
    eigen__6: mu ).

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

thf(ty_eigen__0,type,
    eigen__0: $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 @ eigen__1 @ X1 )
           => ! [X2: mu] :
                ~ ! [X3: mu > $i > $o] :
                    ( ( positive @ X3 @ X1 )
                   => ( X3 @ X2 @ X1 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[eigen__3])]) ).

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] :
          ~ ~ ! [X2: mu > $i > $o] :
                ( ( positive @ X2 @ eigen__3 )
               => ( X2 @ X1 @ eigen__3 ) ) ) ),
    introduced(definition,[new_symbols(definition,[eigen__6])]) ).

thf(sP1,plain,
    ( sP1
  <=> ( ( positive
        @ ^ [X1: mu,X2: $i] :
          ! [X3: mu > $i > $o] :
            ( ( essence @ X3 @ X1 @ X2 )
           => ! [X4: $i] :
                ( ( rel @ X2 @ X4 )
               => ~ ! [X5: mu] :
                      ~ ( X3 @ X5 @ X4 ) ) )
        @ eigen__3 )
     => ! [X1: mu > $i > $o] :
          ( ( essence @ X1 @ eigen__6 @ eigen__3 )
         => ! [X2: $i] :
              ( ( rel @ eigen__3 @ X2 )
             => ~ ! [X3: mu] :
                    ~ ( X1 @ X3 @ X2 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> ! [X1: $i] :
        ( ( rel @ eigen__3 @ X1 )
       => ~ ! [X2: mu] :
              ~ ! [X3: mu > $i > $o] :
                  ( ( positive @ X3 @ X1 )
                 => ( X3 @ X2 @ X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ! [X1: $i] :
        ( positive
        @ ^ [X2: mu,X3: $i] :
          ! [X4: mu > $i > $o] :
            ( ( essence @ X4 @ X2 @ X3 )
           => ! [X5: $i] :
                ( ( rel @ X3 @ X5 )
               => ~ ! [X6: mu] :
                      ~ ( X4 @ X6 @ X5 ) ) )
        @ X1 ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( essence
      @ ^ [X1: mu,X2: $i] :
        ! [X3: mu > $i > $o] :
          ( ( positive @ X3 @ X2 )
         => ( X3 @ X1 @ X2 ) )
      @ eigen__6
      @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ! [X1: mu] :
        ( ! [X2: mu > $i > $o] :
            ( ( positive @ X2 @ eigen__3 )
           => ( X2 @ X1 @ eigen__3 ) )
       => ( essence
          @ ^ [X2: mu,X3: $i] :
            ! [X4: mu > $i > $o] :
              ( ( positive @ X4 @ X3 )
             => ( X4 @ X2 @ X3 ) )
          @ X1
          @ eigen__3 ) ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ( ( rel @ eigen__3 @ eigen__1 )
     => ~ ! [X1: mu] :
            ~ ! [X2: mu > $i > $o] :
                ( ( positive @ X2 @ eigen__1 )
               => ( X2 @ X1 @ eigen__1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: $i,X2: $i] :
        ( ( rel @ X1 @ X2 )
       => ( rel @ X2 @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ! [X1: mu > $i > $o] :
        ( ( essence @ X1 @ eigen__6 @ eigen__3 )
       => ! [X2: $i] :
            ( ( rel @ eigen__3 @ X2 )
           => ~ ! [X3: mu] :
                  ~ ( X1 @ X3 @ X2 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ( rel @ eigen__3 @ eigen__1 ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ( sP4
     => sP2 ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(sP11,plain,
    ( sP11
  <=> ! [X1: mu > $i > $o] :
        ( ( positive @ X1 @ eigen__3 )
       => ( X1 @ eigen__6 @ eigen__3 ) ) ),
    introduced(definition,[new_symbols(definition,[sP11])]) ).

thf(sP12,plain,
    ( sP12
  <=> ! [X1: $i] :
        ( ( rel @ eigen__1 @ X1 )
       => ( rel @ X1 @ eigen__1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP12])]) ).

thf(sP13,plain,
    ( sP13
  <=> ! [X1: $i] :
        ( ( rel @ eigen__1 @ X1 )
       => ! [X2: mu] :
            ~ ! [X3: mu > $i > $o] :
                ( ( positive @ X3 @ X1 )
               => ( X3 @ X2 @ X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP13])]) ).

thf(sP14,plain,
    ( sP14
  <=> ! [X1: mu] :
        ~ ! [X2: mu > $i > $o] :
            ( ( positive @ X2 @ eigen__3 )
           => ( X2 @ X1 @ eigen__3 ) ) ),
    introduced(definition,[new_symbols(definition,[sP14])]) ).

thf(sP15,plain,
    ( sP15
  <=> ! [X1: $i,X2: mu] :
        ( ! [X3: mu > $i > $o] :
            ( ( positive @ X3 @ X1 )
           => ( X3 @ X2 @ X1 ) )
       => ( essence
          @ ^ [X3: mu,X4: $i] :
            ! [X5: mu > $i > $o] :
              ( ( positive @ X5 @ X4 )
             => ( X5 @ X3 @ X4 ) )
          @ X2
          @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP15])]) ).

thf(sP16,plain,
    ( sP16
  <=> ( rel @ eigen__1 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP16])]) ).

thf(sP17,plain,
    ( sP17
  <=> ( sP16
     => sP9 ) ),
    introduced(definition,[new_symbols(definition,[sP17])]) ).

thf(sP18,plain,
    ( sP18
  <=> ! [X1: mu] :
        ~ ! [X2: mu > $i > $o] :
            ( ( positive @ X2 @ eigen__1 )
           => ( X2 @ X1 @ eigen__1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP18])]) ).

thf(sP19,plain,
    ( sP19
  <=> ( positive
      @ ^ [X1: mu,X2: $i] :
        ! [X3: mu > $i > $o] :
          ( ( essence @ X3 @ X1 @ X2 )
         => ! [X4: $i] :
              ( ( rel @ X2 @ X4 )
             => ~ ! [X5: mu] :
                    ~ ( X3 @ X5 @ X4 ) ) )
      @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP19])]) ).

thf(sP20,plain,
    ( sP20
  <=> ( sP11
     => sP4 ) ),
    introduced(definition,[new_symbols(definition,[sP20])]) ).

thf(sP21,plain,
    ( sP21
  <=> ( sP16
     => sP14 ) ),
    introduced(definition,[new_symbols(definition,[sP21])]) ).

thf(sP22,plain,
    ( sP22
  <=> ! [X1: $i] :
        ~ ! [X2: $i] :
            ( ( rel @ X1 @ X2 )
           => ! [X3: mu] :
                ~ ! [X4: mu > $i > $o] :
                    ( ( positive @ X4 @ X2 )
                   => ( X4 @ X3 @ X2 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP22])]) ).

thf(def_meq_ind,definition,
    ( meq_ind
    = ( ^ [X1: mu,X2: mu,X3: $i] : ( X1 = X2 ) ) ) ).

thf(def_mtrue,definition,
    ( mtrue
    = ( ^ [X1: $i] : $true ) ) ).

thf(def_mfalse,definition,
    ( mfalse
    = ( ^ [X1: $i] : $false ) ) ).

thf(def_mnot,definition,
    ( mnot
    = ( ^ [X1: $i > $o,X2: $i] : ( (~) @ ( X1 @ X2 ) ) ) ) ).

thf(def_mor,definition,
    ( mor
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ( X1 @ X3 )
          | ( X2 @ X3 ) ) ) ) ).

thf(def_mand,definition,
    ( mand
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ( X1 @ X3 )
          & ( X2 @ X3 ) ) ) ) ).

thf(def_mimplies,definition,
    ( mimplies
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ^ [X4: $o,X5: $o] :
              ( X4
             => X5 )
          @ ( X1 @ X3 )
          @ ( X2 @ X3 ) ) ) ) ).

thf(def_mimplied,definition,
    ( mimplied
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ^ [X4: $o,X5: $o] :
              ( X4
             => X5 )
          @ ( X2 @ X3 )
          @ ( X1 @ X3 ) ) ) ) ).

thf(def_mequiv,definition,
    ( mequiv
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ( X1 @ X3 )
        <=> ( X2 @ X3 ) ) ) ) ).

thf(def_mxor,definition,
    ( mxor
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ( ( X1 @ X3 )
            & ( (~) @ ( X2 @ X3 ) ) )
          | ( ( (~) @ ( X1 @ X3 ) )
            & ( X2 @ X3 ) ) ) ) ) ).

thf(def_mforall_ind,definition,
    ( mforall_ind
    = ( ^ [X1: mu > $i > $o,X2: $i] :
        ! [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ).

thf(def_mforall_indset,definition,
    ( mforall_indset
    = ( ^ [X1: ( mu > $i > $o ) > $i > $o,X2: $i] :
        ! [X3: mu > $i > $o] : ( 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,X2: $i] :
        ? [X3: mu] : ( X1 @ X3 @ X2 ) ) ) ).

thf(def_mexists_indset,definition,
    ( mexists_indset
    = ( ^ [X1: ( mu > $i > $o ) > $i > $o,X2: $i] :
        ? [X3: mu > $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).

thf(def_mexists_prop,definition,
    ( mexists_prop
    = ( ^ [X1: ( $i > $o ) > $i > $o,X2: $i] :
        ? [X3: $i > $o] : ( X1 @ X3 @ X2 ) ) ) ).

thf(def_mbox_generic,definition,
    ( mbox_generic
    = ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
        ! [X4: $i] :
          ( ( (~) @ ( X1 @ X3 @ X4 ) )
          | ( X2 @ X4 ) ) ) ) ).

thf(def_mdia_generic,definition,
    ( mdia_generic
    = ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i] :
        ? [X4: $i] :
          ( ( X1 @ X3 @ X4 )
          & ( X2 @ X4 ) ) ) ) ).

thf(def_mbox,definition,
    ( mbox
    = ( mbox_generic @ rel ) ) ).

thf(def_mdia,definition,
    ( mdia
    = ( mdia_generic @ rel ) ) ).

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_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_god,definition,
    ( god
    = ( ^ [X1: mu] :
          ( mforall_indset
          @ ^ [X2: mu > $i > $o] : ( mimplies @ ( positive @ X2 ) @ ( X2 @ X1 ) ) ) ) ) ).

thf(def_necessary_existence,definition,
    ( necessary_existence
    = ( ^ [X1: mu] :
          ( mforall_indset
          @ ^ [X2: mu > $i > $o] :
              ( mimplies @ ( essence @ X2 @ X1 )
              @ ( mbox
                @ ( mexists_ind
                  @ ^ [X3: mu] : ( X2 @ X3 ) ) ) ) ) ) ) ).

thf(thmT3,conjecture,
    ! [X1: $i,X2: $i] :
      ( ( rel @ X1 @ X2 )
     => ~ ! [X3: mu] :
            ~ ! [X4: mu > $i > $o] :
                ( ( positive @ X4 @ X2 )
               => ( X4 @ X3 @ X2 ) ) ) ).

thf(h2,negated_conjecture,
    ~ ! [X1: $i,X2: $i] :
        ( ( rel @ X1 @ X2 )
       => ~ ! [X3: mu] :
              ~ ! [X4: mu > $i > $o] :
                  ( ( positive @ X4 @ X2 )
                 => ( X4 @ X3 @ X2 ) ) ),
    inference(assume_negation,[status(cth)],[thmT3]) ).

thf(h3,assumption,
    ~ ! [X1: $i] :
        ( ( rel @ eigen__0 @ X1 )
       => ~ ! [X2: mu] :
              ~ ! [X3: mu > $i > $o] :
                  ( ( positive @ X3 @ X1 )
                 => ( X3 @ X2 @ X1 ) ) ),
    introduced(assumption,[]) ).

thf(h4,assumption,
    ~ ( ( rel @ eigen__0 @ eigen__1 )
     => ~ sP18 ),
    introduced(assumption,[]) ).

thf(h5,assumption,
    rel @ eigen__0 @ eigen__1,
    introduced(assumption,[]) ).

thf(h6,assumption,
    sP18,
    introduced(assumption,[]) ).

thf(1,plain,
    ( ~ sP20
    | ~ sP11
    | sP4 ),
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP5
    | sP20 ),
    inference(all_rule,[status(thm)],]) ).

thf(3,plain,
    ( ~ sP6
    | ~ sP9
    | ~ sP18 ),
    inference(prop_rule,[status(thm)],]) ).

thf(4,plain,
    ( ~ sP2
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ( ~ sP17
    | ~ sP16
    | sP9 ),
    inference(prop_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP10
    | ~ sP4
    | sP2 ),
    inference(prop_rule,[status(thm)],]) ).

thf(7,plain,
    ( ~ sP15
    | sP5 ),
    inference(all_rule,[status(thm)],]) ).

thf(8,plain,
    ( ~ sP3
    | sP19 ),
    inference(all_rule,[status(thm)],]) ).

thf(9,plain,
    ( ~ sP12
    | sP17 ),
    inference(all_rule,[status(thm)],]) ).

thf(10,plain,
    ( ~ sP8
    | sP10 ),
    inference(all_rule,[status(thm)],]) ).

thf(11,plain,
    ( ~ sP1
    | ~ sP19
    | sP8 ),
    inference(prop_rule,[status(thm)],]) ).

thf(12,plain,
    ( ~ sP11
    | sP1 ),
    inference(all_rule,[status(thm)],]) ).

thf(13,plain,
    ( sP14
    | sP11 ),
    inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__6]) ).

thf(14,plain,
    ( sP21
    | ~ sP14 ),
    inference(prop_rule,[status(thm)],]) ).

thf(15,plain,
    ( sP21
    | sP16 ),
    inference(prop_rule,[status(thm)],]) ).

thf(16,plain,
    ( sP13
    | ~ sP21 ),
    inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).

thf(17,plain,
    ( ~ sP7
    | sP12 ),
    inference(all_rule,[status(thm)],]) ).

thf(18,plain,
    ( ~ sP22
    | ~ sP13 ),
    inference(all_rule,[status(thm)],]) ).

thf(sym,axiom,
    sP7 ).

thf(axA5,axiom,
    sP3 ).

thf(corC,axiom,
    sP22 ).

thf(thmT2,axiom,
    sP15 ).

thf(19,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,sym,axA5,corC,thmT2,h6]) ).

thf(20,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,19,h5,h6]) ).

thf(21,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,20,h4]) ).

thf(22,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,21,h3]) ).

thf(23,plain,
    $false,
    inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[22,h1]) ).

thf(24,plain,
    $false,
    inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[23,h0]) ).

thf(0,theorem,
    ! [X1: $i,X2: $i] :
      ( ( rel @ X1 @ X2 )
     => ~ ! [X3: mu] :
            ~ ! [X4: mu > $i > $o] :
                ( ( positive @ X4 @ X2 )
               => ( X4 @ X3 @ X2 ) ) ),
    inference(contra,[status(thm),contra(discharge,[h2])],[22,h2]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13  % Problem  : PHI005^2 : TPTP v8.1.2. Released v6.1.0.
% 0.00/0.13  % Command  : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34  % Computer : n028.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 : Sun Aug 27 09:14:08 EDT 2023
% 0.13/0.34  % CPUTime  : 
% 23.70/24.07  % SZS status Theorem
% 23.70/24.07  % Mode: cade22grackle2x798d
% 23.70/24.07  % Steps: 575
% 23.70/24.07  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------