TSTP Solution File: SET669^3 by Satallax---3.5

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Satallax---3.5
% Problem  : SET669^3 : TPTP v8.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s

% Computer : n008.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  : 600s
% DateTime : Tue Jul 19 04:54:56 EDT 2022

% Result   : Theorem 0.20s 0.39s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_eigen__1,type,
    eigen__1: $i ).

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

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

thf(sP1,plain,
    ( sP1
  <=> ! [X1: $i,X2: $i] :
        ( ~ ( ~ $false
           => ( X1 != X2 ) )
       => ( eigen__0 @ X1 @ X2 ) ) ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> $false ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ! [X1: $i] :
        ( ~ ( ~ sP2
           => ( eigen__1 != X1 ) )
       => ( eigen__0 @ eigen__1 @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ! [X1: $i] :
        ( ~ ( ~ sP2
           => ( eigen__3 != X1 ) )
       => ( eigen__0 @ eigen__3 @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

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

thf(sP6,plain,
    ( sP6
  <=> ( eigen__0 @ eigen__3 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: $i] :
        ~ ( eigen__0 @ X1 @ eigen__3 ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( ~ sP2
     => ( eigen__1 != eigen__1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ( ~ sP8
     => ( eigen__0 @ eigen__1 @ eigen__1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ! [X1: $i] :
        ~ ( eigen__0 @ eigen__1 @ X1 ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(sP11,plain,
    ( sP11
  <=> ( ~ ( ~ sP2
         => ~ sP5 )
     => sP6 ) ),
    introduced(definition,[new_symbols(definition,[sP11])]) ).

thf(sP12,plain,
    ( sP12
  <=> ( ~ sP2
     => ~ sP5 ) ),
    introduced(definition,[new_symbols(definition,[sP12])]) ).

thf(sP13,plain,
    ( sP13
  <=> ( eigen__1 = eigen__1 ) ),
    introduced(definition,[new_symbols(definition,[sP13])]) ).

thf(sP14,plain,
    ( sP14
  <=> ( eigen__0 @ eigen__1 @ eigen__1 ) ),
    introduced(definition,[new_symbols(definition,[sP14])]) ).

thf(def_in,definition,
    ( in
    = ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).

thf(def_is_a,definition,
    ( is_a
    = ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).

thf(def_emptyset,definition,
    ( emptyset
    = ( ^ [X1: $i] : sP2 ) ) ).

thf(def_unord_pair,definition,
    ( unord_pair
    = ( ^ [X1: $i,X2: $i,X3: $i] :
          ( ( X3 != X1 )
         => ( X3 = X2 ) ) ) ) ).

thf(def_singleton,definition,
    ( singleton
    = ( ^ [X1: $i,X2: $i] : ( X2 = X1 ) ) ) ).

thf(def_union,definition,
    ( union
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ~ ( X1 @ X3 )
         => ( X2 @ X3 ) ) ) ) ).

thf(def_excl_union,definition,
    ( excl_union
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ( ( ( X1 @ X3 )
           => ( X2 @ X3 ) )
         => ~ ( ~ ( X1 @ X3 )
             => ~ ( X2 @ X3 ) ) ) ) ) ).

thf(def_intersection,definition,
    ( intersection
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ~ ( ( X1 @ X3 )
           => ~ ( X2 @ X3 ) ) ) ) ).

thf(def_setminus,definition,
    ( setminus
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
          ~ ( ( X1 @ X3 )
           => ( X2 @ X3 ) ) ) ) ).

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

thf(def_disjoint,definition,
    ( disjoint
    = ( ^ [X1: $i > $o,X2: $i > $o] :
          ( ( intersection @ X1 @ X2 )
          = emptyset ) ) ) ).

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

thf(def_meets,definition,
    ( meets
    = ( ^ [X1: $i > $o,X2: $i > $o] :
          ~ ! [X3: $i] :
              ( ( X1 @ X3 )
             => ~ ( X2 @ X3 ) ) ) ) ).

thf(def_misses,definition,
    ( misses
    = ( ^ [X1: $i > $o,X2: $i > $o] :
        ! [X3: $i] :
          ( ( X1 @ X3 )
         => ~ ( X2 @ X3 ) ) ) ) ).

thf(def_cartesian_product,definition,
    ( cartesian_product
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i,X4: $i] :
          ~ ( ( X1 @ X3 )
           => ~ ( X2 @ X4 ) ) ) ) ).

thf(def_pair_rel,definition,
    ( pair_rel
    = ( ^ [X1: $i,X2: $i,X3: $i,X4: $i] :
          ( ( X3 != X1 )
         => ( X4 = X2 ) ) ) ) ).

thf(def_id_rel,definition,
    ( id_rel
    = ( ^ [X1: $i > $o,X2: $i,X3: $i] :
          ~ ( ( X1 @ X2 )
           => ( X2 != X3 ) ) ) ) ).

thf(def_sub_rel,definition,
    ( sub_rel
    = ( ^ [X1: $i > $i > $o,X2: $i > $i > $o] :
        ! [X3: $i,X4: $i] :
          ( ( X1 @ X3 @ X4 )
         => ( X2 @ X3 @ X4 ) ) ) ) ).

thf(def_is_rel_on,definition,
    ( is_rel_on
    = ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i > $o] :
        ! [X4: $i,X5: $i] :
          ( ( X1 @ X4 @ X5 )
         => ~ ( ( X2 @ X4 )
             => ~ ( X3 @ X5 ) ) ) ) ) ).

thf(def_restrict_rel_domain,definition,
    ( restrict_rel_domain
    = ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i,X4: $i] :
          ~ ( ( X2 @ X3 )
           => ~ ( X1 @ X3 @ X4 ) ) ) ) ).

thf(def_rel_diagonal,definition,
    rel_diagonal = (=) ).

thf(def_rel_composition,definition,
    ( rel_composition
    = ( ^ [X1: $i > $i > $o,X2: $i > $i > $o,X3: $i,X4: $i] :
          ~ ! [X5: $i] :
              ( ( X1 @ X3 @ X5 )
             => ~ ( X2 @ X5 @ X4 ) ) ) ) ).

thf(def_reflexive,definition,
    ( reflexive
    = ( ^ [X1: $i > $i > $o] :
        ! [X2: $i] : ( X1 @ X2 @ X2 ) ) ) ).

thf(def_irreflexive,definition,
    ( irreflexive
    = ( ^ [X1: $i > $i > $o] :
        ! [X2: $i] :
          ~ ( X1 @ X2 @ X2 ) ) ) ).

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

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

thf(def_equiv_rel,definition,
    ( equiv_rel
    = ( ^ [X1: $i > $i > $o] :
          ~ ( ~ ( ( reflexive @ X1 )
               => ~ ( symmetric @ X1 ) )
           => ~ ( transitive @ X1 ) ) ) ) ).

thf(def_rel_codomain,definition,
    ( rel_codomain
    = ( ^ [X1: $i > $i > $o,X2: $i] :
          ~ ! [X3: $i] :
              ~ ( X1 @ X3 @ X2 ) ) ) ).

thf(def_rel_domain,definition,
    ( rel_domain
    = ( ^ [X1: $i > $i > $o,X2: $i] :
          ~ ! [X3: $i] :
              ~ ( X1 @ X2 @ X3 ) ) ) ).

thf(def_rel_inverse,definition,
    ( rel_inverse
    = ( ^ [X1: $i > $i > $o,X2: $i,X3: $i] : ( X1 @ X3 @ X2 ) ) ) ).

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

thf(def_restrict_rel_codomain,definition,
    ( restrict_rel_codomain
    = ( ^ [X1: $i > $i > $o,X2: $i > $o,X3: $i,X4: $i] :
          ~ ( ( X2 @ X4 )
           => ~ ( X1 @ X3 @ X4 ) ) ) ) ).

thf(def_rel_field,definition,
    ( rel_field
    = ( ^ [X1: $i > $i > $o,X2: $i] :
          ( ~ ( rel_domain @ X1 @ X2 )
         => ( rel_codomain @ X1 @ X2 ) ) ) ) ).

thf(def_well_founded,definition,
    ( well_founded
    = ( ^ [X1: $i > $i > $o] :
        ! [X2: $i > $o,X3: $i] :
          ( ( X2 @ X3 )
         => ~ ! [X4: $i] :
                ( ( X2 @ X4 )
               => ~ ! [X5: $i] :
                      ( ( X1 @ X4 @ X5 )
                     => ~ ( X2 @ X5 ) ) ) ) ) ) ).

thf(def_upwards_well_founded,definition,
    ( upwards_well_founded
    = ( ^ [X1: $i > $i > $o] :
        ! [X2: $i > $o,X3: $i] :
          ( ( X2 @ X3 )
         => ~ ! [X4: $i] :
                ( ( X2 @ X4 )
               => ~ ! [X5: $i] :
                      ( ( X1 @ X4 @ X4 )
                     => ~ ( X2 @ X5 ) ) ) ) ) ) ).

thf(thm,conjecture,
    ! [X1: $i > $i > $o] :
      ( ! [X2: $i,X3: $i] :
          ( ~ ( ~ sP2
             => ( X2 != X3 ) )
         => ( X1 @ X2 @ X3 ) )
     => ~ ( ! [X2: $i] :
              ( ~ sP2
             => ~ ! [X3: $i] :
                    ~ ( X1 @ X2 @ X3 ) )
         => ( ( ^ [X2: $i] : ~ sP2 )
           != ( ^ [X2: $i] :
                  ~ ! [X3: $i] :
                      ~ ( X1 @ X3 @ X2 ) ) ) ) ) ).

thf(h0,negated_conjecture,
    ~ ! [X1: $i > $i > $o] :
        ( ! [X2: $i,X3: $i] :
            ( ~ ( ~ sP2
               => ( X2 != X3 ) )
           => ( X1 @ X2 @ X3 ) )
       => ~ ( ! [X2: $i] :
                ( ~ sP2
               => ~ ! [X3: $i] :
                      ~ ( X1 @ X2 @ X3 ) )
           => ( ( ^ [X2: $i] : ~ sP2 )
             != ( ^ [X2: $i] :
                    ~ ! [X3: $i] :
                        ~ ( X1 @ X3 @ X2 ) ) ) ) ),
    inference(assume_negation,[status(cth)],[thm]) ).

thf(h1,assumption,
    ~ ( sP1
     => ~ ( ! [X1: $i] :
              ( ~ sP2
             => ~ ! [X2: $i] :
                    ~ ( eigen__0 @ X1 @ X2 ) )
         => ( ( ^ [X1: $i] : ~ sP2 )
           != ( ^ [X1: $i] :
                  ~ ! [X2: $i] :
                      ~ ( eigen__0 @ X2 @ X1 ) ) ) ) ),
    introduced(assumption,[]) ).

thf(h2,assumption,
    sP1,
    introduced(assumption,[]) ).

thf(h3,assumption,
    ( ! [X1: $i] :
        ( ~ sP2
       => ~ ! [X2: $i] :
              ~ ( eigen__0 @ X1 @ X2 ) )
   => ( ( ^ [X1: $i] : ~ sP2 )
     != ( ^ [X1: $i] :
            ~ ! [X2: $i] :
                ~ ( eigen__0 @ X2 @ X1 ) ) ) ),
    introduced(assumption,[]) ).

thf(h4,assumption,
    ~ ! [X1: $i] :
        ( ~ sP2
       => ~ ! [X2: $i] :
              ~ ( eigen__0 @ X1 @ X2 ) ),
    introduced(assumption,[]) ).

thf(h5,assumption,
    ( ^ [X1: $i] : ~ sP2 )
 != ( ^ [X1: $i] :
        ~ ! [X2: $i] :
            ~ ( eigen__0 @ X2 @ X1 ) ),
    introduced(assumption,[]) ).

thf(h6,assumption,
    ~ ( ~ sP2
     => ~ sP10 ),
    introduced(assumption,[]) ).

thf(h7,assumption,
    ~ sP2,
    introduced(assumption,[]) ).

thf(h8,assumption,
    sP10,
    introduced(assumption,[]) ).

thf(1,plain,
    sP13,
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP9
    | sP8
    | sP14 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( ~ sP8
    | sP2
    | ~ sP13 ),
    inference(prop_rule,[status(thm)],]) ).

thf(4,plain,
    ( ~ sP3
    | sP9 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ( ~ sP10
    | ~ sP14 ),
    inference(all_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP1
    | sP3 ),
    inference(all_rule,[status(thm)],]) ).

thf(7,plain,
    ~ sP2,
    inference(prop_rule,[status(thm)],]) ).

thf(8,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h7,h8,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,h2,h8]) ).

thf(9,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,8,h7,h8]) ).

thf(10,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__1)],[h4,9,h6]) ).

thf(h9,assumption,
    ~ ! [X1: $i] :
        ( ( ~ sP2 )
        = ( ~ ! [X2: $i] :
                ~ ( eigen__0 @ X2 @ X1 ) ) ),
    introduced(assumption,[]) ).

thf(h10,assumption,
    ( ~ sP2 )
 != ( ~ sP7 ),
    introduced(assumption,[]) ).

thf(h11,assumption,
    ~ sP7,
    introduced(assumption,[]) ).

thf(h12,assumption,
    sP2,
    introduced(assumption,[]) ).

thf(h13,assumption,
    sP7,
    introduced(assumption,[]) ).

thf(11,plain,
    sP5,
    inference(prop_rule,[status(thm)],]) ).

thf(12,plain,
    ( ~ sP11
    | sP12
    | sP6 ),
    inference(prop_rule,[status(thm)],]) ).

thf(13,plain,
    ( ~ sP12
    | sP2
    | ~ sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(14,plain,
    ( ~ sP4
    | sP11 ),
    inference(all_rule,[status(thm)],]) ).

thf(15,plain,
    ( ~ sP7
    | ~ sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(16,plain,
    ( ~ sP1
    | sP4 ),
    inference(all_rule,[status(thm)],]) ).

thf(17,plain,
    ~ sP2,
    inference(prop_rule,[status(thm)],]) ).

thf(18,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h7,h11,h10,h9,h5,h2,h3,h1,h0])],[11,12,13,14,15,16,17,h2,h11]) ).

thf(19,plain,
    $false,
    inference(tab_false,[status(thm),assumptions([h12,h13,h10,h9,h5,h2,h3,h1,h0])],[h12]) ).

thf(20,plain,
    $false,
    inference(tab_be,[status(thm),assumptions([h10,h9,h5,h2,h3,h1,h0]),tab_be(discharge,[h7,h11]),tab_be(discharge,[h12,h13])],[h10,18,19,h7,h11,h12,h13]) ).

thf(21,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h9,h5,h2,h3,h1,h0]),tab_negall(discharge,[h10]),tab_negall(eigenvar,eigen__3)],[h9,20,h10]) ).

thf(22,plain,
    $false,
    inference(tab_fe,[status(thm),assumptions([h5,h2,h3,h1,h0]),tab_fe(discharge,[h9])],[h5,21,h9]) ).

thf(23,plain,
    $false,
    inference(tab_imp,[status(thm),assumptions([h2,h3,h1,h0]),tab_imp(discharge,[h4]),tab_imp(discharge,[h5])],[h3,10,22,h4,h5]) ).

thf(24,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,23,h2,h3]) ).

thf(25,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,24,h1]) ).

thf(0,theorem,
    ! [X1: $i > $i > $o] :
      ( ! [X2: $i,X3: $i] :
          ( ~ ( ~ sP2
             => ( X2 != X3 ) )
         => ( X1 @ X2 @ X3 ) )
     => ~ ( ! [X2: $i] :
              ( ~ sP2
             => ~ ! [X3: $i] :
                    ~ ( X1 @ X2 @ X3 ) )
         => ( ( ^ [X2: $i] : ~ sP2 )
           != ( ^ [X2: $i] :
                  ~ ! [X3: $i] :
                      ~ ( X1 @ X3 @ X2 ) ) ) ) ),
    inference(contra,[status(thm),contra(discharge,[h0])],[25,h0]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13  % Problem  : SET669^3 : TPTP v8.1.0. Released v3.6.0.
% 0.03/0.13  % Command  : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.35  % Computer : n008.cluster.edu
% 0.13/0.35  % Model    : x86_64 x86_64
% 0.13/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35  % Memory   : 8042.1875MB
% 0.13/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35  % CPULimit : 300
% 0.13/0.35  % WCLimit  : 600
% 0.13/0.35  % DateTime : Sat Jul  9 17:26:08 EDT 2022
% 0.13/0.35  % CPUTime  : 
% 0.20/0.39  % SZS status Theorem
% 0.20/0.39  % Mode: mode213
% 0.20/0.39  % Inferences: 56
% 0.20/0.39  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------