%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : QUA012^1 : TPTP v8.1.2. Released v4.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : lash -P picomus -M modes -p tstp -t %d %s

% Computer : n018.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 13:31:55 EDT 2023

% Result   : Theorem 20.23s 20.65s
% Output   : Proof 20.23s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_multiplication,type,
    multiplication: $i > $i > $i ).

thf(ty_sup,type,
    sup: ( $i > $o ) > $i ).

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

thf(ty_zero,type,
    zero: $i ).

thf(sP1,plain,
    ( sP1
  <=> $false ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> ( ( multiplication
        @ ( sup
          @ ^ [X1: $i] : sP1 )
        @ ( sup @ eigen__0 ) )
      = ( sup
        @ ^ [X1: $i] : sP1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ( ( multiplication
        @ ( sup
          @ ^ [X1: $i] : sP1 )
        @ ( sup @ eigen__0 ) )
      = ( multiplication @ zero @ ( sup @ eigen__0 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( ( multiplication @ zero @ ( sup @ eigen__0 ) )
      = zero ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ( ( sup
        @ ^ [X1: $i] : sP1 )
      = zero ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ! [X1: $i > $o] :
        ( ( multiplication
          @ ( sup
            @ ^ [X2: $i] : sP1 )
          @ ( sup @ X1 ) )
        = ( sup
          @ ^ [X2: $i] : sP1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: $i > $o,X2: $i > $o] :
        ( ( multiplication @ ( sup @ X1 ) @ ( sup @ X2 ) )
        = ( sup
          @ ^ [X3: $i] :
              ~ ! [X4: $i,X5: $i] :
                  ( ~ ( ( X1 @ X4 )
                     => ~ ( X2 @ X5 ) )
                 => ( X3
                   != ( multiplication @ X4 @ X5 ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(def_emptyset,definition,
    ( emptyset
    = ( ^ [X1: $i] : sP1 ) ) ).

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

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

thf(def_supset,definition,
    ( supset
    = ( ^ [X1: ( $i > $o ) > $o,X2: $i] :
        ? [X3: $i > $o] :
          ( ( X1 @ X3 )
          & ( ( sup @ X3 )
            = X2 ) ) ) ) ).

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

thf(def_addition,definition,
    ( addition
    = ( ^ [X1: $i,X2: $i] : ( sup @ ( union @ ( singleton @ X1 ) @ ( singleton @ X2 ) ) ) ) ) ).

thf(def_crossmult,definition,
    ( crossmult
    = ( ^ [X1: $i > $o,X2: $i > $o,X3: $i] :
        ? [X4: $i,X5: $i] :
          ( ( X1 @ X4 )
          & ( X2 @ X5 )
          & ( X3
            = ( multiplication @ X4 @ X5 ) ) ) ) ) ).

thf(multiplication_anni,conjecture,
    ! [X1: $i > $o] :
      ( ( multiplication @ zero @ ( sup @ X1 ) )
      = zero ) ).

thf(h0,negated_conjecture,
    ~ ! [X1: $i > $o] :
        ( ( multiplication @ zero @ ( sup @ X1 ) )
        = zero ),
    inference(assume_negation,[status(cth)],[multiplication_anni]) ).

thf(h1,assumption,
    ~ sP4,
    introduced(assumption,[]) ).

thf(1,plain,
    ( sP3
    | sP1
    | ~ sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP2
    | sP4
    | ~ sP3
    | ~ sP5 ),
    inference(confrontation_rule,[status(thm)],]) ).

thf(3,plain,
    ~ sP1,
    inference(prop_rule,[status(thm)],]) ).

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

thf(5,plain,
    ( ~ sP7
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(sup_es,axiom,
    sP5 ).

thf(multiplication_def,axiom,
    sP7 ).

thf(6,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,sup_es,multiplication_def,h1]) ).

thf(7,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,6,h1]) ).

thf(0,theorem,
    ! [X1: $i > $o] :
      ( ( multiplication @ zero @ ( sup @ X1 ) )
      = zero ),
    inference(contra,[status(thm),contra(discharge,[h0])],[7,h0]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13  % Problem  : QUA012^1 : TPTP v8.1.2. Released v4.1.0.
% 0.00/0.14  % Command  : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.36  % Computer : n018.cluster.edu
% 0.14/0.36  % Model    : x86_64 x86_64
% 0.14/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36  % Memory   : 8042.1875MB
% 0.14/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36  % CPULimit : 300
% 0.14/0.36  % WCLimit  : 300
% 0.14/0.36  % DateTime : Sat Aug 26 16:54:30 EDT 2023
% 0.14/0.36  % CPUTime  : 
% 20.23/20.65  % SZS status Theorem
% 20.23/20.65  % Mode: cade22grackle2x798d
% 20.23/20.65  % Steps: 557
% 20.23/20.65  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------