%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : ITP112^1 : TPTP v8.1.2. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : lash -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  : 300s
% DateTime : Thu Aug 31 04:02:10 EDT 2023

% Result   : Theorem 20.26s 20.63s
% Output   : Proof 20.26s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   47
% Syntax   : Number of formulae    :   52 (  11 unt;  13 typ;   1 def)
%            Number of atoms       :   75 (  12 equ;   0 cnn)
%            Maximal formula atoms :    5 (   1 avg)
%            Number of connectives :  157 (  17   ~;  16   |;   0   &; 105   @)
%                                         (  16 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   3 avg)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  :   20 (  20   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   29 (  27 usr;  22 con; 0-3 aty)
%            Number of variables   :   23 (  16   ^;   7   !;   0   ?;  23   :)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_nat,type,
    nat: $tType ).

thf(ty_filter_nat,type,
    filter_nat: $tType ).

thf(ty_a,type,
    a: $tType ).

thf(ty_extended_ereal,type,
    extended_ereal: $tType ).

thf(ty_a2,type,
    a2: extended_ereal ).

thf(ty_x,type,
    x: nat > a ).

thf(ty_f,type,
    f: a > extended_ereal ).

thf(ty_eigen__3,type,
    eigen__3: nat ).

thf(ty_uminus1208298309_ereal,type,
    uminus1208298309_ereal: extended_ereal > extended_ereal ).

thf(ty_comp_a1112243075al_nat,type,
    comp_a1112243075al_nat: ( a > extended_ereal ) > ( nat > a ) > nat > extended_ereal ).

thf(ty_topolo2140997059_ereal,type,
    topolo2140997059_ereal: extended_ereal > filter2049122004_ereal ).

thf(ty_at_top_nat,type,
    at_top_nat: filter_nat ).

thf(ty_filter1531173832_ereal,type,
    filter1531173832_ereal: ( nat > extended_ereal ) > filter2049122004_ereal > filter_nat > $o ).

thf(h0,assumption,
    ! [X1: nat > $o,X2: nat] :
      ( ( X1 @ X2 )
     => ( X1 @ ( eps__0 @ X1 ) ) ),
    introduced(assumption,[]) ).

thf(eigendef_eigen__3,definition,
    ( eigen__3
    = ( eps__0
      @ ^ [X1: nat] :
          ( ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
         != ( uminus1208298309_ereal @ ( f @ ( x @ X1 ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[eigen__3])]) ).

thf(sP1,plain,
    ( sP1
  <=> ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat ) ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> ( sP1
     => ( filter1531173832_ereal
        @ ^ [X1: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
        @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
        @ at_top_nat ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ( comp_a1112243075al_nat
      = ( ^ [X1: a > extended_ereal,X2: nat > a,X3: nat] : ( X1 @ ( X2 @ X3 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( filter1531173832_ereal
      @ ^ [X1: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
      @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
      @ at_top_nat ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

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

thf(sP6,plain,
    ( sP6
  <=> ( ( comp_a1112243075al_nat @ f )
      = ( ^ [X1: nat > a,X2: nat] : ( f @ ( X1 @ X2 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: a > extended_ereal] :
        ( ( comp_a1112243075al_nat @ X1 )
        = ( ^ [X2: nat > a,X3: nat] : ( X1 @ ( X2 @ X3 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( filter1531173832_ereal
      @ ^ [X1: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ X1 ) ) )
      @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
      @ at_top_nat ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ! [X1: nat] :
        ( ( comp_a1112243075al_nat @ f @ x @ X1 )
        = ( f @ ( x @ X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ! [X1: filter_nat] :
        ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X1 )
       => ( filter1531173832_ereal
          @ ^ [X2: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X2 ) )
          @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
          @ X1 ) ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(sP11,plain,
    ( sP11
  <=> ! [X1: nat > a] :
        ( ( comp_a1112243075al_nat @ f @ X1 )
        = ( ^ [X2: nat] : ( f @ ( X1 @ X2 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP11])]) ).

thf(sP12,plain,
    ( sP12
  <=> ! [X1: nat] :
        ( ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
        = ( uminus1208298309_ereal @ ( f @ ( x @ X1 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP12])]) ).

thf(sP13,plain,
    ( sP13
  <=> ( ( ^ [X1: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) ) )
      = ( ^ [X1: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ X1 ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP13])]) ).

thf(sP14,plain,
    ( sP14
  <=> ( ( comp_a1112243075al_nat @ f @ x )
      = ( ^ [X1: nat] : ( f @ ( x @ X1 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP14])]) ).

thf(sP15,plain,
    ( sP15
  <=> ( ( comp_a1112243075al_nat @ f @ x @ eigen__3 )
      = ( f @ ( x @ eigen__3 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP15])]) ).

thf(sP16,plain,
    ( sP16
  <=> ( ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ eigen__3 ) )
      = ( uminus1208298309_ereal @ ( f @ ( x @ eigen__3 ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP16])]) ).

thf(conj_0,conjecture,
    sP8 ).

thf(h1,negated_conjecture,
    ~ sP8,
    inference(assume_negation,[status(cth)],[conj_0]) ).

thf(1,plain,
    ( ~ sP9
    | sP15 ),
    inference(all_rule,[status(thm)],]) ).

thf(2,plain,
    ( sP16
    | ~ sP15 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( sP12
    | ~ sP16 ),
    inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).

thf(4,plain,
    ( ~ sP14
    | sP9 ),
    inference(prop_rule,[status(thm)],]) ).

thf(5,plain,
    ( sP13
    | ~ sP12 ),
    inference(prop_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP11
    | sP14 ),
    inference(all_rule,[status(thm)],]) ).

thf(7,plain,
    ( ~ sP4
    | sP8
    | sP5
    | sP5
    | ~ sP13 ),
    inference(mating_rule,[status(thm)],]) ).

thf(8,plain,
    ~ sP5,
    inference(prop_rule,[status(thm)],]) ).

thf(9,plain,
    ( ~ sP6
    | sP11 ),
    inference(prop_rule,[status(thm)],]) ).

thf(10,plain,
    ( ~ sP7
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(11,plain,
    ( ~ sP2
    | ~ sP1
    | sP4 ),
    inference(prop_rule,[status(thm)],]) ).

thf(12,plain,
    ( ~ sP3
    | sP7 ),
    inference(prop_rule,[status(thm)],]) ).

thf(13,plain,
    ( ~ sP10
    | sP2 ),
    inference(all_rule,[status(thm)],]) ).

thf(fact_36_comp__apply,axiom,
    sP3 ).

thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
    sP10 ).

thf(fact_1_x__def_I2_J,axiom,
    sP1 ).

thf(14,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,h1,fact_36_comp__apply,fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,fact_1_x__def_I2_J]) ).

thf(15,plain,
    $false,
    inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[14,h0]) ).

thf(0,theorem,
    sP8,
    inference(contra,[status(thm),contra(discharge,[h1])],[14,h1]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13  % Problem  : ITP112^1 : TPTP v8.1.2. Released v7.5.0.
% 0.00/0.14  % Command  : lash -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  : 300
% 0.13/0.35  % DateTime : Sun Aug 27 11:00:47 EDT 2023
% 0.13/0.36  % CPUTime  : 
% 20.26/20.63  % SZS status Theorem
% 20.26/20.63  % Mode: cade22sinegrackle2xfaf3
% 20.26/20.63  % Steps: 2328
% 20.26/20.63  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------