%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : ITP095^1 : 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 : n021.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 04:02:00 EDT 2023

% Result   : Theorem 0.20s 0.47s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)

% Comments : 
%------------------------------------------------------------------------------
thf(ty_int,type,
    int: $tType ).

thf(ty_poly_real,type,
    poly_real: $tType ).

thf(ty_nat,type,
    nat: $tType ).

thf(ty_real,type,
    real: $tType ).

thf(ty_set_real,type,
    set_real: $tType ).

thf(ty_one_one_real,type,
    one_one_real: real ).

thf(ty_ord_less_int,type,
    ord_less_int: int > int > $o ).

thf(ty_divide_divide_real,type,
    divide_divide_real: real > real > real ).

thf(ty_ord_less_eq_real,type,
    ord_less_eq_real: real > real > $o ).

thf(ty_a2,type,
    a2: int ).

thf(ty_degree_real,type,
    degree_real: poly_real > nat ).

thf(ty_ring_1_of_int_real,type,
    ring_1_of_int_real: int > real ).

thf(ty_b,type,
    b: int ).

thf(ty_power_power_real,type,
    power_power_real: real > nat > real ).

thf(ty_zero_zero_real,type,
    zero_zero_real: real ).

thf(ty_coeff_real,type,
    coeff_real: poly_real > nat > real ).

thf(ty_poly_real2,type,
    poly_real2: poly_real > real > real ).

thf(ty_abs_abs_real,type,
    abs_abs_real: real > real ).

thf(ty_zero_zero_poly_real,type,
    zero_zero_poly_real: poly_real ).

thf(ty_p,type,
    p: poly_real ).

thf(ty_zero_zero_int,type,
    zero_zero_int: int ).

thf(ty_member_real,type,
    member_real: real > set_real > $o ).

thf(ty_x,type,
    x: real ).

thf(ty_ring_1_Ints_real,type,
    ring_1_Ints_real: set_real ).

thf(ty_eigen__0,type,
    eigen__0: poly_real ).

thf(sP1,plain,
    ( sP1
  <=> ! [X1: nat] : ( member_real @ ( coeff_real @ p @ X1 ) @ ring_1_Ints_real ) ),
    introduced(definition,[new_symbols(definition,[sP1])]) ).

thf(sP2,plain,
    ( sP2
  <=> ( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) )
      = zero_zero_real ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ! [X1: poly_real,X2: int,X3: int] :
        ( ! [X4: nat] : ( member_real @ ( coeff_real @ X1 @ X4 ) @ ring_1_Ints_real )
       => ( ( ord_less_int @ zero_zero_int @ X2 )
         => ( ( ( poly_real2 @ X1 @ ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ X2 ) ) )
             != zero_zero_real )
           => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X2 ) @ ( degree_real @ X1 ) ) ) @ ( abs_abs_real @ ( poly_real2 @ X1 @ ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ X2 ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ! [X1: int,X2: int] :
        ( sP1
       => ( ( ord_less_int @ zero_zero_int @ X1 )
         => ( ( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) )
             != zero_zero_real )
           => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ X1 ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ X1 ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ( ( ord_less_int @ zero_zero_int @ b )
     => ( ~ sP2
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ( sP1
     => sP5 ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ! [X1: int] :
        ( sP1
       => ( ( ord_less_int @ zero_zero_int @ b )
         => ( ( ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ X1 ) @ ( ring_1_of_int_real @ b ) ) )
             != zero_zero_real )
           => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ X1 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( ~ sP2
     => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(sP9,plain,
    ( sP9
  <=> ( ord_less_int @ zero_zero_int @ b ) ),
    introduced(definition,[new_symbols(definition,[sP9])]) ).

thf(sP10,plain,
    ( sP10
  <=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( ring_1_of_int_real @ b ) @ ( degree_real @ p ) ) ) @ ( abs_abs_real @ ( poly_real2 @ p @ ( divide_divide_real @ ( ring_1_of_int_real @ a2 ) @ ( ring_1_of_int_real @ b ) ) ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP10])]) ).

thf(conj_0,conjecture,
    sP10 ).

thf(h0,negated_conjecture,
    ~ sP10,
    inference(assume_negation,[status(cth)],[conj_0]) ).

thf(h1,assumption,
    ~ ( ! [X1: nat] : ( member_real @ ( coeff_real @ eigen__0 @ X1 ) @ ring_1_Ints_real )
     => ( ( eigen__0 != zero_zero_poly_real )
       => ( ( poly_real2 @ eigen__0 @ x )
         != zero_zero_real ) ) ),
    introduced(assumption,[]) ).

thf(h2,assumption,
    ! [X1: nat] : ( member_real @ ( coeff_real @ eigen__0 @ X1 ) @ ring_1_Ints_real ),
    introduced(assumption,[]) ).

thf(h3,assumption,
    ~ ( ( eigen__0 != zero_zero_poly_real )
     => ( ( poly_real2 @ eigen__0 @ x )
       != zero_zero_real ) ),
    introduced(assumption,[]) ).

thf(h4,assumption,
    eigen__0 != zero_zero_poly_real,
    introduced(assumption,[]) ).

thf(h5,assumption,
    ( ( poly_real2 @ eigen__0 @ x )
    = zero_zero_real ),
    introduced(assumption,[]) ).

thf(1,plain,
    ( ~ sP8
    | sP2
    | sP10 ),
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP5
    | ~ sP9
    | sP8 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( ~ sP6
    | ~ sP1
    | sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(4,plain,
    ( ~ sP7
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ( ~ sP4
    | sP7 ),
    inference(all_rule,[status(thm)],]) ).

thf(6,plain,
    ( ~ sP3
    | sP4 ),
    inference(all_rule,[status(thm)],]) ).

thf(fact_154_int__poly__rat__no__root__ge,axiom,
    sP3 ).

thf(fact_5_no__root,axiom,
    ~ sP2 ).

thf(fact_3_b,axiom,
    sP9 ).

thf(fact_2_p_I1_J,axiom,
    sP1 ).

thf(7,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h4,h5,h2,h3,h1,h0])],[1,2,3,4,5,6,h0,fact_154_int__poly__rat__no__root__ge,fact_5_no__root,fact_3_b,fact_2_p_I1_J]) ).

thf(8,plain,
    $false,
    inference(tab_negimp,[status(thm),assumptions([h2,h3,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,7,h4,h5]) ).

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

thf(fact_36__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,axiom,
    ~ ! [X1: poly_real] :
        ( ! [X2: nat] : ( member_real @ ( coeff_real @ X1 @ X2 ) @ ring_1_Ints_real )
       => ( ( X1 != zero_zero_poly_real )
         => ( ( poly_real2 @ X1 @ x )
           != zero_zero_real ) ) ) ).

thf(10,plain,
    $false,
    inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[fact_36__092_060open_062_092_060And_062thesisa_O_A_I_092_060And_062p_O_A_092_060lbrakk_062_092_060And_062i_O_Acoeff_Ap_Ai_A_092_060in_062_A_092_060int_062_059_Ap_A_092_060noteq_062_A0_059_Apoly_Ap_Ax_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesisa_J_A_092_060Longrightarrow_062_Athesisa_092_060close_062,9,h1]) ).

thf(0,theorem,
    sP10,
    inference(contra,[status(thm),contra(discharge,[h0])],[10,h0]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11  % Problem  : ITP095^1 : TPTP v8.1.2. Released v7.5.0.
% 0.11/0.12  % Command  : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33  % Computer : n021.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Sun Aug 27 11:44:12 EDT 2023
% 0.12/0.33  % CPUTime  : 
% 0.20/0.47  % SZS status Theorem
% 0.20/0.47  % Mode: cade22sinegrackle2x6978
% 0.20/0.47  % Steps: 928
% 0.20/0.47  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------