%------------------------------------------------------------------------------
% File     : Lash---1.13
% Problem  : ITP132^1 : 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 : n011.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 04:02:24 EDT 2023

% Result   : Theorem 0.18s 0.46s
% Output   : Proof 0.18s
% Verified : 
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)

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

thf(ty_times_times_nat,type,
    times_times_nat: nat > nat > nat ).

thf(ty_minus_minus_nat,type,
    minus_minus_nat: nat > nat > nat ).

thf(ty_k,type,
    k: nat ).

thf(ty_one_one_nat,type,
    one_one_nat: nat ).

thf(ty_ord_less_eq_nat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(ty_groups1842438620at_nat,type,
    groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).

thf(ty_number1551313001itions,type,
    number1551313001itions: ( nat > nat ) > nat > $o ).

thf(ty_p,type,
    p: nat > nat ).

thf(ty_n,type,
    n: nat ).

thf(ty_zero_zero_nat,type,
    zero_zero_nat: nat ).

thf(ty_set_ord_atMost_nat,type,
    set_ord_atMost_nat: nat > set_nat ).

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

thf(sP2,plain,
    ( sP2
  <=> ( ( minus_minus_nat
        @ ( groups1842438620at_nat
          @ ^ [X1: nat] : ( times_times_nat @ ( p @ X1 ) @ X1 )
          @ ( set_ord_atMost_nat @ n ) )
        @ k )
      = ( minus_minus_nat @ n @ k ) ) ),
    introduced(definition,[new_symbols(definition,[sP2])]) ).

thf(sP3,plain,
    ( sP3
  <=> ! [X1: nat > nat,X2: nat] :
        ( ( number1551313001itions @ X1 @ X2 )
       => ~ ( ! [X3: nat] :
                ( ( ( X1 @ X3 )
                 != zero_zero_nat )
               => ~ ( ( ord_less_eq_nat @ one_one_nat @ X3 )
                   => ~ ( ord_less_eq_nat @ X3 @ X2 ) ) )
           => ( ( groups1842438620at_nat
                @ ^ [X3: nat] : ( times_times_nat @ ( X1 @ X3 ) @ X3 )
                @ ( set_ord_atMost_nat @ X2 ) )
             != X2 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP3])]) ).

thf(sP4,plain,
    ( sP4
  <=> ( number1551313001itions @ p @ n ) ),
    introduced(definition,[new_symbols(definition,[sP4])]) ).

thf(sP5,plain,
    ( sP5
  <=> ( ( groups1842438620at_nat
        @ ^ [X1: nat] : ( times_times_nat @ ( p @ X1 ) @ X1 )
        @ ( set_ord_atMost_nat @ n ) )
      = n ) ),
    introduced(definition,[new_symbols(definition,[sP5])]) ).

thf(sP6,plain,
    ( sP6
  <=> ! [X1: nat] :
        ( ( number1551313001itions @ p @ X1 )
       => ~ ( ! [X2: nat] :
                ( ( ( p @ X2 )
                 != zero_zero_nat )
               => ~ ( ( ord_less_eq_nat @ one_one_nat @ X2 )
                   => ~ ( ord_less_eq_nat @ X2 @ X1 ) ) )
           => ( ( groups1842438620at_nat
                @ ^ [X2: nat] : ( times_times_nat @ ( p @ X2 ) @ X2 )
                @ ( set_ord_atMost_nat @ X1 ) )
             != X1 ) ) ) ),
    introduced(definition,[new_symbols(definition,[sP6])]) ).

thf(sP7,plain,
    ( sP7
  <=> ( sP4
     => ~ ( ! [X1: nat] :
              ( ( ( p @ X1 )
               != zero_zero_nat )
             => ~ ( ( ord_less_eq_nat @ one_one_nat @ X1 )
                 => ~ ( ord_less_eq_nat @ X1 @ n ) ) )
         => ~ sP5 ) ) ),
    introduced(definition,[new_symbols(definition,[sP7])]) ).

thf(sP8,plain,
    ( sP8
  <=> ( ! [X1: nat] :
          ( ( ( p @ X1 )
           != zero_zero_nat )
         => ~ ( ( ord_less_eq_nat @ one_one_nat @ X1 )
             => ~ ( ord_less_eq_nat @ X1 @ n ) ) )
     => ~ sP5 ) ),
    introduced(definition,[new_symbols(definition,[sP8])]) ).

thf(conj_0,conjecture,
    sP2 ).

thf(h0,negated_conjecture,
    ~ sP2,
    inference(assume_negation,[status(cth)],[conj_0]) ).

thf(1,plain,
    ( sP8
    | sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(2,plain,
    ( ~ sP7
    | ~ sP4
    | ~ sP8 ),
    inference(prop_rule,[status(thm)],]) ).

thf(3,plain,
    ( ~ sP6
    | sP7 ),
    inference(all_rule,[status(thm)],]) ).

thf(4,plain,
    ( ~ sP3
    | sP6 ),
    inference(all_rule,[status(thm)],]) ).

thf(5,plain,
    ~ sP1,
    inference(prop_rule,[status(thm)],]) ).

thf(6,plain,
    ( sP2
    | sP1
    | ~ sP5 ),
    inference(prop_rule,[status(thm)],]) ).

thf(fact_63_partitionsE,axiom,
    sP3 ).

thf(fact_0_partitions,axiom,
    sP4 ).

thf(7,plain,
    $false,
    inference(prop_unsat,[status(thm),assumptions([h0])],[1,2,3,4,5,6,h0,fact_63_partitionsE,fact_0_partitions]) ).

thf(0,theorem,
    sP2,
    inference(contra,[status(thm),contra(discharge,[h0])],[7,h0]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : ITP132^1 : 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 : n011.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 12:30:40 EDT 2023
% 0.12/0.33  % CPUTime  : 
% 0.18/0.46  % SZS status Theorem
% 0.18/0.46  % Mode: cade22sinegrackle2x6978
% 0.18/0.46  % Steps: 1021
% 0.18/0.46  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------