TPTP Problem File: NUM793^4.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : NUM793^4 : TPTP v7.4.0. Released v7.1.0.
% Domain   : Number theory
% Problem  : Grundlagen problem satz87d
% Version  : [Bro17] axioms : Especial.
% English  :

% Refs     : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source   : [Br017]
% Names    :

% Status   : Theorem
% Rating   : 0.86 v7.4.0, 0.89 v7.2.0, 0.88 v7.1.0
% Syntax   : Number of formulae    :  707 (   0 unit; 199 type; 192 defn)
%            Number of atoms       : 7460 ( 222 equality;3129 variable)
%            Maximal formula depth :   22 (   8 average)
%            Number of connectives : 6515 (   7   ~;   4   |;  14   &;6142   @)
%                                         (   3 <=>; 345  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  580 ( 580   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  202 ( 199   :;   0   =)
%            Number of variables   : 2022 (  32 sgn; 151   !;   8   ?;1863   ^)
%                                         (2022   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
include('Axioms/NUM007^0.ax').
include('Axioms/NUM007^1.ax').
%------------------------------------------------------------------------------
thf(typ_inf,type,(
    inf: $i > $i > $o )).

thf(def_inf,definition,
    ( inf
    = ( esti @ frac ) )).

thf(typ_rat,type,(
    rat: $i )).

thf(def_rat,definition,
    ( rat
    = ( ect @ frac @ n_eq ) )).

thf(typ_rt_is,type,(
    rt_is: $i > $i > $o )).

thf(def_rt_is,definition,
    ( rt_is
    = ( e_is @ rat ) )).

thf(typ_rt_nis,type,(
    rt_nis: $i > $i > $o )).

thf(def_rt_nis,definition,
    ( rt_nis
    = ( ^ [X0: $i,X1: $i] :
          ( d_not @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(typ_rt_some,type,(
    rt_some: ( $i > $o ) > $o )).

thf(def_rt_some,definition,
    ( rt_some
    = ( l_some @ rat ) )).

thf(typ_rt_all,type,(
    rt_all: ( $i > $o ) > $o )).

thf(def_rt_all,definition,
    ( rt_all
    = ( all @ rat ) )).

thf(typ_rt_one,type,(
    rt_one: ( $i > $o ) > $o )).

thf(def_rt_one,definition,
    ( rt_one
    = ( one @ rat ) )).

thf(typ_rt_in,type,(
    rt_in: $i > $i > $o )).

thf(def_rt_in,definition,
    ( rt_in
    = ( esti @ rat ) )).

thf(typ_ratof,type,(
    ratof: $i > $i )).

thf(def_ratof,definition,
    ( ratof
    = ( ectelt @ frac @ n_eq ) )).

thf(typ_class,type,(
    class: $i > $i )).

thf(def_class,definition,
    ( class
    = ( ecect @ frac @ n_eq ) )).

thf(typ_fixf,type,(
    fixf: $i > $i > $o )).

thf(def_fixf,definition,
    ( fixf
    = ( fixfu2 @ frac @ n_eq ) )).

thf(typ_indrat,type,(
    indrat: $i > $i > $i > $i > $i )).

thf(def_indrat,definition,
    ( indrat
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( indeq2 @ frac @ n_eq @ X2 @ X3 @ X0 @ X1 ) ) )).

thf(satz78,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( rt_is @ X0 @ X0 ) )).

thf(satz79,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_is @ X0 @ X1 )
           => ( rt_is @ X1 @ X0 ) ) ) )).

thf(satz80,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_is @ X0 @ X1 )
               => ( ( rt_is @ X1 @ X2 )
                 => ( rt_is @ X0 @ X2 ) ) ) ) ) )).

thf(typ_rt_more,type,(
    rt_more: $i > $i > $o )).

thf(def_rt_more,definition,
    ( rt_more
    = ( ^ [X0: $i,X1: $i] :
          ( l_some @ frac
          @ ^ [X2: $i] :
              ( l_some @ frac
              @ ^ [X3: $i] :
                  ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) ) ) )).

thf(typ_propm,type,(
    propm: $i > $i > $i > $i > $o )).

thf(def_propm,definition,
    ( propm
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( moref @ X2 @ X3 ) ) ) )).

thf(typ_rt_less,type,(
    rt_less: $i > $i > $o )).

thf(def_rt_less,definition,
    ( rt_less
    = ( ^ [X0: $i,X1: $i] :
          ( l_some @ frac
          @ ^ [X2: $i] :
              ( l_some @ frac
              @ ^ [X3: $i] :
                  ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) ) ) )).

thf(typ_propl,type,(
    propl: $i > $i > $i > $i > $o )).

thf(def_propl,definition,
    ( propl
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( and3 @ ( inf @ X2 @ ( class @ X0 ) ) @ ( inf @ X3 @ ( class @ X1 ) ) @ ( lessf @ X2 @ X3 ) ) ) )).

thf(satz81,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( orec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( or3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ec3 @ ( rt_is @ X0 @ X1 ) @ ( rt_more @ X0 @ X1 ) @ ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz82,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( rt_less @ X1 @ X0 ) ) ) )).

thf(satz83,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_less @ X0 @ X1 )
           => ( rt_more @ X1 @ X0 ) ) ) )).

thf(typ_rt_moreis,type,(
    rt_moreis: $i > $i > $o )).

thf(def_rt_moreis,definition,
    ( rt_moreis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( rt_more @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(typ_rt_lessis,type,(
    rt_lessis: $i > $i > $o )).

thf(def_rt_lessis,definition,
    ( rt_lessis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( rt_less @ X0 @ X1 ) @ ( rt_is @ X0 @ X1 ) ) ) )).

thf(satz81c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_moreis @ X0 @ X1 )
           => ( d_not @ ( rt_less @ X0 @ X1 ) ) ) ) )).

thf(satz81d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_lessis @ X0 @ X1 )
           => ( d_not @ ( rt_more @ X0 @ X1 ) ) ) ) )).

thf(satz81e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_more @ X0 @ X1 ) )
           => ( rt_lessis @ X0 @ X1 ) ) ) )).

thf(satz81f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_less @ X0 @ X1 ) )
           => ( rt_moreis @ X0 @ X1 ) ) ) )).

thf(satz81g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_more @ X0 @ X1 )
           => ( d_not @ ( rt_lessis @ X0 @ X1 ) ) ) ) )).

thf(satz81h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_less @ X0 @ X1 )
           => ( d_not @ ( rt_moreis @ X0 @ X1 ) ) ) ) )).

thf(satz81j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_moreis @ X0 @ X1 ) )
           => ( rt_less @ X0 @ X1 ) ) ) )).

thf(satz81k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( rt_lessis @ X0 @ X1 ) )
           => ( rt_more @ X0 @ X1 ) ) ) )).

thf(satz84,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_moreis @ X0 @ X1 )
           => ( rt_lessis @ X1 @ X0 ) ) ) )).

thf(satz85,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( ( rt_lessis @ X0 @ X1 )
           => ( rt_moreis @ X1 @ X0 ) ) ) )).

thf(satz86,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( ( rt_less @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_lessis @ X0 @ X1 )
               => ( ( rt_less @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_less @ X0 @ X1 )
               => ( ( rt_lessis @ X1 @ X2 )
                 => ( rt_less @ X0 @ X2 ) ) ) ) ) )).

thf(satz87c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_moreis @ X0 @ X1 )
               => ( ( rt_more @ X1 @ X2 )
                 => ( rt_more @ X0 @ X2 ) ) ) ) ) )).

thf(satz87d,conjecture,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ rat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ rat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ rat )
            @ ^ [X2: $i] :
                ( ( rt_more @ X0 @ X1 )
               => ( ( rt_moreis @ X1 @ X2 )
                 => ( rt_more @ X0 @ X2 ) ) ) ) ) )).

%------------------------------------------------------------------------------