TPTP Problem File: NUM657^4.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : NUM657^4 : TPTP v8.1.2. Released v7.1.0.
% Domain   : Number theory
% Problem  : Grundlagen problem satz10h
% Version  : [Bro17] axioms : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 0.69 v8.1.0, 0.64 v7.5.0, 0.86 v7.4.0, 0.67 v7.2.0, 0.62 v7.1.0
% Syntax   : Number of formulae    :  356 ( 140 unt; 131 typ; 124 def)
%            Number of atoms       :  927 ( 154 equ;   0 cnn)
%            Maximal formula atoms :   18 (   4 avg)
%            Number of connectives : 1857 (   7   ~;   4   |;  14   &;1735   @)
%                                         (   3 <=>;  94  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  443 ( 443   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  156 ( 154 usr;  29 con; 0-7 aty)
%            Number of variables   :  684 ( 527   ^ 149   !;   8   ?; 684   :)
% SPC      : TH0_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
include('Axioms/NUM007^0.ax').
%------------------------------------------------------------------------------
thf(satz1,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( nis @ X0 @ X1 )
           => ( nis @ ( ordsucc @ X0 ) @ ( ordsucc @ X1 ) ) ) ) ) ).

thf(typ_d_22_prop1,type,
    d_22_prop1: $i > $o ).

thf(def_d_22_prop1,definition,
    ( d_22_prop1
    = ( ^ [X0: $i] : ( nis @ ( ordsucc @ X0 ) @ X0 ) ) ) ).

thf(satz2,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] : ( nis @ ( ordsucc @ X0 ) @ X0 ) ) ).

thf(typ_d_23_prop1,type,
    d_23_prop1: $i > $o ).

thf(def_d_23_prop1,definition,
    ( d_23_prop1
    = ( ^ [X0: $i] :
          ( l_or @ ( n_is @ X0 @ n_1 )
          @ ( n_some
            @ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ) ).

thf(satz3,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( ( nis @ X0 @ n_1 )
       => ( n_some
          @ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).

thf(satz3a,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( ( nis @ X0 @ n_1 )
       => ( n_one
          @ ^ [X1: $i] : ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).

thf(typ_d_24_prop1,type,
    d_24_prop1: $i > $o ).

thf(def_d_24_prop1,definition,
    ( d_24_prop1
    = ( ^ [X0: $i] :
          ( n_all
          @ ^ [X1: $i] : ( n_is @ ( ap @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) ) ) ).

thf(typ_d_24_prop2,type,
    d_24_prop2: $i > $i > $o ).

thf(def_d_24_prop2,definition,
    ( d_24_prop2
    = ( ^ [X0: $i,X1: $i] : ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) ) @ ( d_24_prop1 @ X1 ) ) ) ) ).

thf(typ_prop3,type,
    prop3: $i > $i > $i > $o ).

thf(def_prop3,definition,
    ( prop3
    = ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( ap @ X0 @ X2 ) @ ( ap @ X1 @ X2 ) ) ) ) ).

thf(typ_prop4,type,
    prop4: $i > $o ).

thf(def_prop4,definition,
    ( prop4
    = ( ^ [X0: $i] :
          ( l_some
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_24_prop2 @ X0 ) ) ) ) ).

thf(typ_d_24_g,type,
    d_24_g: $i > $i ).

thf(def_d_24_g,definition,
    ( d_24_g
    = ( ^ [X0: $i] :
          ( d_Sigma @ nat
          @ ^ [X1: $i] : ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) ) ).

thf(satz4,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( one
        @ ( d_Pi @ nat
          @ ^ [X1: $i] : nat )
        @ ^ [X1: $i] :
            ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) )
            @ ( n_all
              @ ^ [X2: $i] : ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( ordsucc @ ( ap @ X1 @ X2 ) ) ) ) ) ) ) ).

thf(typ_plus,type,
    plus: $i > $i ).

thf(def_plus,definition,
    ( plus
    = ( ^ [X0: $i] :
          ( ind
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_24_prop2 @ X0 ) ) ) ) ).

thf(typ_n_pl,type,
    n_pl: $i > $i > $i ).

thf(def_n_pl,definition,
    ( n_pl
    = ( ^ [X0: $i] : ( ap @ ( plus @ X0 ) ) ) ) ).

thf(satz4a,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] : ( n_is @ ( n_pl @ X0 @ n_1 ) @ ( ordsucc @ X0 ) ) ) ).

thf(satz4b,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( n_is @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) ) ).

thf(satz4c,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] : ( n_is @ ( n_pl @ n_1 @ X0 ) @ ( ordsucc @ X0 ) ) ) ).

thf(satz4d,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( n_is @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) ) ).

thf(satz4e,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] : ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ X0 @ n_1 ) ) ) ).

thf(satz4f,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) ) ) ) ).

thf(satz4g,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] : ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ n_1 @ X0 ) ) ) ).

thf(satz4h,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) ) ) ) ).

thf(typ_d_25_prop1,type,
    d_25_prop1: $i > $i > $i > $o ).

thf(def_d_25_prop1,definition,
    ( d_25_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ).

thf(satz5,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] : ( in @ X2 @ nat )
            @ ^ [X2: $i] : ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ) ).

thf(typ_d_26_prop1,type,
    d_26_prop1: $i > $i > $o ).

thf(def_d_26_prop1,definition,
    ( d_26_prop1
    = ( ^ [X0: $i,X1: $i] : ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) ) ).

thf(satz6,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) ) ).

thf(typ_d_27_prop1,type,
    d_27_prop1: $i > $i > $o ).

thf(def_d_27_prop1,definition,
    ( d_27_prop1
    = ( ^ [X0: $i,X1: $i] : ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) ) ).

thf(satz7,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) ) ).

thf(typ_d_28_prop1,type,
    d_28_prop1: $i > $i > $i > $o ).

thf(def_d_28_prop1,definition,
    ( d_28_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] : ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) ) ).

thf(satz8,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] : ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( nis @ X1 @ X2 )
               => ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz8a,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] : ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) )
               => ( n_is @ X1 @ X2 ) ) ) ) ) ).

thf(typ_diffprop,type,
    diffprop: $i > $i > $i > $o ).

thf(def_diffprop,definition,
    ( diffprop
    = ( ^ [X0: $i,X1: $i,X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ).

thf(satz8b,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( amone @ nat
            @ ^ [X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) ).

thf(typ_d_29_ii,type,
    d_29_ii: $i > $i > $o ).

thf(def_d_29_ii,definition,
    ( d_29_ii
    = ( ^ [X0: $i,X1: $i] : ( n_some @ ( diffprop @ X0 @ X1 ) ) ) ) ).

thf(typ_iii,type,
    iii: $i > $i > $o ).

thf(def_iii,definition,
    ( iii
    = ( ^ [X0: $i,X1: $i] : ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ).

thf(typ_d_29_prop1,type,
    d_29_prop1: $i > $i > $o ).

thf(def_d_29_prop1,definition,
    ( d_29_prop1
    = ( ^ [X0: $i,X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).

thf(satz9,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( orec3 @ ( n_is @ X0 @ X1 )
            @ ( n_some
              @ ^ [X2: $i] : ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) )
            @ ( n_some
              @ ^ [X2: $i] : ( n_is @ X1 @ ( n_pl @ X0 @ X2 ) ) ) ) ) ) ).

thf(satz9a,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ) ).

thf(satz9b,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( ec3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) ) ).

thf(satz10,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( orec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).

thf(satz10a,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).

thf(satz10b,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] : ( ec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) ) ).

thf(satz11,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X0 @ X1 )
           => ( iii @ X1 @ X0 ) ) ) ) ).

thf(satz12,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X0 @ X1 )
           => ( d_29_ii @ X1 @ X0 ) ) ) ) ).

thf(typ_moreis,type,
    moreis: $i > $i > $o ).

thf(def_moreis,definition,
    ( moreis
    = ( ^ [X0: $i,X1: $i] : ( l_or @ ( d_29_ii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) ) ).

thf(typ_lessis,type,
    lessis: $i > $i > $o ).

thf(def_lessis,definition,
    ( lessis
    = ( ^ [X0: $i,X1: $i] : ( l_or @ ( iii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) ) ).

thf(satz13,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( moreis @ X0 @ X1 )
           => ( lessis @ X1 @ X0 ) ) ) ) ).

thf(satz14,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( lessis @ X0 @ X1 )
           => ( moreis @ X1 @ X0 ) ) ) ) ).

thf(satz10c,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( moreis @ X0 @ X1 )
           => ( d_not @ ( iii @ X0 @ X1 ) ) ) ) ) ).

thf(satz10d,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( lessis @ X0 @ X1 )
           => ( d_not @ ( d_29_ii @ X0 @ X1 ) ) ) ) ) ).

thf(satz10e,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( d_29_ii @ X0 @ X1 ) )
           => ( lessis @ X0 @ X1 ) ) ) ) ).

thf(satz10f,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( iii @ X0 @ X1 ) )
           => ( moreis @ X0 @ X1 ) ) ) ) ).

thf(satz10g,axiom,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X0 @ X1 )
           => ( d_not @ ( lessis @ X0 @ X1 ) ) ) ) ) ).

thf(satz10h,conjecture,
    ( all_of
    @ ^ [X0: $i] : ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] : ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X0 @ X1 )
           => ( d_not @ ( moreis @ X0 @ X1 ) ) ) ) ) ).

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