## TPTP Problem File: NUM657^4.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : NUM657^4 : TPTP v8.0.0. 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   : 0.64 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

%------------------------------------------------------------------------------
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 ) ) ) ) ) ).

%------------------------------------------------------------------------------
```