TPTP Problem File: NUM788^4.p
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%------------------------------------------------------------------------------
% File : NUM788^4 : TPTP v8.2.0. Released v7.1.0.
% Domain : Number theory
% Problem : Grundlagen problem satz81g
% Version : [Bro17] axioms : Especial.
% English :
% Refs : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source : [Br017]
% Names :
% Status : Theorem
% Rating : 0.80 v8.2.0, 0.77 v8.1.0, 0.82 v7.5.0, 0.86 v7.4.0, 0.67 v7.2.0, 0.62 v7.1.0
% Syntax : Number of formulae : 697 ( 208 unt; 199 typ; 192 def)
% Number of atoms : 2913 ( 222 equ; 0 cnn)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 6347 ( 7 ~; 4 |; 14 &;5989 @)
% ( 3 <=>; 330 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 580 ( 580 >; 0 *; 0 +; 0 <<)
% Number of symbols : 223 ( 221 usr; 35 con; 0-7 aty)
% Number of variables : 1972 (1813 ^; 151 !; 8 ?;1972 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
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include('Axioms/NUM007^0.ax').
include('Axioms/NUM007^1.ax').
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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,conjecture,
( 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 ) ) ) ) ) ).
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