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% File     : NUM736^1 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Number Theory
% Problem  : Landau theorem 42
% Version  : Especial.
% English  : less (ts (num y) (den x)) (ts (num x) (den y))

% Refs     : [Lan30] Landau (1930), Grundlagen der Analysis
%          : [vBJ79] van Benthem Jutting (1979), Checking Landau's "Grundla
%          : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : satz42 [Lan30]

% Status   : Theorem
%          : Without extensionality : Theorem
% Rating   : 0.00 v5.3.0, 0.25 v5.2.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :   12 (   2 unt;   9 typ;   0 def)
%            Number of atoms       :    4 (   0 equ;   0 cnn)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :   25 (   0   ~;   0   |;   0   &;  24   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   6 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   7 usr;   2 con; 0-2 aty)
%            Number of variables   :    2 (   0   ^;   2   !;   0   ?;   2   :)
% SPC      : TH0_THM_NEQ_NAR

% Comments : 
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thf(frac_type,type,
    frac: $tType ).

thf(x,type,
    x: frac ).

thf(y,type,
    y: frac ).

thf(nat_type,type,
    nat: $tType ).

thf(more,type,
    more: nat > nat > $o ).

thf(ts,type,
    ts: nat > nat > nat ).

thf(num,type,
    num: frac > nat ).

thf(den,type,
    den: frac > nat ).

thf(m,axiom,
    more @ ( ts @ ( num @ x ) @ ( den @ y ) ) @ ( ts @ ( num @ y ) @ ( den @ x ) ) ).

thf(less,type,
    less: nat > nat > $o ).

thf(satz11,axiom,
    ! [Xx: nat,Xy: nat] :
      ( ( more @ Xx @ Xy )
     => ( less @ Xy @ Xx ) ) ).

thf(satz42,conjecture,
    less @ ( ts @ ( num @ y ) @ ( den @ x ) ) @ ( ts @ ( num @ x ) @ ( den @ y ) ) ).

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