%------------------------------------------------------------------------------
% File     : NUM766^1 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Number Theory
% Problem  : Landau theorem 67a
% Version  : Especial.
% English  : ~(forall x_0:frac.~(eq (pf y x_0) x))

% 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    : satz67a [Lan30]

% Status   : Theorem
%          : Without extensionality : Theorem
% Rating   : 0.38 v9.0.0, 0.50 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.3.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v5.3.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   24 (   8 unt;  13 typ;   0 def)
%            Number of atoms       :   16 (   3 equ;   0 cnn)
%            Maximal formula atoms :    3 (   1 avg)
%            Number of connectives :   89 (   8   ~;   0   |;   0   &;  75   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   7 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   17 (  17   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  11 usr;   2 con; 0-2 aty)
%            Number of variables   :   24 (   1   ^;  23   !;   0   ?;  24   :)
% SPC      : TH0_THM_EQU_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(ts,type,
    ts: nat > nat > nat ).

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

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

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

thf(m,axiom,
    ~ ! [Xx_0: nat] :
        ( ( ts @ ( num @ x ) @ ( den @ y ) )
       != ( pl @ ( ts @ ( num @ y ) @ ( den @ x ) ) @ Xx_0 ) ) ).

thf(eq,type,
    eq: frac > frac > $o ).

thf(pf,type,
    pf: frac > frac > frac ).

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

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

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

thf(satz8b,axiom,
    ! [Xx: nat,Xy: nat] :
      ( amone
      @ ^ [Xz: nat] :
          ( Xx
          = ( pl @ Xy @ Xz ) ) ) ).

thf(satz39,axiom,
    ! [Xx: frac,Xy: frac,Xz: frac] :
      ( ( eq @ Xx @ Xy )
     => ( ( eq @ Xy @ Xz )
       => ( eq @ Xx @ Xz ) ) ) ).

thf(satz56,axiom,
    ! [Xx: frac,Xy: frac,Xz: frac,Xu: frac] :
      ( ( eq @ Xx @ Xy )
     => ( ( eq @ Xz @ Xu )
       => ( eq @ ( pf @ Xx @ Xz ) @ ( pf @ Xy @ Xu ) ) ) ) ).

thf(satz40,axiom,
    ! [Xx: frac,Xn: nat] : ( eq @ Xx @ ( fr @ ( ts @ ( num @ Xx ) @ Xn ) @ ( ts @ ( den @ Xx ) @ Xn ) ) ) ).

thf(satz37,axiom,
    ! [Xx: frac] : ( eq @ Xx @ Xx ) ).

thf(satz29,axiom,
    ! [Xx: nat,Xy: nat] :
      ( ( ts @ Xx @ Xy )
      = ( ts @ Xy @ Xx ) ) ).

thf(satz57,axiom,
    ! [Xx1: nat,Xx2: nat,Xn: nat] : ( eq @ ( pf @ ( fr @ Xx1 @ Xn ) @ ( fr @ Xx2 @ Xn ) ) @ ( fr @ ( pl @ Xx1 @ Xx2 ) @ Xn ) ) ).

thf(oneax,axiom,
    ! [Xp: nat > $o] :
      ( ~ ( ( amone @ Xp )
         => ~ ~ ! [Xx: nat] :
                  ~ ( Xp @ Xx ) )
     => ( Xp @ ( ind @ Xp ) ) ) ).

thf(satz40a,axiom,
    ! [Xx: frac,Xn: nat] : ( eq @ ( fr @ ( ts @ ( num @ Xx ) @ Xn ) @ ( ts @ ( den @ Xx ) @ Xn ) ) @ Xx ) ).

thf(satz67a,conjecture,
    ~ ! [Xx_0: frac] :
        ~ ( eq @ ( pf @ y @ Xx_0 ) @ x ) ).

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