%------------------------------------------------------------------------------
% File     : NUM696^1 : TPTP v8.2.0. Released v3.7.0.
% Domain   : Number Theory
% Problem  : Landau theorem 25
% Version  : Especial.
% English  : ~(~(forall x_0:nat.~(y = pl (pl x n_1) x_0))) -> y = pl x n_1

% 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    : satz25 [Lan30]

% Status   : Theorem
%          : Without extensionality : Theorem
% Rating   : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   11 (   2 unt;   5 typ;   0 def)
%            Number of atoms       :    8 (   8 equ;   0 cnn)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :   40 (  14   ~;   0   |;   0   &;  22   @)
%                                         (   0 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   6 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :    2 (   2   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    5 (   4 usr;   3 con; 0-2 aty)
%            Number of variables   :   12 (   0   ^;  12   !;   0   ?;  12   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : 
%------------------------------------------------------------------------------
thf(nat_type,type,
    nat: $tType ).

thf(x,type,
    x: nat ).

thf(y,type,
    y: nat ).

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

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

thf(n_1,type,
    n_1: nat ).

thf(et,axiom,
    ! [Xa: $o] :
      ( ~ ~ Xa
     => Xa ) ).

thf(satz24,axiom,
    ! [Xx: nat] :
      ( ~ ~ ! [Xx_0: nat] :
              ( Xx
             != ( pl @ n_1 @ Xx_0 ) )
     => ( Xx = n_1 ) ) ).

thf(satz19a,axiom,
    ! [Xx: nat,Xy: nat,Xz: nat] :
      ( ~ ! [Xx_0: nat] :
            ( Xx
           != ( pl @ Xy @ Xx_0 ) )
     => ~ ! [Xx_0: nat] :
            ( ( pl @ Xx @ Xz )
           != ( pl @ ( pl @ Xy @ Xz ) @ Xx_0 ) ) ) ).

thf(satz6,axiom,
    ! [Xx: nat,Xy: nat] :
      ( ( pl @ Xx @ Xy )
      = ( pl @ Xy @ Xx ) ) ).

thf(satz25,conjecture,
    ( ~ ~ ! [Xx_0: nat] :
            ( y
           != ( pl @ ( pl @ x @ n_1 ) @ Xx_0 ) )
   => ( y
      = ( pl @ x @ n_1 ) ) ) ).

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