## TPTP Problem File: NUM727^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : NUM727^1 : TPTP v7.1.0. Released v3.7.0.
% Domain   : Number Theory
% Problem  : Landau theorem 39
% Version  : Especial.
% English  : ts (num x) (den z) = ts (num z) (den 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    : satz39 [Lan30]

% Status   : Theorem
%          : Without extensionality : Theorem
% Rating   : 0.50 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 v5.2.0, 1.00 v5.0.0, 0.80 v4.1.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   14 (   0 unit;   8 type;   0 defn)
%            Number of atoms       :   61 (   7 equality;  16 variable)
%            Maximal formula depth :    9 (   4 average)
%            Number of connectives :   41 (   0   ~;   0   |;   0   &;  40   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8   :;   0   =)
%            Number of variables   :    8 (   0 sgn;   8   !;   0   ?;   0   ^)
%                                         (   8   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(x,type,(
x: frac )).

thf(y,type,(
y: frac )).

thf(z,type,(
z: 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(e,axiom,
( ( ts @ ( num @ x ) @ ( den @ y ) )
= ( ts @ ( num @ y ) @ ( den @ x ) ) )).

thf(f,axiom,
( ( ts @ ( num @ y ) @ ( den @ z ) )
= ( ts @ ( num @ z ) @ ( den @ y ) ) )).

thf(satz33b,axiom,(
! [Xx: nat,Xy: nat,Xz: nat] :
( ( ( ts @ Xx @ Xz )
= ( ts @ Xy @ Xz ) )
=> ( Xx = Xy ) ) )).

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

thf(satz31,axiom,(
! [Xx: nat,Xy: nat,Xz: nat] :
( ( ts @ ( ts @ Xx @ Xy ) @ Xz )
= ( ts @ Xx @ ( ts @ Xy @ Xz ) ) ) )).

thf(satz39,conjecture,
( ( ts @ ( num @ x ) @ ( den @ z ) )
= ( ts @ ( num @ z ) @ ( den @ x ) ) )).

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