## TPTP Problem File: NUM810^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : NUM810^5 : TPTP v7.2.0. Bugfixed v5.2.0.
% Domain   : Number Theory (Induction on naturals)
% Problem  : TPS problem THM140
% Version  : Especial.
% English  : Existence of doubles of naturals.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0347 [Bro09]
%          : THM140 [TPS]
%          : THM579 [TPS]

% Status   : Theorem
% Rating   : 0.67 v7.2.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.60 v5.3.0, 0.80 v5.2.0
% Syntax   : Number of formulae    :    6 (   0 unit;   4 type;   1 defn)
%            Number of atoms       :   27 (   1 equality;  13 variable)
%            Maximal formula depth :   10 (   5 average)
%            Number of connectives :   23 (   0   ~;   0   |;   3   &;  16   @)
%                                         (   0 <=>;   4  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   4   :;   0   =)
%            Number of variables   :    7 (   0 sgn;   6   !;   1   ?;   0   ^)
%                                         (   7   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%          :
% Bugfixes : v5.2.0 - Added missing type declarations.
%------------------------------------------------------------------------------
thf(c0_type,type,(
c0: \$i )).

thf(cDOUBLE_type,type,(
cDOUBLE: \$i > \$i > \$o )).

thf(cS_type,type,(
cS: \$i > \$i )).

thf(cIND_type,type,(
cIND: \$o )).

thf(cIND_def,definition,
( cIND
= ( ! [Xp: \$i > \$o] :
( ( ( Xp @ c0 )
& ! [Xx: \$i] :
( ( Xp @ Xx )
=> ( Xp @ ( cS @ Xx ) ) ) )
=> ! [Xx: \$i] :
( Xp @ Xx ) ) ) )).

thf(cTHM140,conjecture,
( ( cIND
& ( cDOUBLE @ c0 @ c0 )
& ! [Xx: \$i,Xy: \$i] :
( ( cDOUBLE @ Xx @ Xy )
=> ( cDOUBLE @ ( cS @ Xx ) @ ( cS @ ( cS @ Xy ) ) ) ) )
=> ! [Xx: \$i] :
? [Xy: \$i] :
( cDOUBLE @ Xx @ Xy ) )).

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