## TPTP Problem File: NUM814^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : NUM814^5 : TPTP v7.1.0. Bugfixed v5.2.0.
% Domain   : Number Theory (Induction on naturals)
% Problem  : TPS problem from IND-THMS
% Version  : Especial.
% English  :

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0606 [Bro09]

% Status   : Theorem
% Rating   : 0.62 v7.0.0, 0.71 v6.4.0, 0.67 v6.3.0, 0.60 v6.2.0, 0.71 v6.1.0, 0.57 v5.5.0, 1.00 v5.2.0
% Syntax   : Number of formulae    :   11 (   0 unit;   6 type;   4 defn)
%            Number of atoms       :   45 (   7 equality;  21 variable)
%            Maximal formula depth :   11 (   5 average)
%            Number of connectives :   28 (   2   ~;   0   |;   4   &;  16   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   6   :;   0   =)
%            Number of variables   :   11 (   0 sgn;   8   !;   1   ?;   2   ^)
%                                         (  11   :;   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(cS_type,type,(
cS: \$i > \$i )).

thf(cEVEN1_type,type,(
cEVEN1: \$i > \$o )).

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

thf(cODD1_type,type,(
cODD1: \$i > \$o )).

thf(cPEANO1_type,type,(
cPEANO1: \$o )).

thf(cEVEN1_def,definition,
( cEVEN1
= ( ^ [Xn: \$i] :
! [Xp: \$i > \$o] :
( ( ( Xp @ c0 )
& ! [Xx: \$i] :
( ( Xp @ Xx )
=> ( Xp @ ( cS @ ( cS @ Xx ) ) ) ) )
=> ( Xp @ Xn ) ) ) )).

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

thf(cODD1_def,definition,
( cODD1
= ( ^ [Xn: \$i] :
~ ( cEVEN1 @ Xn ) ) )).

thf(cPEANO1_def,definition,
( cPEANO1
= ( ! [Xu: \$i] :
( ( cS @ Xu )
!= c0 )
& ! [Xv: \$i,Xw: \$i] :
( ( ( cS @ Xv )
= ( cS @ Xw ) )
=> ( Xv = Xw ) )
& cIND ) )).

thf(cTHM404,conjecture,
( cPEANO1
=> ? [Xn: \$i] :
( cODD1 @ Xn ) )).

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