TPTP Problem File: SEV295^5.p

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% File     : SEV295^5 : TPTP v7.2.0. Bugfixed v6.2.0.
% Domain   : Set Theory
% Problem  : TPS problem THM130-NAT
% Version  : Especial.
% English  :

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0548 [Bro09]
%          : THM130-NAT [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
% Syntax   : Number of formulae    :   10 (   0 unit;   5 type;   4 defn)
%            Number of atoms       :   55 (   5 equality;  31 variable)
%            Maximal formula depth :   11 (   7 average)
%            Number of connectives :   42 (   2   ~;   0   |;   6   &;  26   @)
%                                         (   0 <=>;   8  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   42 (  42   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5   :;   0   =)
%            Number of variables   :   16 (   0 sgn;   8   !;   3   ?;   5   ^)
%                                         (  16   :;   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
%            Mellon University. Distributed under the Creative Commons copyleft
%            license: http://creativecommons.org/licenses/by-sa/3.0/
%          : 
% Bugfixes : v5.2.0 - Added missing type declarations.
%          : v6.2.0 - Reordered definitions.
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thf(r_type,type,(
    r: ( ( $i > $o ) > $o ) > ( ( $i > $o ) > $o ) > $o )).

thf(cINDUCTION_type,type,(
    cINDUCTION: $o )).

thf(cNAT_type,type,(
    cNAT: ( ( $i > $o ) > $o ) > $o )).

thf(cSUCC_type,type,(
    cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o )).

thf(cZERO_type,type,(
    cZERO: ( $i > $o ) > $o )).

thf(cZERO_def,definition,
    ( cZERO
    = ( ^ [Xp: $i > $o] :
          ~ ( ? [Xx: $i] :
                ( Xp @ Xx ) ) ) )).

thf(cSUCC_def,definition,
    ( cSUCC
    = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
        ? [Xx: $i] :
          ( ( Xp @ Xx )
          & ( Xn
            @ ^ [Xt: $i] :
                ( ( Xt != Xx )
                & ( Xp @ Xt ) ) ) ) ) )).

thf(cNAT_def,definition,
    ( cNAT
    = ( ^ [Xn: ( $i > $o ) > $o] :
        ! [Xp: ( ( $i > $o ) > $o ) > $o] :
          ( ( ( Xp @ cZERO )
            & ! [Xx: ( $i > $o ) > $o] :
                ( ( Xp @ Xx )
               => ( Xp @ ( cSUCC @ Xx ) ) ) )
         => ( Xp @ Xn ) ) ) )).

thf(cINDUCTION_def,definition,
    ( cINDUCTION
    = ( ! [P: ( ( $i > $o ) > $o ) > $o] :
          ( ( ( P @ cZERO )
            & ! [X: ( $i > $o ) > $o] :
                ( ( P @ X )
               => ( P @ ( cSUCC @ X ) ) ) )
         => ! [M: ( $i > $o ) > $o] :
              ( ( cNAT @ M )
             => ( P @ M ) ) ) ) )).

thf(cTHM130_NAT,conjecture,
    ( ( cINDUCTION
      & ( r @ cZERO @ cZERO )
      & ! [Xx: ( $i > $o ) > $o,Xy: ( $i > $o ) > $o] :
          ( ( r @ Xx @ Xy )
         => ( r @ ( cSUCC @ Xx ) @ ( cSUCC @ Xy ) ) ) )
   => ! [Xx: ( $i > $o ) > $o] :
        ( ( cNAT @ Xx )
       => ? [Xy: ( $i > $o ) > $o] :
            ( r @ Xx @ Xy ) ) )).

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