%------------------------------------------------------------------------------
% File     : NUM812^5 : TPTP v9.0.0. Bugfixed v5.2.0.
% Domain   : Number Theory (Induction on naturals)
% Problem  : TPS problem THM578
% Version  : Especial.
% English  : Variant of THM6104, including induction in the hypothesis.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0443 [Bro09]
%          : THM578 [TPS]

% Status   : Theorem
% Rating   : 0.25 v9.0.0, 0.30 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.29 v7.4.0, 0.11 v7.2.0, 0.12 v7.0.0, 0.00 v6.2.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0
% Syntax   : Number of formulae    :    5 (   1 unt;   3 typ;   1 def)
%            Number of atoms       :    5 (   3 equ;   0 cnn)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   11 (   0   ~;   1   |;   1   &;   6   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    2 (   2   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    4 (   3 usr;   2 con; 0-2 aty)
%            Number of variables   :    5 (   0   ^;   4   !;   1   ?;   5   :)
% 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.
%------------------------------------------------------------------------------
thf(c0_type,type,
    c0: $i ).

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(cTHM578,conjecture,
    ( cIND
   => ! [Xn: $i] :
        ( ( Xn = c0 )
        | ? [Xm: $i] :
            ( Xn
            = ( cS @ Xm ) ) ) ) ).

%------------------------------------------------------------------------------