%------------------------------------------------------------------------------
% File     : SEV206^5 : TPTP v9.0.0. Released v4.0.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem from S-THMS
% Version  : Especial.
% English  :

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_1203 [Bro09]

% Status   : Theorem
% Rating   : 0.88 v9.0.0, 0.90 v8.2.0, 1.00 v8.1.0, 0.91 v7.5.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :    7 (   0 unt;   6 typ;   0 def)
%            Number of atoms       :   20 (  18 equ;   0 cnn)
%            Maximal formula atoms :   20 (  20 avg)
%            Number of connectives :   88 (   1   ~;   4   |;  19   &;  55   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (  25 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5 usr;   5 con; 0-2 aty)
%            Number of variables   :   30 (   0   ^;  18   !;  12   ?;  30   :)
% 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/
%------------------------------------------------------------------------------
thf(iS_type,type,
    iS: $tType ).

thf(y,type,
    y: iS ).

thf(z,type,
    z: iS ).

thf(cP,type,
    cP: iS > iS > iS ).

thf(x,type,
    x: iS ).

thf(c0,type,
    c0: iS ).

thf(cS_INCL_LEM8_pme,conjecture,
    ( ( ! [Xx0: iS,Xy0: iS] :
          ( ( cP @ Xx0 @ Xy0 )
         != c0 )
      & ! [Xx0: iS,Xy0: iS,Xu: iS,Xv: iS] :
          ( ( ( cP @ Xx0 @ Xu )
            = ( cP @ Xy0 @ Xv ) )
         => ( ( Xx0 = Xy0 )
            & ( Xu = Xv ) ) )
      & ! [X: iS > $o] :
          ( ( ( X @ c0 )
            & ! [Xx0: iS,Xy0: iS] :
                ( ( ( X @ Xx0 )
                  & ( X @ Xy0 ) )
               => ( X @ ( cP @ Xx0 @ Xy0 ) ) ) )
         => ! [Xx0: iS] : ( X @ Xx0 ) ) )
   => ( ! [R: iS > iS > iS > $o] :
          ( ( $true
            & ! [Xa: iS,Xb: iS,Xc: iS] :
                ( ( ( ( Xa = c0 )
                    & ( Xb = Xc ) )
                  | ( ( Xb = c0 )
                    & ( Xa = Xc ) )
                  | ? [Xx1: iS,Xx2: iS,Xy1: iS,Xy2: iS,Xz1: iS,Xz2: iS] :
                      ( ( Xa
                        = ( cP @ Xx1 @ Xx2 ) )
                      & ( Xb
                        = ( cP @ Xy1 @ Xy2 ) )
                      & ( Xc
                        = ( cP @ Xz1 @ Xz2 ) )
                      & ( R @ Xx1 @ Xy1 @ Xz1 )
                      & ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
               => ( R @ Xa @ Xb @ Xc ) ) )
         => ( R @ x @ y @ y ) )
     => ! [R: iS > iS > iS > $o] :
          ( ( $true
            & ! [Xa: iS,Xb: iS,Xc: iS] :
                ( ( ( ( Xa = c0 )
                    & ( Xb = Xc ) )
                  | ( ( Xb = c0 )
                    & ( Xa = Xc ) )
                  | ? [Xx1: iS,Xx2: iS,Xy1: iS,Xy2: iS,Xz1: iS,Xz2: iS] :
                      ( ( Xa
                        = ( cP @ Xx1 @ Xx2 ) )
                      & ( Xb
                        = ( cP @ Xy1 @ Xy2 ) )
                      & ( Xc
                        = ( cP @ Xz1 @ Xz2 ) )
                      & ( R @ Xx1 @ Xy1 @ Xz1 )
                      & ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
               => ( R @ Xa @ Xb @ Xc ) ) )
         => ( R @ ( cP @ z @ x ) @ ( cP @ z @ y ) @ ( cP @ z @ y ) ) ) ) ) ).

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