%--------------------------------------------------------------------------
% File     : SET776+4 : TPTP v8.2.0. Released v2.2.0.
% Domain   : Set Theory (Equivalence relations)
% Problem  : Property of pre-order
% Version  : [Pas99] axioms.
% English  : If P is a pre-order relation,and R defined by R(A,B) iff P(A,B)
%            and P(B,A), then R(X1,Y1) and R(X2,Y2) and P(X1,X2) implies
%            P(Y1,Y2).

% Refs     : [Pas99] Pastre (1999), Email to G. Sutcliffe
% Source   : [Pas99]
% Names    :

% Status   : Theorem
% Rating   : 0.22 v8.2.0, 0.17 v8.1.0, 0.11 v7.5.0, 0.16 v7.4.0, 0.13 v7.3.0, 0.17 v7.0.0, 0.20 v6.4.0, 0.19 v6.3.0, 0.25 v6.2.0, 0.24 v6.1.0, 0.27 v6.0.0, 0.22 v5.5.0, 0.26 v5.4.0, 0.32 v5.3.0, 0.37 v5.2.0, 0.20 v5.1.0, 0.19 v5.0.0, 0.25 v4.1.0, 0.26 v4.0.0, 0.29 v3.7.0, 0.35 v3.5.0, 0.37 v3.4.0, 0.32 v3.3.0, 0.14 v3.2.0, 0.09 v3.1.0, 0.22 v2.7.0, 0.17 v2.6.0, 0.43 v2.5.0, 0.62 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1
% Syntax   : Number of formulae    :   17 (   1 unt;   0 def)
%            Number of atoms       :   82 (   4 equ)
%            Maximal formula atoms :   14 (   4 avg)
%            Number of connectives :   68 (   3   ~;   2   |;  29   &)
%                                         (  16 <=>;  18  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 2-3 aty)
%            Number of functors    :   10 (  10 usr;   1 con; 0-3 aty)
%            Number of variables   :   66 (  62   !;   4   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Include set theory definitions
include('Axioms/SET006+0.ax').
%----Include equivalence relation axioms
include('Axioms/SET006+2.ax').
%--------------------------------------------------------------------------
fof(thIII12,conjecture,
    ! [E,P,R] :
      ( ( pre_order(P,E)
        & ! [A,B] :
            ( ( member(A,E)
              & member(B,E) )
           => ( apply(R,A,B)
            <=> ( apply(P,A,B)
                & apply(P,B,A) ) ) ) )
     => ! [X1,X2,Y1,Y2] :
          ( ( member(X1,E)
            & member(X2,E)
            & member(Y1,E)
            & member(Y2,E) )
         => ( ( apply(R,X1,Y1)
              & apply(R,X2,Y2)
              & apply(P,X1,X2) )
           => apply(P,Y1,Y2) ) ) ) ).

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