%--------------------------------------------------------------%
%   Author: Dragan Stosic                                      %
%--------------------------------------------------------------%

relation_extension_props[T, U: TYPE+]: THEORY
 BEGIN
   IMPORTING relation_extension

 le_T: VAR equivalence[T]

 le_U: VAR equivalence[U]

 m, n: VAR [T, U] 

% NON-EMPTY LEMMAS

 extended_equiv_class_non_empty: LEMMA
   nonempty?[[T,U]](extended_equiv_class(rel_extension(le_T, le_U))(m))

 nonempty_equivClass_impl_extended_equiv_class_non_empty: LEMMA
   nonempty?[T](EquivClass(le_T)(m`1))  IFF 
     nonempty?[[T,U]](extended_equiv_class(rel_extension(le_T, le_U))(m))

 nonempty_equivClass_iff_extended_equiv_classT_non_empty: LEMMA
   nonempty?[T](EquivClass(le_T)(m`1))  IFF 
     nonempty?[T](extended_equiv_classT(
      rel_extension(le_T, le_U),proj_tuple_1, m, m))

 nonempty_equivClass_iff_extended_equiv_classU_non_empty: LEMMA
   nonempty?[T](EquivClass(le_T)(m`1))  IFF 
     nonempty?[U](extended_equiv_classU(
      rel_extension(le_T, le_U),proj_tuple_2, m, m))

% CLASS REPRESENTATIVE LEMMAS
  
 proj_tuple_1_equiv_class_representative: LEMMA
   m`1 = repEC_ext(le_T, le_U, m)`1 IMPLIES 
   le_T( proj_tuple_1(repEC_ext(le_T, le_U, m)), repEC(le_T)(m`1))
 
 proj_tuple_2_equiv_class_representative: LEMMA
   m`2 = repEC_ext(le_T, le_U, m)`2 IMPLIES 
   le_U( proj_tuple_2(repEC_ext(le_T, le_U, m)), repEC(le_U)(m`2));

 END relation_extension_props