Article ORDERS_4, MML version 4.99.1005

:: ORDERS_4:modenot 1 => ORDERS_4:mode 1
definition
  mode Chain -> RelStr means
    (it is not non empty reflexive transitive antisymmetric connected RelStr) implies it is empty;
end;

:: ORDERS_4:dfs 1
definiens
  let a1 be RelStr;
To prove
     a1 is Chain
it is sufficient to prove
  thus (a1 is not non empty reflexive transitive antisymmetric connected RelStr) implies a1 is empty;

:: ORDERS_4:def 1
theorem
for b1 being RelStr holds
      b1 is Chain
   iff
      (b1 is non empty reflexive transitive antisymmetric connected RelStr or b1 is empty);

:: ORDERS_4:condreg 1
registration
  cluster empty -> reflexive transitive antisymmetric (RelStr);
end;

:: ORDERS_4:condreg 2
registration
  cluster -> reflexive transitive antisymmetric (Chain);
end;

:: ORDERS_4:exreg 1
registration
  cluster non empty reflexive transitive antisymmetric Chain;
end;

:: ORDERS_4:condreg 3
registration
  cluster non empty -> connected (Chain);
end;

:: ORDERS_4:attrnot 1 => ORDERS_4:attr 1
definition
  let a1 be 1-sorted;
  attr a1 is countable means
    the carrier of a1 is countable;
end;

:: ORDERS_4:dfs 2
definiens
  let a1 be 1-sorted;
To prove
     a1 is countable
it is sufficient to prove
  thus the carrier of a1 is countable;

:: ORDERS_4:def 2
theorem
for b1 being 1-sorted holds
      b1 is countable
   iff
      the carrier of b1 is countable;

:: ORDERS_4:exreg 2
registration
  cluster non empty finite reflexive transitive antisymmetric Chain;
end;

:: ORDERS_4:exreg 3
registration
  cluster reflexive transitive antisymmetric countable Chain;
end;

:: ORDERS_4:condreg 4
registration
  let a1 be non empty connected RelStr;
  cluster non empty full -> connected (SubRelStr of a1);
end;

:: ORDERS_4:condreg 5
registration
  let a1 be finite RelStr;
  cluster -> finite (SubRelStr of a1);
end;

:: ORDERS_4:th 1
theorem
for b1, b2 being natural set
      st b1 <= b2
   holds InclPoset b1 is full SubRelStr of InclPoset b2;

:: ORDERS_4:prednot 1 => ORDERS_4:pred 1
definition
  let a1 be RelStr;
  let a2, a3 be set;
  pred A2,A3 form_upper_lower_partition_of A1 means
    a2 \/ a3 = the carrier of a1 &
     (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a3
        holds b1 < b2);
end;

:: ORDERS_4:dfs 3
definiens
  let a1 be RelStr;
  let a2, a3 be set;
To prove
     a2,a3 form_upper_lower_partition_of a1
it is sufficient to prove
  thus a2 \/ a3 = the carrier of a1 &
     (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a3
        holds b1 < b2);

:: ORDERS_4:def 3
theorem
for b1 being RelStr
for b2, b3 being set holds
   b2,b3 form_upper_lower_partition_of b1
iff
   b2 \/ b3 = the carrier of b1 &
    (for b4, b5 being Element of the carrier of b1
          st b4 in b2 & b5 in b3
       holds b4 < b5);

:: ORDERS_4:th 2
theorem
for b1 being RelStr
for b2, b3 being set
      st b2,b3 form_upper_lower_partition_of b1
   holds b2 misses b3;

:: ORDERS_4:th 3
theorem
for b1 being non empty antisymmetric upper-bounded RelStr holds
   (the carrier of b1) \ {Top b1},{Top b1} form_upper_lower_partition_of b1;

:: ORDERS_4:th 4
theorem
for b1, b2 being RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is isomorphic(b1, b2)
   holds (the carrier of b1 <> {} implies the carrier of b2 <> {}) &
    (the carrier of b2 <> {} implies the carrier of b1 <> {}) &
    (the carrier of b2 = {} implies the carrier of b1 = {}) &
    (the carrier of b1 = {} implies the carrier of b2 = {}) &
    (the carrier of b2 = {} implies the carrier of b1 = {});

:: ORDERS_4:th 5
theorem
for b1, b2 being antisymmetric RelStr
for b3, b4 being Element of bool the carrier of b1
   st b3,b4 form_upper_lower_partition_of b1
for b5, b6 being Element of bool the carrier of b2
   st b5,b6 form_upper_lower_partition_of b2
for b7 being Function-like quasi_total Relation of the carrier of subrelstr b3,the carrier of subrelstr b5
   st b7 is isomorphic(subrelstr b3, subrelstr b5)
for b8 being Function-like quasi_total Relation of the carrier of subrelstr b4,the carrier of subrelstr b6
      st b8 is isomorphic(subrelstr b4, subrelstr b6)
   holds ex b9 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
      b9 = b7 +* b8 & b9 is isomorphic(b1, b2);

:: ORDERS_4:th 6
theorem
for b1 being finite Chain
for b2 being natural set
      st Card the carrier of b1 = b2
   holds b1,InclPoset b2 are_isomorphic;