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;