Article WAYBEL35, MML version 4.99.1005
:: WAYBEL35:exreg 1
registration
let a1 be set;
cluster trivial Element of bool a1;
end;
:: WAYBEL35:condreg 1
registration
let a1 be trivial set;
cluster -> trivial (Element of bool a1);
end;
:: WAYBEL35:exreg 2
registration
let a1 be 1-sorted;
cluster trivial Element of bool the carrier of a1;
end;
:: WAYBEL35:exreg 3
registration
let a1 be RelStr;
cluster trivial Element of bool the carrier of a1;
end;
:: WAYBEL35:exreg 4
registration
let a1 be non empty 1-sorted;
cluster non empty trivial Element of bool the carrier of a1;
end;
:: WAYBEL35:exreg 5
registration
let a1 be non empty RelStr;
cluster non empty trivial Element of bool the carrier of a1;
end;
:: WAYBEL35:exreg 6
registration
let a1 be RelStr;
cluster Relation-like auxiliary(i) Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL35:exreg 7
registration
let a1 be transitive RelStr;
cluster Relation-like auxiliary(i) auxiliary(ii) Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL35:exreg 8
registration
let a1 be antisymmetric with_suprema RelStr;
cluster Relation-like auxiliary(iii) Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL35:exreg 9
registration
let a1 be non empty antisymmetric lower-bounded RelStr;
cluster Relation-like auxiliary(iv) Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL35:attrnot 1 => WAYBEL35:attr 1
definition
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
attr a2 is extra-order means
a2 is auxiliary(i)(a1) & a2 is auxiliary(ii)(a1) & a2 is auxiliary(iv)(a1);
end;
:: WAYBEL35:dfs 1
definiens
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
a2 is extra-order
it is sufficient to prove
thus a2 is auxiliary(i)(a1) & a2 is auxiliary(ii)(a1) & a2 is auxiliary(iv)(a1);
:: WAYBEL35:def 1
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
b2 is extra-order(b1)
iff
b2 is auxiliary(i)(b1) & b2 is auxiliary(ii)(b1) & b2 is auxiliary(iv)(b1);
:: WAYBEL35:condreg 2
registration
let a1 be non empty RelStr;
cluster extra-order -> auxiliary(i) auxiliary(ii) auxiliary(iv) (Relation of the carrier of a1,the carrier of a1);
end;
:: WAYBEL35:condreg 3
registration
let a1 be non empty RelStr;
cluster auxiliary(i) auxiliary(ii) auxiliary(iv) -> extra-order (Relation of the carrier of a1,the carrier of a1);
end;
:: WAYBEL35:condreg 4
registration
let a1 be non empty RelStr;
cluster auxiliary(iii) extra-order -> auxiliary (Relation of the carrier of a1,the carrier of a1);
end;
:: WAYBEL35:condreg 5
registration
let a1 be non empty RelStr;
cluster auxiliary -> extra-order (Relation of the carrier of a1,the carrier of a1);
end;
:: WAYBEL35:exreg 10
registration
let a1 be non empty transitive antisymmetric lower-bounded RelStr;
cluster Relation-like extra-order Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL35:funcnot 1 => WAYBEL35:func 1
definition
let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
let a2 be auxiliary(ii) Relation of the carrier of a1,the carrier of a1;
func A2 -LowerMap -> Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset LOWER a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = a2 -below b1;
end;
:: WAYBEL35:def 2
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset LOWER b1 holds
b3 = b2 -LowerMap
iff
for b4 being Element of the carrier of b1 holds
b3 . b4 = b2 -below b4;
:: WAYBEL35:funcreg 1
registration
let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
let a2 be auxiliary(ii) Relation of the carrier of a1,the carrier of a1;
cluster a2 -LowerMap -> Function-like quasi_total monotone;
end;
:: WAYBEL35:modenot 1 => WAYBEL35:mode 1
definition
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
mode strict_chain of A2 -> Element of bool the carrier of a1 means
for b1, b2 being set
st b1 in it & b2 in it & not [b1,b2] in a2 & b1 <> b2
holds [b2,b1] in a2;
end;
:: WAYBEL35:dfs 3
definiens
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is strict_chain of a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a3 & b2 in a3 & not [b1,b2] in a2 & b1 <> b2
holds [b2,b1] in a2;
:: WAYBEL35:def 3
theorem
for b1 being 1-sorted
for b2 being Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 is strict_chain of b2
iff
for b4, b5 being set
st b4 in b3 & b5 in b3 & not [b4,b5] in b2 & b4 <> b5
holds [b5,b4] in b2;
:: WAYBEL35:th 4
theorem
for b1 being 1-sorted
for b2 being trivial Element of bool the carrier of b1
for b3 being Relation of the carrier of b1,the carrier of b1 holds
b2 is strict_chain of b3;
:: WAYBEL35:exreg 11
registration
let a1 be non empty 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
cluster non empty trivial strict_chain of a2;
end;
:: WAYBEL35:th 5
theorem
for b1 being antisymmetric RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
for b4, b5 being Element of the carrier of b1
st b4 in b3 & b5 in b3 & b4 < b5
holds [b4,b5] in b2;
:: WAYBEL35:th 6
theorem
for b1 being antisymmetric RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
st [b3,b4] in b2 & [b4,b3] in b2
holds b3 = b4;
:: WAYBEL35:th 7
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being Element of the carrier of b1
for b3 being auxiliary(iv) Relation of the carrier of b1,the carrier of b1 holds
{Bottom b1,b2} is strict_chain of b3;
:: WAYBEL35:th 8
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being auxiliary(iv) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2 holds
b3 \/ {Bottom b1} is strict_chain of b2;
:: WAYBEL35:attrnot 2 => WAYBEL35:attr 2
definition
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
attr a3 is maximal means
for b1 being strict_chain of a2
st a3 c= b1
holds a3 = b1;
end;
:: WAYBEL35:dfs 4
definiens
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
To prove
a3 is maximal
it is sufficient to prove
thus for b1 being strict_chain of a2
st a3 c= b1
holds a3 = b1;
:: WAYBEL35:def 4
theorem
for b1 being 1-sorted
for b2 being Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2 holds
b3 is maximal(b1, b2)
iff
for b4 being strict_chain of b2
st b3 c= b4
holds b3 = b4;
:: WAYBEL35:funcnot 2 => WAYBEL35:func 2
definition
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be set;
func Strict_Chains(A2,A3) -> set means
for b1 being set holds
b1 in it
iff
b1 is strict_chain of a2 & a3 c= b1;
end;
:: WAYBEL35:def 5
theorem
for b1 being 1-sorted
for b2 being Relation of the carrier of b1,the carrier of b1
for b3, b4 being set holds
b4 = Strict_Chains(b2,b3)
iff
for b5 being set holds
b5 in b4
iff
b5 is strict_chain of b2 & b3 c= b5;
:: WAYBEL35:funcreg 2
registration
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
cluster Strict_Chains(a2,a3) -> non empty;
end;
:: WAYBEL35:prednot 1 => ORDERS_1:pred 4
notation
let a1 be Relation-like set;
let a2 be set;
synonym a2 is_inductive_wrt a1 for a2 has_upper_Zorn_property_wrt a1;
end;
:: WAYBEL35:th 9
theorem
for b1 being 1-sorted
for b2 being Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2 holds
Strict_Chains(b2,b3) has_upper_Zorn_property_wrt RelIncl Strict_Chains(b2,b3) &
(ex b4 being set st
b4 is_maximal_in RelIncl Strict_Chains(b2,b3) & b3 c= b4);
:: WAYBEL35:th 10
theorem
for b1 being non empty transitive RelStr
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of bool b2
st ex_sup_of b3,b1 & "\/"(b3,b1) in b2
holds ex_sup_of b3,subrelstr b2 & "\/"(b3,b1) = "\/"(b3,subrelstr b2);
:: WAYBEL35:th 11
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being non empty strict_chain of b2
for b4 being Element of bool b3
st ex_sup_of b4,b1 & b3 is maximal(b1, b2)
holds ex_sup_of b4,subrelstr b3;
:: WAYBEL35:th 12
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being non empty strict_chain of b2
for b4 being Element of bool b3
st ex_inf_of (uparrow "\/"(b4,b1)) /\ b3,b1 & ex_sup_of b4,b1 & b3 is maximal(b1, b2)
holds "\/"(b4,subrelstr b3) = "/\"((uparrow "\/"(b4,b1)) /\ b3,b1) &
("\/"(b4,b1) in b3 or "\/"(b4,b1) < "/\"((uparrow "\/"(b4,b1)) /\ b3,b1));
:: WAYBEL35:th 13
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being non empty strict_chain of b2
st b3 is maximal(b1, b2)
holds subrelstr b3 is complete;
:: WAYBEL35:th 14
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being auxiliary(iv) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
st b3 is maximal(b1, b2)
holds Bottom b1 in b3;
:: WAYBEL35:th 15
theorem
for b1 being non empty reflexive transitive antisymmetric upper-bounded RelStr
for b2 being auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
for b4 being Element of the carrier of b1
st b3 is maximal(b1, b2) & b4 is_maximum_of b3 & [b4,Top b1] in b2
holds [Top b1,Top b1] in b2 & b4 = Top b1;
:: WAYBEL35:prednot 2 => WAYBEL35:pred 1
definition
let a1 be RelStr;
let a2 be set;
let a3 be Relation of the carrier of a1,the carrier of a1;
pred A3 satisfies_SIC_on A2 means
for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2 & [b1,b2] in a3 & b1 <> b2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 & [b1,b3] in a3 & [b3,b2] in a3 & b1 <> b3;
end;
:: WAYBEL35:dfs 6
definiens
let a1 be RelStr;
let a2 be set;
let a3 be Relation of the carrier of a1,the carrier of a1;
To prove
a3 satisfies_SIC_on a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2 & [b1,b2] in a3 & b1 <> b2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 & [b1,b3] in a3 & [b3,b2] in a3 & b1 <> b3;
:: WAYBEL35:def 6
theorem
for b1 being RelStr
for b2 being set
for b3 being Relation of the carrier of b1,the carrier of b1 holds
b3 satisfies_SIC_on b2
iff
for b4, b5 being Element of the carrier of b1
st b4 in b2 & b5 in b2 & [b4,b5] in b3 & b4 <> b5
holds ex b6 being Element of the carrier of b1 st
b6 in b2 & [b4,b6] in b3 & [b6,b5] in b3 & b4 <> b6;
:: WAYBEL35:attrnot 3 => WAYBEL35:attr 3
definition
let a1 be RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
attr a3 is satisfying_SIC means
a2 satisfies_SIC_on a3;
end;
:: WAYBEL35:dfs 7
definiens
let a1 be RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
To prove
a3 is satisfying_SIC
it is sufficient to prove
thus a2 satisfies_SIC_on a3;
:: WAYBEL35:def 7
theorem
for b1 being RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2 holds
b3 is satisfying_SIC(b1, b2)
iff
b2 satisfies_SIC_on b3;
:: WAYBEL35:attrnot 4 => WAYBEL35:attr 3
notation
let a1 be RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
synonym satisfying_the_interpolation_property for satisfying_SIC;
end;
:: WAYBEL35:prednot 3 => WAYBEL35:attr 3
notation
let a1 be RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be strict_chain of a2;
synonym a3 satisfies_the_interpolation_property for satisfying_SIC;
end;
:: WAYBEL35:th 16
theorem
for b1 being RelStr
for b2 being set
for b3 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
st b3 satisfies_SIC_on b2
for b4, b5 being Element of the carrier of b1
st b4 in b2 & b5 in b2 & [b4,b5] in b3 & b4 <> b5
holds ex b6 being Element of the carrier of b1 st
b6 in b2 & [b4,b6] in b3 & [b6,b5] in b3 & b4 < b6;
:: WAYBEL35:condreg 6
registration
let a1 be RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
cluster trivial -> satisfying_SIC (strict_chain of a2);
end;
:: WAYBEL35:exreg 12
registration
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
cluster non empty trivial strict_chain of a2;
end;
:: WAYBEL35:th 17
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
st b3 is maximal(b1, b2) & b2 is satisfying_SI(b1)
holds b2 satisfies_SIC_on b3;
:: WAYBEL35:funcnot 3 => WAYBEL35:func 3
definition
let a1 be Relation-like set;
let a2, a3 be set;
func SetBelow(A1,A2,A3) -> set equals
(a1 " {a3}) /\ a2;
end;
:: WAYBEL35:def 8
theorem
for b1 being Relation-like set
for b2, b3 being set holds
SetBelow(b1,b2,b3) = (b1 " {b3}) /\ b2;
:: WAYBEL35:th 18
theorem
for b1 being Relation-like set
for b2, b3, b4 being set holds
b3 in SetBelow(b1,b2,b4)
iff
[b3,b4] in b1 & b3 in b2;
:: WAYBEL35:funcnot 4 => WAYBEL35:func 4
definition
let a1 be 1-sorted;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3, a4 be set;
redefine func SetBelow(a2,a3,a4) -> Element of bool the carrier of a1;
end;
:: WAYBEL35:th 19
theorem
for b1 being RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3 being set
for b4 being Element of the carrier of b1 holds
SetBelow(b2,b3,b4) is_<=_than b4;
:: WAYBEL35:th 20
theorem
for b1 being reflexive transitive RelStr
for b2 being auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
st b4 <= b5
holds SetBelow(b2,b3,b4) c= SetBelow(b2,b3,b5);
:: WAYBEL35:th 21
theorem
for b1 being RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3 being set
for b4 being Element of the carrier of b1
st b4 in b3 & [b4,b4] in b2 & ex_sup_of SetBelow(b2,b3,b4),b1
holds b4 = "\/"(SetBelow(b2,b3,b4),b1);
:: WAYBEL35:attrnot 5 => WAYBEL35:attr 4
definition
let a1 be RelStr;
let a2 be Element of bool the carrier of a1;
attr a2 is sup-closed means
for b1 being Element of bool a2
st ex_sup_of b1,a1
holds "\/"(b1,a1) = "\/"(b1,subrelstr a2);
end;
:: WAYBEL35:dfs 9
definiens
let a1 be RelStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is sup-closed
it is sufficient to prove
thus for b1 being Element of bool a2
st ex_sup_of b1,a1
holds "\/"(b1,a1) = "\/"(b1,subrelstr a2);
:: WAYBEL35:def 9
theorem
for b1 being RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is sup-closed(b1)
iff
for b3 being Element of bool b2
st ex_sup_of b3,b1
holds "\/"(b3,b1) = "\/"(b3,subrelstr b2);
:: WAYBEL35:th 22
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being extra-order Relation of the carrier of b1,the carrier of b1
for b3 being satisfying_SIC strict_chain of b2
for b4, b5 being Element of the carrier of b1
st b4 in b3 & b5 in b3 & b4 < b5
holds ex b6 being Element of the carrier of b1 st
b4 < b6 & [b6,b5] in b2 & b6 = "\/"(SetBelow(b2,b3,b6),b1);
:: WAYBEL35:th 23
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being extra-order Relation of the carrier of b1,the carrier of b1
for b3 being non empty strict_chain of b2
st b3 is sup-closed(b1) &
(for b4 being Element of the carrier of b1
st b4 in b3
holds ex_sup_of SetBelow(b2,b3,b4),b1) &
b2 satisfies_SIC_on b3
for b4 being Element of the carrier of b1
st b4 in b3
holds b4 = "\/"(SetBelow(b2,b3,b4),b1);
:: WAYBEL35:th 24
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being auxiliary(i) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
st for b4 being Element of the carrier of b1
st b4 in b3
holds ex_sup_of SetBelow(b2,b3,b4),b1 & b4 = "\/"(SetBelow(b2,b3,b4),b1)
holds b2 satisfies_SIC_on b3;
:: WAYBEL35:funcnot 5 => WAYBEL35:func 5
definition
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be set;
func SupBelow(A2,A3) -> set means
for b1 being set holds
b1 in it
iff
b1 = "\/"(SetBelow(a2,a3,b1),a1);
end;
:: WAYBEL35:def 10
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1
for b3, b4 being set holds
b4 = SupBelow(b2,b3)
iff
for b5 being set holds
b5 in b4
iff
b5 = "\/"(SetBelow(b2,b3,b5),b1);
:: WAYBEL35:funcnot 6 => WAYBEL35:func 6
definition
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
let a3 be set;
redefine func SupBelow(a2,a3) -> Element of bool the carrier of a1;
end;
:: WAYBEL35:th 25
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being strict_chain of b2
st for b4 being Element of the carrier of b1 holds
ex_sup_of SetBelow(b2,b3,b4),b1
holds SupBelow(b2,b3) is strict_chain of b2;
:: WAYBEL35:th 26
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being auxiliary(i) auxiliary(ii) Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st for b4 being Element of the carrier of b1 holds
ex_sup_of SetBelow(b2,b3,b4),b1
holds SupBelow(b2,b3) is sup-closed(b1);
:: WAYBEL35:th 27
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being extra-order Relation of the carrier of b1,the carrier of b1
for b3 being satisfying_SIC strict_chain of b2
for b4 being Element of the carrier of b1
st b4 in SupBelow(b2,b3)
holds b4 = "\/"({b5 where b5 is Element of the carrier of b1: b5 in SupBelow(b2,b3) & [b5,b4] in b2},b1);
:: WAYBEL35:th 28
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being extra-order Relation of the carrier of b1,the carrier of b1
for b3 being satisfying_SIC strict_chain of b2 holds
b2 satisfies_SIC_on SupBelow(b2,b3);
:: WAYBEL35:th 29
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being extra-order Relation of the carrier of b1,the carrier of b1
for b3 being satisfying_SIC strict_chain of b2
for b4, b5 being Element of the carrier of b1
st b4 in b3 & b5 in b3 & b4 < b5
holds ex b6 being Element of the carrier of b1 st
b6 in SupBelow(b2,b3) & b4 < b6 & [b6,b5] in b2;