Article WAYBEL_6, MML version 4.99.1005
:: WAYBEL_6:sch 1
scheme WAYBEL_6:sch 1
{F1 -> non empty RelStr,
F2 -> Element of bool the carrier of F1(),
F3 -> non empty Element of bool the carrier of F1()}:
ex b1 being Function-like quasi_total Relation of F2(),F3() st
for b2 being Element of the carrier of F1()
st b2 in F2()
holds ex b3 being Element of the carrier of F1() st
b3 in F3() & b3 = b1 . b2 & P1[b2, b3]
provided
for b1 being Element of the carrier of F1()
st b1 in F2()
holds ex b2 being Element of the carrier of F1() st
b2 in F3() & P1[b1, b2];
:: WAYBEL_6:funcnot 1 => WAYBEL_6:func 1
definition
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
let a2 be non empty Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of a2,a2;
let a4 be Element of NAT;
redefine func iter(a3,a4) -> Function-like quasi_total Relation of a2,a2;
end;
:: WAYBEL_6:funcnot 2 => WAYBEL_6:func 2
definition
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
let a2, a3 be non empty Element of bool the carrier of a1;
let a4 be Function-like quasi_total Relation of a2,a3;
let a5 be Element of a2;
redefine func a4 . a5 -> Element of the carrier of a1;
end;
:: WAYBEL_6:condreg 1
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
cluster strongly_connected -> directed filtered (Element of bool the carrier of a1);
end;
:: WAYBEL_6:exreg 1
registration
cluster non empty strict total reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima continuous RelStr;
end;
:: WAYBEL_6:th 1
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is meet-preserving(b1, b2)
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 "/\" b5) = (b3 . b4) "/\" (b3 . b5);
:: WAYBEL_6:th 2
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is join-preserving(b1, b2)
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 "\/" b5) = (b3 . b4) "\/" (b3 . b5);
:: WAYBEL_6:th 3
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b2 is distributive & b3 is meet-preserving(b1, b2) & b3 is join-preserving(b1, b2) & b3 is one-to-one
holds b1 is distributive;
:: WAYBEL_6:exreg 2
registration
let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
cluster Relation-like Function-like quasi_total sups-preserving Relation of the carrier of a1,the carrier of a2;
end;
:: WAYBEL_6:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total sups-preserving Relation of the carrier of b1,the carrier of b2
st b2 is meet-continuous & b3 is meet-preserving(b1, b2) & b3 is one-to-one
holds b1 is meet-continuous;
:: WAYBEL_6:attrnot 1 => WAYBEL_6:attr 1
definition
let a1 be non empty reflexive RelStr;
let a2 be Element of bool the carrier of a1;
attr a2 is Open means
for b1 being Element of the carrier of a1
st b1 in a2
holds ex b2 being Element of the carrier of a1 st
b2 in a2 & b2 is_way_below b1;
end;
:: WAYBEL_6:dfs 1
definiens
let a1 be non empty reflexive RelStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is Open
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 in a2
holds ex b2 being Element of the carrier of a1 st
b2 in a2 & b2 is_way_below b1;
:: WAYBEL_6:def 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is Open(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being Element of the carrier of b1 st
b4 in b2 & b4 is_way_below b3;
:: WAYBEL_6:th 5
theorem
for b1 being reflexive transitive antisymmetric up-complete with_suprema with_infima RelStr
for b2 being upper Element of bool the carrier of b1 holds
b2 is Open(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds waybelow b3 meets b2;
:: WAYBEL_6:th 6
theorem
for b1 being reflexive transitive antisymmetric up-complete with_suprema with_infima RelStr
for b2 being upper Element of bool the carrier of b1 holds
b2 is Open(b1)
iff
b2 = union {wayabove b3 where b3 is Element of the carrier of b1: b3 in b2};
:: WAYBEL_6:exreg 3
registration
let a1 be reflexive transitive antisymmetric lower-bounded up-complete with_suprema with_infima RelStr;
cluster non empty filtered upper Open Element of bool the carrier of a1;
end;
:: WAYBEL_6:th 7
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1 holds
wayabove b2 is Open(b1);
:: WAYBEL_6:th 8
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2, b3 being Element of the carrier of b1
st b2 is_way_below b3
holds ex b4 being non empty filtered upper Open Element of bool the carrier of b1 st
b3 in b4 & b4 c= wayabove b2;
:: WAYBEL_6:th 9
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being upper Open Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2 `
holds ex b4 being Element of the carrier of b1 st
b3 <= b4 & b4 is_maximal_in b2 `;
:: WAYBEL_6:attrnot 2 => WAYBEL_6:attr 2
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
attr a2 is meet-irreducible means
for b1, b2 being Element of the carrier of a1
st a2 = b1 "/\" b2 & b1 <> a2
holds b2 = a2;
end;
:: WAYBEL_6:dfs 2
definiens
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is meet-irreducible
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st a2 = b1 "/\" b2 & b1 <> a2
holds b2 = a2;
:: WAYBEL_6:def 2
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
b2 is meet-irreducible(b1)
iff
for b3, b4 being Element of the carrier of b1
st b2 = b3 "/\" b4 & b3 <> b2
holds b4 = b2;
:: WAYBEL_6:attrnot 3 => WAYBEL_6:attr 2
notation
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
synonym irreducible for meet-irreducible;
end;
:: WAYBEL_6:attrnot 4 => WAYBEL_6:attr 3
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
attr a2 is join-irreducible means
for b1, b2 being Element of the carrier of a1
st a2 = b1 "\/" b2 & b1 <> a2
holds b2 = a2;
end;
:: WAYBEL_6:dfs 3
definiens
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is join-irreducible
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st a2 = b1 "\/" b2 & b1 <> a2
holds b2 = a2;
:: WAYBEL_6:def 3
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
b2 is join-irreducible(b1)
iff
for b3, b4 being Element of the carrier of b1
st b2 = b3 "\/" b4 & b3 <> b2
holds b4 = b2;
:: WAYBEL_6:funcnot 3 => WAYBEL_6:func 3
definition
let a1 be non empty RelStr;
func IRR A1 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
b1 is meet-irreducible(a1);
end;
:: WAYBEL_6:def 4
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
b2 = IRR b1
iff
for b3 being Element of the carrier of b1 holds
b3 in b2
iff
b3 is meet-irreducible(b1);
:: WAYBEL_6:th 10
theorem
for b1 being non empty antisymmetric upper-bounded with_infima RelStr holds
Top b1 is meet-irreducible(b1);
:: WAYBEL_6:exreg 4
registration
let a1 be non empty antisymmetric upper-bounded with_infima RelStr;
cluster meet-irreducible Element of the carrier of a1;
end;
:: WAYBEL_6:th 11
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 is meet-irreducible(b1)
iff
for b3 being non empty finite Element of bool the carrier of b1
st b2 = "/\"(b3,b1)
holds b2 in b3;
:: WAYBEL_6:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st (uparrow b2) \ {b2} is non empty filtered upper Element of bool the carrier of b1
holds b2 is meet-irreducible(b1);
:: WAYBEL_6:th 13
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
st b2 is_maximal_in b3 `
holds b2 is meet-irreducible(b1);
:: WAYBEL_6:th 14
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2, b3 being Element of the carrier of b1
st not b3 <= b2
holds ex b4 being Element of the carrier of b1 st
b4 is meet-irreducible(b1) & b2 <= b4 & not b3 <= b4;
:: WAYBEL_6:attrnot 5 => WAYBEL_6:attr 4
definition
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
attr a2 is order-generating means
for b1 being Element of the carrier of a1 holds
ex_inf_of (uparrow b1) /\ a2,a1 & b1 = "/\"((uparrow b1) /\ a2,a1);
end;
:: WAYBEL_6:dfs 5
definiens
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is order-generating
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex_inf_of (uparrow b1) /\ a2,a1 & b1 = "/\"((uparrow b1) /\ a2,a1);
:: WAYBEL_6:def 5
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is order-generating(b1)
iff
for b3 being Element of the carrier of b1 holds
ex_inf_of (uparrow b3) /\ b2,b1 & b3 = "/\"((uparrow b3) /\ b2,b1);
:: WAYBEL_6:th 15
theorem
for b1 being reflexive transitive antisymmetric lower-bounded up-complete with_suprema with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is order-generating(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being Element of bool b2 st
b3 = "/\"(b4,b1);
:: WAYBEL_6:th 16
theorem
for b1 being reflexive transitive antisymmetric lower-bounded up-complete with_suprema with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is order-generating(b1)
iff
for b3 being Element of bool the carrier of b1
st b2 c= b3 &
(for b4 being Element of bool b3 holds
"/\"(b4,b1) in b3)
holds the carrier of b1 = b3;
:: WAYBEL_6:th 17
theorem
for b1 being reflexive transitive antisymmetric lower-bounded up-complete with_suprema with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is order-generating(b1)
iff
for b3, b4 being Element of the carrier of b1
st not b4 <= b3
holds ex b5 being Element of the carrier of b1 st
b5 in b2 & b3 <= b5 & not b4 <= b5;
:: WAYBEL_6:th 18
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being Element of bool the carrier of b1
st b2 = (IRR b1) \ {Top b1}
holds b2 is order-generating(b1);
:: WAYBEL_6:th 19
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 is order-generating(b1) & b2 c= b3
holds b3 is order-generating(b1);
:: WAYBEL_6:attrnot 6 => WAYBEL_6:attr 5
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
attr a2 is prime means
for b1, b2 being Element of the carrier of a1
st b1 "/\" b2 <= a2 & not b1 <= a2
holds b2 <= a2;
end;
:: WAYBEL_6:dfs 6
definiens
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is prime
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 "/\" b2 <= a2 & not b1 <= a2
holds b2 <= a2;
:: WAYBEL_6:def 6
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
b2 is prime(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 "/\" b4 <= b2 & not b3 <= b2
holds b4 <= b2;
:: WAYBEL_6:funcnot 4 => WAYBEL_6:func 4
definition
let a1 be non empty RelStr;
func PRIME A1 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
b1 is prime(a1);
end;
:: WAYBEL_6:def 7
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
b2 = PRIME b1
iff
for b3 being Element of the carrier of b1 holds
b3 in b2
iff
b3 is prime(b1);
:: WAYBEL_6:attrnot 7 => WAYBEL_6:attr 6
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
attr a2 is co-prime means
a2 ~ is prime(a1 ~);
end;
:: WAYBEL_6:dfs 8
definiens
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is co-prime
it is sufficient to prove
thus a2 ~ is prime(a1 ~);
:: WAYBEL_6:def 8
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
b2 is co-prime(b1)
iff
b2 ~ is prime(b1 ~);
:: WAYBEL_6:th 20
theorem
for b1 being non empty antisymmetric upper-bounded RelStr holds
Top b1 is prime(b1);
:: WAYBEL_6:th 21
theorem
for b1 being non empty antisymmetric lower-bounded RelStr holds
Bottom b1 is co-prime(b1);
:: WAYBEL_6:exreg 5
registration
let a1 be non empty antisymmetric upper-bounded RelStr;
cluster prime Element of the carrier of a1;
end;
:: WAYBEL_6:th 22
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 is prime(b1)
iff
for b3 being non empty finite Element of bool the carrier of b1
st "/\"(b3,b1) <= b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 <= b2;
:: WAYBEL_6:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
b2 is co-prime(b1)
iff
for b3 being non empty finite Element of bool the carrier of b1
st b2 <= "\/"(b3,b1)
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b2 <= b4;
:: WAYBEL_6:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st b2 is prime(b1)
holds b2 is meet-irreducible(b1);
:: WAYBEL_6:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 is prime(b1)
iff
for b3 being set
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of BoolePoset {b3}
st for b5 being Element of the carrier of b1 holds
b4 . b5 = {}
iff
b5 <= b2
holds b4 is meet-preserving(b1, BoolePoset {b3}) & b4 is join-preserving(b1, BoolePoset {b3});
:: WAYBEL_6:th 26
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st b2 <> Top b1
holds b2 is prime(b1)
iff
(downarrow b2) ` is non empty filtered upper Element of bool the carrier of b1;
:: WAYBEL_6:th 27
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 is prime(b1)
iff
b2 is meet-irreducible(b1);
:: WAYBEL_6:th 28
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr holds
PRIME b1 = IRR b1;
:: WAYBEL_6:th 29
theorem
for b1 being reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st b2 <> Top b1
holds b2 is prime(b1)
iff
for b3 being Element of the carrier of b1
st b2 < b3
holds b3 = Top b1;
:: WAYBEL_6:th 30
theorem
for b1 being reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
st b2 <> Top b1
holds b2 is prime(b1)
iff
ex b3 being non empty filtered upper Open Element of bool the carrier of b1 st
b2 is_maximal_in b3 `;
:: WAYBEL_6:th 31
theorem
for b1 being RelStr
for b2 being Element of bool the carrier of b1 holds
chi(b2,the carrier of b1) is Function-like quasi_total Relation of the carrier of b1,the carrier of BoolePoset {{}};
:: WAYBEL_6:th 32
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1 holds
(chi((downarrow b2) `,the carrier of b1)) . b3 = {}
iff
b3 <= b2;
:: WAYBEL_6:th 33
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_suprema with_infima RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of BoolePoset {{}}
for b3 being prime Element of the carrier of b1
st chi((downarrow b3) `,the carrier of b1) = b2
holds b2 is meet-preserving(b1, BoolePoset {{}}) & b2 is join-preserving(b1, BoolePoset {{}});
:: WAYBEL_6:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
st PRIME b1 is order-generating(b1)
holds b1 is distributive & b1 is meet-continuous;
:: WAYBEL_6:th 35
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr holds
b1 is distributive
iff
PRIME b1 is order-generating(b1);
:: WAYBEL_6:th 36
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr holds
b1 is distributive
iff
b1 is Heyting;
:: WAYBEL_6:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr
st for b2 being Element of the carrier of b1 holds
ex b3 being Element of bool the carrier of b1 st
b2 = "\/"(b3,b1) &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b4 is co-prime(b1))
for b2 being Element of the carrier of b1 holds
b2 = "\/"((waybelow b2) /\ PRIME (b1 ~),b1);
:: WAYBEL_6:th 38
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
b1 is completely-distributive
iff
b1 is continuous &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of bool the carrier of b1 st
b2 = "\/"(b3,b1) &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b4 is co-prime(b1)));
:: WAYBEL_6:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
b1 is completely-distributive
iff
b1 is distributive & b1 is continuous & b1 ~ is continuous;