Article KNASTER, MML version 4.99.1005
:: KNASTER:th 3
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 = b2 \/ b3 & proj1 b2 misses proj1 b3
holds b1 is one-to-one
iff
b2 is one-to-one & b3 is one-to-one & proj2 b2 misses proj2 b3;
:: KNASTER:th 4
theorem
for b1, b2 being set
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of b1,b1
st b2 is_a_fixpoint_of iter(b4,b3)
holds b4 . b2 is_a_fixpoint_of iter(b4,b3);
:: KNASTER:th 5
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b1
st ex b4 being Element of NAT st
b2 is_a_fixpoint_of iter(b3,b4) &
(for b5 being set
st b5 is_a_fixpoint_of iter(b3,b4)
holds b2 = b5)
holds b2 is_a_fixpoint_of b3;
:: KNASTER:attrnot 1 => KNASTER:attr 1
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,a2;
redefine attr a3 is c=-monotone means
for b1, b2 being Element of a1
st b1 c= b2
holds a3 . b1 c= a3 . b2;
end;
:: KNASTER:dfs 1
definiens
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,a2;
To prove
a1 is c=-monotone
it is sufficient to prove
thus for b1, b2 being Element of a1
st b1 c= b2
holds a3 . b1 c= a3 . b2;
:: KNASTER:def 3
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2 holds
b3 is c=-monotone
iff
for b4, b5 being Element of b1
st b4 c= b5
holds b3 . b4 c= b3 . b5;
:: KNASTER:exreg 1
registration
let a1 be set;
let a2 be non empty set;
cluster Relation-like Function-like quasi_total c=-monotone Relation of a1,a2;
end;
:: KNASTER:funcnot 1 => KNASTER:func 1
definition
let a1 be set;
let a2 be Function-like quasi_total c=-monotone Relation of bool a1,bool a1;
func lfp(A1,A2) -> Element of bool a1 equals
meet {b1 where b1 is Element of bool a1: a2 . b1 c= b1};
end;
:: KNASTER:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 holds
lfp(b1,b2) = meet {b3 where b3 is Element of bool b1: b2 . b3 c= b3};
:: KNASTER:funcnot 2 => KNASTER:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total c=-monotone Relation of bool a1,bool a1;
func gfp(A1,A2) -> Element of bool a1 equals
union {b1 where b1 is Element of bool a1: b1 c= a2 . b1};
end;
:: KNASTER:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 holds
gfp(b1,b2) = union {b3 where b3 is Element of bool b1: b3 c= b2 . b3};
:: KNASTER:th 6
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 holds
lfp(b1,b2) is_a_fixpoint_of b2;
:: KNASTER:th 7
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 holds
gfp(b1,b2) is_a_fixpoint_of b2;
:: KNASTER:th 8
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1
for b3 being Element of bool b1
st b2 . b3 c= b3
holds lfp(b1,b2) c= b3;
:: KNASTER:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1
for b3 being Element of bool b1
st b3 c= b2 . b3
holds b3 c= gfp(b1,b2);
:: KNASTER:th 10
theorem
for b1 being set
for b2 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1
for b3 being Element of bool b1
st b3 is_a_fixpoint_of b2
holds lfp(b1,b2) c= b3 & b3 c= gfp(b1,b2);
:: KNASTER:sch 1
scheme KNASTER:sch 1
{F1 -> set,
F2 -> set}:
ex b1 being set st
F2(b1) = b1 & b1 c= F1()
provided
for b1, b2 being set
st b1 c= b2
holds F2(b1) c= F2(b2)
and
F2(F1()) c= F1();
:: KNASTER:th 11
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of b2,b1 holds
ex b5, b6, b7, b8 being set st
b5 misses b6 & b7 misses b8 & b5 \/ b6 = b1 & b7 \/ b8 = b2 & b3 .: b5 = b7 & b4 .: b8 = b6;
:: KNASTER:th 12
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of b2,b1
st b3 is one-to-one & b4 is one-to-one
holds ex b5 being Function-like quasi_total Relation of b1,b2 st
b5 is bijective(b1, b2);
:: KNASTER:th 13
theorem
for b1, b2 being non empty set
st ex b3 being Function-like quasi_total Relation of b2,b1 st
b3 is bijective(b2, b1)
holds b2,b1 are_equipotent;
:: KNASTER:th 14
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of b2,b1
st b3 is one-to-one & b4 is one-to-one
holds b1,b2 are_equipotent;
:: KNASTER:funcnot 3 => KNASTER:func 3
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the carrier of a1;
let a4 be ordinal set;
func (A2,A4)+. A3 -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a4 &
b1 . {} = a3 &
(for b2 being ordinal set
st succ b2 in succ a4
holds b1 . succ b2 = a2 . (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a4 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = "\/"(proj2 (b1 | b2),a1));
end;
:: KNASTER:def 6
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set
for b5 being set holds
b5 = (b2,b4)+. b3
iff
ex b6 being Relation-like Function-like T-Sequence-like set st
b5 = last b6 &
proj1 b6 = succ b4 &
b6 . {} = b3 &
(for b7 being ordinal set
st succ b7 in succ b4
holds b6 . succ b7 = b2 . (b6 . b7)) &
(for b7 being ordinal set
st b7 in succ b4 & b7 <> {} & b7 is being_limit_ordinal
holds b6 . b7 = "\/"(proj2 (b6 | b7),b1));
:: KNASTER:funcnot 4 => KNASTER:func 4
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the carrier of a1;
let a4 be ordinal set;
func (A2,A4)-. A3 -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a4 &
b1 . {} = a3 &
(for b2 being ordinal set
st succ b2 in succ a4
holds b1 . succ b2 = a2 . (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a4 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = "/\"(proj2 (b1 | b2),a1));
end;
:: KNASTER:def 7
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set
for b5 being set holds
b5 = (b2,b4)-. b3
iff
ex b6 being Relation-like Function-like T-Sequence-like set st
b5 = last b6 &
proj1 b6 = succ b4 &
b6 . {} = b3 &
(for b7 being ordinal set
st succ b7 in succ b4
holds b6 . succ b7 = b2 . (b6 . b7)) &
(for b7 being ordinal set
st b7 in succ b4 & b7 <> {} & b7 is being_limit_ordinal
holds b6 . b7 = "/\"(proj2 (b6 | b7),b1));
:: KNASTER:th 16
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1 holds
(b2,{})+. b3 = b3;
:: KNASTER:th 17
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1 holds
(b2,{})-. b3 = b3;
:: KNASTER:th 18
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set holds
(b2,succ b4)+. b3 = b2 . ((b2,b4)+. b3);
:: KNASTER:th 19
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set holds
(b2,succ b4)-. b3 = b2 . ((b2,b4)-. b3);
:: KNASTER:th 20
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set
for b5 being Relation-like Function-like T-Sequence-like set
st b4 <> {} &
b4 is being_limit_ordinal &
proj1 b5 = b4 &
(for b6 being ordinal set
st b6 in b4
holds b5 . b6 = (b2,b6)+. b3)
holds (b2,b4)+. b3 = "\/"(proj2 b5,b1);
:: KNASTER:th 21
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being ordinal set
for b5 being Relation-like Function-like T-Sequence-like set
st b4 <> {} &
b4 is being_limit_ordinal &
proj1 b5 = b4 &
(for b6 being ordinal set
st b6 in b4
holds b5 . b6 = (b2,b6)-. b3)
holds (b2,b4)-. b3 = "/\"(proj2 b5,b1);
:: KNASTER:th 22
theorem
for b1 being Element of NAT
for b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
for b4 being Element of the carrier of b2 holds
(iter(b3,b1)) . b4 = (b3,b1)+. b4;
:: KNASTER:th 23
theorem
for b1 being Element of NAT
for b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
for b4 being Element of the carrier of b2 holds
(iter(b3,b1)) . b4 = (b3,b1)-. b4;
:: KNASTER:funcnot 5 => KNASTER:func 5
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the carrier of a1;
let a4 be ordinal set;
redefine func (a2,a4)+. a3 -> Element of the carrier of a1;
end;
:: KNASTER:funcnot 6 => KNASTER:func 6
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the carrier of a1;
let a4 be ordinal set;
redefine func (a2,a4)-. a3 -> Element of the carrier of a1;
end;
:: KNASTER:attrnot 2 => KNASTER:attr 2
definition
let a1 be non empty LattStr;
let a2 be Element of bool the carrier of a1;
attr a2 is with_suprema means
for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 &
b1 [= b3 &
b2 [= b3 &
(for b4 being Element of the carrier of a1
st b4 in a2 & b1 [= b4 & b2 [= b4
holds b3 [= b4);
end;
:: KNASTER:dfs 6
definiens
let a1 be non empty LattStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is with_suprema
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 &
b1 [= b3 &
b2 [= b3 &
(for b4 being Element of the carrier of a1
st b4 in a2 & b1 [= b4 & b2 [= b4
holds b3 [= b4);
:: KNASTER:def 8
theorem
for b1 being non empty LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is with_suprema(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds ex b5 being Element of the carrier of b1 st
b5 in b2 &
b3 [= b5 &
b4 [= b5 &
(for b6 being Element of the carrier of b1
st b6 in b2 & b3 [= b6 & b4 [= b6
holds b5 [= b6);
:: KNASTER:attrnot 3 => KNASTER:attr 3
definition
let a1 be non empty LattStr;
let a2 be Element of bool the carrier of a1;
attr a2 is with_infima means
for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 &
b3 [= b1 &
b3 [= b2 &
(for b4 being Element of the carrier of a1
st b4 in a2 & b4 [= b1 & b4 [= b2
holds b4 [= b3);
end;
:: KNASTER:dfs 7
definiens
let a1 be non empty LattStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is with_infima
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds ex b3 being Element of the carrier of a1 st
b3 in a2 &
b3 [= b1 &
b3 [= b2 &
(for b4 being Element of the carrier of a1
st b4 in a2 & b4 [= b1 & b4 [= b2
holds b4 [= b3);
:: KNASTER:def 9
theorem
for b1 being non empty LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is with_infima(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds ex b5 being Element of the carrier of b1 st
b5 in b2 &
b5 [= b3 &
b5 [= b4 &
(for b6 being Element of the carrier of b1
st b6 in b2 & b6 [= b3 & b6 [= b4
holds b6 [= b5);
:: KNASTER:exreg 2
registration
let a1 be non empty Lattice-like LattStr;
cluster non empty with_suprema with_infima Element of bool the carrier of a1;
end;
:: KNASTER:funcnot 7 => KNASTER:func 7
definition
let a1 be non empty Lattice-like LattStr;
let a2 be non empty with_suprema with_infima Element of bool the carrier of a1;
func latt A2 -> non empty strict Lattice-like LattStr means
the carrier of it = a2 &
(for b1, b2 being Element of the carrier of it holds
ex b3, b4 being Element of the carrier of a1 st
b1 = b3 & b2 = b4 & (b1 [= b2 implies b3 [= b4) & (b3 [= b4 implies b1 [= b2));
end;
:: KNASTER:def 10
theorem
for b1 being non empty Lattice-like LattStr
for b2 being non empty with_suprema with_infima Element of bool the carrier of b1
for b3 being non empty strict Lattice-like LattStr holds
b3 = latt b2
iff
the carrier of b3 = b2 &
(for b4, b5 being Element of the carrier of b3 holds
ex b6, b7 being Element of the carrier of b1 st
b4 = b6 & b5 = b7 & (b4 [= b5 implies b6 [= b7) & (b6 [= b7 implies b4 [= b5));
:: KNASTER:condreg 1
registration
cluster non empty Lattice-like complete -> bounded (LattStr);
end;
:: KNASTER:th 24
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b3 is_a_fixpoint_of b2;
:: KNASTER:th 25
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 [= b2 . b3
for b4 being ordinal set holds
b3 [= (b2,b4)+. b3;
:: KNASTER:th 26
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b2 . b3 [= b3
for b4 being ordinal set holds
(b2,b4)-. b3 [= b3;
:: KNASTER:th 27
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 [= b2 . b3
for b4, b5 being ordinal set
st b4 c= b5
holds (b2,b4)+. b3 [= (b2,b5)+. b3;
:: KNASTER:th 28
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b2 . b3 [= b3
for b4, b5 being ordinal set
st b4 c= b5
holds (b2,b5)-. b3 [= (b2,b4)-. b3;
:: KNASTER:th 29
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 [= b2 . b3
for b4, b5 being ordinal set
st b4 c< b5 & not (b2,b5)+. b3 is_a_fixpoint_of b2
holds (b2,b4)+. b3 <> (b2,b5)+. b3;
:: KNASTER:th 30
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b2 . b3 [= b3
for b4, b5 being ordinal set
st b4 c< b5 & not (b2,b5)-. b3 is_a_fixpoint_of b2
holds (b2,b4)-. b3 <> (b2,b5)-. b3;
:: KNASTER:th 31
theorem
for b1 being ordinal set
for b2 being non empty Lattice-like complete LattStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b2,the carrier of b2
for b4 being Element of the carrier of b2
st b4 [= b3 . b4 & (b3,b1)+. b4 is_a_fixpoint_of b3
for b5 being ordinal set
st b1 c= b5
holds (b3,b1)+. b4 = (b3,b5)+. b4;
:: KNASTER:th 32
theorem
for b1 being ordinal set
for b2 being non empty Lattice-like complete LattStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b2,the carrier of b2
for b4 being Element of the carrier of b2
st b3 . b4 [= b4 & (b3,b1)-. b4 is_a_fixpoint_of b3
for b5 being ordinal set
st b1 c= b5
holds (b3,b1)-. b4 = (b3,b5)-. b4;
:: KNASTER:th 33
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 [= b2 . b3
holds ex b4 being ordinal set st
Card b4 c= Card the carrier of b1 & (b2,b4)+. b3 is_a_fixpoint_of b2;
:: KNASTER:th 34
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b2 . b3 [= b3
holds ex b4 being ordinal set st
Card b4 c= Card the carrier of b1 & (b2,b4)-. b3 is_a_fixpoint_of b2;
:: KNASTER:th 35
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 is_a_fixpoint_of b2 & b4 is_a_fixpoint_of b2
holds ex b5 being ordinal set st
Card b5 c= Card the carrier of b1 & (b2,b5)+. (b3 "\/" b4) is_a_fixpoint_of b2 & b3 [= (b2,b5)+. (b3 "\/" b4) & b4 [= (b2,b5)+. (b3 "\/" b4);
:: KNASTER:th 36
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 is_a_fixpoint_of b2 & b4 is_a_fixpoint_of b2
holds ex b5 being ordinal set st
Card b5 c= Card the carrier of b1 & (b2,b5)-. (b3 "/\" b4) is_a_fixpoint_of b2 & (b2,b5)-. (b3 "/\" b4) [= b3 & (b2,b5)-. (b3 "/\" b4) [= b4;
:: KNASTER:th 37
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 [= b2 . b3 & b3 [= b4 & b4 is_a_fixpoint_of b2
for b5 being ordinal set holds
(b2,b5)+. b3 [= b4;
:: KNASTER:th 38
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b2 . b3 [= b3 & b4 [= b3 & b4 is_a_fixpoint_of b2
for b5 being ordinal set holds
b4 [= (b2,b5)-. b3;
:: KNASTER:funcnot 8 => KNASTER:func 8
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
assume a2 is monotone(a1);
func FixPoints A2 -> non empty strict Lattice-like LattStr means
ex b1 being non empty with_suprema with_infima Element of bool the carrier of a1 st
b1 = {b2 where b2 is Element of the carrier of a1: b2 is_a_fixpoint_of a2} &
it = latt b1;
end;
:: KNASTER:def 11
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is monotone(b1)
for b3 being non empty strict Lattice-like LattStr holds
b3 = FixPoints b2
iff
ex b4 being non empty with_suprema with_infima Element of bool the carrier of b1 st
b4 = {b5 where b5 is Element of the carrier of b1: b5 is_a_fixpoint_of b2} &
b3 = latt b4;
:: KNASTER:th 39
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
the carrier of FixPoints b2 = {b3 where b3 is Element of the carrier of b1: b3 is_a_fixpoint_of b2};
:: KNASTER:th 40
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
the carrier of FixPoints b2 c= the carrier of b1;
:: KNASTER:th 41
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in the carrier of FixPoints b2
iff
b3 is_a_fixpoint_of b2;
:: KNASTER:th 42
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of FixPoints b2
for b5, b6 being Element of the carrier of b1
st b3 = b5 & b4 = b6
holds b3 [= b4
iff
b5 [= b6;
:: KNASTER:th 43
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
FixPoints b2 is complete;
:: KNASTER:funcnot 9 => KNASTER:func 9
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Function-like quasi_total monotone Relation of the carrier of a1,the carrier of a1;
func lfp A2 -> Element of the carrier of a1 equals
(a2,nextcard the carrier of a1)+. Bottom a1;
end;
:: KNASTER:def 12
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
lfp b2 = (b2,nextcard the carrier of b1)+. Bottom b1;
:: KNASTER:funcnot 10 => KNASTER:func 10
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Function-like quasi_total monotone Relation of the carrier of a1,the carrier of a1;
func gfp A2 -> Element of the carrier of a1 equals
(a2,nextcard the carrier of a1)-. Top a1;
end;
:: KNASTER:def 13
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
gfp b2 = (b2,nextcard the carrier of b1)-. Top b1;
:: KNASTER:th 44
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
lfp b2 is_a_fixpoint_of b2 &
(ex b3 being ordinal set st
Card b3 c= Card the carrier of b1 & (b2,b3)+. Bottom b1 = lfp b2);
:: KNASTER:th 45
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1 holds
gfp b2 is_a_fixpoint_of b2 &
(ex b3 being ordinal set st
Card b3 c= Card the carrier of b1 & (b2,b3)-. Top b1 = gfp b2);
:: KNASTER:th 46
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 is_a_fixpoint_of b2
holds lfp b2 [= b3 & b3 [= gfp b2;
:: KNASTER:th 47
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b2 . b3 [= b3
holds lfp b2 [= b3;
:: KNASTER:th 48
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
st b3 [= b2 . b3
holds b3 [= gfp b2;
:: KNASTER:th 49
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of the carrier of BooleLatt b1,the carrier of BooleLatt b1 holds
b2 is monotone(BooleLatt b1)
iff
b2 is c=-monotone;
:: KNASTER:th 50
theorem
for b1 being non empty set
for b2 being Function-like quasi_total monotone Relation of the carrier of BooleLatt b1,the carrier of BooleLatt b1 holds
ex b3 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 st
lfp(b1,b3) = lfp b2;
:: KNASTER:th 51
theorem
for b1 being non empty set
for b2 being Function-like quasi_total monotone Relation of the carrier of BooleLatt b1,the carrier of BooleLatt b1 holds
ex b3 being Function-like quasi_total c=-monotone Relation of bool b1,bool b1 st
gfp(b1,b3) = gfp b2;