Article COHSP_1, MML version 4.99.1005
:: COHSP_1:attrnot 1 => COH_SP:attr 1
definition
let a1 be set;
attr a1 is binary_complete means
for b1 being set
st for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 \/ b3 in a1
holds union b1 in a1;
end;
:: COHSP_1:dfs 1
definiens
let a1 be set;
To prove
a1 is binary_complete
it is sufficient to prove
thus for b1 being set
st for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 \/ b3 in a1
holds union b1 in a1;
:: COHSP_1:def 1
theorem
for b1 being set holds
b1 is binary_complete
iff
for b2 being set
st for b3, b4 being set
st b3 in b2 & b4 in b2
holds b3 \/ b4 in b1
holds union b2 in b1;
:: COHSP_1:exreg 1
registration
cluster non empty finite subset-closed binary_complete set;
end;
:: COHSP_1:funcnot 1 => COHSP_1:func 1
definition
let a1 be set;
func FlatCoh A1 -> set equals
CohSp id a1;
end;
:: COHSP_1:def 2
theorem
for b1 being set holds
FlatCoh b1 = CohSp id b1;
:: COHSP_1:funcnot 2 => COHSP_1:func 2
definition
let a1 be set;
func Sub_of_Fin A1 -> Element of bool a1 means
for b1 being set holds
b1 in it
iff
b1 in a1 & b1 is finite;
end;
:: COHSP_1:def 3
theorem
for b1 being set
for b2 being Element of bool b1 holds
b2 = Sub_of_Fin b1
iff
for b3 being set holds
b3 in b2
iff
b3 in b1 & b3 is finite;
:: COHSP_1:th 1
theorem
for b1, b2 being set holds
b2 in FlatCoh b1
iff
(b2 <> {} implies ex b3 being set st
b2 = {b3} & b3 in b1);
:: COHSP_1:th 2
theorem
for b1 being set holds
union FlatCoh b1 = b1;
:: COHSP_1:th 3
theorem
for b1 being finite subset-closed set holds
Sub_of_Fin b1 = b1;
:: COHSP_1:funcreg 1
registration
cluster {{}} -> subset-closed binary_complete;
end;
:: COHSP_1:funcreg 2
registration
let a1 be set;
cluster bool a1 -> subset-closed binary_complete;
end;
:: COHSP_1:funcreg 3
registration
let a1 be set;
cluster FlatCoh a1 -> non empty subset-closed binary_complete;
end;
:: COHSP_1:funcreg 4
registration
let a1 be non empty subset-closed set;
cluster Sub_of_Fin a1 -> non empty subset-closed;
end;
:: COHSP_1:th 4
theorem
Web {{}} = {};
:: COHSP_1:sch 1
scheme COHSP_1:sch 1
{F1 -> set,
F2 -> set}:
ex b1 being set st
b1 in F2() &
P1[b1] &
(for b2 being set
st b2 in F2() & P1[b2] & b2 c= b1
holds b2 = b1)
provided
F1() in F2()
and
P1[F1()]
and
F1() is finite;
:: COHSP_1:exreg 2
registration
let a1 be non empty subset-closed binary_complete set;
cluster finite Element of a1;
end;
:: COHSP_1:attrnot 2 => COHSP_1:attr 1
definition
let a1 be set;
attr a1 is c=directed means
for b1 being finite Element of bool a1 holds
ex b2 being set st
union b1 c= b2 & b2 in a1;
end;
:: COHSP_1:dfs 4
definiens
let a1 be set;
To prove
a1 is c=directed
it is sufficient to prove
thus for b1 being finite Element of bool a1 holds
ex b2 being set st
union b1 c= b2 & b2 in a1;
:: COHSP_1:def 4
theorem
for b1 being set holds
b1 is c=directed
iff
for b2 being finite Element of bool b1 holds
ex b3 being set st
union b2 c= b3 & b3 in b1;
:: COHSP_1:attrnot 3 => COHSP_1:attr 2
definition
let a1 be set;
attr a1 is c=filtered means
for b1 being finite Element of bool a1 holds
ex b2 being set st
(for b3 being set
st b3 in b1
holds b2 c= b3) &
b2 in a1;
end;
:: COHSP_1:dfs 5
definiens
let a1 be set;
To prove
a1 is c=filtered
it is sufficient to prove
thus for b1 being finite Element of bool a1 holds
ex b2 being set st
(for b3 being set
st b3 in b1
holds b2 c= b3) &
b2 in a1;
:: COHSP_1:def 5
theorem
for b1 being set holds
b1 is c=filtered
iff
for b2 being finite Element of bool b1 holds
ex b3 being set st
(for b4 being set
st b4 in b2
holds b3 c= b4) &
b3 in b1;
:: COHSP_1:condreg 1
registration
cluster c=directed -> non empty (set);
end;
:: COHSP_1:condreg 2
registration
cluster c=filtered -> non empty (set);
end;
:: COHSP_1:th 5
theorem
for b1 being set
st b1 is c=directed
for b2, b3 being set
st b2 in b1 & b3 in b1
holds ex b4 being set st
b2 \/ b3 c= b4 & b4 in b1;
:: COHSP_1:th 6
theorem
for b1 being non empty set
st for b2, b3 being set
st b2 in b1 & b3 in b1
holds ex b4 being set st
b2 \/ b3 c= b4 & b4 in b1
holds b1 is c=directed;
:: COHSP_1:th 7
theorem
for b1 being set
st b1 is c=filtered
for b2, b3 being set
st b2 in b1 & b3 in b1
holds ex b4 being set st
b4 c= b2 /\ b3 & b4 in b1;
:: COHSP_1:th 8
theorem
for b1 being non empty set
st for b2, b3 being set
st b2 in b1 & b3 in b1
holds ex b4 being set st
b4 c= b2 /\ b3 & b4 in b1
holds b1 is c=filtered;
:: COHSP_1:th 9
theorem
for b1 being set holds
{b1} is c=directed & {b1} is c=filtered;
:: COHSP_1:th 10
theorem
for b1, b2 being set holds
{b1,b2,b1 \/ b2} is c=directed;
:: COHSP_1:th 11
theorem
for b1, b2 being set holds
{b1,b2,b1 /\ b2} is c=filtered;
:: COHSP_1:exreg 3
registration
cluster finite c=directed c=filtered set;
end;
:: COHSP_1:exreg 4
registration
let a1 be non empty set;
cluster finite c=directed c=filtered Element of bool a1;
end;
:: COHSP_1:th 12
theorem
for b1 being set holds
Fin b1 is c=directed & Fin b1 is c=filtered;
:: COHSP_1:funcreg 5
registration
let a1 be set;
cluster Fin a1 -> preBoolean c=directed c=filtered;
end;
:: COHSP_1:exreg 5
registration
let a1 be non empty subset-closed set;
cluster non empty preBoolean Element of bool a1;
end;
:: COHSP_1:funcnot 3 => COHSP_1:func 3
definition
let a1 be non empty subset-closed set;
let a2 be Element of a1;
redefine func Fin a2 -> non empty preBoolean Element of bool a1;
end;
:: COHSP_1:th 13
theorem
for b1 being non empty set
for b2 being set
st b1 is c=directed & b2 c= union b1 & b2 is finite
holds ex b3 being set st
b3 in b1 & b2 c= b3;
:: COHSP_1:attrnot 4 => FINSUB_1:attr 2
notation
let a1 be set;
synonym multiplicative for cap-closed;
end;
:: COHSP_1:attrnot 5 => COHSP_1:attr 3
definition
let a1 be set;
attr a1 is d.union-closed means
for b1 being Element of bool a1
st b1 is c=directed
holds union b1 in a1;
end;
:: COHSP_1:dfs 6
definiens
let a1 be set;
To prove
a1 is d.union-closed
it is sufficient to prove
thus for b1 being Element of bool a1
st b1 is c=directed
holds union b1 in a1;
:: COHSP_1:def 7
theorem
for b1 being set holds
b1 is d.union-closed
iff
for b2 being Element of bool b1
st b2 is c=directed
holds union b2 in b1;
:: COHSP_1:condreg 3
registration
cluster subset-closed -> cap-closed (set);
end;
:: COHSP_1:th 15
theorem
for b1 being non empty subset-closed binary_complete set
for b2 being c=directed Element of bool b1 holds
union b2 in b1;
:: COHSP_1:condreg 4
registration
cluster non empty subset-closed binary_complete -> d.union-closed (set);
end;
:: COHSP_1:exreg 6
registration
cluster non empty cap-closed subset-closed binary_complete d.union-closed set;
end;
:: COHSP_1:funcnot 4 => COHSP_1:func 4
definition
let a1 be non empty d.union-closed set;
let a2 be c=directed Element of bool a1;
redefine func union a2 -> Element of a1;
end;
:: COHSP_1:prednot 1 => COHSP_1:pred 1
definition
let a1, a2 be set;
pred A1 includes_lattice_of A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2
holds b1 /\ b2 in a1 & b1 \/ b2 in a1;
end;
:: COHSP_1:dfs 7
definiens
let a1, a2 be set;
To prove
a1 includes_lattice_of a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2
holds b1 /\ b2 in a1 & b1 \/ b2 in a1;
:: COHSP_1:def 8
theorem
for b1, b2 being set holds
b1 includes_lattice_of b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2
holds b3 /\ b4 in b1 & b3 \/ b4 in b1;
:: COHSP_1:th 16
theorem
for b1 being non empty set
st b1 includes_lattice_of b1
holds b1 is c=directed & b1 is c=filtered;
:: COHSP_1:prednot 2 => COHSP_1:pred 2
definition
let a1, a2, a3 be set;
pred A1 includes_lattice_of A2,A3 means
a1 includes_lattice_of {a2,a3};
end;
:: COHSP_1:dfs 8
definiens
let a1, a2, a3 be set;
To prove
a1 includes_lattice_of a2,a3
it is sufficient to prove
thus a1 includes_lattice_of {a2,a3};
:: COHSP_1:def 9
theorem
for b1, b2, b3 being set holds
b1 includes_lattice_of b2,b3
iff
b1 includes_lattice_of {b2,b3};
:: COHSP_1:th 17
theorem
for b1, b2, b3 being set holds
b1 includes_lattice_of b2,b3
iff
b2 in b1 & b3 in b1 & b2 /\ b3 in b1 & b2 \/ b3 in b1;
:: COHSP_1:attrnot 6 => COHSP_1:attr 4
definition
let a1 be Relation-like Function-like set;
attr a1 is union-distributive means
for b1 being Element of bool proj1 a1
st union b1 in proj1 a1
holds a1 . union b1 = union (a1 .: b1);
end;
:: COHSP_1:dfs 9
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is union-distributive
it is sufficient to prove
thus for b1 being Element of bool proj1 a1
st union b1 in proj1 a1
holds a1 . union b1 = union (a1 .: b1);
:: COHSP_1:def 10
theorem
for b1 being Relation-like Function-like set holds
b1 is union-distributive
iff
for b2 being Element of bool proj1 b1
st union b2 in proj1 b1
holds b1 . union b2 = union (b1 .: b2);
:: COHSP_1:attrnot 7 => COHSP_1:attr 5
definition
let a1 be Relation-like Function-like set;
attr a1 is d.union-distributive means
for b1 being Element of bool proj1 a1
st b1 is c=directed & union b1 in proj1 a1
holds a1 . union b1 = union (a1 .: b1);
end;
:: COHSP_1:dfs 10
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is d.union-distributive
it is sufficient to prove
thus for b1 being Element of bool proj1 a1
st b1 is c=directed & union b1 in proj1 a1
holds a1 . union b1 = union (a1 .: b1);
:: COHSP_1:def 11
theorem
for b1 being Relation-like Function-like set holds
b1 is d.union-distributive
iff
for b2 being Element of bool proj1 b1
st b2 is c=directed & union b2 in proj1 b1
holds b1 . union b2 = union (b1 .: b2);
:: COHSP_1:attrnot 8 => COHSP_1:attr 6
definition
let a1 be Relation-like Function-like set;
attr a1 is c=-monotone means
for b1, b2 being set
st b1 in proj1 a1 & b2 in proj1 a1 & b1 c= b2
holds a1 . b1 c= a1 . b2;
end;
:: COHSP_1:dfs 11
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is c=-monotone
it is sufficient to prove
thus for b1, b2 being set
st b1 in proj1 a1 & b2 in proj1 a1 & b1 c= b2
holds a1 . b1 c= a1 . b2;
:: COHSP_1:def 12
theorem
for b1 being Relation-like Function-like set holds
b1 is c=-monotone
iff
for b2, b3 being set
st b2 in proj1 b1 & b3 in proj1 b1 & b2 c= b3
holds b1 . b2 c= b1 . b3;
:: COHSP_1:attrnot 9 => COHSP_1:attr 7
definition
let a1 be Relation-like Function-like set;
attr a1 is cap-distributive means
for b1, b2 being set
st proj1 a1 includes_lattice_of b1,b2
holds a1 . (b1 /\ b2) = (a1 . b1) /\ (a1 . b2);
end;
:: COHSP_1:dfs 12
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is cap-distributive
it is sufficient to prove
thus for b1, b2 being set
st proj1 a1 includes_lattice_of b1,b2
holds a1 . (b1 /\ b2) = (a1 . b1) /\ (a1 . b2);
:: COHSP_1:def 13
theorem
for b1 being Relation-like Function-like set holds
b1 is cap-distributive
iff
for b2, b3 being set
st proj1 b1 includes_lattice_of b2,b3
holds b1 . (b2 /\ b3) = (b1 . b2) /\ (b1 . b3);
:: COHSP_1:condreg 5
registration
cluster Relation-like Function-like d.union-distributive -> c=-monotone (set);
end;
:: COHSP_1:condreg 6
registration
cluster Relation-like Function-like union-distributive -> d.union-distributive (set);
end;
:: COHSP_1:th 18
theorem
for b1 being Relation-like Function-like set
st b1 is union-distributive
for b2, b3 being set
st b2 in proj1 b1 & b3 in proj1 b1 & b2 \/ b3 in proj1 b1
holds b1 . (b2 \/ b3) = (b1 . b2) \/ (b1 . b3);
:: COHSP_1:th 19
theorem
for b1 being Relation-like Function-like set
st b1 is union-distributive
holds b1 . {} = {};
:: COHSP_1:exreg 7
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster Relation-like Function-like quasi_total union-distributive cap-distributive Relation of a1,a2;
end;
:: COHSP_1:exreg 8
registration
let a1 be non empty subset-closed binary_complete set;
cluster Relation-like Function-like union-distributive cap-distributive ManySortedSet of a1;
end;
:: COHSP_1:attrnot 10 => COHSP_1:attr 8
definition
let a1 be Relation-like Function-like set;
attr a1 is U-continuous means
proj1 a1 is d.union-closed & a1 is d.union-distributive;
end;
:: COHSP_1:dfs 13
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is U-continuous
it is sufficient to prove
thus proj1 a1 is d.union-closed & a1 is d.union-distributive;
:: COHSP_1:def 14
theorem
for b1 being Relation-like Function-like set holds
b1 is U-continuous
iff
proj1 b1 is d.union-closed & b1 is d.union-distributive;
:: COHSP_1:attrnot 11 => COHSP_1:attr 9
definition
let a1 be Relation-like Function-like set;
attr a1 is U-stable means
proj1 a1 is cap-closed & a1 is U-continuous & a1 is cap-distributive;
end;
:: COHSP_1:dfs 14
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is U-stable
it is sufficient to prove
thus proj1 a1 is cap-closed & a1 is U-continuous & a1 is cap-distributive;
:: COHSP_1:def 15
theorem
for b1 being Relation-like Function-like set holds
b1 is U-stable
iff
proj1 b1 is cap-closed & b1 is U-continuous & b1 is cap-distributive;
:: COHSP_1:attrnot 12 => COHSP_1:attr 10
definition
let a1 be Relation-like Function-like set;
attr a1 is U-linear means
a1 is U-stable & a1 is union-distributive;
end;
:: COHSP_1:dfs 15
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is U-linear
it is sufficient to prove
thus a1 is U-stable & a1 is union-distributive;
:: COHSP_1:def 16
theorem
for b1 being Relation-like Function-like set holds
b1 is U-linear
iff
b1 is U-stable & b1 is union-distributive;
:: COHSP_1:condreg 7
registration
cluster Relation-like Function-like U-continuous -> d.union-distributive (set);
end;
:: COHSP_1:condreg 8
registration
cluster Relation-like Function-like U-stable -> cap-distributive U-continuous (set);
end;
:: COHSP_1:condreg 9
registration
cluster Relation-like Function-like U-linear -> union-distributive U-stable (set);
end;
:: COHSP_1:condreg 10
registration
let a1 be d.union-closed set;
cluster d.union-distributive -> U-continuous (ManySortedSet of a1);
end;
:: COHSP_1:condreg 11
registration
let a1 be cap-closed set;
cluster cap-distributive U-continuous -> U-stable (ManySortedSet of a1);
end;
:: COHSP_1:condreg 12
registration
cluster Relation-like Function-like union-distributive U-stable -> U-linear (set);
end;
:: COHSP_1:exreg 9
registration
cluster Relation-like Function-like U-linear set;
end;
:: COHSP_1:exreg 10
registration
let a1 be non empty subset-closed binary_complete set;
cluster Relation-like Function-like U-linear ManySortedSet of a1;
end;
:: COHSP_1:exreg 11
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster Relation-like Function-like quasi_total U-linear Relation of a2,a1;
end;
:: COHSP_1:funcreg 6
registration
let a1 be Relation-like Function-like U-continuous set;
cluster proj1 a1 -> d.union-closed;
end;
:: COHSP_1:funcreg 7
registration
let a1 be Relation-like Function-like U-stable set;
cluster proj1 a1 -> cap-closed;
end;
:: COHSP_1:th 20
theorem
for b1 being set holds
union Fin b1 = b1;
:: COHSP_1:th 21
theorem
for b1 being Relation-like Function-like U-continuous set
st proj1 b1 is subset-closed
for b2 being set
st b2 in proj1 b1
holds b1 . b2 = union (b1 .: Fin b2);
:: COHSP_1:th 22
theorem
for b1 being Relation-like Function-like set
st proj1 b1 is subset-closed
holds b1 is U-continuous
iff
proj1 b1 is d.union-closed &
b1 is c=-monotone &
(for b2, b3 being set
st b2 in proj1 b1 & b3 in b1 . b2
holds ex b4 being set st
b4 is finite & b4 c= b2 & b3 in b1 . b4);
:: COHSP_1:th 23
theorem
for b1 being Relation-like Function-like set
st proj1 b1 is subset-closed & proj1 b1 is d.union-closed
holds b1 is U-stable
iff
b1 is c=-monotone &
(for b2, b3 being set
st b2 in proj1 b1 & b3 in b1 . b2
holds ex b4 being set st
b4 is finite &
b4 c= b2 &
b3 in b1 . b4 &
(for b5 being set
st b5 c= b2 & b3 in b1 . b5
holds b4 c= b5));
:: COHSP_1:th 24
theorem
for b1 being Relation-like Function-like set
st proj1 b1 is subset-closed & proj1 b1 is d.union-closed
holds b1 is U-linear
iff
b1 is c=-monotone &
(for b2, b3 being set
st b2 in proj1 b1 & b3 in b1 . b2
holds ex b4 being set st
b4 in b2 &
b3 in b1 . {b4} &
(for b5 being set
st b5 c= b2 & b3 in b1 . b5
holds b4 in b5));
:: COHSP_1:funcnot 5 => COHSP_1:func 5
definition
let a1 be Relation-like Function-like set;
func graph A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being finite set st
ex b3 being set st
b1 = [b2,b3] & b2 in proj1 a1 & b3 in a1 . b2;
end;
:: COHSP_1:def 17
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
b2 = graph b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being finite set st
ex b5 being set st
b3 = [b4,b5] & b4 in proj1 b1 & b5 in b1 . b4;
:: COHSP_1:funcnot 6 => COHSP_1:func 6
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,a2;
redefine func graph a3 -> Element of bool [:a1,union a2:];
end;
:: COHSP_1:funcreg 8
registration
let a1 be Relation-like Function-like set;
cluster graph a1 -> Relation-like;
end;
:: COHSP_1:th 25
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set holds
[b2,b3] in graph b1
iff
b2 is finite & b2 in proj1 b1 & b3 in b1 . b2;
:: COHSP_1:th 26
theorem
for b1 being Relation-like Function-like c=-monotone set
for b2, b3 being set
st b3 in proj1 b1 & b2 c= b3 & b3 is finite
for b4 being set
st [b2,b4] in graph b1
holds [b3,b4] in graph b1;
:: COHSP_1:th 27
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of b1
for b5, b6 being set
st [b4,b5] in graph b3 & [b4,b6] in graph b3
holds {b5,b6} in b2;
:: COHSP_1:th 28
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total c=-monotone Relation of b1,b2
for b4, b5 being Element of b1
st b4 \/ b5 in b1
for b6, b7 being set
st [b4,b6] in graph b3 & [b5,b7] in graph b3
holds {b6,b7} in b2;
:: COHSP_1:th 29
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being Function-like quasi_total U-continuous Relation of b1,b2
st graph b3 = graph b4
holds b3 = b4;
:: COHSP_1:th 30
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Element of bool [:b1,union b2:]
st (for b4 being set
st b4 in b3
holds b4 `1 is finite) &
(for b4, b5 being finite Element of b1
st b4 c= b5
for b6 being set
st [b4,b6] in b3
holds [b5,b6] in b3) &
(for b4 being finite Element of b1
for b5, b6 being set
st [b4,b5] in b3 & [b4,b6] in b3
holds {b5,b6} in b2)
holds ex b4 being Function-like quasi_total U-continuous Relation of b1,b2 st
b3 = graph b4;
:: COHSP_1:th 31
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-continuous Relation of b1,b2
for b4 being Element of b1 holds
b3 . b4 = (graph b3) .: Fin b4;
:: COHSP_1:funcnot 7 => COHSP_1:func 7
definition
let a1 be Relation-like Function-like set;
func Trace A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2, b3 being set st
b1 = [b2,b3] &
b2 in proj1 a1 &
b3 in a1 . b2 &
(for b4 being set
st b4 in proj1 a1 & b4 c= b2 & b3 in a1 . b4
holds b2 = b4);
end;
:: COHSP_1:def 18
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
b2 = Trace b1
iff
for b3 being set holds
b3 in b2
iff
ex b4, b5 being set st
b3 = [b4,b5] &
b4 in proj1 b1 &
b5 in b1 . b4 &
(for b6 being set
st b6 in proj1 b1 & b6 c= b4 & b5 in b1 . b6
holds b4 = b6);
:: COHSP_1:th 32
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set holds
[b2,b3] in Trace b1
iff
b2 in proj1 b1 &
b3 in b1 . b2 &
(for b4 being set
st b4 in proj1 b1 & b4 c= b2 & b3 in b1 . b4
holds b2 = b4);
:: COHSP_1:funcnot 8 => COHSP_1:func 8
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,a2;
redefine func Trace a3 -> Element of bool [:a1,union a2:];
end;
:: COHSP_1:funcreg 9
registration
let a1 be Relation-like Function-like set;
cluster Trace a1 -> Relation-like;
end;
:: COHSP_1:th 33
theorem
for b1 being Relation-like Function-like U-continuous set
st proj1 b1 is subset-closed
holds Trace b1 c= graph b1;
:: COHSP_1:th 34
theorem
for b1 being Relation-like Function-like U-continuous set
st proj1 b1 is subset-closed
for b2, b3 being set
st [b2,b3] in Trace b1
holds b2 is finite;
:: COHSP_1:th 35
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total c=-monotone Relation of b1,b2
for b4, b5 being set
st b4 \/ b5 in b1
for b6, b7 being set
st [b4,b6] in Trace b3 & [b5,b7] in Trace b3
holds {b6,b7} in b2;
:: COHSP_1:th 36
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total cap-distributive Relation of b1,b2
for b4, b5 being set
st b4 \/ b5 in b1
for b6 being set
st [b4,b6] in Trace b3 & [b5,b6] in Trace b3
holds b4 = b5;
:: COHSP_1:th 37
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being Function-like quasi_total U-stable Relation of b1,b2
st Trace b3 c= Trace b4
for b5 being Element of b1 holds
b3 . b5 c= b4 . b5;
:: COHSP_1:th 38
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being Function-like quasi_total U-stable Relation of b1,b2
st Trace b3 = Trace b4
holds b3 = b4;
:: COHSP_1:th 39
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Element of bool [:b1,union b2:]
st (for b4 being set
st b4 in b3
holds b4 `1 is finite) &
(for b4, b5 being Element of b1
st b4 \/ b5 in b1
for b6, b7 being set
st [b4,b6] in b3 & [b5,b7] in b3
holds {b6,b7} in b2) &
(for b4, b5 being Element of b1
st b4 \/ b5 in b1
for b6 being set
st [b4,b6] in b3 & [b5,b6] in b3
holds b4 = b5)
holds ex b4 being Function-like quasi_total U-stable Relation of b1,b2 st
b3 = Trace b4;
:: COHSP_1:th 40
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-stable Relation of b1,b2
for b4 being Element of b1 holds
b3 . b4 = (Trace b3) .: Fin b4;
:: COHSP_1:th 41
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-stable Relation of b1,b2
for b4 being Element of b1
for b5 being set holds
b5 in b3 . b4
iff
ex b6 being Element of b1 st
[b6,b5] in Trace b3 & b6 c= b4;
:: COHSP_1:th 42
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
ex b3 being Function-like quasi_total U-stable Relation of b1,b2 st
Trace b3 = {};
:: COHSP_1:th 43
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being finite Element of b1
for b4 being set
st b4 in union b2
holds ex b5 being Function-like quasi_total U-stable Relation of b1,b2 st
Trace b5 = {[b3,b4]};
:: COHSP_1:th 44
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Element of b1
for b4 being set
for b5 being Function-like quasi_total U-stable Relation of b1,b2
st Trace b5 = {[b3,b4]}
for b6 being Element of b1 holds
(b3 c= b6 implies b5 . b6 = {b4}) & (b3 c= b6 or b5 . b6 = {});
:: COHSP_1:th 45
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-stable Relation of b1,b2
for b4 being Element of bool Trace b3 holds
ex b5 being Function-like quasi_total U-stable Relation of b1,b2 st
Trace b5 = b4;
:: COHSP_1:th 46
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set
st for b4, b5 being set
st b4 in b3 & b5 in b3
holds ex b6 being Function-like quasi_total U-stable Relation of b1,b2 st
b4 \/ b5 = Trace b6
holds ex b4 being Function-like quasi_total U-stable Relation of b1,b2 st
union b3 = Trace b4;
:: COHSP_1:funcnot 9 => COHSP_1:func 9
definition
let a1, a2 be non empty subset-closed binary_complete set;
func StabCoh(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Function-like quasi_total U-stable Relation of a1,a2 st
b1 = Trace b2;
end;
:: COHSP_1:def 19
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set holds
b3 = StabCoh(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Function-like quasi_total U-stable Relation of b1,b2 st
b4 = Trace b5;
:: COHSP_1:funcreg 10
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster StabCoh(a1,a2) -> non empty subset-closed binary_complete;
end;
:: COHSP_1:th 47
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-stable Relation of b1,b2 holds
Trace b3 c= [:Sub_of_Fin b1,union b2:];
:: COHSP_1:th 48
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
union StabCoh(b1,b2) = [:Sub_of_Fin b1,union b2:];
:: COHSP_1:th 49
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being finite Element of b1
for b5, b6 being set holds
[[b3,b5],[b4,b6]] in Web StabCoh(b1,b2)
iff
(not b3 \/ b4 in b1 & b5 in union b2 & b6 in union b2 or [b5,b6] in Web b2 & (b5 = b6 implies b3 = b4));
:: COHSP_1:th 50
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-stable Relation of b1,b2 holds
b3 is U-linear
iff
for b4, b5 being set
st [b4,b5] in Trace b3
holds ex b6 being set st
b4 = {b6};
:: COHSP_1:funcnot 10 => COHSP_1:func 10
definition
let a1 be Relation-like Function-like set;
func LinTrace A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2, b3 being set st
b1 = [b2,b3] & [{b2},b3] in Trace a1;
end;
:: COHSP_1:def 20
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
b2 = LinTrace b1
iff
for b3 being set holds
b3 in b2
iff
ex b4, b5 being set st
b3 = [b4,b5] & [{b4},b5] in Trace b1;
:: COHSP_1:th 51
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set holds
[b2,b3] in LinTrace b1
iff
[{b2},b3] in Trace b1;
:: COHSP_1:th 52
theorem
for b1 being Relation-like Function-like set
st b1 . {} = {}
for b2, b3 being set
st {b2} in proj1 b1 & b3 in b1 . {b2}
holds [b2,b3] in LinTrace b1;
:: COHSP_1:th 53
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
st [b2,b3] in LinTrace b1
holds {b2} in proj1 b1 & b3 in b1 . {b2};
:: COHSP_1:funcnot 11 => COHSP_1:func 11
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,a2;
redefine func LinTrace a3 -> Element of bool [:union a1,union a2:];
end;
:: COHSP_1:funcreg 11
registration
let a1 be Relation-like Function-like set;
cluster LinTrace a1 -> Relation-like;
end;
:: COHSP_1:funcnot 12 => COHSP_1:func 12
definition
let a1, a2 be non empty subset-closed binary_complete set;
func LinCoh(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Function-like quasi_total U-linear Relation of a1,a2 st
b1 = LinTrace b2;
end;
:: COHSP_1:def 21
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set holds
b3 = LinCoh(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Function-like quasi_total U-linear Relation of b1,b2 st
b4 = LinTrace b5;
:: COHSP_1:th 54
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total c=-monotone Relation of b1,b2
for b4, b5 being set
st {b4,b5} in b1
for b6, b7 being set
st [b4,b6] in LinTrace b3 & [b5,b7] in LinTrace b3
holds {b6,b7} in b2;
:: COHSP_1:th 55
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total cap-distributive Relation of b1,b2
for b4, b5 being set
st {b4,b5} in b1
for b6 being set
st [b4,b6] in LinTrace b3 & [b5,b6] in LinTrace b3
holds b4 = b5;
:: COHSP_1:th 56
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being Function-like quasi_total U-linear Relation of b1,b2
st LinTrace b3 = LinTrace b4
holds b3 = b4;
:: COHSP_1:th 57
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Element of bool [:union b1,union b2:]
st (for b4, b5 being set
st {b4,b5} in b1
for b6, b7 being set
st [b4,b6] in b3 & [b5,b7] in b3
holds {b6,b7} in b2) &
(for b4, b5 being set
st {b4,b5} in b1
for b6 being set
st [b4,b6] in b3 & [b5,b6] in b3
holds b4 = b5)
holds ex b4 being Function-like quasi_total U-linear Relation of b1,b2 st
b3 = LinTrace b4;
:: COHSP_1:th 58
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-linear Relation of b1,b2
for b4 being Element of b1 holds
b3 . b4 = (LinTrace b3) .: b4;
:: COHSP_1:th 59
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
ex b3 being Function-like quasi_total U-linear Relation of b1,b2 st
LinTrace b3 = {};
:: COHSP_1:th 60
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set
st b3 in union b1 & b4 in union b2
holds ex b5 being Function-like quasi_total U-linear Relation of b1,b2 st
LinTrace b5 = {[b3,b4]};
:: COHSP_1:th 61
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set
st b3 in union b1 & b4 in union b2
for b5 being Function-like quasi_total U-linear Relation of b1,b2
st LinTrace b5 = {[b3,b4]}
for b6 being Element of b1 holds
(b3 in b6 implies b5 . b6 = {b4}) & (b3 in b6 or b5 . b6 = {});
:: COHSP_1:th 62
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Function-like quasi_total U-linear Relation of b1,b2
for b4 being Element of bool LinTrace b3 holds
ex b5 being Function-like quasi_total U-linear Relation of b1,b2 st
LinTrace b5 = b4;
:: COHSP_1:th 63
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set
st for b4, b5 being set
st b4 in b3 & b5 in b3
holds ex b6 being Function-like quasi_total U-linear Relation of b1,b2 st
b4 \/ b5 = LinTrace b6
holds ex b4 being Function-like quasi_total U-linear Relation of b1,b2 st
union b3 = LinTrace b4;
:: COHSP_1:funcreg 12
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster LinCoh(a1,a2) -> non empty subset-closed binary_complete;
end;
:: COHSP_1:th 64
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
union LinCoh(b1,b2) = [:union b1,union b2:];
:: COHSP_1:th 65
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4, b5, b6 being set holds
[[b3,b5],[b4,b6]] in Web LinCoh(b1,b2)
iff
b3 in union b1 &
b4 in union b1 &
(not [b3,b4] in Web b1 & b5 in union b2 & b6 in union b2 or [b5,b6] in Web b2 & (b5 = b6 implies b3 = b4));
:: COHSP_1:funcnot 13 => COHSP_1:func 13
definition
let a1 be non empty subset-closed binary_complete set;
func 'not' A1 -> set equals
{b1 where b1 is Element of bool union a1: for b2 being Element of a1 holds
ex b3 being set st
b1 /\ b2 c= {b3}};
end;
:: COHSP_1:def 22
theorem
for b1 being non empty subset-closed binary_complete set holds
'not' b1 = {b2 where b2 is Element of bool union b1: for b3 being Element of b1 holds
ex b4 being set st
b2 /\ b3 c= {b4}};
:: COHSP_1:th 66
theorem
for b1 being non empty subset-closed binary_complete set
for b2 being set holds
b2 in 'not' b1
iff
b2 c= union b1 &
(for b3 being Element of b1 holds
ex b4 being set st
b2 /\ b3 c= {b4});
:: COHSP_1:funcreg 13
registration
let a1 be non empty subset-closed binary_complete set;
cluster 'not' a1 -> non empty subset-closed binary_complete;
end;
:: COHSP_1:th 67
theorem
for b1 being non empty subset-closed binary_complete set holds
union 'not' b1 = union b1;
:: COHSP_1:th 68
theorem
for b1 being non empty subset-closed binary_complete set
for b2, b3 being set
st b2 <> b3 & {b2,b3} in b1
holds not {b2,b3} in 'not' b1;
:: COHSP_1:th 69
theorem
for b1 being non empty subset-closed binary_complete set
for b2, b3 being set
st {b2,b3} c= union b1 & not {b2,b3} in b1
holds {b2,b3} in 'not' b1;
:: COHSP_1:th 70
theorem
for b1 being non empty subset-closed binary_complete set
for b2, b3 being set holds
[b2,b3] in Web 'not' b1
iff
b2 in union b1 &
b3 in union b1 &
(b2 <> b3 implies not [b2,b3] in Web b1);
:: COHSP_1:th 71
theorem
for b1 being non empty subset-closed binary_complete set holds
'not' 'not' b1 = b1;
:: COHSP_1:th 72
theorem
'not' {{}} = {{}};
:: COHSP_1:th 73
theorem
for b1 being set holds
'not' FlatCoh b1 = bool b1 & 'not' bool b1 = FlatCoh b1;
:: COHSP_1:funcnot 14 => COHSP_1:func 14
definition
let a1, a2 be set;
func A1 U+ A2 -> set equals
Union disjoin <*a1,a2*>;
end;
:: COHSP_1:def 23
theorem
for b1, b2 being set holds
b1 U+ b2 = Union disjoin <*b1,b2*>;
:: COHSP_1:th 74
theorem
for b1, b2 being set holds
b1 U+ b2 = [:b1,{1}:] \/ [:b2,{2}:];
:: COHSP_1:th 75
theorem
for b1 being set holds
b1 U+ {} = [:b1,{1}:] &
{} U+ b1 = [:b1,{2}:];
:: COHSP_1:th 76
theorem
for b1, b2, b3 being set
st b3 in b1 U+ b2
holds b3 = [b3 `1,b3 `2] &
(b3 `2 = 1 & b3 `1 in b1 or b3 `2 = 2 & b3 `1 in b2);
:: COHSP_1:th 77
theorem
for b1, b2, b3 being set holds
[b3,1] in b1 U+ b2
iff
b3 in b1;
:: COHSP_1:th 78
theorem
for b1, b2, b3 being set holds
[b3,2] in b1 U+ b2
iff
b3 in b2;
:: COHSP_1:th 79
theorem
for b1, b2, b3, b4 being set holds
b1 U+ b2 c= b3 U+ b4
iff
b1 c= b3 & b2 c= b4;
:: COHSP_1:th 80
theorem
for b1, b2, b3 being set
st b3 c= b1 U+ b2
holds ex b4, b5 being set st
b3 = b4 U+ b5 & b4 c= b1 & b5 c= b2;
:: COHSP_1:th 81
theorem
for b1, b2, b3, b4 being set holds
b1 U+ b2 = b3 U+ b4
iff
b1 = b3 & b2 = b4;
:: COHSP_1:th 82
theorem
for b1, b2, b3, b4 being set holds
(b1 U+ b2) \/ (b3 U+ b4) = (b1 \/ b3) U+ (b2 \/ b4);
:: COHSP_1:th 83
theorem
for b1, b2, b3, b4 being set holds
(b1 U+ b2) /\ (b3 U+ b4) = (b1 /\ b3) U+ (b2 /\ b4);
:: COHSP_1:funcnot 15 => COHSP_1:func 15
definition
let a1, a2 be non empty subset-closed binary_complete set;
func A1 "/\" A2 -> set equals
{b1 U+ b2 where b1 is Element of a1, b2 is Element of a2: TRUE};
end;
:: COHSP_1:def 24
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
b1 "/\" b2 = {b3 U+ b4 where b3 is Element of b1, b4 is Element of b2: TRUE};
:: COHSP_1:funcnot 16 => COHSP_1:func 16
definition
let a1, a2 be non empty subset-closed binary_complete set;
func A1 "\/" A2 -> set equals
{b1 U+ {} where b1 is Element of a1: TRUE} \/ {{} U+ b1 where b1 is Element of a2: TRUE};
end;
:: COHSP_1:def 25
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
b1 "\/" b2 = {b3 U+ {} where b3 is Element of b1: TRUE} \/ {{} U+ b3 where b3 is Element of b2: TRUE};
:: COHSP_1:th 84
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set holds
b3 in b1 "/\" b2
iff
ex b4 being Element of b1 st
ex b5 being Element of b2 st
b3 = b4 U+ b5;
:: COHSP_1:th 85
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
b3 U+ b4 in b1 "/\" b2
iff
b3 in b1 & b4 in b2;
:: COHSP_1:th 86
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
union (b1 "/\" b2) = (union b1) U+ union b2;
:: COHSP_1:th 87
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
b3 U+ b4 in b1 "\/" b2
iff
(b3 in b1 & b4 = {} or b3 = {} & b4 in b2);
:: COHSP_1:th 88
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set
st b3 in b1 "\/" b2
holds ex b4 being Element of b1 st
ex b5 being Element of b2 st
b3 = b4 U+ b5 & (b4 = {} or b5 = {});
:: COHSP_1:th 89
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
union (b1 "\/" b2) = (union b1) U+ union b2;
:: COHSP_1:funcreg 14
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster a1 "/\" a2 -> non empty subset-closed binary_complete;
end;
:: COHSP_1:funcreg 15
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster a1 "\/" a2 -> non empty subset-closed binary_complete;
end;
:: COHSP_1:th 90
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
[[b3,1],[b4,1]] in Web (b1 "/\" b2)
iff
[b3,b4] in Web b1;
:: COHSP_1:th 91
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
[[b3,2],[b4,2]] in Web (b1 "/\" b2)
iff
[b3,b4] in Web b2;
:: COHSP_1:th 92
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set
st b3 in union b1 & b4 in union b2
holds [[b3,1],[b4,2]] in Web (b1 "/\" b2) &
[[b4,2],[b3,1]] in Web (b1 "/\" b2);
:: COHSP_1:th 93
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
[[b3,1],[b4,1]] in Web (b1 "\/" b2)
iff
[b3,b4] in Web b1;
:: COHSP_1:th 94
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
[[b3,2],[b4,2]] in Web (b1 "\/" b2)
iff
[b3,b4] in Web b2;
:: COHSP_1:th 95
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4 being set holds
not [[b3,1],[b4,2]] in Web (b1 "\/" b2) &
not [[b4,2],[b3,1]] in Web (b1 "\/" b2);
:: COHSP_1:th 96
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
'not' (b1 "/\" b2) = ('not' b1) "\/" 'not' b2;
:: COHSP_1:funcnot 17 => COHSP_1:func 17
definition
let a1, a2 be non empty subset-closed binary_complete set;
func A1 [*] A2 -> set equals
union {bool [:b1,b2:] where b1 is Element of a1, b2 is Element of a2: TRUE};
end;
:: COHSP_1:def 26
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
b1 [*] b2 = union {bool [:b3,b4:] where b3 is Element of b1, b4 is Element of b2: TRUE};
:: COHSP_1:th 97
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being set holds
b3 in b1 [*] b2
iff
ex b4 being Element of b1 st
ex b5 being Element of b2 st
b3 c= [:b4,b5:];
:: COHSP_1:funcreg 16
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster a1 [*] a2 -> non empty;
end;
:: COHSP_1:th 98
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3 being Element of b1 [*] b2 holds
proj1 b3 in b1 & proj2 b3 in b2 & b3 c= [:proj1 b3,proj2 b3:];
:: COHSP_1:funcreg 17
registration
let a1, a2 be non empty subset-closed binary_complete set;
cluster a1 [*] a2 -> subset-closed binary_complete;
end;
:: COHSP_1:th 99
theorem
for b1, b2 being non empty subset-closed binary_complete set holds
union (b1 [*] b2) = [:union b1,union b2:];
:: COHSP_1:th 100
theorem
for b1, b2 being non empty subset-closed binary_complete set
for b3, b4, b5, b6 being set holds
[[b3,b5],[b4,b6]] in Web (b1 [*] b2)
iff
[b3,b4] in Web b1 & [b5,b6] in Web b2;