Article LATTICE5, MML version 4.99.1005
:: LATTICE5:sch 1
scheme LATTICE5:sch 1
{F1 -> set}:
ex b1 being Relation-like Function-like set st
proj1 b1 = NAT &
b1 . 0 = F1() &
(for b2 being Element of NAT holds
P1[b2, b1 . b2, b1 . (b2 + 1)])
provided
for b1 being Element of NAT
for b2 being set holds
ex b3 being set st
P1[b1, b2, b3];
:: LATTICE5:th 1
theorem
for b1 being Relation-like Function-like set
for b2 being Relation-like Function-like Function-yielding set
st b1 = union proj2 b2
holds proj1 b1 = union proj2 doms b2;
:: LATTICE5:th 2
theorem
for b1, b2 being non empty set holds
[:union b1,union b2:] = union {[:b3,b4:] where b3 is Element of b1, b4 is Element of b2: b3 in b1 & b4 in b2};
:: LATTICE5:th 3
theorem
for b1 being non empty set
st b1 is c=-linear
holds [:union b1,union b1:] = union {[:b2,b2:] where b2 is Element of b1: b2 in b1};
:: LATTICE5:funcnot 1 => LATTICE5:func 1
definition
let a1 be set;
func EqRelLATT A1 -> reflexive transitive antisymmetric RelStr equals
LattPOSet EqRelLatt a1;
end;
:: LATTICE5:def 1
theorem
for b1 being set holds
EqRelLATT b1 = LattPOSet EqRelLatt b1;
:: LATTICE5:funcreg 1
registration
let a1 be set;
cluster EqRelLATT a1 -> reflexive transitive antisymmetric with_suprema with_infima;
end;
:: LATTICE5:th 4
theorem
for b1, b2 being set holds
b2 in the carrier of EqRelLATT b1
iff
b2 is total symmetric transitive Relation of b1,b1;
:: LATTICE5:th 5
theorem
for b1 being set
for b2, b3 being Element of the carrier of EqRelLatt b1 holds
b2 [= b3
iff
b2 c= b3;
:: LATTICE5:th 6
theorem
for b1 being set
for b2, b3 being Element of the carrier of EqRelLATT b1 holds
b2 <= b3
iff
b2 c= b3;
:: LATTICE5:th 7
theorem
for b1 being non empty Lattice-like LattStr
for b2, b3 being Element of the carrier of LattPOSet b1 holds
b2 "/\" b3 = (% b2) "/\" % b3;
:: LATTICE5:th 8
theorem
for b1 being set
for b2, b3 being Element of the carrier of EqRelLATT b1 holds
b2 "/\" b3 = b2 /\ b3;
:: LATTICE5:th 9
theorem
for b1 being non empty Lattice-like LattStr
for b2, b3 being Element of the carrier of LattPOSet b1 holds
b2 "\/" b3 = (% b2) "\/" % b3;
:: LATTICE5:th 10
theorem
for b1 being set
for b2, b3 being Element of the carrier of EqRelLATT b1
for b4, b5 being total symmetric transitive Relation of b1,b1
st b2 = b4 & b3 = b5
holds b2 "\/" b3 = b4 "\/" b5;
:: LATTICE5:attrnot 1 => LATTICE3:attr 3
definition
let a1 be RelStr;
attr a1 is complete means
for b1 being Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_<=_than b1 &
(for b3 being Element of the carrier of a1
st b3 is_<=_than b1
holds b3 <= b2);
end;
:: LATTICE5:dfs 2
definiens
let a1 be non empty RelStr;
To prove
a1 is complete
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_<=_than b1 &
(for b3 being Element of the carrier of a1
st b3 is_<=_than b1
holds b3 <= b2);
:: LATTICE5:def 2
theorem
for b1 being non empty RelStr holds
b1 is complete
iff
for b2 being Element of bool the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b3 is_<=_than b2 &
(for b4 being Element of the carrier of b1
st b4 is_<=_than b2
holds b4 <= b3);
:: LATTICE5:funcreg 2
registration
let a1 be set;
cluster EqRelLATT a1 -> reflexive transitive antisymmetric complete;
end;
:: LATTICE5:exreg 1
registration
let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
cluster Relation-like Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of a2;
end;
:: LATTICE5:modenot 1
definition
let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
mode Homomorphism of a1,a2 is Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of a2;
end;
:: LATTICE5:exreg 2
registration
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
cluster strict meet-inheriting join-inheriting SubRelStr of a1;
end;
:: LATTICE5:modenot 2
definition
let a1 be non empty RelStr;
mode Sublattice of a1 is meet-inheriting join-inheriting SubRelStr of a1;
end;
:: LATTICE5:funcnot 2 => LATTICE5:func 2
definition
let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
let a3 be Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of a2;
redefine func Image a3 -> strict full meet-inheriting join-inheriting SubRelStr of a2;
end;
:: LATTICE5:prednot 1 => LATTICE5:pred 1
definition
let a1 be non empty set;
let a2 be non empty FinSequence of a1;
let a3, a4 be set;
let a5, a6 be Relation-like set;
pred A3,A4 are_joint_by A2,A5,A6 means
a2 . 1 = a3 &
a2 . len a2 = a4 &
(for b1 being Element of NAT
st 1 <= b1 & b1 < len a2
holds (b1 is even or [a2 . b1,a2 . (b1 + 1)] in a5) &
(b1 is even implies [a2 . b1,a2 . (b1 + 1)] in a6));
end;
:: LATTICE5:dfs 3
definiens
let a1 be non empty set;
let a2 be non empty FinSequence of a1;
let a3, a4 be set;
let a5, a6 be Relation-like set;
To prove
a3,a4 are_joint_by a2,a5,a6
it is sufficient to prove
thus a2 . 1 = a3 &
a2 . len a2 = a4 &
(for b1 being Element of NAT
st 1 <= b1 & b1 < len a2
holds (b1 is even or [a2 . b1,a2 . (b1 + 1)] in a5) &
(b1 is even implies [a2 . b1,a2 . (b1 + 1)] in a6));
:: LATTICE5:def 3
theorem
for b1 being non empty set
for b2 being non empty FinSequence of b1
for b3, b4 being set
for b5, b6 being Relation-like set holds
b3,b4 are_joint_by b2,b5,b6
iff
b2 . 1 = b3 &
b2 . len b2 = b4 &
(for b7 being Element of NAT
st 1 <= b7 & b7 < len b2
holds (b7 is even or [b2 . b7,b2 . (b7 + 1)] in b5) &
(b7 is even implies [b2 . b7,b2 . (b7 + 1)] in b6));
:: LATTICE5:th 12
theorem
for b1 being non empty set
for b2 being set
for b3 being Element of NAT
for b4, b5 being Relation-like set
for b6 being non empty FinSequence of b1
st b4 is_reflexive_in b1 & b5 is_reflexive_in b1 & b6 = b3 |-> b2
holds b2,b2 are_joint_by b6,b4,b5;
:: LATTICE5:th 14
theorem
for b1 being non empty set
for b2, b3 being set
for b4, b5 being Relation-like set
for b6, b7 being Element of NAT
st b6 <= b7 &
b4 is_reflexive_in b1 &
b5 is_reflexive_in b1 &
(ex b8 being non empty FinSequence of b1 st
len b8 = b6 & b2,b3 are_joint_by b8,b4,b5)
holds ex b8 being non empty FinSequence of b1 st
len b8 = b7 & b2,b3 are_joint_by b8,b4,b5;
:: LATTICE5:funcnot 3 => LATTICE5:func 3
definition
let a1 be non empty set;
let a2 be meet-inheriting join-inheriting SubRelStr of EqRelLATT a1;
assume (ex b1 being total symmetric transitive Relation of a1,a1 st
b1 in the carrier of a2 & b1 <> id a1) &
(ex b1 being Element of NAT st
for b2, b3 being total symmetric transitive Relation of a1,a1
for b4, b5 being set
st b2 in the carrier of a2 & b3 in the carrier of a2 & [b4,b5] in b2 "\/" b3
holds ex b6 being non empty FinSequence of a1 st
len b6 = b1 & b4,b5 are_joint_by b6,b2,b3);
func type_of A2 -> Element of NAT means
(for b1, b2 being total symmetric transitive Relation of a1,a1
for b3, b4 being set
st b1 in the carrier of a2 & b2 in the carrier of a2 & [b3,b4] in b1 "\/" b2
holds ex b5 being non empty FinSequence of a1 st
len b5 = it + 2 & b3,b4 are_joint_by b5,b1,b2) &
(ex b1, b2 being total symmetric transitive Relation of a1,a1 st
ex b3, b4 being set st
b1 in the carrier of a2 &
b2 in the carrier of a2 &
[b3,b4] in b1 "\/" b2 &
(for b5 being non empty FinSequence of a1
st len b5 = it + 1
holds not b3,b4 are_joint_by b5,b1,b2));
end;
:: LATTICE5:def 4
theorem
for b1 being non empty set
for b2 being meet-inheriting join-inheriting SubRelStr of EqRelLATT b1
st (ex b3 being total symmetric transitive Relation of b1,b1 st
b3 in the carrier of b2 & b3 <> id b1) &
(ex b3 being Element of NAT st
for b4, b5 being total symmetric transitive Relation of b1,b1
for b6, b7 being set
st b4 in the carrier of b2 & b5 in the carrier of b2 & [b6,b7] in b4 "\/" b5
holds ex b8 being non empty FinSequence of b1 st
len b8 = b3 & b6,b7 are_joint_by b8,b4,b5)
for b3 being Element of NAT holds
b3 = type_of b2
iff
(for b4, b5 being total symmetric transitive Relation of b1,b1
for b6, b7 being set
st b4 in the carrier of b2 & b5 in the carrier of b2 & [b6,b7] in b4 "\/" b5
holds ex b8 being non empty FinSequence of b1 st
len b8 = b3 + 2 & b6,b7 are_joint_by b8,b4,b5) &
(ex b4, b5 being total symmetric transitive Relation of b1,b1 st
ex b6, b7 being set st
b4 in the carrier of b2 &
b5 in the carrier of b2 &
[b6,b7] in b4 "\/" b5 &
(for b8 being non empty FinSequence of b1
st len b8 = b3 + 1
holds not b6,b7 are_joint_by b8,b4,b5));
:: LATTICE5:th 15
theorem
for b1 being non empty set
for b2 being meet-inheriting join-inheriting SubRelStr of EqRelLATT b1
for b3 being Element of NAT
st (ex b4 being total symmetric transitive Relation of b1,b1 st
b4 in the carrier of b2 & b4 <> id b1) &
(for b4, b5 being total symmetric transitive Relation of b1,b1
for b6, b7 being set
st b4 in the carrier of b2 & b5 in the carrier of b2 & [b6,b7] in b4 "\/" b5
holds ex b8 being non empty FinSequence of b1 st
len b8 = b3 + 2 & b6,b7 are_joint_by b8,b4,b5)
holds type_of b2 <= b3;
:: LATTICE5:modenot 3
definition
let a1 be set;
let a2 be 1-sorted;
mode BiFunction of a1,a2 is Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
end;
:: LATTICE5:funcnot 4 => LATTICE5:func 4
definition
let a1 be non empty set;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4, a5 be Element of a1;
redefine func a3 .(a4,a5) -> Element of the carrier of a2;
end;
:: LATTICE5:attrnot 2 => LATTICE5:attr 1
definition
let a1 be non empty set;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
attr a3 is symmetric means
for b1, b2 being Element of a1 holds
a3 .(b1,b2) = a3 .(b2,b1);
end;
:: LATTICE5:dfs 5
definiens
let a1 be non empty set;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
To prove
a3 is symmetric
it is sufficient to prove
thus for b1, b2 being Element of a1 holds
a3 .(b1,b2) = a3 .(b2,b1);
:: LATTICE5:def 6
theorem
for b1 being non empty set
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
b3 is symmetric(b1, b2)
iff
for b4, b5 being Element of b1 holds
b3 .(b4,b5) = b3 .(b5,b4);
:: LATTICE5:attrnot 3 => LATTICE5:attr 2
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
attr a3 is zeroed means
for b1 being Element of a1 holds
a3 .(b1,b1) = Bottom a2;
end;
:: LATTICE5:dfs 6
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
To prove
a3 is zeroed
it is sufficient to prove
thus for b1 being Element of a1 holds
a3 .(b1,b1) = Bottom a2;
:: LATTICE5:def 7
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
b3 is zeroed(b1, b2)
iff
for b4 being Element of b1 holds
b3 .(b4,b4) = Bottom b2;
:: LATTICE5:attrnot 4 => LATTICE5:attr 3
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
attr a3 is u.t.i. means
for b1, b2, b3 being Element of a1 holds
a3 .(b1,b3) <= (a3 .(b1,b2)) "\/" (a3 .(b2,b3));
end;
:: LATTICE5:dfs 7
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
To prove
a3 is u.t.i.
it is sufficient to prove
thus for b1, b2, b3 being Element of a1 holds
a3 .(b1,b3) <= (a3 .(b1,b2)) "\/" (a3 .(b2,b3));
:: LATTICE5:def 8
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
b3 is u.t.i.(b1, b2)
iff
for b4, b5, b6 being Element of b1 holds
b3 .(b4,b6) <= (b3 .(b4,b5)) "\/" (b3 .(b5,b6));
:: LATTICE5:exreg 3
registration
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
cluster Relation-like Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
end;
:: LATTICE5:modenot 4
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
mode distance_function of a1,a2 is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
end;
:: LATTICE5:funcnot 5 => LATTICE5:func 5
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
func alpha A3 -> Function-like quasi_total Relation of the carrier of a2,the carrier of EqRelLATT a1 means
for b1 being Element of the carrier of a2 holds
ex b2 being total symmetric transitive Relation of a1,a1 st
b2 = it . b1 &
(for b3, b4 being Element of a1 holds
[b3,b4] in b2
iff
a3 .(b3,b4) <= b1);
end;
:: LATTICE5:def 9
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of EqRelLATT b1 holds
b4 = alpha b3
iff
for b5 being Element of the carrier of b2 holds
ex b6 being total symmetric transitive Relation of b1,b1 st
b6 = b4 . b5 &
(for b7, b8 being Element of b1 holds
[b7,b8] in b6
iff
b3 .(b7,b8) <= b5);
:: LATTICE5:th 16
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2 holds
alpha b3 is meet-preserving(b2, EqRelLATT b1);
:: LATTICE5:th 17
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
st b3 is onto([:b1,b1:], the carrier of b2)
holds alpha b3 is one-to-one;
:: LATTICE5:funcnot 6 => LATTICE5:func 6
definition
let a1 be set;
func new_set A1 -> set equals
a1 \/ {{a1},{{a1}},{{{a1}}}};
end;
:: LATTICE5:def 10
theorem
for b1 being set holds
new_set b1 = b1 \/ {{b1},{{b1}},{{{b1}}}};
:: LATTICE5:funcreg 3
registration
let a1 be set;
cluster new_set a1 -> non empty;
end;
:: LATTICE5:funcnot 7 => LATTICE5:func 7
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be Element of [:a1,a1,the carrier of a2,the carrier of a2:];
func new_bi_fun(A3,A4) -> Function-like quasi_total Relation of [:new_set a1,new_set a1:],the carrier of a2 means
(for b1, b2 being Element of a1 holds
it .(b1,b2) = a3 .(b1,b2)) &
it .({a1},{a1}) = Bottom a2 &
it .({{a1}},{{a1}}) = Bottom a2 &
it .({{{a1}}},{{{a1}}}) = Bottom a2 &
it .({{a1}},{{{a1}}}) = a4 `3 &
it .({{{a1}}},{{a1}}) = a4 `3 &
it .({a1},{{a1}}) = a4 `4 &
it .({{a1}},{a1}) = a4 `4 &
it .({a1},{{{a1}}}) = a4 `3 "\/" (a4 `4) &
it .({{{a1}}},{a1}) = a4 `3 "\/" (a4 `4) &
(for b1 being Element of a1 holds
it .(b1,{a1}) = (a3 .(b1,a4 `1)) "\/" (a4 `3) &
it .({a1},b1) = (a3 .(b1,a4 `1)) "\/" (a4 `3) &
it .(b1,{{a1}}) = ((a3 .(b1,a4 `1)) "\/" (a4 `3)) "\/" (a4 `4) &
it .({{a1}},b1) = ((a3 .(b1,a4 `1)) "\/" (a4 `3)) "\/" (a4 `4) &
it .(b1,{{{a1}}}) = (a3 .(b1,a4 `2)) "\/" (a4 `4) &
it .({{{a1}}},b1) = (a3 .(b1,a4 `2)) "\/" (a4 `4));
end;
:: LATTICE5:def 11
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:]
for b5 being Function-like quasi_total Relation of [:new_set b1,new_set b1:],the carrier of b2 holds
b5 = new_bi_fun(b3,b4)
iff
(for b6, b7 being Element of b1 holds
b5 .(b6,b7) = b3 .(b6,b7)) &
b5 .({b1},{b1}) = Bottom b2 &
b5 .({{b1}},{{b1}}) = Bottom b2 &
b5 .({{{b1}}},{{{b1}}}) = Bottom b2 &
b5 .({{b1}},{{{b1}}}) = b4 `3 &
b5 .({{{b1}}},{{b1}}) = b4 `3 &
b5 .({b1},{{b1}}) = b4 `4 &
b5 .({{b1}},{b1}) = b4 `4 &
b5 .({b1},{{{b1}}}) = b4 `3 "\/" (b4 `4) &
b5 .({{{b1}}},{b1}) = b4 `3 "\/" (b4 `4) &
(for b6 being Element of b1 holds
b5 .(b6,{b1}) = (b3 .(b6,b4 `1)) "\/" (b4 `3) &
b5 .({b1},b6) = (b3 .(b6,b4 `1)) "\/" (b4 `3) &
b5 .(b6,{{b1}}) = ((b3 .(b6,b4 `1)) "\/" (b4 `3)) "\/" (b4 `4) &
b5 .({{b1}},b6) = ((b3 .(b6,b4 `1)) "\/" (b4 `3)) "\/" (b4 `4) &
b5 .(b6,{{{b1}}}) = (b3 .(b6,b4 `2)) "\/" (b4 `4) &
b5 .({{{b1}}},b6) = (b3 .(b6,b4 `2)) "\/" (b4 `4));
:: LATTICE5:th 18
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is zeroed(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
new_bi_fun(b3,b4) is zeroed(new_set b1, b2);
:: LATTICE5:th 19
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is symmetric(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
new_bi_fun(b3,b4) is symmetric(new_set b1, b2);
:: LATTICE5:th 20
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is symmetric(b1, b2) & b3 is u.t.i.(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:]
st b3 .(b4 `1,b4 `2) <= b4 `3 "\/" (b4 `4)
holds new_bi_fun(b3,b4) is u.t.i.(new_set b1, b2);
:: LATTICE5:th 22
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
b3 c= new_bi_fun(b3,b4);
:: LATTICE5:funcnot 8 => LATTICE5:func 8
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
func DistEsti A3 -> cardinal set means
it,{[b1,b2,b3,b4] where b1 is Element of a1, b2 is Element of a1, b3 is Element of the carrier of a2, b4 is Element of the carrier of a2: a3 .(b1,b2) <= b3 "\/" b4} are_equipotent;
end;
:: LATTICE5:def 12
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being cardinal set holds
b4 = DistEsti b3
iff
b4,{[b5,b6,b7,b8] where b5 is Element of b1, b6 is Element of b1, b7 is Element of the carrier of b2, b8 is Element of the carrier of b2: b3 .(b5,b6) <= b7 "\/" b8} are_equipotent;
:: LATTICE5:th 23
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2 holds
DistEsti b3 <> {};
:: LATTICE5:funcnot 9 => LATTICE5:func 9
definition
let a1 be non empty set;
let a2 be ordinal set;
func ConsecutiveSet(A1,A2) -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a2 &
b1 . {} = a1 &
(for b2 being ordinal set
st succ b2 in succ a2
holds b1 . succ b2 = new_set (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a2 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = union proj2 (b1 | b2));
end;
:: LATTICE5:def 13
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being set holds
b3 = ConsecutiveSet(b1,b2)
iff
ex b4 being Relation-like Function-like T-Sequence-like set st
b3 = last b4 &
proj1 b4 = succ b2 &
b4 . {} = b1 &
(for b5 being ordinal set
st succ b5 in succ b2
holds b4 . succ b5 = new_set (b4 . b5)) &
(for b5 being ordinal set
st b5 in succ b2 & b5 <> {} & b5 is being_limit_ordinal
holds b4 . b5 = union proj2 (b4 | b5));
:: LATTICE5:th 24
theorem
for b1 being non empty set holds
ConsecutiveSet(b1,{}) = b1;
:: LATTICE5:th 25
theorem
for b1 being non empty set
for b2 being ordinal set holds
ConsecutiveSet(b1,succ b2) = new_set ConsecutiveSet(b1,b2);
:: LATTICE5:th 26
theorem
for b1 being non empty set
for b2 being Relation-like Function-like T-Sequence-like set
for b3 being ordinal set
st b3 <> {} &
b3 is being_limit_ordinal &
proj1 b2 = b3 &
(for b4 being ordinal set
st b4 in b3
holds b2 . b4 = ConsecutiveSet(b1,b4))
holds ConsecutiveSet(b1,b3) = union proj2 b2;
:: LATTICE5:funcreg 4
registration
let a1 be non empty set;
let a2 be ordinal set;
cluster ConsecutiveSet(a1,a2) -> non empty;
end;
:: LATTICE5:th 27
theorem
for b1 being non empty set
for b2 being ordinal set holds
b1 c= ConsecutiveSet(b1,b2);
:: LATTICE5:modenot 5 => LATTICE5:mode 1
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
mode QuadrSeq of A3 -> T-Sequence of [:a1,a1,the carrier of a2,the carrier of a2:] means
proj1 it is cardinal set &
it is one-to-one &
proj2 it = {[b1,b2,b3,b4] where b1 is Element of a1, b2 is Element of a1, b3 is Element of the carrier of a2, b4 is Element of the carrier of a2: a3 .(b1,b2) <= b3 "\/" b4};
end;
:: LATTICE5:dfs 13
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be T-Sequence of [:a1,a1,the carrier of a2,the carrier of a2:];
To prove
a4 is QuadrSeq of a3
it is sufficient to prove
thus proj1 a4 is cardinal set &
a4 is one-to-one &
proj2 a4 = {[b1,b2,b3,b4] where b1 is Element of a1, b2 is Element of a1, b3 is Element of the carrier of a2, b4 is Element of the carrier of a2: a3 .(b1,b2) <= b3 "\/" b4};
:: LATTICE5:def 14
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being T-Sequence of [:b1,b1,the carrier of b2,the carrier of b2:] holds
b4 is QuadrSeq of b3
iff
proj1 b4 is cardinal set &
b4 is one-to-one &
proj2 b4 = {[b5,b6,b7,b8] where b5 is Element of b1, b6 is Element of b1, b7 is Element of the carrier of b2, b8 is Element of the carrier of b2: b3 .(b5,b6) <= b7 "\/" b8};
:: LATTICE5:funcnot 10 => LATTICE5:func 10
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
let a5 be ordinal set;
assume a5 in proj1 a4;
func Quadr(A4,A5) -> Element of [:ConsecutiveSet(a1,a5),ConsecutiveSet(a1,a5),the carrier of a2,the carrier of a2:] equals
a4 . a5;
end;
:: LATTICE5:def 15
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set
st b5 in proj1 b4
holds Quadr(b4,b5) = b4 . b5;
:: LATTICE5:th 28
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b5 being QuadrSeq of b4 holds
b3 in DistEsti b4
iff
b3 in proj1 b5;
:: LATTICE5:funcnot 11 => LATTICE5:func 11
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be set;
assume a3 is Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
func BiFun(A3,A1,A2) -> Function-like quasi_total Relation of [:a1,a1:],the carrier of a2 equals
a3;
end;
:: LATTICE5:def 16
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being set
st b3 is Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
holds BiFun(b3,b1,b2) = b3;
:: LATTICE5:funcnot 12 => LATTICE5:func 12
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
let a5 be ordinal set;
func ConsecutiveDelta(A4,A5) -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a5 &
b1 . {} = a3 &
(for b2 being ordinal set
st succ b2 in succ a5
holds b1 . succ b2 = new_bi_fun(BiFun(b1 . b2,ConsecutiveSet(a1,b2),a2),Quadr(a4,b2))) &
(for b2 being ordinal set
st b2 in succ a5 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = union proj2 (b1 | b2));
end;
:: LATTICE5:def 17
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set
for b6 being set holds
b6 = ConsecutiveDelta(b4,b5)
iff
ex b7 being Relation-like Function-like T-Sequence-like set st
b6 = last b7 &
proj1 b7 = succ b5 &
b7 . {} = b3 &
(for b8 being ordinal set
st succ b8 in succ b5
holds b7 . succ b8 = new_bi_fun(BiFun(b7 . b8,ConsecutiveSet(b1,b8),b2),Quadr(b4,b8))) &
(for b8 being ordinal set
st b8 in succ b5 & b8 <> {} & b8 is being_limit_ordinal
holds b7 . b8 = union proj2 (b7 | b8));
:: LATTICE5:th 29
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
ConsecutiveDelta(b4,{}) = b3;
:: LATTICE5:th 30
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b5 being QuadrSeq of b4 holds
ConsecutiveDelta(b5,succ b3) = new_bi_fun(BiFun(ConsecutiveDelta(b5,b3),ConsecutiveSet(b1,b3),b2),Quadr(b5,b3));
:: LATTICE5:th 31
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Relation-like Function-like T-Sequence-like set
for b4 being ordinal set
for b5 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b6 being QuadrSeq of b5
st b4 <> {} &
b4 is being_limit_ordinal &
proj1 b3 = b4 &
(for b7 being ordinal set
st b7 in b4
holds b3 . b7 = ConsecutiveDelta(b6,b7))
holds ConsecutiveDelta(b6,b4) = union proj2 b3;
:: LATTICE5:th 32
theorem
for b1 being non empty set
for b2, b3 being ordinal set
st b2 c= b3
holds ConsecutiveSet(b1,b2) c= ConsecutiveSet(b1,b3);
:: LATTICE5:th 33
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b5 being QuadrSeq of b4 holds
ConsecutiveDelta(b5,b3) is Function-like quasi_total Relation of [:ConsecutiveSet(b1,b3),ConsecutiveSet(b1,b3):],the carrier of b2;
:: LATTICE5:funcnot 13 => LATTICE5:func 13
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
let a5 be ordinal set;
redefine func ConsecutiveDelta(a4,a5) -> Function-like quasi_total Relation of [:ConsecutiveSet(a1,a5),ConsecutiveSet(a1,a5):],the carrier of a2;
end;
:: LATTICE5:th 34
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b5 being QuadrSeq of b4 holds
b4 c= ConsecutiveDelta(b5,b3);
:: LATTICE5:th 35
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3, b4 being ordinal set
for b5 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b6 being QuadrSeq of b5
st b3 c= b4
holds ConsecutiveDelta(b6,b3) c= ConsecutiveDelta(b6,b4);
:: LATTICE5:th 36
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b4 is zeroed(b1, b2)
for b5 being QuadrSeq of b4 holds
ConsecutiveDelta(b5,b3) is zeroed(ConsecutiveSet(b1,b3), b2);
:: LATTICE5:th 37
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b4 is symmetric(b1, b2)
for b5 being QuadrSeq of b4 holds
ConsecutiveDelta(b5,b3) is symmetric(ConsecutiveSet(b1,b3), b2);
:: LATTICE5:th 38
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b4 is symmetric(b1, b2) & b4 is u.t.i.(b1, b2)
for b5 being QuadrSeq of b4
st b3 c= DistEsti b4
holds ConsecutiveDelta(b5,b3) is u.t.i.(ConsecutiveSet(b1,b3), b2);
:: LATTICE5:th 39
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being ordinal set
for b4 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b5 being QuadrSeq of b4
st b3 c= DistEsti b4
holds ConsecutiveDelta(b5,b3) is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:ConsecutiveSet(b1,b3),ConsecutiveSet(b1,b3):],the carrier of b2;
:: LATTICE5:funcnot 14 => LATTICE5:func 14
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
func NextSet A3 -> set equals
ConsecutiveSet(a1,DistEsti a3);
end;
:: LATTICE5:def 18
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
NextSet b3 = ConsecutiveSet(b1,DistEsti b3);
:: LATTICE5:funcreg 5
registration
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
cluster NextSet a3 -> non empty;
end;
:: LATTICE5:funcnot 15 => LATTICE5:func 15
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
func NextDelta A4 -> set equals
ConsecutiveDelta(a4,DistEsti a3);
end;
:: LATTICE5:def 19
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
NextDelta b4 = ConsecutiveDelta(b4,DistEsti b3);
:: LATTICE5:funcnot 16 => LATTICE5:func 16
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
redefine func NextDelta a4 -> Function-like quasi_total symmetric zeroed u.t.i. Relation of [:NextSet a3,NextSet a3:],the carrier of a2;
end;
:: LATTICE5:prednot 2 => LATTICE5:pred 2
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
let a4 be non empty set;
let a5 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a4,a4:],the carrier of a2;
pred A4,A5 is_extension_of A1,A3 means
ex b1 being QuadrSeq of a3 st
a4 = NextSet a3 & a5 = NextDelta b1;
end;
:: LATTICE5:dfs 19
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
let a4 be non empty set;
let a5 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a4,a4:],the carrier of a2;
To prove
a4,a5 is_extension_of a1,a3
it is sufficient to prove
thus ex b1 being QuadrSeq of a3 st
a4 = NextSet a3 & a5 = NextDelta b1;
:: LATTICE5:def 20
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being non empty set
for b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of b2 holds
b4,b5 is_extension_of b1,b3
iff
ex b6 being QuadrSeq of b3 st
b4 = NextSet b3 & b5 = NextDelta b6;
:: LATTICE5:th 40
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being non empty set
for b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of b2
st b4,b5 is_extension_of b1,b3
for b6, b7 being Element of b1
for b8, b9 being Element of the carrier of b2
st b3 .(b6,b7) <= b8 "\/" b9
holds ex b10, b11, b12 being Element of b4 st
b5 .(b6,b10) = b8 & b5 .(b11,b12) = b8 & b5 .(b10,b11) = b9 & b5 .(b12,b7) = b9;
:: LATTICE5:modenot 6 => LATTICE5:mode 2
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
mode ExtensionSeq of A1,A3 -> Relation-like Function-like set means
proj1 it = NAT &
it . 0 = [a1,a3] &
(for b1 being Element of NAT holds
ex b2 being non empty set st
ex b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b2,b2:],the carrier of a2 st
ex b4 being non empty set st
ex b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of a2 st
b4,b5 is_extension_of b2,b3 & it . b1 = [b2,b3] & it . (b1 + 1) = [b4,b5]);
end;
:: LATTICE5:dfs 20
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
let a4 be Relation-like Function-like set;
To prove
a4 is ExtensionSeq of a1,a3
it is sufficient to prove
thus proj1 a4 = NAT &
a4 . 0 = [a1,a3] &
(for b1 being Element of NAT holds
ex b2 being non empty set st
ex b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b2,b2:],the carrier of a2 st
ex b4 being non empty set st
ex b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of a2 st
b4,b5 is_extension_of b2,b3 & a4 . b1 = [b2,b3] & a4 . (b1 + 1) = [b4,b5]);
:: LATTICE5:def 21
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being Relation-like Function-like set holds
b4 is ExtensionSeq of b1,b3
iff
proj1 b4 = NAT &
b4 . 0 = [b1,b3] &
(for b5 being Element of NAT holds
ex b6 being non empty set st
ex b7 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b6,b6:],the carrier of b2 st
ex b8 being non empty set st
ex b9 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b8,b8:],the carrier of b2 st
b8,b9 is_extension_of b6,b7 & b4 . b5 = [b6,b7] & b4 . (b5 + 1) = [b8,b9]);
:: LATTICE5:th 41
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq of b1,b3
for b5, b6 being Element of NAT
st b5 <= b6
holds (b4 . b5) `1 c= (b4 . b6) `1;
:: LATTICE5:th 42
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq of b1,b3
for b5, b6 being Element of NAT
st b5 <= b6
holds (b4 . b5) `2 c= (b4 . b6) `2;
:: LATTICE5:funcnot 17 => LATTICE5:func 17
definition
let a1 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr;
func BasicDF A1 -> Function-like quasi_total symmetric zeroed u.t.i. Relation of [:the carrier of a1,the carrier of a1:],the carrier of a1 means
for b1, b2 being Element of the carrier of a1 holds
(b1 = b2 or it .(b1,b2) = b1 "\/" b2) & (b1 = b2 implies it .(b1,b2) = Bottom a1);
end;
:: LATTICE5:def 22
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:the carrier of b1,the carrier of b1:],the carrier of b1 holds
b2 = BasicDF b1
iff
for b3, b4 being Element of the carrier of b1 holds
(b3 = b4 or b2 .(b3,b4) = b3 "\/" b4) & (b3 = b4 implies b2 .(b3,b4) = Bottom b1);
:: LATTICE5:th 43
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
BasicDF b1 is onto([:the carrier of b1,the carrier of b1:], the carrier of b1);
:: LATTICE5:th 44
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2 being ExtensionSeq of the carrier of b1,BasicDF b1
for b3 being non empty set
st b3 = union {(b2 . b4) `1 where b4 is Element of NAT: TRUE}
holds union {(b2 . b4) `2 where b4 is Element of NAT: TRUE} is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1;
:: LATTICE5:th 45
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2 being ExtensionSeq of the carrier of b1,BasicDF b1
for b3 being non empty set
for b4 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1
for b5, b6 being Element of b3
for b7, b8 being Element of the carrier of b1
st b3 = union {(b2 . b9) `1 where b9 is Element of NAT: TRUE} &
b4 = union {(b2 . b9) `2 where b9 is Element of NAT: TRUE} &
b4 .(b5,b6) <= b7 "\/" b8
holds ex b9, b10, b11 being Element of b3 st
b4 .(b5,b9) = b7 & b4 .(b10,b11) = b7 & b4 .(b9,b10) = b8 & b4 .(b11,b6) = b8;
:: LATTICE5:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2 being ExtensionSeq of the carrier of b1,BasicDF b1
for b3 being non empty set
for b4 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1
for b5 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b3
for b6, b7 being Element of b3
for b8, b9 being total symmetric transitive Relation of b3,b3
for b10, b11 being set
st b5 = alpha b4 &
b3 = union {(b2 . b12) `1 where b12 is Element of NAT: TRUE} &
b4 = union {(b2 . b12) `2 where b12 is Element of NAT: TRUE} &
b8 in the carrier of Image b5 &
b9 in the carrier of Image b5 &
[b10,b11] in b8 "\/" b9
holds ex b12 being non empty FinSequence of b3 st
len b12 = 3 + 2 & b10,b11 are_joint_by b12,b8,b9;
:: LATTICE5:th 47
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr holds
ex b2 being non empty set st
ex b3 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b2 st
b3 is one-to-one & type_of Image b3 <= 3;