Article LATTICE8, MML version 4.99.1005
:: LATTICE8:attrnot 1 => LATTICE8:attr 1
definition
let a1 be RelStr;
attr a1 is finitely_typed means
ex b1 being non empty set st
(for b2 being set
st b2 in the carrier of a1
holds b2 is total symmetric transitive Relation of b1,b1) &
(ex b2 being Element of NAT st
for b3, b4 being total symmetric transitive Relation of b1,b1
for b5, b6 being set
st b3 in the carrier of a1 & b4 in the carrier of a1 & [b5,b6] in b3 "\/" b4
holds ex b7 being non empty FinSequence of b1 st
len b7 = b2 & b5,b6 are_joint_by b7,b3,b4);
end;
:: LATTICE8:dfs 1
definiens
let a1 be RelStr;
To prove
a1 is finitely_typed
it is sufficient to prove
thus ex b1 being non empty set st
(for b2 being set
st b2 in the carrier of a1
holds b2 is total symmetric transitive Relation of b1,b1) &
(ex b2 being Element of NAT st
for b3, b4 being total symmetric transitive Relation of b1,b1
for b5, b6 being set
st b3 in the carrier of a1 & b4 in the carrier of a1 & [b5,b6] in b3 "\/" b4
holds ex b7 being non empty FinSequence of b1 st
len b7 = b2 & b5,b6 are_joint_by b7,b3,b4);
:: LATTICE8:def 2
theorem
for b1 being RelStr holds
b1 is finitely_typed
iff
ex b2 being non empty set st
(for b3 being set
st b3 in the carrier of b1
holds b3 is total symmetric transitive Relation of b2,b2) &
(ex b3 being Element of NAT st
for b4, b5 being total symmetric transitive Relation of b2,b2
for b6, b7 being set
st b4 in the carrier of b1 & b5 in the carrier of b1 & [b6,b7] in b4 "\/" b5
holds ex b8 being non empty FinSequence of b2 st
len b8 = b3 & b6,b7 are_joint_by b8,b4,b5);
:: LATTICE8:prednot 1 => LATTICE8:pred 1
definition
let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
let a2 be Element of NAT;
pred A1 has_a_representation_of_type<= A2 means
ex b1 being non trivial set st
ex b2 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of EqRelLATT b1 st
b2 is one-to-one &
Image b2 is finitely_typed &
(ex b3 being total symmetric transitive Relation of b1,b1 st
b3 in the carrier of Image b2 & b3 <> id b1) &
type_of Image b2 <= a2;
end;
:: LATTICE8:dfs 2
definiens
let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
let a2 be Element of NAT;
To prove
a1 has_a_representation_of_type<= a2
it is sufficient to prove
thus ex b1 being non trivial set st
ex b2 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of EqRelLATT b1 st
b2 is one-to-one &
Image b2 is finitely_typed &
(ex b3 being total symmetric transitive Relation of b1,b1 st
b3 in the carrier of Image b2 & b3 <> id b1) &
type_of Image b2 <= a2;
:: LATTICE8:def 3
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being Element of NAT holds
b1 has_a_representation_of_type<= b2
iff
ex b3 being non trivial set st
ex b4 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b3 st
b4 is one-to-one &
Image b4 is finitely_typed &
(ex b5 being total symmetric transitive Relation of b3,b3 st
b5 in the carrier of Image b4 & b5 <> id b3) &
type_of Image b4 <= b2;
:: LATTICE8:exreg 1
registration
cluster non empty finite reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima RelStr;
end;
:: LATTICE8:exreg 2
registration
let a1 be non trivial set;
cluster non empty non trivial full meet-inheriting join-inheriting finitely_typed SubRelStr of EqRelLATT a1;
end;
:: LATTICE8:th 1
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2 holds
succ {} c= DistEsti b3;
:: LATTICE8:th 2
theorem
for b1 being trivial reflexive transitive antisymmetric with_infima RelStr holds
b1 is modular;
:: LATTICE8:th 3
theorem
for b1 being non empty set
for b2 being non empty meet-inheriting join-inheriting SubRelStr of EqRelLATT b1
st b2 is not trivial
holds ex b3 being total symmetric transitive Relation of b1,b1 st
b3 in the carrier of b2 & b3 <> id b1;
:: LATTICE8:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is infs-preserving(b1, b2) & b3 is sups-preserving(b1, b2)
holds b3 is meet-preserving(b1, b2) & b3 is join-preserving(b1, b2);
:: LATTICE8:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
st b1,b2 are_isomorphic & b1 is modular
holds b2 is modular;
:: LATTICE8:th 6
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2 holds
Image b3 is lower-bounded;
:: LATTICE8:th 7
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2, b3 being Element of the carrier of b1
for b4 being non empty set
for b5 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b4
st b5 is one-to-one &
(corestr b5) . b2 <= (corestr b5) . b3
holds b2 <= b3;
:: LATTICE8:th 8
theorem
for b1 being non trivial set
for b2 being non empty full meet-inheriting join-inheriting finitely_typed SubRelStr of EqRelLATT b1
for b3 being total symmetric transitive Relation of b1,b1
st b3 in the carrier of b2 & b3 <> id b1 & type_of b2 <= 2
holds b2 is modular;
:: LATTICE8:th 9
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
st b1 has_a_representation_of_type<= 2
holds b1 is modular;
:: LATTICE8:funcnot 1 => LATTICE8:func 1
definition
let a1 be set;
func new_set2 A1 -> set equals
a1 \/ {{a1},{{a1}}};
end;
:: LATTICE8:def 4
theorem
for b1 being set holds
new_set2 b1 = b1 \/ {{b1},{{b1}}};
:: LATTICE8:funcreg 1
registration
let a1 be set;
cluster new_set2 a1 -> non empty;
end;
:: LATTICE8:funcnot 2 => LATTICE8:func 2
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_fun2(A3,A4) -> Function-like quasi_total Relation of [:new_set2 a1,new_set2 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}}) = ((a3 .(a4 `1,a4 `2)) "\/" (a4 `3)) "/\" (a4 `4) &
it .({{a1}},{a1}) = ((a3 .(a4 `1,a4 `2)) "\/" (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 `2)) "\/" (a4 `3) &
it .({{a1}},b1) = (a3 .(b1,a4 `2)) "\/" (a4 `3));
end;
:: LATTICE8:def 5
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_set2 b1,new_set2 b1:],the carrier of b2 holds
b5 = new_bi_fun2(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}}) = ((b3 .(b4 `1,b4 `2)) "\/" (b4 `3)) "/\" (b4 `4) &
b5 .({{b1}},{b1}) = ((b3 .(b4 `1,b4 `2)) "\/" (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 `2)) "\/" (b4 `3) &
b5 .({{b1}},b6) = (b3 .(b6,b4 `2)) "\/" (b4 `3));
:: LATTICE8:th 10
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_fun2(b3,b4) is zeroed(new_set2 b1, b2);
:: LATTICE8:th 11
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_fun2(b3,b4) is symmetric(new_set2 b1, b2);
:: LATTICE8:th 12
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
st b2 is modular
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_fun2(b3,b4) is u.t.i.(new_set2 b1, b2);
:: LATTICE8:th 14
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_fun2(b3,b4);
:: LATTICE8:funcnot 3 => LATTICE8:func 3
definition
let a1 be non empty set;
let a2 be ordinal set;
func ConsecutiveSet2(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_set2 (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;
:: LATTICE8:def 6
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being set holds
b3 = ConsecutiveSet2(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_set2 (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));
:: LATTICE8:th 15
theorem
for b1 being non empty set holds
ConsecutiveSet2(b1,{}) = b1;
:: LATTICE8:th 16
theorem
for b1 being non empty set
for b2 being ordinal set holds
ConsecutiveSet2(b1,succ b2) = new_set2 ConsecutiveSet2(b1,b2);
:: LATTICE8:th 17
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being Relation-like Function-like T-Sequence-like set
st b2 <> {} &
b2 is being_limit_ordinal &
proj1 b3 = b2 &
(for b4 being ordinal set
st b4 in b2
holds b3 . b4 = ConsecutiveSet2(b1,b4))
holds ConsecutiveSet2(b1,b2) = union proj2 b3;
:: LATTICE8:funcreg 2
registration
let a1 be non empty set;
let a2 be ordinal set;
cluster ConsecutiveSet2(a1,a2) -> non empty;
end;
:: LATTICE8:th 18
theorem
for b1 being non empty set
for b2 being ordinal set holds
b1 c= ConsecutiveSet2(b1,b2);
:: LATTICE8:funcnot 4 => LATTICE8:func 4
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 Quadr2(A4,A5) -> Element of [:ConsecutiveSet2(a1,a5),ConsecutiveSet2(a1,a5),the carrier of a2,the carrier of a2:] equals
a4 . a5;
end;
:: LATTICE8:def 7
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 Quadr2(b4,b5) = b4 . b5;
:: LATTICE8:funcnot 5 => LATTICE8:func 5
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 ConsecutiveDelta2(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_fun2(BiFun(b1 . b2,ConsecutiveSet2(a1,b2),a2),Quadr2(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;
:: LATTICE8:def 8
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 = ConsecutiveDelta2(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_fun2(BiFun(b7 . b8,ConsecutiveSet2(b1,b8),b2),Quadr2(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));
:: LATTICE8:th 19
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
ConsecutiveDelta2(b4,{}) = b3;
:: LATTICE8:th 20
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 holds
ConsecutiveDelta2(b4,succ b5) = new_bi_fun2(BiFun(ConsecutiveDelta2(b4,b5),ConsecutiveSet2(b1,b5),b2),Quadr2(b4,b5));
:: LATTICE8:th 21
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 Relation-like Function-like T-Sequence-like set
for b6 being ordinal set
st b6 <> {} &
b6 is being_limit_ordinal &
proj1 b5 = b6 &
(for b7 being ordinal set
st b7 in b6
holds b5 . b7 = ConsecutiveDelta2(b4,b7))
holds ConsecutiveDelta2(b4,b6) = union proj2 b5;
:: LATTICE8:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being ordinal set
st b3 c= b4
holds ConsecutiveSet2(b1,b3) c= ConsecutiveSet2(b1,b4);
:: LATTICE8:th 23
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 holds
ConsecutiveDelta2(b4,b5) is Function-like quasi_total Relation of [:ConsecutiveSet2(b1,b5),ConsecutiveSet2(b1,b5):],the carrier of b2;
:: LATTICE8:funcnot 6 => LATTICE8:func 6
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 ConsecutiveDelta2(a4,a5) -> Function-like quasi_total Relation of [:ConsecutiveSet2(a1,a5),ConsecutiveSet2(a1,a5):],the carrier of a2;
end;
:: LATTICE8:th 24
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 holds
b3 c= ConsecutiveDelta2(b4,b5);
:: LATTICE8:th 25
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4, b5 being ordinal set
for b6 being QuadrSeq of b3
st b4 c= b5
holds ConsecutiveDelta2(b6,b4) c= ConsecutiveDelta2(b6,b5);
:: LATTICE8:th 26
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 QuadrSeq of b3
for b5 being ordinal set holds
ConsecutiveDelta2(b4,b5) is zeroed(ConsecutiveSet2(b1,b5), b2);
:: LATTICE8:th 27
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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 QuadrSeq of b3
for b5 being ordinal set holds
ConsecutiveDelta2(b4,b5) is symmetric(ConsecutiveSet2(b1,b5), b2);
:: LATTICE8:th 28
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
st b2 is modular
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 ordinal set
for b5 being QuadrSeq of b3
st b4 c= DistEsti b3
holds ConsecutiveDelta2(b5,b4) is u.t.i.(ConsecutiveSet2(b1,b4), b2);
:: LATTICE8:th 29
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ordinal set
for b5 being QuadrSeq of b3
st b4 c= DistEsti b3
holds ConsecutiveDelta2(b5,b4) is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:ConsecutiveSet2(b1,b4),ConsecutiveSet2(b1,b4):],the carrier of b2;
:: LATTICE8:funcnot 7 => LATTICE8:func 7
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
func NextSet2 A3 -> set equals
ConsecutiveSet2(a1,DistEsti a3);
end;
:: LATTICE8:def 9
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
NextSet2 b3 = ConsecutiveSet2(b1,DistEsti b3);
:: LATTICE8:funcreg 3
registration
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
cluster NextSet2 a3 -> non empty;
end;
:: LATTICE8:funcnot 8 => LATTICE8:func 8
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be QuadrSeq of a3;
func NextDelta2 A4 -> set equals
ConsecutiveDelta2(a4,DistEsti a3);
end;
:: LATTICE8:def 10
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
NextDelta2 b4 = ConsecutiveDelta2(b4,DistEsti b3);
:: LATTICE8:funcnot 9 => LATTICE8:func 9
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular 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 NextDelta2 a4 -> Function-like quasi_total symmetric zeroed u.t.i. Relation of [:NextSet2 a3,NextSet2 a3:],the carrier of a2;
end;
:: LATTICE8:prednot 2 => LATTICE8:pred 2
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_extension2_of A1,A3 means
ex b1 being QuadrSeq of a3 st
a4 = NextSet2 a3 & a5 = NextDelta2 b1;
end;
:: LATTICE8:dfs 10
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_extension2_of a1,a3
it is sufficient to prove
thus ex b1 being QuadrSeq of a3 st
a4 = NextSet2 a3 & a5 = NextDelta2 b1;
:: LATTICE8:def 11
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_extension2_of b1,b3
iff
ex b6 being QuadrSeq of b3 st
b4 = NextSet2 b3 & b5 = NextDelta2 b6;
:: LATTICE8:th 30
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima 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_extension2_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 being Element of b4 st
b5 .(b6,b10) = b8 &
b5 .(b10,b11) = ((b3 .(b6,b7)) "\/" b8) "/\" b9 &
b5 .(b11,b7) = b8;
:: LATTICE8:modenot 1 => LATTICE8:mode 1
definition
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr;
let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
mode ExtensionSeq2 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_extension2_of b2,b3 & it . b1 = [b2,b3] & it . (b1 + 1) = [b4,b5]);
end;
:: LATTICE8:dfs 11
definiens
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular 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 ExtensionSeq2 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_extension2_of b2,b3 & a4 . b1 = [b2,b3] & a4 . (b1 + 1) = [b4,b5]);
:: LATTICE8:def 12
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular 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 ExtensionSeq2 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_extension2_of b6,b7 & b4 . b5 = [b6,b7] & b4 . (b5 + 1) = [b8,b9]);
:: LATTICE8:th 31
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq2 of b1,b3
for b5, b6 being Element of NAT
st b5 <= b6
holds (b4 . b5) `1 c= (b4 . b6) `1;
:: LATTICE8:th 32
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq2 of b1,b3
for b5, b6 being Element of NAT
st b5 <= b6
holds (b4 . b5) `2 c= (b4 . b6) `2;
:: LATTICE8:th 33
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 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;
:: LATTICE8:th 34
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 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 being Element of b3 st
b4 .(b5,b9) = b7 &
b4 .(b9,b10) = ((b4 .(b5,b6)) "\/" b7) "/\" b8 &
b4 .(b10,b6) = b7;
:: LATTICE8:th 35
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 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 total symmetric transitive Relation of b3,b3
for b8, b9 being set
st b5 = alpha b4 &
b3 = union {(b2 . b10) `1 where b10 is Element of NAT: TRUE} &
b4 = union {(b2 . b10) `2 where b10 is Element of NAT: TRUE} &
b6 in the carrier of Image b5 &
b7 in the carrier of Image b5 &
[b8,b9] in b6 "\/" b7
holds ex b10 being non empty FinSequence of b3 st
len b10 = 2 + 2 & b8,b9 are_joint_by b10,b6,b7;
:: LATTICE8:th 36
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr holds
b1 has_a_representation_of_type<= 2;
:: LATTICE8:th 37
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr holds
b1 has_a_representation_of_type<= 2
iff
b1 is modular;