Article NECKLA_3, MML version 4.99.1005
:: NECKLA_3:th 1
theorem
for b1, b2 being set holds
(id b1) | b2 = (id b1) /\ [:b2,b2:];
:: NECKLA_3:th 2
theorem
for b1, b2, b3, b4 being set holds
id {b1,b2,b3,b4} = {[b1,b1],[b2,b2],[b3,b3],[b4,b4]};
:: NECKLA_3:th 3
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:{b1,b2,b3,b4},{b5,b6,b7,b8}:] = {[b1,b5],[b1,b6],[b2,b5],[b2,b6],[b1,b7],[b1,b8],[b2,b7],[b2,b8]} \/ {[b3,b5],[b3,b6],[b4,b5],[b4,b6],[b3,b7],[b3,b8],[b4,b7],[b4,b8]};
:: NECKLA_3:condreg 1
registration
let a1, a2 be trivial set;
cluster -> trivial (Relation of a1,a2);
end;
:: NECKLA_3:th 4
theorem
for b1 being trivial set
for b2 being Relation of b1,b1
st b2 is not empty
holds ex b3 being set st
b2 = {[b3,b3]};
:: NECKLA_3:condreg 2
registration
let a1 be trivial set;
cluster -> reflexive symmetric strongly_connected transitive trivial (Relation of a1,a1);
end;
:: NECKLA_3:th 5
theorem
for b1 being non empty trivial set
for b2 being Relation of b1,b1 holds
b2 is_symmetric_in b1;
:: NECKLA_3:exreg 1
registration
cluster non empty finite strict symmetric irreflexive RelStr;
end;
:: NECKLA_3:condreg 3
registration
let a1 be irreflexive RelStr;
cluster full -> irreflexive (SubRelStr of a1);
end;
:: NECKLA_3:condreg 4
registration
let a1 be symmetric RelStr;
cluster full -> symmetric (SubRelStr of a1);
end;
:: NECKLA_3:th 6
theorem
for b1 being symmetric irreflexive RelStr
st Card the carrier of b1 = 2
holds ex b2, b3 being set st
the carrier of b1 = {b2,b3} &
(the InternalRel of b1 = {[b2,b3],[b3,b2]} or the InternalRel of b1 = {});
:: NECKLA_3:funcreg 1
registration
let a1 be non empty RelStr;
let a2 be RelStr;
cluster union_of(a1,a2) -> non empty strict;
end;
:: NECKLA_3:funcreg 2
registration
let a1 be non empty RelStr;
let a2 be RelStr;
cluster sum_of(a1,a2) -> non empty strict;
end;
:: NECKLA_3:funcreg 3
registration
let a1 be RelStr;
let a2 be non empty RelStr;
cluster union_of(a1,a2) -> non empty strict;
end;
:: NECKLA_3:funcreg 4
registration
let a1 be RelStr;
let a2 be non empty RelStr;
cluster sum_of(a1,a2) -> non empty strict;
end;
:: NECKLA_3:funcreg 5
registration
let a1, a2 be finite RelStr;
cluster union_of(a1,a2) -> finite strict;
end;
:: NECKLA_3:funcreg 6
registration
let a1, a2 be finite RelStr;
cluster sum_of(a1,a2) -> finite strict;
end;
:: NECKLA_3:funcreg 7
registration
let a1, a2 be symmetric RelStr;
cluster union_of(a1,a2) -> strict symmetric;
end;
:: NECKLA_3:funcreg 8
registration
let a1, a2 be symmetric RelStr;
cluster sum_of(a1,a2) -> strict symmetric;
end;
:: NECKLA_3:funcreg 9
registration
let a1, a2 be irreflexive RelStr;
cluster union_of(a1,a2) -> strict irreflexive;
end;
:: NECKLA_3:th 7
theorem
for b1, b2 being irreflexive RelStr
st the carrier of b1 misses the carrier of b2
holds sum_of(b1,b2) is irreflexive;
:: NECKLA_3:th 8
theorem
for b1, b2 being RelStr holds
union_of(b1,b2) = union_of(b2,b1) & sum_of(b1,b2) = sum_of(b2,b1);
:: NECKLA_3:th 9
theorem
for b1 being irreflexive RelStr
for b2, b3 being RelStr
st (b1 = union_of(b2,b3) or b1 = sum_of(b2,b3))
holds b2 is irreflexive & b3 is irreflexive;
:: NECKLA_3:th 10
theorem
for b1 being non empty RelStr
for b2, b3 being RelStr
st the carrier of b2 misses the carrier of b3 &
(RelStr(#the carrier of b1,the InternalRel of b1#) = union_of(b2,b3) or RelStr(#the carrier of b1,the InternalRel of b1#) = sum_of(b2,b3))
holds b2 is full SubRelStr of b1 & b3 is full SubRelStr of b1;
:: NECKLA_3:th 11
theorem
the InternalRel of ComplRelStr Necklace 4 = {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]};
:: NECKLA_3:funcreg 10
registration
let a1 be RelStr;
cluster ComplRelStr a1 -> strict irreflexive;
end;
:: NECKLA_3:funcreg 11
registration
let a1 be symmetric RelStr;
cluster ComplRelStr a1 -> strict symmetric;
end;
:: NECKLA_3:th 12
theorem
for b1 being RelStr holds
the InternalRel of b1 misses the InternalRel of ComplRelStr b1;
:: NECKLA_3:th 13
theorem
for b1 being RelStr holds
id the carrier of b1 misses the InternalRel of ComplRelStr b1;
:: NECKLA_3:th 14
theorem
for b1 being RelStr holds
[:the carrier of b1,the carrier of b1:] = ((id the carrier of b1) \/ the InternalRel of b1) \/ the InternalRel of ComplRelStr b1;
:: NECKLA_3:th 15
theorem
for b1 being strict irreflexive RelStr
st b1 is trivial
holds ComplRelStr b1 = b1;
:: NECKLA_3:th 16
theorem
for b1 being strict irreflexive RelStr holds
ComplRelStr ComplRelStr b1 = b1;
:: NECKLA_3:th 17
theorem
for b1, b2 being RelStr
st the carrier of b1 misses the carrier of b2
holds ComplRelStr union_of(b1,b2) = sum_of(ComplRelStr b1,ComplRelStr b2);
:: NECKLA_3:th 18
theorem
for b1, b2 being RelStr
st the carrier of b1 misses the carrier of b2
holds ComplRelStr sum_of(b1,b2) = union_of(ComplRelStr b1,ComplRelStr b2);
:: NECKLA_3:th 19
theorem
for b1 being RelStr
for b2 being full SubRelStr of b1 holds
the InternalRel of ComplRelStr b2 = (the InternalRel of ComplRelStr b1) |_2 the carrier of ComplRelStr b2;
:: NECKLA_3:th 20
theorem
for b1 being non empty irreflexive RelStr
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of ComplRelStr b1
st b2 = b3
holds ComplRelStr subrelstr (([#] b1) \ {b2}) = subrelstr (([#] ComplRelStr b1) \ {b3});
:: NECKLA_3:condreg 5
registration
cluster non empty trivial strict -> N-free (RelStr);
end;
:: NECKLA_3:th 21
theorem
for b1 being reflexive antisymmetric RelStr
for b2 being RelStr holds
ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
for b4, b5 being Element of the carrier of b1 holds
[b4,b5] in the InternalRel of b1
iff
[b3 . b4,b3 . b5] in the InternalRel of b2
iff
b2 embeds b1;
:: NECKLA_3:th 22
theorem
for b1 being non empty RelStr
for b2 being non empty full SubRelStr of b1 holds
b1 embeds b2;
:: NECKLA_3:th 23
theorem
for b1 being non empty RelStr
for b2 being non empty full SubRelStr of b1
st b1 is N-free
holds b2 is N-free;
:: NECKLA_3:th 24
theorem
for b1 being non empty irreflexive RelStr holds
b1 embeds Necklace 4
iff
ComplRelStr b1 embeds Necklace 4;
:: NECKLA_3:th 25
theorem
for b1 being non empty irreflexive RelStr holds
b1 is N-free
iff
ComplRelStr b1 is N-free;
:: NECKLA_3:modenot 1
definition
let a1 be RelStr;
mode path of a1 is RedSequence of the InternalRel of a1;
end;
:: NECKLA_3:attrnot 1 => NECKLA_3:attr 1
definition
let a1 be RelStr;
attr a1 is path-connected means
for b1, b2 being set
st b1 in the carrier of a1 & b2 in the carrier of a1 & b1 <> b2 & not the InternalRel of a1 reduces b1,b2
holds the InternalRel of a1 reduces b2,b1;
end;
:: NECKLA_3:dfs 1
definiens
let a1 be RelStr;
To prove
a1 is path-connected
it is sufficient to prove
thus for b1, b2 being set
st b1 in the carrier of a1 & b2 in the carrier of a1 & b1 <> b2 & not the InternalRel of a1 reduces b1,b2
holds the InternalRel of a1 reduces b2,b1;
:: NECKLA_3:def 1
theorem
for b1 being RelStr holds
b1 is path-connected
iff
for b2, b3 being set
st b2 in the carrier of b1 & b3 in the carrier of b1 & b2 <> b3 & not the InternalRel of b1 reduces b2,b3
holds the InternalRel of b1 reduces b3,b2;
:: NECKLA_3:condreg 6
registration
cluster empty -> path-connected (RelStr);
end;
:: NECKLA_3:condreg 7
registration
cluster non empty connected -> path-connected (RelStr);
end;
:: NECKLA_3:th 26
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1
st the InternalRel of b1 reduces b2,b3
holds [b2,b3] in the InternalRel of b1;
:: NECKLA_3:condreg 8
registration
cluster non empty reflexive transitive path-connected -> connected (RelStr);
end;
:: NECKLA_3:th 27
theorem
for b1 being symmetric RelStr
for b2, b3 being set
st b2 in the carrier of b1 & b3 in the carrier of b1 & the InternalRel of b1 reduces b2,b3
holds the InternalRel of b1 reduces b3,b2;
:: NECKLA_3:attrnot 2 => NECKLA_3:attr 1
definition
let a1 be RelStr;
attr a1 is path-connected means
for b1, b2 being set
st b1 in the carrier of a1 & b2 in the carrier of a1 & b1 <> b2
holds the InternalRel of a1 reduces b1,b2;
end;
:: NECKLA_3:dfs 2
definiens
let a1 be symmetric RelStr;
To prove
a1 is path-connected
it is sufficient to prove
thus for b1, b2 being set
st b1 in the carrier of a1 & b2 in the carrier of a1 & b1 <> b2
holds the InternalRel of a1 reduces b1,b2;
:: NECKLA_3:def 2
theorem
for b1 being symmetric RelStr holds
b1 is path-connected
iff
for b2, b3 being set
st b2 in the carrier of b1 & b3 in the carrier of b1 & b2 <> b3
holds the InternalRel of b1 reduces b2,b3;
:: NECKLA_3:funcnot 1 => NECKLA_3:func 1
definition
let a1 be RelStr;
let a2 be Element of the carrier of a1;
func component A2 -> Element of bool the carrier of a1 equals
Class(EqCl the InternalRel of a1,a2);
end;
:: NECKLA_3:def 3
theorem
for b1 being RelStr
for b2 being Element of the carrier of b1 holds
component b2 = Class(EqCl the InternalRel of b1,b2);
:: NECKLA_3:funcreg 12
registration
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
cluster component a2 -> non empty;
end;
:: NECKLA_3:th 29
theorem
for b1 being RelStr
for b2 being Element of the carrier of b1
for b3 being set
st b3 in component b2
holds [b2,b3] in EqCl the InternalRel of b1;
:: NECKLA_3:th 30
theorem
for b1 being RelStr
for b2 being Element of the carrier of b1
for b3 being set holds
b3 = component b2
iff
for b4 being set holds
b4 in b3
iff
[b2,b4] in EqCl the InternalRel of b1;
:: NECKLA_3:th 31
theorem
for b1 being non empty symmetric irreflexive RelStr
st b1 is not path-connected
holds ex b2, b3 being non empty strict symmetric irreflexive RelStr st
the carrier of b2 misses the carrier of b3 &
RelStr(#the carrier of b1,the InternalRel of b1#) = union_of(b2,b3);
:: NECKLA_3:th 32
theorem
for b1 being non empty symmetric irreflexive RelStr
st ComplRelStr b1 is not path-connected
holds ex b2, b3 being non empty strict symmetric irreflexive RelStr st
the carrier of b2 misses the carrier of b3 &
RelStr(#the carrier of b1,the InternalRel of b1#) = sum_of(b2,b3);
:: NECKLA_3:th 33
theorem
for b1 being irreflexive RelStr
st b1 in fin_RelStr_sp
holds ComplRelStr b1 in fin_RelStr_sp;
:: NECKLA_3:th 34
theorem
for b1 being symmetric irreflexive RelStr
st Card the carrier of b1 = 2 & the carrier of b1 in FinSETS
holds RelStr(#the carrier of b1,the InternalRel of b1#) in fin_RelStr_sp;
:: NECKLA_3:th 35
theorem
for b1 being RelStr
st b1 in fin_RelStr_sp
holds b1 is symmetric;
:: NECKLA_3:th 36
theorem
for b1 being RelStr
for b2, b3 being non empty RelStr
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b3
st b1 = union_of(b2,b3) & the carrier of b2 misses the carrier of b3
holds not [b4,b5] in the InternalRel of b1;
:: NECKLA_3:th 37
theorem
for b1 being RelStr
for b2, b3 being non empty RelStr
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b3
st b1 = sum_of(b2,b3)
holds not [b4,b5] in the InternalRel of ComplRelStr b1;
:: NECKLA_3:th 38
theorem
for b1 being non empty symmetric RelStr
for b2 being Element of the carrier of b1
for b3, b4 being non empty RelStr
st the carrier of b3 misses the carrier of b4 &
subrelstr (([#] b1) \ {b2}) = union_of(b3,b4) &
b1 is path-connected
holds ex b5 being Element of the carrier of b3 st
[b5,b2] in the InternalRel of b1;
:: NECKLA_3:th 39
theorem
for b1 being non empty symmetric irreflexive RelStr
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
st b6 = {b2,b3,b4,b5} & b2,b3,b4,b5 are_mutually_different & [b2,b3] in the InternalRel of b1 & [b3,b4] in the InternalRel of b1 & [b4,b5] in the InternalRel of b1 & not [b2,b4] in the InternalRel of b1 & not [b2,b5] in the InternalRel of b1 & not [b3,b5] in the InternalRel of b1
holds subrelstr b6 embeds Necklace 4;
:: NECKLA_3:th 40
theorem
for b1 being non empty symmetric irreflexive RelStr
for b2 being Element of the carrier of b1
for b3, b4 being non empty RelStr
st the carrier of b3 misses the carrier of b4 &
subrelstr (([#] b1) \ {b2}) = union_of(b3,b4) &
b1 is not trivial &
b1 is path-connected &
ComplRelStr b1 is path-connected
holds b1 embeds Necklace 4;
:: NECKLA_3:th 41
theorem
for b1 being non empty finite strict symmetric irreflexive RelStr
st b1 is N-free & the carrier of b1 in FinSETS
holds RelStr(#the carrier of b1,the InternalRel of b1#) in fin_RelStr_sp;