Article TOLER_1, MML version 4.99.1005
:: TOLER_1:th 2
theorem
{} is reflexive;
:: TOLER_1:th 3
theorem
{} is symmetric;
:: TOLER_1:th 4
theorem
{} is irreflexive;
:: TOLER_1:th 5
theorem
{} is antisymmetric;
:: TOLER_1:th 6
theorem
{} is asymmetric;
:: TOLER_1:th 7
theorem
{} is connected;
:: TOLER_1:th 8
theorem
{} is strongly_connected;
:: TOLER_1:th 9
theorem
{} is transitive;
:: TOLER_1:funcreg 1
registration
cluster {} -> reflexive irreflexive symmetric antisymmetric asymmetric connected strongly_connected transitive;
end;
:: TOLER_1:funcnot 1 => EQREL_1:func 1
notation
let a1 be set;
synonym Total a1 for nabla a1;
end;
:: TOLER_1:funcnot 2 => TOLER_1:func 1
definition
let a1 be Relation-like set;
let a2 be set;
redefine func a1 |_2 a2 -> Relation of a2,a2;
end;
:: TOLER_1:th 13
theorem
for b1 being set holds
rng nabla b1 = b1;
:: TOLER_1:th 15
theorem
for b1, b2, b3 being set
st b2 in b1 & b3 in b1
holds [b2,b3] in nabla b1;
:: TOLER_1:th 16
theorem
for b1, b2, b3 being set
st b2 in field nabla b1 & b3 in field nabla b1
holds [b2,b3] in nabla b1;
:: TOLER_1:th 19
theorem
for b1 being set holds
nabla b1 is strongly_connected;
:: TOLER_1:th 21
theorem
for b1 being set holds
nabla b1 is connected;
:: TOLER_1:th 25
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
rng b2 = b1;
:: TOLER_1:th 27
theorem
for b1, b2 being set
for b3 being reflexive total Relation of b1,b1 holds
b2 in b1
iff
[b2,b2] in b3;
:: TOLER_1:th 28
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
b2 is_reflexive_in b1;
:: TOLER_1:th 29
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
b2 is_symmetric_in b1;
:: TOLER_1:th 32
theorem
for b1, b2, b3 being set
for b4 being Relation of b1,b2
st b4 is symmetric
holds b4 |_2 b3 is symmetric;
:: TOLER_1:funcnot 3 => TOLER_1:func 2
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
let a3 be Element of bool a1;
redefine func a2 |_2 a3 -> reflexive symmetric total Relation of a3,a3;
end;
:: TOLER_1:th 33
theorem
for b1, b2 being set
for b3 being reflexive symmetric total Relation of b2,b2
st b1 c= b2
holds b3 |_2 b1 is reflexive symmetric total Relation of b1,b1;
:: TOLER_1:modenot 1 => TOLER_1:mode 1
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
mode TolSet of A2 means
for b1, b2 being set
st b1 in it & b2 in it
holds [b1,b2] in a2;
end;
:: TOLER_1:dfs 1
definiens
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
let a3 be set;
To prove
a3 is TolSet of a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a3 & b2 in a3
holds [b1,b2] in a2;
:: TOLER_1:def 3
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set holds
b3 is TolSet of b2
iff
for b4, b5 being set
st b4 in b3 & b5 in b3
holds [b4,b5] in b2;
:: TOLER_1:th 34
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
{} is TolSet of b2;
:: TOLER_1:attrnot 1 => TOLER_1:attr 1
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
let a3 be TolSet of a2;
attr a3 is TolClass-like means
for b1 being set
st not b1 in a3 & b1 in a1
holds ex b2 being set st
b2 in a3 & not [b1,b2] in a2;
end;
:: TOLER_1:dfs 2
definiens
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
let a3 be TolSet of a2;
To prove
a3 is TolClass-like
it is sufficient to prove
thus for b1 being set
st not b1 in a3 & b1 in a1
holds ex b2 being set st
b2 in a3 & not [b1,b2] in a2;
:: TOLER_1:def 4
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being TolSet of b2 holds
b3 is TolClass-like(b1, b2)
iff
for b4 being set
st not b4 in b3 & b4 in b1
holds ex b5 being set st
b5 in b3 & not [b4,b5] in b2;
:: TOLER_1:exreg 1
registration
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
cluster TolClass-like TolSet of a2;
end;
:: TOLER_1:modenot 2
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
mode TolClass of a2 is TolClass-like TolSet of a2;
end;
:: TOLER_1:th 38
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
st {} is TolClass-like TolSet of b2
holds b2 = {};
:: TOLER_1:th 39
theorem
{} is reflexive symmetric total Relation of {},{};
:: TOLER_1:th 40
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3, b4 being set
st [b3,b4] in b2
holds {b3,b4} is TolSet of b2;
:: TOLER_1:th 41
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set
st b3 in b1
holds {b3} is TolSet of b2;
:: TOLER_1:th 42
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3, b4 being set
st b3 is TolSet of b2 & b4 is TolSet of b2
holds b3 /\ b4 is TolSet of b2;
:: TOLER_1:th 43
theorem
for b1, b2 being set
for b3 being reflexive symmetric total Relation of b2,b2
st b1 is TolSet of b3
holds b1 c= b2;
:: TOLER_1:th 45
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being TolSet of b2 holds
ex b4 being TolClass-like TolSet of b2 st
b3 c= b4;
:: TOLER_1:th 46
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3, b4 being set
st [b3,b4] in b2
holds ex b5 being TolClass-like TolSet of b2 st
b3 in b5 & b4 in b5;
:: TOLER_1:th 47
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set
st b3 in b1
holds ex b4 being TolClass-like TolSet of b2 st
b3 in b4;
:: TOLER_1:th 49
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
b2 c= nabla b1;
:: TOLER_1:th 50
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
id b1 c= b2;
:: TOLER_1:sch 1
scheme TOLER_1:sch 1
{F1 -> set}:
ex b1 being reflexive symmetric total Relation of F1(),F1() st
for b2, b3 being set
st b2 in F1() & b3 in F1()
holds [b2,b3] in b1
iff
P1[b2, b3]
provided
for b1 being set
st b1 in F1()
holds P1[b1, b1]
and
for b1, b2 being set
st b1 in F1() & b2 in F1() & P1[b1, b2]
holds P1[b2, b1];
:: TOLER_1:th 51
theorem
for b1 being set holds
ex b2 being reflexive symmetric total Relation of union b1,union b1 st
for b3 being set
st b3 in b1
holds b3 is TolSet of b2;
:: TOLER_1:th 52
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of union b1,union b1
st (for b4, b5 being set holds
[b4,b5] in b2
iff
ex b6 being set st
b6 in b1 & b4 in b6 & b5 in b6) &
(for b4, b5 being set holds
[b4,b5] in b3
iff
ex b6 being set st
b6 in b1 & b4 in b6 & b5 in b6)
holds b2 = b3;
:: TOLER_1:th 53
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of b1,b1
st for b4 being set holds
b4 is TolClass-like TolSet of b2
iff
b4 is TolClass-like TolSet of b3
holds b2 = b3;
:: TOLER_1:funcnot 4 => RELAT_1:func 11
notation
let a1, a2 be set;
let a3 be Relation of a1,a2;
let a4 be set;
synonym neighbourhood(a4,a3) for Im(a1,a2);
end;
:: TOLER_1:th 54
theorem
for b1, b2 being set
for b3 being reflexive symmetric total Relation of b1,b1
for b4 being set holds
b4 in Im(b3,b2)
iff
[b2,b4] in b3;
:: TOLER_1:th 58
theorem
for b1, b2 being set
for b3 being reflexive symmetric total Relation of b1,b1
for b4 being set
st for b5 being set holds
b5 in b4
iff
b2 in b5 & b5 is TolClass-like TolSet of b3
holds Im(b3,b2) = union b4;
:: TOLER_1:th 59
theorem
for b1, b2 being set
for b3 being reflexive symmetric total Relation of b1,b1
for b4 being set
st for b5 being set holds
b5 in b4
iff
b2 in b5 & b5 is TolSet of b3
holds Im(b3,b2) = union b4;
:: TOLER_1:funcnot 5 => TOLER_1:func 3
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
func TolSets A2 -> set means
for b1 being set holds
b1 in it
iff
b1 is TolSet of a2;
end;
:: TOLER_1:def 6
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set holds
b3 = TolSets b2
iff
for b4 being set holds
b4 in b3
iff
b4 is TolSet of b2;
:: TOLER_1:funcnot 6 => TOLER_1:func 4
definition
let a1 be set;
let a2 be reflexive symmetric total Relation of a1,a1;
func TolClasses A2 -> set means
for b1 being set holds
b1 in it
iff
b1 is TolClass-like TolSet of a2;
end;
:: TOLER_1:def 7
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set holds
b3 = TolClasses b2
iff
for b4 being set holds
b4 in b3
iff
b4 is TolClass-like TolSet of b2;
:: TOLER_1:th 64
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of b1,b1
st TolClasses b2 c= TolClasses b3
holds b2 c= b3;
:: TOLER_1:th 65
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of b1,b1
st TolClasses b2 = TolClasses b3
holds b2 = b3;
:: TOLER_1:th 66
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
union TolClasses b2 = b1;
:: TOLER_1:th 67
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
union TolSets b2 = b1;
:: TOLER_1:th 68
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
st for b3 being set
st b3 in b1
holds Im(b2,b3) is TolSet of b2
holds b2 is transitive;
:: TOLER_1:th 69
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
st b2 is transitive
for b3 being set
st b3 in b1
holds Im(b2,b3) is TolClass-like TolSet of b2;
:: TOLER_1:th 70
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1
for b3 being set
for b4 being TolClass-like TolSet of b2
st b3 in b4
holds b4 c= Im(b2,b3);
:: TOLER_1:th 71
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of b1,b1 holds
TolSets b2 c= TolSets b3
iff
b2 c= b3;
:: TOLER_1:th 72
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
TolClasses b2 c= TolSets b2;
:: TOLER_1:th 73
theorem
for b1 being set
for b2, b3 being reflexive symmetric total Relation of b1,b1
st for b4 being set
st b4 in b1
holds Im(b2,b4) c= Im(b3,b4)
holds b2 c= b3;
:: TOLER_1:th 74
theorem
for b1 being set
for b2 being reflexive symmetric total Relation of b1,b1 holds
b2 c= b2 * b2;