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;