Article FUZZY_1, MML version 4.99.1005

:: FUZZY_1:th 1
theorem
for b1, b2 being set holds
proj2 chi(b1,b2) c= [.0,1.];

:: FUZZY_1:modenot 1 => FUZZY_1:mode 1
definition
  let a1 be non empty set;
  mode Membership_Func of A1 -> Function-like Relation of a1,REAL means
    dom it = a1 & proj2 it c= [.0,1.];
end;

:: FUZZY_1:dfs 1
definiens
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,REAL;
To prove
     a2 is Membership_Func of a1
it is sufficient to prove
  thus dom a2 = a1 & proj2 a2 c= [.0,1.];

:: FUZZY_1:def 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
      b2 is Membership_Func of b1
   iff
      dom b2 = b1 & proj2 b2 c= [.0,1.];

:: FUZZY_1:th 2
theorem
for b1 being non empty set holds
   chi(b1,b1) is Membership_Func of b1;

:: FUZZY_1:condreg 1
registration
  let a1 be non empty set;
  cluster -> real-valued (Membership_Func of a1);
end;

:: FUZZY_1:prednot 1 => FUZZY_1:pred 1
definition
  let a1, a2 be Relation-like Function-like real-valued set;
  pred A1 is_less_than A2 means
    proj1 a1 c= proj1 a2 &
     (for b1 being set
           st b1 in proj1 a1
        holds a1 . b1 <= a2 . b1);
  reflexivity;
::  for a1 being Relation-like Function-like real-valued set holds
::     a1 is_less_than a1;
end;

:: FUZZY_1:dfs 2
definiens
  let a1, a2 be Relation-like Function-like real-valued set;
To prove
     a1 is_less_than a2
it is sufficient to prove
  thus proj1 a1 c= proj1 a2 &
     (for b1 being set
           st b1 in proj1 a1
        holds a1 . b1 <= a2 . b1);

:: FUZZY_1:def 2
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
   b1 is_less_than b2
iff
   proj1 b1 c= proj1 b2 &
    (for b3 being set
          st b3 in proj1 b1
       holds b1 . b3 <= b2 . b3);

:: FUZZY_1:prednot 2 => FUZZY_1:pred 1
notation
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  synonym a2 c= a3 for a1 is_less_than a2;
end;

:: FUZZY_1:prednot 3 => FUZZY_1:pred 2
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  redefine pred A2 is_less_than A3 means
    for b1 being Element of a1 holds
       a2 . b1 <= a3 . b1;
  reflexivity;
::  for a1 being non empty set
::  for a2 being Membership_Func of a1 holds
::     a2 is_less_than a2;
end;

:: FUZZY_1:dfs 3
definiens
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
To prove
     a2 is_less_than a3
it is sufficient to prove
  thus for b1 being Element of a1 holds
       a2 . b1 <= a3 . b1;

:: FUZZY_1:def 3
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
   b2 is_less_than b3
iff
   for b4 being Element of b1 holds
      b2 . b4 <= b3 . b4;

:: FUZZY_1:th 3
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
   b2 = b3
iff
   b2 is_less_than b3 & b3 is_less_than b2;

:: FUZZY_1:th 4
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 is_less_than b2;

:: FUZZY_1:th 5
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3 & b3 is_less_than b4
   holds b3 is_less_than b4;

:: FUZZY_1:funcnot 1 => FUZZY_1:func 1
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func min(A2,A3) -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = min(a2 . b1,a3 . b1);
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Membership_Func of a1 holds
::  min(a2,a3) = min(a3,a2);
  idempotence;
::  for a1 being non empty set
::  for a2 being Membership_Func of a1 holds
::     min(a2,a2) = a2;
end;

:: FUZZY_1:def 4
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b4 = min(b2,b3)
iff
   for b5 being Element of b1 holds
      b4 . b5 = min(b2 . b5,b3 . b5);

:: FUZZY_1:funcnot 2 => FUZZY_1:func 2
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func max(A2,A3) -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = max(a2 . b1,a3 . b1);
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Membership_Func of a1 holds
::  max(a2,a3) = max(a3,a2);
  idempotence;
::  for a1 being non empty set
::  for a2 being Membership_Func of a1 holds
::     max(a2,a2) = a2;
end;

:: FUZZY_1:def 5
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b4 = max(b2,b3)
iff
   for b5 being Element of b1 holds
      b4 . b5 = max(b2 . b5,b3 . b5);

:: FUZZY_1:funcnot 3 => FUZZY_1:func 3
definition
  let a1 be non empty set;
  let a2 be Membership_Func of a1;
  func 1_minus A2 -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = 1 - (a2 . b1);
  involutiveness;
::  for a1 being non empty set
::  for a2 being Membership_Func of a1 holds
::     1_minus 1_minus a2 = a2;
end;

:: FUZZY_1:def 6
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
   b3 = 1_minus b2
iff
   for b4 being Element of b1 holds
      b3 . b4 = 1 - (b2 . b4);

:: FUZZY_1:th 6
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Membership_Func of b1 holds
min(b3 . b2,b4 . b2) = (min(b3,b4)) . b2 &
 max(b3 . b2,b4 . b2) = (max(b3,b4)) . b2;

:: FUZZY_1:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
max(b2,b2) = b2 & min(b2,b2) = b2 & max(b2,b2) = min(b2,b2) & min(b3,b4) = min(b4,b3) & max(b3,b4) = max(b4,b3);

:: FUZZY_1:th 8
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
max(max(b2,b3),b4) = max(b2,max(b3,b4)) &
 min(min(b2,b3),b4) = min(b2,min(b3,b4));

:: FUZZY_1:th 9
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(b2,min(b2,b3)) = b2 & min(b2,max(b2,b3)) = b2;

:: FUZZY_1:th 10
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
min(b2,max(b3,b4)) = max(min(b2,b3),min(b2,b4)) &
 max(b2,min(b3,b4)) = min(max(b2,b3),max(b2,b4));

:: FUZZY_1:th 11
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   1_minus 1_minus b2 = b2;

:: FUZZY_1:th 12
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus max(b2,b3) = min(1_minus b2,1_minus b3) &
 1_minus min(b2,b3) = max(1_minus b2,1_minus b3);

:: FUZZY_1:th 13
theorem
for b1 being non empty set holds
   chi({},b1) is Membership_Func of b1;

:: FUZZY_1:funcnot 4 => FUZZY_1:func 4
definition
  let a1 be non empty set;
  func EMF A1 -> Membership_Func of a1 equals
    chi({},a1);
end;

:: FUZZY_1:def 7
theorem
for b1 being non empty set holds
   EMF b1 = chi({},b1);

:: FUZZY_1:funcnot 5 => FUZZY_1:func 5
definition
  let a1 be non empty set;
  func UMF A1 -> Membership_Func of a1 equals
    chi(a1,a1);
end;

:: FUZZY_1:def 8
theorem
for b1 being non empty set holds
   UMF b1 = chi(b1,b1);

:: FUZZY_1:th 14
theorem
for b1 being non empty set
for b2, b3 being Element of REAL
for b4 being Function-like Relation of b1,REAL
   st proj2 b4 c= [.b2,b3.] & b2 <= b3
for b5 being Element of b1
      st b5 in dom b4
   holds b2 <= b4 . b5 & b4 . b5 <= b3;

:: FUZZY_1:th 15
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   EMF b1 is_less_than b2;

:: FUZZY_1:th 16
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 is_less_than UMF b1;

:: FUZZY_1:th 17
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Membership_Func of b1 holds
   (EMF b1) . b2 <= b3 . b2 & b3 . b2 <= (UMF b1) . b2;

:: FUZZY_1:th 18
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
min(b2,b3) is_less_than b2 & b2 is_less_than max(b2,b3);

:: FUZZY_1:th 19
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   max(b2,UMF b1) = UMF b1 & min(b2,UMF b1) = b2 & max(b2,EMF b1) = b2 & min(b2,EMF b1) = EMF b1;

:: FUZZY_1:th 20
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3 & b4 is_less_than b3
   holds max(b2,b4) is_less_than b3;

:: FUZZY_1:th 21
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3
   holds max(b2,b4) is_less_than max(b3,b4);

:: FUZZY_1:th 22
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Membership_Func of b1
      st b2 is_less_than b3 & b4 is_less_than b5
   holds max(b2,b4) is_less_than max(b3,b5);

:: FUZZY_1:th 23
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st b2 is_less_than b3
   holds max(b2,b3) = b3;

:: FUZZY_1:th 24
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
min(b2,b3) is_less_than max(b2,b3);

:: FUZZY_1:th 25
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3 & b2 is_less_than b4
   holds b2 is_less_than min(b3,b4);

:: FUZZY_1:th 28
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3
   holds min(b2,b4) is_less_than min(b3,b4);

:: FUZZY_1:th 29
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Membership_Func of b1
      st b2 is_less_than b3 & b4 is_less_than b5
   holds min(b2,b4) is_less_than min(b3,b5);

:: FUZZY_1:th 30
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st b2 is_less_than b3
   holds min(b2,b3) = b2;

:: FUZZY_1:th 31
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3 & b2 is_less_than b4 & min(b3,b4) = EMF b1
   holds b2 = EMF b1;

:: FUZZY_1:th 32
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st max(min(b2,b3),min(b2,b4)) = b2
   holds b2 is_less_than max(b3,b4);

:: FUZZY_1:th 33
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3 & min(b3,b4) = EMF b1
   holds min(b2,b4) = EMF b1;

:: FUZZY_1:th 34
theorem
for b1 being non empty set
for b2 being Membership_Func of b1
      st b2 is_less_than EMF b1
   holds b2 = EMF b1;

:: FUZZY_1:th 35
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
   max(b2,b3) = EMF b1
iff
   b2 = EMF b1 & b3 = EMF b1;

:: FUZZY_1:th 36
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b2 = max(b3,b4)
iff
   b3 is_less_than b2 &
    b4 is_less_than b2 &
    (for b5 being Membership_Func of b1
          st b3 is_less_than b5 & b4 is_less_than b5
       holds b2 is_less_than b5);

:: FUZZY_1:th 37
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b2 = min(b3,b4)
iff
   b2 is_less_than b3 &
    b2 is_less_than b4 &
    (for b5 being Membership_Func of b1
          st b5 is_less_than b3 & b5 is_less_than b4
       holds b5 is_less_than b2);

:: FUZZY_1:th 38
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than max(b3,b4) & min(b2,b4) = EMF b1
   holds b2 is_less_than b3;

:: FUZZY_1:th 39
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
   b2 is_less_than b3
iff
   1_minus b3 is_less_than 1_minus b2;

:: FUZZY_1:th 40
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st b2 is_less_than 1_minus b3
   holds b3 is_less_than 1_minus b2;

:: FUZZY_1:th 41
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus max(b2,b3) is_less_than 1_minus b2;

:: FUZZY_1:th 42
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus b2 is_less_than 1_minus min(b2,b3);

:: FUZZY_1:th 44
theorem
for b1 being non empty set holds
   1_minus EMF b1 = UMF b1 & 1_minus UMF b1 = EMF b1;

:: FUZZY_1:funcnot 6 => FUZZY_1:func 6
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func A2 \+\ A3 -> Membership_Func of a1 equals
    max(min(a2,1_minus a3),min(1_minus a2,a3));
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Membership_Func of a1 holds
::  a2 \+\ a3 = a3 \+\ a2;
end;

:: FUZZY_1:def 9
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \+\ b3 = max(min(b2,1_minus b3),min(1_minus b2,b3));

:: FUZZY_1:th 45
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 \+\ EMF b1 = b2;

:: FUZZY_1:th 46
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 \+\ UMF b1 = 1_minus b2;

:: FUZZY_1:th 47
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
min(min(max(b2,b3),max(b3,b4)),max(b4,b2)) = max(max(min(b2,b3),min(b3,b4)),min(b4,b2));

:: FUZZY_1:th 48
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(min(b2,b3),min(1_minus b2,1_minus b3)) is_less_than 1_minus (b2 \+\ b3);

:: FUZZY_1:th 49
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(b2 \+\ b3,min(b2,b3)) is_less_than max(b2,b3);

:: FUZZY_1:th 50
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 \+\ b2 = min(b2,1_minus b2);

:: FUZZY_1:funcnot 7 => FUZZY_1:func 7
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func ab_difMF(A2,A3) -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = abs ((a2 . b1) - (a3 . b1));
end;

:: FUZZY_1:def 10
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b4 = ab_difMF(b2,b3)
iff
   for b5 being Element of b1 holds
      b4 . b5 = abs ((b2 . b5) - (b3 . b5));