Article ASYMPT_0, MML version 4.99.1005

:: ASYMPT_0:sch 1
scheme ASYMPT_0:sch 1
{F1 -> natural set,
  F2 -> natural set,
  F3 -> non empty set,
  F4 -> Element of F3()}:
{F4(b1) where b1 is Element of NAT: F1() <= b1 & b1 <= F2()} is non empty finite Element of bool F3()
provided
   F1() <= F2();


:: ASYMPT_0:sch 2
scheme ASYMPT_0:sch 2
{F1 -> natural set,
  F2 -> non empty set,
  F3 -> Element of F2()}:
{F3(b1) where b1 is Element of NAT: b1 <= F1()} is non empty finite Element of bool F2()


:: ASYMPT_0:sch 3
scheme ASYMPT_0:sch 3
{F1 -> natural set,
  F2 -> non empty set,
  F3 -> Element of F2()}:
{F3(b1) where b1 is Element of NAT: b1 < F1()} is non empty finite Element of bool F2()
provided
   0 < F1();


:: ASYMPT_0:attrnot 1 => ASYMPT_0:attr 1
definition
  let a1 be real set;
  attr a1 is logbase means
    0 < a1 & a1 <> 1;
end;

:: ASYMPT_0:dfs 1
definiens
  let a1 be real set;
To prove
     a1 is logbase
it is sufficient to prove
  thus 0 < a1 & a1 <> 1;

:: ASYMPT_0:def 3
theorem
for b1 being real set holds
      b1 is logbase
   iff
      0 < b1 & b1 <> 1;

:: ASYMPT_0:exreg 1
registration
  cluster complex real ext-real positive Element of REAL;
end;

:: ASYMPT_0:exreg 2
registration
  cluster complex real ext-real negative Element of REAL;
end;

:: ASYMPT_0:exreg 3
registration
  cluster complex real ext-real logbase Element of REAL;
end;

:: ASYMPT_0:exreg 4
registration
  cluster complex real ext-real non negative Element of REAL;
end;

:: ASYMPT_0:exreg 5
registration
  cluster complex real ext-real non positive Element of REAL;
end;

:: ASYMPT_0:exreg 6
registration
  cluster complex real ext-real non logbase Element of REAL;
end;

:: ASYMPT_0:attrnot 2 => ASYMPT_0:attr 2
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is eventually-nonnegative means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds 0 <= a1 . b2;
end;

:: ASYMPT_0:dfs 2
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is eventually-nonnegative
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds 0 <= a1 . b2;

:: ASYMPT_0:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is eventually-nonnegative
   iff
      ex b2 being Element of NAT st
         for b3 being Element of NAT
               st b2 <= b3
            holds 0 <= b1 . b3;

:: ASYMPT_0:attrnot 3 => ASYMPT_0:attr 3
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is positive means
    for b1 being Element of NAT holds
       0 < a1 . b1;
end;

:: ASYMPT_0:dfs 3
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is positive
it is sufficient to prove
  thus for b1 being Element of NAT holds
       0 < a1 . b1;

:: ASYMPT_0:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is positive
   iff
      for b2 being Element of NAT holds
         0 < b1 . b2;

:: ASYMPT_0:attrnot 4 => ASYMPT_0:attr 4
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is eventually-positive means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds 0 < a1 . b2;
end;

:: ASYMPT_0:dfs 4
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is eventually-positive
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds 0 < a1 . b2;

:: ASYMPT_0:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is eventually-positive
   iff
      ex b2 being Element of NAT st
         for b3 being Element of NAT
               st b2 <= b3
            holds 0 < b1 . b3;

:: ASYMPT_0:attrnot 5 => ASYMPT_0:attr 5
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is eventually-nonzero means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <> 0;
end;

:: ASYMPT_0:dfs 5
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is eventually-nonzero
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <> 0;

:: ASYMPT_0:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is eventually-nonzero
   iff
      ex b2 being Element of NAT st
         for b3 being Element of NAT
               st b2 <= b3
            holds b1 . b3 <> 0;

:: ASYMPT_0:attrnot 6 => ASYMPT_0:attr 6
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is eventually-nondecreasing means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <= a1 . (b2 + 1);
end;

:: ASYMPT_0:dfs 6
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is eventually-nondecreasing
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <= a1 . (b2 + 1);

:: ASYMPT_0:def 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is eventually-nondecreasing
   iff
      ex b2 being Element of NAT st
         for b3 being Element of NAT
               st b2 <= b3
            holds b1 . b3 <= b1 . (b3 + 1);

:: ASYMPT_0:exreg 7
registration
  cluster Relation-like Function-like non empty quasi_total complex-valued ext-real-valued real-valued total eventually-nonnegative positive eventually-positive eventually-nonzero eventually-nondecreasing Relation of NAT,REAL;
end;

:: ASYMPT_0:funcnot 1 => ASYMPT_0:func 1
definition
  let a1, a2 be Element of REAL;
  redefine func min(a1,a2) -> Element of REAL;
  commutativity;
::  for a1, a2 being Element of REAL holds
::  min(a1,a2) = min(a2,a1);
  idempotence;
::  for a1 being Element of REAL holds
::     min(a1,a1) = a1;
end;

:: ASYMPT_0:funcnot 2 => ASYMPT_0:func 2
definition
  let a1, a2 be Element of REAL;
  redefine func max(a1,a2) -> Element of REAL;
  commutativity;
::  for a1, a2 being Element of REAL holds
::  max(a1,a2) = max(a2,a1);
  idempotence;
::  for a1 being Element of REAL holds
::     max(a1,a1) = a1;
end;

:: ASYMPT_0:funcnot 3 => ASYMPT_0:func 3
definition
  let a1, a2 be Element of NAT;
  redefine func min(a1,a2) -> Element of NAT;
  commutativity;
::  for a1, a2 being Element of NAT holds
::  min(a1,a2) = min(a2,a1);
  idempotence;
::  for a1 being Element of NAT holds
::     min(a1,a1) = a1;
end;

:: ASYMPT_0:funcnot 4 => ASYMPT_0:func 4
definition
  let a1, a2 be Element of NAT;
  redefine func max(a1,a2) -> Element of NAT;
  commutativity;
::  for a1, a2 being Element of NAT holds
::  max(a1,a2) = max(a2,a1);
  idempotence;
::  for a1 being Element of NAT holds
::     max(a1,a1) = a1;
end;

:: ASYMPT_0:condreg 1
registration
  cluster Function-like quasi_total positive -> eventually-positive (Relation of NAT,REAL);
end;

:: ASYMPT_0:condreg 2
registration
  cluster Function-like quasi_total eventually-positive -> eventually-nonnegative eventually-nonzero (Relation of NAT,REAL);
end;

:: ASYMPT_0:condreg 3
registration
  cluster Function-like quasi_total eventually-nonnegative eventually-nonzero -> eventually-positive (Relation of NAT,REAL);
end;

:: ASYMPT_0:funcnot 5 => ASYMPT_0:func 5
definition
  let a1, a2 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  redefine func a1 + a2 -> Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  commutativity;
::  for a1, a2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
::  a1 + a2 = a2 + a1;
end;

:: ASYMPT_0:funcnot 6 => ASYMPT_0:func 6
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be positive Element of REAL;
  redefine func a2 (#) a1 -> Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
end;

:: ASYMPT_0:funcnot 7 => ASYMPT_0:func 7
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be non negative Element of REAL;
  redefine func a2 + a1 -> Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
end;

:: ASYMPT_0:funcnot 8 => ASYMPT_0:func 8
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be positive Element of REAL;
  redefine func a2 + a1 -> Function-like quasi_total eventually-positive Relation of NAT,REAL;
end;

:: ASYMPT_0:funcnot 9 => ASYMPT_0:func 9
definition
  let a1, a2 be Function-like quasi_total Relation of NAT,REAL;
  func max(A1,A2) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = max(a1 . b1,a2 . b1);
  commutativity;
::  for a1, a2 being Function-like quasi_total Relation of NAT,REAL holds
::  max(a1,a2) = max(a2,a1);
end;

:: ASYMPT_0:def 10
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
   b3 = max(b1,b2)
iff
   for b4 being Element of NAT holds
      b3 . b4 = max(b1 . b4,b2 . b4);

:: ASYMPT_0:funcreg 1
registration
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  cluster max(a1,a2) -> Function-like quasi_total eventually-nonnegative;
end;

:: ASYMPT_0:funcreg 2
registration
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Function-like quasi_total eventually-positive Relation of NAT,REAL;
  cluster max(a1,a2) -> Function-like quasi_total eventually-positive;
end;

:: ASYMPT_0:prednot 1 => ASYMPT_0:pred 1
definition
  let a1, a2 be Function-like quasi_total Relation of NAT,REAL;
  pred A2 majorizes A1 means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <= a2 . b2;
end;

:: ASYMPT_0:dfs 8
definiens
  let a1, a2 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a2 majorizes a1
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a1 . b2 <= a2 . b2;

:: ASYMPT_0:def 11
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 majorizes b1
iff
   ex b3 being Element of NAT st
      for b4 being Element of NAT
            st b3 <= b4
         holds b1 . b4 <= b2 . b4;

:: ASYMPT_0:th 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
   st for b3 being Element of NAT
           st b2 <= b3
        holds b1 . b3 <= b1 . (b3 + 1)
for b3, b4 being Element of NAT
      st b2 <= b3 & b3 <= b4
   holds b1 . b3 <= b1 . b4;

:: ASYMPT_0:th 2
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & lim (b1 /" b2) <> 0
   holds b2 /" b1 is convergent &
    lim (b2 /" b1) = (lim (b1 /" b2)) ";

:: ASYMPT_0:th 3
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 is convergent
   holds 0 <= lim b1;

:: ASYMPT_0:th 4
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent & b2 is convergent & b2 majorizes b1
   holds lim b1 <= lim b2;

:: ASYMPT_0:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonzero Relation of NAT,REAL
      st b1 /" b2 is divergent_to+infty
   holds b2 /" b1 is convergent & lim (b2 /" b1) = 0;

:: ASYMPT_0:funcnot 10 => ASYMPT_0:func 10
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  func Big_Oh A1 -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2 being Element of REAL st
       ex b3 being Element of NAT st
          0 < b2 &
           (for b4 being Element of NAT
                 st b3 <= b4
              holds b1 . b4 <= b2 * (a1 . b4) & 0 <= b1 . b4)};
end;

:: ASYMPT_0:def 12
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Oh b1 = {b2 where b2 is Element of Funcs(NAT,REAL): ex b3 being Element of REAL st
      ex b4 being Element of NAT st
         0 < b3 &
          (for b5 being Element of NAT
                st b4 <= b5
             holds b2 . b5 <= b3 * (b1 . b5) & 0 <= b2 . b5)};

:: ASYMPT_0:th 6
theorem
for b1 being set
for b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Oh b2
   holds b1 is Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;

:: ASYMPT_0:th 7
theorem
for b1 being Function-like quasi_total positive Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
      b2 in Big_Oh b1
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          (for b4 being Element of NAT holds
             b2 . b4 <= b3 * (b1 . b4));

:: ASYMPT_0:th 8
theorem
for b1 being Function-like quasi_total eventually-positive Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b3 being Element of NAT
      st b2 in Big_Oh b1 &
         (for b4 being Element of NAT
               st b3 <= b4
            holds 0 < b1 . b4)
   holds ex b4 being Element of REAL st
      0 < b4 &
       (for b5 being Element of NAT
             st b3 <= b5
          holds b2 . b5 <= b4 * (b1 . b5));

:: ASYMPT_0:th 9
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
Big_Oh (b1 + b2) = Big_Oh max(b1,b2);

:: ASYMPT_0:th 10
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Oh b1;

:: ASYMPT_0:th 11
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Oh b2
   holds Big_Oh b1 c= Big_Oh b2;

:: ASYMPT_0:th 12
theorem
for b1, b2, b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Oh b2 & b2 in Big_Oh b3
   holds b1 in Big_Oh b3;

:: ASYMPT_0:th 13
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being positive Element of REAL holds
   Big_Oh b1 = Big_Oh (b2 (#) b1);

:: ASYMPT_0:th 14
theorem
for b1 being non negative Element of REAL
for b2, b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b2 in Big_Oh b3
   holds b2 in Big_Oh (b1 + b3);

:: ASYMPT_0:th 15
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & 0 < lim (b1 /" b2)
   holds Big_Oh b1 = Big_Oh b2;

:: ASYMPT_0:th 16
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & lim (b1 /" b2) = 0
   holds b1 in Big_Oh b2 & not b2 in Big_Oh b1;

:: ASYMPT_0:th 17
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is divergent_to+infty
   holds not b1 in Big_Oh b2 & b2 in Big_Oh b1;

:: ASYMPT_0:funcnot 11 => ASYMPT_0:func 11
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  func Big_Omega A1 -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2 being Element of REAL st
       ex b3 being Element of NAT st
          0 < b2 &
           (for b4 being Element of NAT
                 st b3 <= b4
              holds b2 * (a1 . b4) <= b1 . b4 & 0 <= b1 . b4)};
end;

:: ASYMPT_0:def 13
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Omega b1 = {b2 where b2 is Element of Funcs(NAT,REAL): ex b3 being Element of REAL st
      ex b4 being Element of NAT st
         0 < b3 &
          (for b5 being Element of NAT
                st b4 <= b5
             holds b3 * (b1 . b5) <= b2 . b5 & 0 <= b2 . b5)};

:: ASYMPT_0:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Omega b2
   holds b1 is Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;

:: ASYMPT_0:th 19
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Omega b2
iff
   b2 in Big_Oh b1;

:: ASYMPT_0:th 20
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Omega b1;

:: ASYMPT_0:th 21
theorem
for b1, b2, b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Omega b2 & b2 in Big_Omega b3
   holds b1 in Big_Omega b3;

:: ASYMPT_0:th 22
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & 0 < lim (b1 /" b2)
   holds Big_Omega b1 = Big_Omega b2;

:: ASYMPT_0:th 23
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & lim (b1 /" b2) = 0
   holds b2 in Big_Omega b1 & not b1 in Big_Omega b2;

:: ASYMPT_0:th 24
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is divergent_to+infty
   holds not b2 in Big_Omega b1 & b1 in Big_Omega b2;

:: ASYMPT_0:th 25
theorem
for b1, b2 being Function-like quasi_total positive Relation of NAT,REAL holds
   b2 in Big_Omega b1
iff
   ex b3 being Element of REAL st
      0 < b3 &
       (for b4 being Element of NAT holds
          b3 * (b1 . b4) <= b2 . b4);

:: ASYMPT_0:th 26
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
Big_Omega (b1 + b2) = Big_Omega max(b1,b2);

:: ASYMPT_0:funcnot 12 => ASYMPT_0:func 12
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  func Big_Theta A1 -> FUNCTION_DOMAIN of NAT,REAL equals
    (Big_Oh a1) /\ Big_Omega a1;
end;

:: ASYMPT_0:def 14
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Theta b1 = (Big_Oh b1) /\ Big_Omega b1;

:: ASYMPT_0:th 27
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Theta b1 = {b2 where b2 is Element of Funcs(NAT,REAL): ex b3, b4 being Element of REAL st
      ex b5 being Element of NAT st
         0 < b3 &
          0 < b4 &
          (for b6 being Element of NAT
                st b5 <= b6
             holds b4 * (b1 . b6) <= b2 . b6 & b2 . b6 <= b3 * (b1 . b6))};

:: ASYMPT_0:th 28
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Theta b1;

:: ASYMPT_0:th 29
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Theta b2
   holds b2 in Big_Theta b1;

:: ASYMPT_0:th 30
theorem
for b1, b2, b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 in Big_Theta b2 & b2 in Big_Theta b3
   holds b1 in Big_Theta b3;

:: ASYMPT_0:th 31
theorem
for b1, b2 being Function-like quasi_total positive Relation of NAT,REAL holds
   b2 in Big_Theta b1
iff
   ex b3, b4 being Element of REAL st
      0 < b3 &
       0 < b4 &
       (for b5 being Element of NAT holds
          b4 * (b1 . b5) <= b2 . b5 & b2 . b5 <= b3 * (b1 . b5));

:: ASYMPT_0:th 32
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
Big_Theta (b1 + b2) = Big_Theta max(b1,b2);

:: ASYMPT_0:th 33
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & 0 < lim (b1 /" b2)
   holds b1 in Big_Theta b2;

:: ASYMPT_0:th 34
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is convergent & lim (b1 /" b2) = 0
   holds b1 in Big_Oh b2 & not b1 in Big_Theta b2;

:: ASYMPT_0:th 35
theorem
for b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL
      st b1 /" b2 is divergent_to+infty
   holds b1 in Big_Omega b2 & not b1 in Big_Theta b2;

:: ASYMPT_0:funcnot 13 => ASYMPT_0:func 13
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be set;
  func Big_Oh(A1,A2) -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2 being Element of REAL st
       ex b3 being Element of NAT st
          0 < b2 &
           (for b4 being Element of NAT
                 st b3 <= b4 & b4 in a2
              holds b1 . b4 <= b2 * (a1 . b4) & 0 <= b1 . b4)};
end;

:: ASYMPT_0:def 15
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being set holds
   Big_Oh(b1,b2) = {b3 where b3 is Element of Funcs(NAT,REAL): ex b4 being Element of REAL st
      ex b5 being Element of NAT st
         0 < b4 &
          (for b6 being Element of NAT
                st b5 <= b6 & b6 in b2
             holds b3 . b6 <= b4 * (b1 . b6) & 0 <= b3 . b6)};

:: ASYMPT_0:funcnot 14 => ASYMPT_0:func 14
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be set;
  func Big_Omega(A1,A2) -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2 being Element of REAL st
       ex b3 being Element of NAT st
          0 < b2 &
           (for b4 being Element of NAT
                 st b3 <= b4 & b4 in a2
              holds b2 * (a1 . b4) <= b1 . b4 & 0 <= b1 . b4)};
end;

:: ASYMPT_0:def 16
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being set holds
   Big_Omega(b1,b2) = {b3 where b3 is Element of Funcs(NAT,REAL): ex b4 being Element of REAL st
      ex b5 being Element of NAT st
         0 < b4 &
          (for b6 being Element of NAT
                st b5 <= b6 & b6 in b2
             holds b4 * (b1 . b6) <= b3 . b6 & 0 <= b3 . b6)};

:: ASYMPT_0:funcnot 15 => ASYMPT_0:func 15
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be set;
  func Big_Theta(A1,A2) -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2, b3 being Element of REAL st
       ex b4 being Element of NAT st
          0 < b2 &
           0 < b3 &
           (for b5 being Element of NAT
                 st b4 <= b5 & b5 in a2
              holds b3 * (a1 . b5) <= b1 . b5 & b1 . b5 <= b2 * (a1 . b5))};
end;

:: ASYMPT_0:def 17
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being set holds
   Big_Theta(b1,b2) = {b3 where b3 is Element of Funcs(NAT,REAL): ex b4, b5 being Element of REAL st
      ex b6 being Element of NAT st
         0 < b4 &
          0 < b5 &
          (for b7 being Element of NAT
                st b6 <= b7 & b7 in b2
             holds b5 * (b1 . b7) <= b3 . b7 & b3 . b7 <= b4 * (b1 . b7))};

:: ASYMPT_0:th 36
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being set holds
   Big_Theta(b1,b2) = (Big_Oh(b1,b2)) /\ Big_Omega(b1,b2);

:: ASYMPT_0:funcnot 16 => ASYMPT_0:func 16
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Element of NAT;
  func A1 taken_every A2 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = a1 . (a2 * b1);
end;

:: ASYMPT_0:def 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = b1 taken_every b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 = b1 . (b2 * b4);

:: ASYMPT_0:prednot 2 => ASYMPT_0:pred 2
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be Element of NAT;
  pred A1 is_smooth_wrt A2 means
    a1 is eventually-nondecreasing & a1 taken_every a2 in Big_Oh a1;
end;

:: ASYMPT_0:dfs 16
definiens
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  let a2 be Element of NAT;
To prove
     a1 is_smooth_wrt a2
it is sufficient to prove
  thus a1 is eventually-nondecreasing & a1 taken_every a2 in Big_Oh a1;

:: ASYMPT_0:def 19
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being Element of NAT holds
      b1 is_smooth_wrt b2
   iff
      b1 is eventually-nondecreasing & b1 taken_every b2 in Big_Oh b1;

:: ASYMPT_0:attrnot 7 => ASYMPT_0:attr 7
definition
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
  attr a1 is smooth means
    for b1 being Element of NAT
          st 2 <= b1
       holds a1 is_smooth_wrt b1;
end;

:: ASYMPT_0:dfs 17
definiens
  let a1 be Function-like quasi_total eventually-nonnegative Relation of NAT,REAL;
To prove
     a1 is smooth
it is sufficient to prove
  thus for b1 being Element of NAT
          st 2 <= b1
       holds a1 is_smooth_wrt b1;

:: ASYMPT_0:def 20
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
      b1 is smooth
   iff
      for b2 being Element of NAT
            st 2 <= b2
         holds b1 is_smooth_wrt b2;

:: ASYMPT_0:th 37
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st ex b2 being Element of NAT st
           2 <= b2 & b1 is_smooth_wrt b2
   holds b1 is smooth;

:: ASYMPT_0:th 38
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonnegative eventually-nondecreasing Relation of NAT,REAL
for b3 being Element of NAT
      st b1 is smooth &
         2 <= b3 &
         b2 in Big_Oh(b1,{b3 |^ b4 where b4 is Element of NAT: TRUE})
   holds b2 in Big_Oh b1;

:: ASYMPT_0:th 39
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonnegative eventually-nondecreasing Relation of NAT,REAL
for b3 being Element of NAT
      st b1 is smooth &
         2 <= b3 &
         b2 in Big_Omega(b1,{b3 |^ b4 where b4 is Element of NAT: TRUE})
   holds b2 in Big_Omega b1;

:: ASYMPT_0:th 40
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
for b2 being Function-like quasi_total eventually-nonnegative eventually-nondecreasing Relation of NAT,REAL
for b3 being Element of NAT
      st b1 is smooth &
         2 <= b3 &
         b2 in Big_Theta(b1,{b3 |^ b4 where b4 is Element of NAT: TRUE})
   holds b2 in Big_Theta b1;

:: ASYMPT_0:funcnot 17 => ASYMPT_0:func 17
definition
  let a1 be non empty set;
  let a2, a3 be FUNCTION_DOMAIN of a1,REAL;
  func A2 + A3 -> FUNCTION_DOMAIN of a1,REAL equals
    {b1 where b1 is Element of Funcs(a1,REAL): ex b2, b3 being Element of Funcs(a1,REAL) st
       b2 in a2 &
        b3 in a3 &
        (for b4 being Element of a1 holds
           b1 . b4 = (b2 . b4) + (b3 . b4))};
end;

:: ASYMPT_0:def 21
theorem
for b1 being non empty set
for b2, b3 being FUNCTION_DOMAIN of b1,REAL holds
b2 + b3 = {b4 where b4 is Element of Funcs(b1,REAL): ex b5, b6 being Element of Funcs(b1,REAL) st
   b5 in b2 &
    b6 in b3 &
    (for b7 being Element of b1 holds
       b4 . b7 = (b5 . b7) + (b6 . b7))};

:: ASYMPT_0:funcnot 18 => ASYMPT_0:func 18
definition
  let a1 be non empty set;
  let a2, a3 be FUNCTION_DOMAIN of a1,REAL;
  func max(A2,A3) -> FUNCTION_DOMAIN of a1,REAL equals
    {b1 where b1 is Element of Funcs(a1,REAL): ex b2, b3 being Element of Funcs(a1,REAL) st
       b2 in a2 &
        b3 in a3 &
        (for b4 being Element of a1 holds
           b1 . b4 = max(b2 . b4,b3 . b4))};
end;

:: ASYMPT_0:def 22
theorem
for b1 being non empty set
for b2, b3 being FUNCTION_DOMAIN of b1,REAL holds
max(b2,b3) = {b4 where b4 is Element of Funcs(b1,REAL): ex b5, b6 being Element of Funcs(b1,REAL) st
   b5 in b2 &
    b6 in b3 &
    (for b7 being Element of b1 holds
       b4 . b7 = max(b5 . b7,b6 . b7))};

:: ASYMPT_0:th 41
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
(Big_Oh b1) + Big_Oh b2 = Big_Oh (b1 + b2);

:: ASYMPT_0:th 42
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
max(Big_Oh b1,Big_Oh b2) = Big_Oh max(b1,b2);

:: ASYMPT_0:funcnot 19 => ASYMPT_0:func 19
definition
  let a1, a2 be FUNCTION_DOMAIN of NAT,REAL;
  func A1 to_power A2 -> FUNCTION_DOMAIN of NAT,REAL equals
    {b1 where b1 is Element of Funcs(NAT,REAL): ex b2, b3 being Element of Funcs(NAT,REAL) st
       ex b4 being Element of NAT st
          b2 in a1 &
           b3 in a2 &
           (for b5 being Element of NAT
                 st b4 <= b5
              holds b1 . b5 = (b2 . b5) to_power (b3 . b5))};
end;

:: ASYMPT_0:def 23
theorem
for b1, b2 being FUNCTION_DOMAIN of NAT,REAL holds
b1 to_power b2 = {b3 where b3 is Element of Funcs(NAT,REAL): ex b4, b5 being Element of Funcs(NAT,REAL) st
   ex b6 being Element of NAT st
      b4 in b1 &
       b5 in b2 &
       (for b7 being Element of NAT
             st b6 <= b7
          holds b3 . b7 = (b4 . b7) to_power (b5 . b7))};