Article ASYMPT_1, MML version 4.99.1005

:: ASYMPT_1:th 1
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b1 . b3 = ((((12 * (b3 to_power 3)) * log(2,b3)) - (5 * (b3 ^2))) + ((log(2,b3)) ^2)) + 36) &
         b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = (b3 to_power 3) * log(2,b3))
   holds ex b3, b4 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b3 = b1 & b4 = b2 & b3 in Big_Oh b4;

:: ASYMPT_1:th 2
theorem
for b1, b2 being logbase Element of REAL
for b3, b4 being Function-like quasi_total Relation of NAT,REAL
      st 1 < b1 &
         1 < b2 &
         b3 . 0 = 0 &
         (for b5 being Element of NAT
               st 0 < b5
            holds b3 . b5 = log(b1,b5)) &
         b4 . 0 = 0 &
         (for b5 being Element of NAT
               st 0 < b5
            holds b4 . b5 = log(b2,b5))
   holds ex b5, b6 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b5 = b3 & b6 = b4 & Big_Oh b5 = Big_Oh b6;

:: ASYMPT_1:funcnot 1 => ASYMPT_1:func 1
definition
  let a1, a2, a3 be Element of REAL;
  func seq_a^(A1,A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = a1 to_power ((a2 * b1) + a3);
end;

:: ASYMPT_1:def 1
theorem
for b1, b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of NAT,REAL holds
      b4 = seq_a^(b1,b2,b3)
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b1 to_power ((b2 * b5) + b3);

:: ASYMPT_1:funcreg 1
registration
  let a1 be positive Element of REAL;
  let a2, a3 be Element of REAL;
  cluster seq_a^(a1,a2,a3) -> Function-like quasi_total eventually-positive;
end;

:: ASYMPT_1:th 3
theorem
for b1, b2 being positive Element of REAL
      st b1 < b2
   holds not seq_a^(b2,1,0) in Big_Oh seq_a^(b1,1,0);

:: ASYMPT_1:funcnot 2 => ASYMPT_1:func 2
definition
  func seq_logn -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = 0 &
     (for b1 being Element of NAT
           st 0 < b1
        holds it . b1 = log(2,b1));
end;

:: ASYMPT_1:def 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 = seq_logn
   iff
      b1 . 0 = 0 &
       (for b2 being Element of NAT
             st 0 < b2
          holds b1 . b2 = log(2,b2));

:: ASYMPT_1:funcnot 3 => ASYMPT_1:func 3
definition
  let a1 be Element of REAL;
  func seq_n^ A1 -> Function-like quasi_total Relation of NAT,REAL means
    it . 0 = 0 &
     (for b1 being Element of NAT
           st 0 < b1
        holds it . b1 = b1 to_power a1);
end;

:: ASYMPT_1:def 3
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = seq_n^ b1
   iff
      b2 . 0 = 0 &
       (for b3 being Element of NAT
             st 0 < b3
          holds b2 . b3 = b3 to_power b1);

:: ASYMPT_1:funcreg 2
registration
  cluster seq_logn -> Function-like quasi_total eventually-positive;
end;

:: ASYMPT_1:funcreg 3
registration
  let a1 be Element of REAL;
  cluster seq_n^ a1 -> Function-like quasi_total eventually-positive;
end;

:: ASYMPT_1:th 4
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Oh b1 c= Big_Oh b2 & Big_Oh b1 <> Big_Oh b2
iff
   b1 in Big_Oh b2 & not b1 in Big_Omega b2;

:: ASYMPT_1:th 5
theorem
Big_Oh seq_logn c= Big_Oh seq_n^ (1 / 2) &
 Big_Oh seq_logn <> Big_Oh seq_n^ (1 / 2);

:: ASYMPT_1:th 6
theorem
seq_n^ (1 / 2) in Big_Omega seq_logn &
 not seq_logn in Big_Omega seq_n^ (1 / 2);

:: ASYMPT_1:th 7
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 holds
           b1 . b3 = Sum(seq_n^ b2,b3)
   holds b1 in Big_Theta seq_n^ (b2 + 1);

:: ASYMPT_1:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b2 being Element of NAT
               st 0 < b2
            holds b1 . b2 = b2 to_power log(2,b2))
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & b2 is not smooth;

:: ASYMPT_1:funcnot 4 => ASYMPT_1:func 4
definition
  let a1 be Element of REAL;
  func seq_const A1 -> Function-like quasi_total Relation of NAT,REAL equals
    NAT --> a1;
end;

:: ASYMPT_1:def 4
theorem
for b1 being Element of REAL holds
   seq_const b1 = NAT --> b1;

:: ASYMPT_1:funcreg 4
registration
  cluster seq_const 1 -> Function-like quasi_total eventually-nonnegative;
end;

:: ASYMPT_1:th 9
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   ex b2 being FUNCTION_DOMAIN of NAT,REAL st
      b2 = {seq_n^ 1} &
       (b1 in b2 to_power Big_Oh seq_const 1 implies ex b3 being Element of NAT st
          ex b4 being Element of REAL st
             ex b5 being Element of NAT st
                0 < b4 &
                 (for b6 being Element of NAT
                       st b3 <= b6
                    holds 1 <= b1 . b6 & b1 . b6 <= b4 * ((seq_n^ b5) . b6))) &
       (for b3 being Element of NAT
       for b4 being Element of REAL
       for b5 being Element of NAT
             st 0 < b4
          holds ex b6 being Element of NAT st
             b3 <= b6 &
              (1 <= b1 . b6 implies b4 * ((seq_n^ b5) . b6) < b1 . b6) or b1 in b2 to_power Big_Oh seq_const 1);

:: ASYMPT_1:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = ((3 * (10 to_power 6)) - ((18 * (10 to_power 3)) * b2)) + (27 * (b2 ^2))
   holds b1 in Big_Oh seq_n^ 2;

:: ASYMPT_1:th 11
theorem
seq_n^ 2 in Big_Oh seq_n^ 3;

:: ASYMPT_1:th 12
theorem
not seq_n^ 2 in Big_Omega seq_n^ 3;

:: ASYMPT_1:th 13
theorem
ex b1 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
   b1 = seq_a^(2,1,1) & seq_a^(2,1,0) in Big_Theta b1;

:: ASYMPT_1:funcnot 5 => ASYMPT_1:func 5
definition
  let a1 be Element of NAT;
  func seq_n! A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (b1 + a1) !;
end;

:: ASYMPT_1:def 5
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b2 = seq_n! b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = (b3 + b1) !;

:: ASYMPT_1:funcreg 5
registration
  let a1 be Element of NAT;
  cluster seq_n! a1 -> Function-like quasi_total eventually-positive;
end;

:: ASYMPT_1:th 14
theorem
not seq_n! 0 in Big_Theta seq_n! 1;

:: ASYMPT_1:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 in Big_Oh seq_n^ 1
   holds b1 (#) b1 in Big_Oh seq_n^ 2;

:: ASYMPT_1:th 16
theorem
ex b1 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
   b1 = seq_a^(2,1,0) &
    2 (#) seq_n^ 1 in Big_Oh seq_n^ 1 &
    not seq_a^(2,2,0) in Big_Oh b1;

:: ASYMPT_1:th 17
theorem
(159 / 100 <= log(2,3) or seq_n^ log(2,3) in Big_Oh seq_n^ (159 / 100) &
 not seq_n^ log(2,3) in Big_Omega seq_n^ (159 / 100) &
 not seq_n^ log(2,3) in Big_Theta seq_n^ (159 / 100));

:: ASYMPT_1:th 18
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st (for b3 being Element of NAT holds
            b1 . b3 = b3 mod 2) &
         (for b3 being Element of NAT holds
            b2 . b3 = (b3 + 1) mod 2)
   holds ex b3, b4 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL st
      b3 = b1 & b4 = b2 & not b3 in Big_Oh b4 & not b4 in Big_Oh b3;

:: ASYMPT_1:th 19
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   Big_Oh b1 = Big_Oh b2
iff
   b1 in Big_Theta b2;

:: ASYMPT_1:th 20
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Theta b2
iff
   Big_Theta b1 = Big_Theta b2;

:: ASYMPT_1:th 21
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st 0 < b1 &
         b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = b3 * log(2,b3))
   holds ex b3 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b3 = b2 &
       Big_Oh b3 c= Big_Oh seq_n^ (1 + b1) &
       Big_Oh b3 <> Big_Oh seq_n^ (1 + b1);

:: ASYMPT_1:th 22
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st 0 < b1 &
         b1 < 1 &
         b2 . 0 = 0 &
         b2 . 1 = 0 &
         (for b3 being Element of NAT
               st 1 < b3
            holds b2 . b3 = (b3 to_power 2) / log(2,b3))
   holds ex b3 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b3 = b2 &
       Big_Oh seq_n^ (1 + b1) c= Big_Oh b3 &
       Big_Oh seq_n^ (1 + b1) <> Big_Oh b3;

:: ASYMPT_1:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         b1 . 1 = 0 &
         (for b2 being Element of NAT
               st 1 < b2
            holds b1 . b2 = (b2 to_power 2) / log(2,b2))
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & Big_Oh b2 c= Big_Oh seq_n^ 8 & Big_Oh b2 <> Big_Oh seq_n^ 8;

:: ASYMPT_1:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = ((b2 ^2 - b2) + 1) to_power 4
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & Big_Oh seq_n^ 8 = Big_Oh b2;

:: ASYMPT_1:th 25
theorem
for b1 being Element of REAL
      st 0 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = seq_a^(1 + b1,1,0) & Big_Oh seq_n^ 8 c= Big_Oh b2 & Big_Oh seq_n^ 8 <> Big_Oh b2;

:: ASYMPT_1:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b1 . b3 = b3 to_power log(2,b3)) &
         b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = b3 to_power sqrt b3)
   holds ex b3, b4 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b3 = b1 & b4 = b2 & Big_Oh b3 c= Big_Oh b4 & Big_Oh b3 <> Big_Oh b4;

:: ASYMPT_1:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b2 being Element of NAT
               st 0 < b2
            holds b1 . b2 = b2 to_power sqrt b2)
   holds ex b2, b3 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & b3 = seq_a^(2,1,0) & Big_Oh b2 c= Big_Oh b3 & Big_Oh b2 <> Big_Oh b3;

:: ASYMPT_1:th 28
theorem
ex b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
   b1 = seq_a^(2,1,0) & b2 = seq_a^(2,1,1) & Big_Oh b1 = Big_Oh b2;

:: ASYMPT_1:th 29
theorem
ex b1, b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
   b1 = seq_a^(2,1,0) & b2 = seq_a^(2,2,0) & Big_Oh b1 c= Big_Oh b2 & Big_Oh b1 <> Big_Oh b2;

:: ASYMPT_1:th 30
theorem
ex b1 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
   b1 = seq_a^(2,2,0) & Big_Oh b1 c= Big_Oh seq_n! 0 & Big_Oh b1 <> Big_Oh seq_n! 0;

:: ASYMPT_1:th 31
theorem
Big_Oh seq_n! 0 c= Big_Oh seq_n! 1 &
 Big_Oh seq_n! 0 <> Big_Oh seq_n! 1;

:: ASYMPT_1:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b2 being Element of NAT
               st 0 < b2
            holds b1 . b2 = b2 to_power b2)
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & Big_Oh seq_n! 1 c= Big_Oh b2 & Big_Oh seq_n! 1 <> Big_Oh b2;

:: ASYMPT_1:th 33
theorem
for b1 being Element of NAT
   st 1 <= b1
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT
      st for b4 being Element of NAT holds
           b2 . b4 = Sum(seq_n^ b3,b4)
   holds (b1 to_power (b3 + 1)) / (b3 + 1) <= b2 . b1;

:: ASYMPT_1:th 34
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = b3 * log(2,b3)) &
         (for b3 being Element of NAT holds
            b1 . b3 = log(2,b3 !))
   holds ex b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL st
      b3 = b2 & b1 in Big_Theta b3;

:: ASYMPT_1:th 35
theorem
for b1 being Function-like quasi_total eventually-nonnegative eventually-nondecreasing Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           (b3 mod 2 = 0 implies b2 . b3 = 1) &
            (b3 mod 2 = 1 implies b2 . b3 = b3)
   holds not b2 in Big_Theta b1;

:: ASYMPT_1:funcnot 6 => ASYMPT_1:func 6
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL *;
  let a2 be Element of NAT;
  redefine func a1 . a2 -> FinSequence of REAL;
end;

:: ASYMPT_1:funcnot 7 => ASYMPT_1:func 7
definition
  func POWEROF2SET -> non empty Element of bool NAT equals
    {2 to_power b1 where b1 is Element of NAT: TRUE};
end;

:: ASYMPT_1:def 8
theorem
POWEROF2SET = {2 to_power b1 where b1 is Element of NAT: TRUE};

:: ASYMPT_1:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           (b2 in POWEROF2SET implies b1 . b2 = b2) &
            (b2 in POWEROF2SET or b1 . b2 = 2 to_power b2)
   holds b1 in Big_Theta(seq_n^ 1,POWEROF2SET) & not b1 in Big_Theta seq_n^ 1 & seq_n^ 1 is smooth & b1 is not eventually-nondecreasing;

:: ASYMPT_1:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b1 . b3 = b3 to_power (2 to_power [\log(2,b3)/])) &
         b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = b3 to_power b3)
   holds ex b3 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b3 = b2 & b1 in Big_Theta(b3,POWEROF2SET) & not b1 in Big_Theta b3 & b1 is eventually-nondecreasing & b3 is eventually-nondecreasing & not b3 is_smooth_wrt 2;

:: ASYMPT_1:th 39
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           (b2 in POWEROF2SET implies b1 . b2 = b2) &
            (b2 in POWEROF2SET or b1 . b2 = b2 to_power 2)
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 & seq_n^ 1 in Big_Theta(b2,POWEROF2SET) & not seq_n^ 1 in Big_Theta b2 & b2 taken_every 2 in Big_Oh b2 & seq_n^ 1 is eventually-nondecreasing & b2 is not eventually-nondecreasing;

:: ASYMPT_1:funcnot 8 => ASYMPT_1:func 8
definition
  let a1 be Element of NAT;
  func Step1 A1 -> Element of NAT means
    ex b1 being Element of NAT st
       b1 ! <= a1 & a1 < (b1 + 1) ! & it = b1 !
    if a1 <> 0
    otherwise it = 0;
end;

:: ASYMPT_1:def 9
theorem
for b1, b2 being Element of NAT holds
(b1 = 0 or    (b2 = Step1 b1
 iff
    ex b3 being Element of NAT st
       b3 ! <= b1 & b1 < (b3 + 1) ! & b2 = b3 !)) &
 (b1 = 0 implies    (b2 = Step1 b1
 iff
    b2 = 0));

:: ASYMPT_1:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = Step1 b2
   holds ex b2 being Function-like quasi_total eventually-positive Relation of NAT,REAL st
      b2 = b1 &
       b1 is eventually-nondecreasing &
       (for b3 being Element of NAT holds
          b1 . b3 <= (seq_n^ 1) . b3) &
       b2 is not smooth;

:: ASYMPT_1:th 41
theorem
for b1 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st b1 = (seq_n^ 1) - seq_const 1
   holds (Big_Theta b1) + Big_Theta seq_n^ 1 = Big_Theta seq_n^ 1;

:: ASYMPT_1:th 42
theorem
ex b1 being FUNCTION_DOMAIN of NAT,REAL st
   b1 = {seq_n^ 1} &
    (for b2 being Element of NAT holds
       (seq_n^ - 1) . b2 <= (seq_n^ 1) . b2) &
    not seq_n^ - 1 in b1 to_power Big_Oh seq_const 1;

:: ASYMPT_1:th 43
theorem
for b1 being non negative Element of REAL
for b2, b3 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL
      st (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
                   holds b4 <= b3 . b6)) &
         b2 in Big_Oh (b1 + b3)
   holds b2 in Big_Oh b3;

:: ASYMPT_1:th 44
theorem
2 to_power 2 = 4;

:: ASYMPT_1:th 45
theorem
2 to_power 3 = 8;

:: ASYMPT_1:th 46
theorem
2 to_power 4 = 16;

:: ASYMPT_1:th 47
theorem
2 to_power 5 = 32;

:: ASYMPT_1:th 48
theorem
2 to_power 6 = 64;

:: ASYMPT_1:th 49
theorem
2 to_power 12 = 4096;

:: ASYMPT_1:th 50
theorem
for b1 being Element of NAT
      st 3 <= b1
   holds (2 * b1) + 1 < b1 ^2;

:: ASYMPT_1:th 51
theorem
for b1 being Element of NAT
      st 10 <= b1
   holds (2 * b1) ^2 < 2 to_power (b1 - 1);

:: ASYMPT_1:th 52
theorem
for b1 being Element of NAT
      st 9 <= b1
   holds (b1 + 1) to_power 6 < 2 * (b1 to_power 6);

:: ASYMPT_1:th 53
theorem
for b1 being Element of NAT
      st 30 <= b1
   holds b1 to_power 6 < 2 to_power b1;

:: ASYMPT_1:th 54
theorem
for b1 being Element of REAL
      st 9 < b1
   holds (2 * b1) ^2 < 2 to_power b1;

:: ASYMPT_1:th 55
theorem
ex b1 being Element of NAT st
   for b2 being Element of NAT
         st b1 <= b2
      holds 1 < (sqrt b2) - log(2,b2);

:: ASYMPT_1:th 56
theorem
for b1, b2, b3 being Element of REAL
      st 0 < b1 & 0 < b3 & b3 <> 1
   holds b1 to_power b2 = b3 to_power (b2 * log(b3,b1));

:: ASYMPT_1:th 57
theorem
(4 + 1) ! = 120;

:: ASYMPT_1:th 58
theorem
5 to_power 5 = 3125;

:: ASYMPT_1:th 59
theorem
4 to_power 4 = 256;

:: ASYMPT_1:th 60
theorem
for b1 being Element of NAT holds
   0 < (b1 ^2 - b1) + 1;

:: ASYMPT_1:th 61
theorem
for b1 being Element of NAT
      st 2 <= b1
   holds 1 < b1 !;

:: ASYMPT_1:th 62
theorem
for b1, b2 being Element of NAT
      st b2 <= b1
   holds b2 ! <= b1 !;

:: ASYMPT_1:th 63
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds ex b2 being Element of NAT st
      b2 ! <= b1 &
       b1 < (b2 + 1) ! &
       (for b3 being Element of NAT
             st b3 ! <= b1 & b1 < (b3 + 1) !
          holds b3 = b2);

:: ASYMPT_1:th 64
theorem
for b1 being Element of NAT
      st 2 <= b1
   holds [/b1 / 2\] < b1;

:: ASYMPT_1:th 65
theorem
for b1 being Element of NAT
      st 3 <= b1
   holds b1 < b1 !;

:: ASYMPT_1:th 66
theorem
(seq_n^ 1) - seq_const 1 is eventually-positive;

:: ASYMPT_1:th 67
theorem
for b1 being Element of NAT
      st 2 <= b1
   holds b1 + 1 < 2 to_power b1;

:: ASYMPT_1:th 68
theorem
for b1 being logbase Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st 1 < b1 &
         b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = log(b1,b3))
   holds b2 is eventually-positive;

:: ASYMPT_1:th 69
theorem
for b1, b2 being Function-like quasi_total eventually-nonnegative Relation of NAT,REAL holds
   b1 in Big_Oh b2 & b2 in Big_Oh b1
iff
   Big_Oh b1 = Big_Oh b2;

:: ASYMPT_1:th 70
theorem
for b1, b2, b3 being Element of REAL
      st 0 < b1 & b1 <= b2 & 0 <= b3
   holds b1 to_power b3 <= b2 to_power b3;

:: ASYMPT_1:th 71
theorem
for b1 being Element of NAT
      st 4 <= b1
   holds (2 * b1) + 3 < 2 to_power b1;

:: ASYMPT_1:th 72
theorem
for b1 being Element of NAT
      st 6 <= b1
   holds (b1 + 1) ^2 < 2 to_power b1;

:: ASYMPT_1:th 73
theorem
for b1 being Element of REAL
      st 6 < b1
   holds b1 ^2 < 2 to_power b1;

:: ASYMPT_1:th 74
theorem
for b1 being positive Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 . 0 = 0 &
         (for b3 being Element of NAT
               st 0 < b3
            holds b2 . b3 = log(2,b3 to_power b1))
   holds b2 /" seq_n^ b1 is convergent & lim (b2 /" seq_n^ b1) = 0;

:: ASYMPT_1:th 75
theorem
for b1 being Element of REAL
      st 0 < b1
   holds seq_logn /" seq_n^ b1 is convergent & lim (seq_logn /" seq_n^ b1) = 0;

:: ASYMPT_1:th 76
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 b3 <= b2
           holds 0 <= b1 . b3
   holds 0 <= Sum(b1,b2);

:: ASYMPT_1:th 77
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT
      st for b4 being Element of NAT
              st b4 <= b3
           holds b1 . b4 <= b2 . b4
   holds Sum(b1,b3) <= Sum(b2,b3);

:: ASYMPT_1:th 78
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
   st b1 . 0 = 0 &
      (for b3 being Element of NAT
            st 0 < b3
         holds b1 . b3 = b2)
for b3 being Element of NAT holds
   Sum(b1,b3) = b2 * b3;

:: ASYMPT_1:th 79
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Element of NAT holds
(Sum(b1,b2,b3)) + (b1 . (b2 + 1)) = Sum(b1,b2 + 1,b3);

:: ASYMPT_1:th 80
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3, b4 being Element of NAT
      st b3 + 1 <= b4 &
         (for b5 being Element of NAT
               st b3 + 1 <= b5 & b5 <= b4
            holds b1 . b5 <= b2 . b5)
   holds Sum(b1,b4,b3) <= Sum(b2,b4,b3);

:: ASYMPT_1:th 81
theorem
for b1 being Element of NAT holds
   [/b1 / 2\] <= b1;

:: ASYMPT_1:th 82
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
for b3 being Element of NAT
   st b1 . 0 = 0 &
      (for b4 being Element of NAT
            st 0 < b4
         holds b1 . b4 = b2)
for b4 being Element of NAT holds
   Sum(b1,b3,b4) = b2 * (b3 - b4);

:: ASYMPT_1:th 83
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT
for b4 being Element of REAL
      st b1 is convergent &
         lim b1 = b4 &
         (for b5 being Element of NAT
               st b3 <= b5
            holds b1 . b5 = b2 . b5)
   holds b2 is convergent & lim b2 = b4;

:: ASYMPT_1:th 84
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds (b1 ^2 - b1) + 1 <= b1 ^2;

:: ASYMPT_1:th 85
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds b1 ^2 <= 2 * ((b1 ^2 - b1) + 1);

:: ASYMPT_1:th 86
theorem
for b1 being Element of REAL
      st 0 < b1 & b1 < 1
   holds ex b2 being Element of NAT st
      for b3 being Element of NAT
            st b2 <= b3
         holds 8 * log(2,b3) < (b3 * log(2,1 + b1)) - (8 * log(2,b3));

:: ASYMPT_1:th 87
theorem
for b1 being Element of NAT
      st 10 <= b1
   holds (2 to_power (2 * b1)) / (b1 !) < 1 / (2 to_power (b1 - 9));

:: ASYMPT_1:th 88
theorem
for b1 being Element of NAT
      st 3 <= b1
   holds b1 - 1 <= 2 * (b1 - 2);

:: ASYMPT_1:th 89
theorem
for b1 being real set
      st 0 <= b1
   holds b1 to_power (1 / 2) = sqrt b1;

:: ASYMPT_1:th 90
theorem
ex b1 being Element of NAT st
   for b2 being Element of NAT
         st b1 <= b2
      holds b2 / 2 < b2 - ((sqrt b2) * log(2,b2));

:: ASYMPT_1:th 91
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = (1 + (1 / (b2 + 1))) to_power (b2 + 1)
   holds b1 is non-decreasing;

:: ASYMPT_1:th 92
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds ((b1 + 1) / b1) to_power b1 <= ((b1 + 2) / (b1 + 1)) to_power (b1 + 1);

:: ASYMPT_1:th 93
theorem
for b1, b2 being Element of NAT
      st b1 <= b2
   holds ((b2 + 1) choose b1) / (b2 + 1) <= b2 choose b1;

:: ASYMPT_1:th 94
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = log(2,b2 !)
for b2 being Element of NAT holds
   b1 . b2 = Sum(seq_logn,b2);

:: ASYMPT_1:th 95
theorem
for b1 being Element of NAT
      st 4 <= b1
   holds 2 * b1 <= b1 * log(2,b1);

:: ASYMPT_1:th 97
theorem
for b1 being Element of NAT
      st 2 <= b1
   holds b1 + 1 < b1 ^2;

:: ASYMPT_1:th 98
theorem
for b1 being Element of NAT
      st 1 <= b1
   holds 1 < (2 to_power (b1 + 1)) - (2 to_power b1);

:: ASYMPT_1:th 99
theorem
for b1 being Element of NAT
      st 2 <= b1
   holds not (2 to_power b1) - 1 in POWEROF2SET;

:: ASYMPT_1:th 100
theorem
for b1, b2 being Element of NAT
      st 1 <= b2 & b1 ! <= b2 & b2 < (b1 + 1) !
   holds Step1 b2 = b1 !;

:: ASYMPT_1:th 101
theorem
for b1, b2, b3 being Element of REAL
      st 1 < b1 & b1 <= b2 & 1 <= b3
   holds log(b2,b3) <= log(b1,b3);