Article TAYLOR_1, MML version 4.99.1005

:: TAYLOR_1:funcnot 1 => TAYLOR_1:func 1
definition
  let a1 be integer set;
  func #Z A1 -> Function-like quasi_total Relation of REAL,REAL means
    for b1 being real set holds
       it . b1 = b1 #Z a1;
end;

:: TAYLOR_1:def 1
theorem
for b1 being integer set
for b2 being Function-like quasi_total Relation of REAL,REAL holds
      b2 = #Z b1
   iff
      for b3 being real set holds
         b2 . b3 = b3 #Z b1;

:: TAYLOR_1:th 1
theorem
for b1 being real set
for b2, b3 being natural set holds
b1 #Z (b3 + b2) = (b1 #Z b3) * (b1 #Z b2);

:: TAYLOR_1:th 2
theorem
for b1 being natural set
for b2 being real set holds
   #Z b1 is_differentiable_in b2 &
    diff(#Z b1,b2) = b1 * (b2 #Z (b1 - 1));

:: TAYLOR_1:th 3
theorem
for b1 being natural set
for b2 being real set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_in b2
   holds (#Z b1) * b3 is_differentiable_in b2 &
    diff((#Z b1) * b3,b2) = (b1 * ((b3 . b2) #Z (b1 - 1))) * diff(b3,b2);

:: TAYLOR_1:th 4
theorem
for b1 being real set holds
   exp_R - b1 = 1 / exp_R b1;

:: TAYLOR_1:th 5
theorem
for b1 being integer set
for b2 being real set holds
   (exp_R b2) #R (1 / b1) = exp_R (b2 / b1);

:: TAYLOR_1:th 6
theorem
for b1 being real set
for b2, b3 being integer set holds
(exp_R b1) #R (b2 / b3) = exp_R ((b2 / b3) * b1);

:: TAYLOR_1:th 7
theorem
for b1 being real set
for b2 being rational set holds
   (exp_R b1) #Q b2 = exp_R (b2 * b1);

:: TAYLOR_1:th 8
theorem
for b1, b2 being real set holds
(exp_R b1) #R b2 = exp_R (b2 * b1);

:: TAYLOR_1:th 9
theorem
for b1 being real set holds
   (exp_R 1) #R b1 = exp_R b1 & (exp_R 1) to_power b1 = exp_R b1 & number_e to_power b1 = exp_R b1 & number_e #R b1 = exp_R b1;

:: TAYLOR_1:th 10
theorem
for b1 being real set holds
   (exp_R . 1) #R b1 = exp_R . b1 &
    (exp_R . 1) to_power b1 = exp_R . b1 &
    number_e to_power b1 = exp_R . b1 &
    number_e #R b1 = exp_R . b1;

:: TAYLOR_1:th 11
theorem
2 <= number_e;

:: TAYLOR_1:th 12
theorem
for b1 being real set holds
   log(number_e,exp_R b1) = b1;

:: TAYLOR_1:th 13
theorem
for b1 being real set holds
   log(number_e,exp_R . b1) = b1;

:: TAYLOR_1:th 14
theorem
for b1 being real set
      st 0 < b1
   holds exp_R log(number_e,b1) = b1;

:: TAYLOR_1:th 15
theorem
for b1 being real set
      st 0 < b1
   holds exp_R . log(number_e,b1) = b1;

:: TAYLOR_1:th 16
theorem
exp_R is one-to-one &
 exp_R is_differentiable_on REAL &
 exp_R is_differentiable_on [#] REAL &
 (for b1 being Element of REAL holds
    diff(exp_R,b1) = exp_R . b1) &
 (for b1 being Element of REAL holds
    0 < diff(exp_R,b1)) &
 dom exp_R = REAL &
 dom exp_R = [#] REAL &
 rng exp_R = right_open_halfline 0;

:: TAYLOR_1:funcreg 1
registration
  cluster exp_R -> Function-like one-to-one quasi_total;
end;

:: TAYLOR_1:th 17
theorem
exp_R " is_differentiable_on dom (exp_R ") &
 (for b1 being real set
       st b1 in dom (exp_R ")
    holds diff(exp_R ",b1) = 1 / b1);

:: TAYLOR_1:funcreg 2
registration
  cluster right_open_halfline 0 -> non empty;
end;

:: TAYLOR_1:funcnot 2 => TAYLOR_1:func 2
definition
  let a1 be real set;
  func log_ A1 -> Function-like Relation of REAL,REAL means
    dom it = right_open_halfline 0 &
     (for b1 being Element of right_open_halfline 0 holds
        it . b1 = log(a1,b1));
end;

:: TAYLOR_1:def 2
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL holds
      b2 = log_ b1
   iff
      dom b2 = right_open_halfline 0 &
       (for b3 being Element of right_open_halfline 0 holds
          b2 . b3 = log(b1,b3));

:: TAYLOR_1:funcnot 3 => TAYLOR_1:func 3
definition
  func ln -> Function-like Relation of REAL,REAL equals
    log_ number_e;
end;

:: TAYLOR_1:def 3
theorem
ln = log_ number_e;

:: TAYLOR_1:th 18
theorem
ln = exp_R " &
 ln is one-to-one &
 dom ln = right_open_halfline 0 &
 rng ln = REAL &
 ln is_differentiable_on right_open_halfline 0 &
 (for b1 being Element of REAL
       st 0 < b1
    holds ln is_differentiable_in b1) &
 (for b1 being Element of right_open_halfline 0 holds
    diff(ln,b1) = 1 / b1) &
 (for b1 being Element of right_open_halfline 0 holds
    0 < diff(ln,b1));

:: TAYLOR_1:th 19
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1
   holds exp_R * b2 is_differentiable_in b1 &
    diff(exp_R * b2,b1) = (exp_R . (b2 . b1)) * diff(b2,b1);

:: TAYLOR_1:th 20
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1 & 0 < b2 . b1
   holds ln * b2 is_differentiable_in b1 &
    diff(ln * b2,b1) = (diff(b2,b1)) / (b2 . b1);

:: TAYLOR_1:funcnot 4 => TAYLOR_1:func 4
definition
  let a1 be real set;
  func #R A1 -> Function-like Relation of REAL,REAL means
    dom it = right_open_halfline 0 &
     (for b1 being Element of right_open_halfline 0 holds
        it . b1 = b1 #R a1);
end;

:: TAYLOR_1:def 4
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL holds
      b2 = #R b1
   iff
      dom b2 = right_open_halfline 0 &
       (for b3 being Element of right_open_halfline 0 holds
          b2 . b3 = b3 #R b1);

:: TAYLOR_1:th 21
theorem
for b1, b2 being real set
      st 0 < b1
   holds #R b2 is_differentiable_in b1 &
    diff(#R b2,b1) = b2 * (b1 #R (b2 - 1));

:: TAYLOR_1:th 22
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_in b1 & 0 < b3 . b1
   holds (#R b2) * b3 is_differentiable_in b1 &
    diff((#R b2) * b3,b1) = (b2 * ((b3 . b1) #R (b2 - 1))) * diff(b3,b1);

:: TAYLOR_1:funcnot 5 => TAYLOR_1:func 5
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of bool REAL;
  func diff(A1,A2) -> Functional_Sequence of REAL,REAL means
    it . 0 = a1 | a2 &
     (for b1 being natural set holds
        it . (b1 + 1) = (it . b1) `| a2);
end;

:: TAYLOR_1:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being Functional_Sequence of REAL,REAL holds
      b3 = diff(b1,b2)
   iff
      b3 . 0 = b1 | b2 &
       (for b4 being natural set holds
          b3 . (b4 + 1) = (b3 . b4) `| b2);

:: TAYLOR_1:prednot 1 => TAYLOR_1:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be natural set;
  let a3 be Element of bool REAL;
  pred A1 is_differentiable_on A2,A3 means
    for b1 being Element of NAT
          st b1 <= a2 - 1
       holds (diff(a1,a3)) . b1 is_differentiable_on a3;
end;

:: TAYLOR_1:dfs 6
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be natural set;
  let a3 be Element of bool REAL;
To prove
     a1 is_differentiable_on a2,a3
it is sufficient to prove
  thus for b1 being Element of NAT
          st b1 <= a2 - 1
       holds (diff(a1,a3)) . b1 is_differentiable_on a3;

:: TAYLOR_1:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being natural set
for b3 being Element of bool REAL holds
      b1 is_differentiable_on b2,b3
   iff
      for b4 being Element of NAT
            st b4 <= b2 - 1
         holds (diff(b1,b3)) . b4 is_differentiable_on b3;

:: TAYLOR_1:th 23
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being natural set
   st b1 is_differentiable_on b3,b2
for b4 being natural set
      st b4 <= b3
   holds b1 is_differentiable_on b4,b2;

:: TAYLOR_1:funcnot 6 => TAYLOR_1:func 6
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of bool REAL;
  let a3, a4 be real set;
  func Taylor(A1,A2,A3,A4) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being natural set holds
       it . b1 = ((((diff(a1,a2)) . b1) . a3) * ((a4 - a3) |^ b1)) / (b1 !);
end;

:: TAYLOR_1:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3, b4 being real set
for b5 being Function-like quasi_total Relation of NAT,REAL holds
      b5 = Taylor(b1,b2,b3,b4)
   iff
      for b6 being natural set holds
         b5 . b6 = ((((diff(b1,b2)) . b6) . b3) * ((b4 - b3) |^ b6)) / (b6 !);

:: TAYLOR_1:th 24
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being Element of NAT
   st b1 is_differentiable_on b3,b2
for b4, b5 being Element of REAL
      st b4 < b5 & ].b4,b5.[ c= b2
   holds ((diff(b1,b2)) . b3) | ].b4,b5.[ = (diff(b1,].b4,b5.[)) . b3;

:: TAYLOR_1:th 25
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of bool REAL
   st b2 is_differentiable_on b1,b3
for b4, b5 being Element of REAL
   st b4 < b5 &
      [.b4,b5.] c= b3 &
      (diff(b2,b3)) . b1 is_continuous_on [.b4,b5.] &
      b2 is_differentiable_on b1 + 1,].b4,b5.[
for b6 being Element of REAL
for b7 being Function-like Relation of REAL,REAL
      st dom b7 = REAL &
         (for b8 being Element of REAL holds
            b7 . b8 = ((b2 . b5) - ((Partial_Sums Taylor(b2,b3,b8,b5)) . b1)) - ((b6 * ((b5 - b8) |^ (b1 + 1))) / ((b1 + 1) !))) &
         ((b2 . b5) - ((Partial_Sums Taylor(b2,b3,b4,b5)) . b1)) - ((b6 * ((b5 - b4) |^ (b1 + 1))) / ((b1 + 1) !)) = 0
   holds b7 is_differentiable_on ].b4,b5.[ &
    b7 . b4 = 0 &
    b7 . b5 = 0 &
    b7 is_continuous_on [.b4,b5.] &
    (for b8 being Element of REAL
          st b8 in ].b4,b5.[
       holds diff(b7,b8) = (- (((((diff(b2,].b4,b5.[)) . (b1 + 1)) . b8) * ((b5 - b8) |^ b1)) / (b1 !))) + ((b6 * ((b5 - b8) |^ b1)) / (b1 !)));

:: TAYLOR_1:th 26
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of bool REAL
for b4, b5 being Element of REAL holds
ex b6 being Function-like quasi_total Relation of REAL,REAL st
   for b7 being Element of REAL holds
      b6 . b7 = ((b2 . b4) - ((Partial_Sums Taylor(b2,b3,b7,b4)) . b1)) - ((b5 * ((b4 - b7) |^ (b1 + 1))) / ((b1 + 1) !));

:: TAYLOR_1:th 27
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of bool REAL
   st b2 is_differentiable_on b1,b3
for b4, b5 being Element of REAL
      st b4 < b5 &
         [.b4,b5.] c= b3 &
         (diff(b2,b3)) . b1 is_continuous_on [.b4,b5.] &
         b2 is_differentiable_on b1 + 1,].b4,b5.[
   holds ex b6 being Element of REAL st
      b6 in ].b4,b5.[ &
       b2 . b5 = ((Partial_Sums Taylor(b2,b3,b4,b5)) . b1) + (((((diff(b2,].b4,b5.[)) . (b1 + 1)) . b6) * ((b5 - b4) |^ (b1 + 1))) / ((b1 + 1) !));

:: TAYLOR_1:th 28
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of bool REAL
   st b2 is_differentiable_on b1,b3
for b4, b5 being Element of REAL
   st b4 < b5 &
      [.b4,b5.] c= b3 &
      (diff(b2,b3)) . b1 is_continuous_on [.b4,b5.] &
      b2 is_differentiable_on b1 + 1,].b4,b5.[
for b6 being Element of REAL
for b7 being Function-like Relation of REAL,REAL
      st dom b7 = REAL &
         (for b8 being Element of REAL holds
            b7 . b8 = ((b2 . b4) - ((Partial_Sums Taylor(b2,b3,b8,b4)) . b1)) - ((b6 * ((b4 - b8) |^ (b1 + 1))) / ((b1 + 1) !))) &
         ((b2 . b4) - ((Partial_Sums Taylor(b2,b3,b5,b4)) . b1)) - ((b6 * ((b4 - b5) |^ (b1 + 1))) / ((b1 + 1) !)) = 0
   holds b7 is_differentiable_on ].b4,b5.[ &
    b7 . b5 = 0 &
    b7 . b4 = 0 &
    b7 is_continuous_on [.b4,b5.] &
    (for b8 being Element of REAL
          st b8 in ].b4,b5.[
       holds diff(b7,b8) = (- (((((diff(b2,].b4,b5.[)) . (b1 + 1)) . b8) * ((b4 - b8) |^ b1)) / (b1 !))) + ((b6 * ((b4 - b8) |^ b1)) / (b1 !)));

:: TAYLOR_1:th 29
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of bool REAL
   st b2 is_differentiable_on b1,b3
for b4, b5 being Element of REAL
      st b4 < b5 &
         [.b4,b5.] c= b3 &
         (diff(b2,b3)) . b1 is_continuous_on [.b4,b5.] &
         b2 is_differentiable_on b1 + 1,].b4,b5.[
   holds ex b6 being Element of REAL st
      b6 in ].b4,b5.[ &
       b2 . b4 = ((Partial_Sums Taylor(b2,b3,b5,b4)) . b1) + (((((diff(b2,].b4,b5.[)) . (b1 + 1)) . b6) * ((b4 - b5) |^ (b1 + 1))) / ((b1 + 1) !));

:: TAYLOR_1:th 30
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being open Element of bool REAL
   st b3 c= b2
for b4 being Element of NAT
      st b1 is_differentiable_on b4,b2
   holds ((diff(b1,b2)) . b4) | b3 = (diff(b1,b3)) . b4;

:: TAYLOR_1:th 31
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being open Element of bool REAL
   st b3 c= b2
for b4 being natural set
      st b1 is_differentiable_on b4 + 1,b2
   holds b1 is_differentiable_on b4 + 1,b3;

:: TAYLOR_1:th 32
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being Element of REAL
   st b3 in b2
for b4 being Element of NAT holds
   b1 . b3 = (Partial_Sums Taylor(b1,b2,b3,b3)) . b4;

:: TAYLOR_1:th 33
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3, b4 being Element of REAL
   st 0 < b4 &
      b2 is_differentiable_on b1 + 1,].b3 - b4,b3 + b4.[
for b5 being Element of REAL
      st b5 in ].b3 - b4,b3 + b4.[
   holds ex b6 being Element of REAL st
      0 < b6 &
       b6 < 1 &
       b2 . b5 = ((Partial_Sums Taylor(b2,].b3 - b4,b3 + b4.[,b3,b5)) . b1) + (((((diff(b2,].b3 - b4,b3 + b4.[)) . (b1 + 1)) . (b3 + (b6 * (b5 - b3)))) * ((b5 - b3) |^ (b1 + 1))) / ((b1 + 1) !));