Theory Taylor

Up to index of Isabelle/HOL/HOL-Complex

theory Taylor
imports MacLaurin
begin

(*  Title:      HOL/Hyperreal/Taylor.thy
    ID:         $Id: Taylor.thy,v 1.4 2007/10/23 21:27:24 nipkow Exp $
    Author:     Lukas Bulwahn, Bernhard Haeupler, Technische Universitaet Muenchen
*)

header {* Taylor series *}

theory Taylor
imports MacLaurin
begin

text {*
We use MacLaurin and the translation of the expansion point @{text c} to @{text 0}
to prove Taylor's theorem.
*}

lemma taylor_up: 
  assumes INIT: "n>0" "diff 0 = f"
  and DERIV: "(∀ m t. m < n & a ≤ t & t ≤ b --> DERIV (diff m) t :> (diff (Suc m) t))"
  and INTERV: "a ≤ c" "c < b" 
  shows "∃ t. c < t & t < b & 
    f b = setsum (%m. (diff m c / real (fact m)) * (b - c)^m) {0..<n} +
      (diff n t / real (fact n)) * (b - c)^n"
proof -
  from INTERV have "0 < b-c" by arith
  moreover 
  from INIT have "n>0" "((λm x. diff m (x + c)) 0) = (λx. f (x + c))" by auto
  moreover
  have "ALL m t. m < n & 0 <= t & t <= b - c --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
  proof (intro strip)
    fix m t
    assume "m < n & 0 <= t & t <= b - c"
    with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto
    moreover
    from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
    ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)"
      by (rule DERIV_chain2)
    thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
  qed
  ultimately 
  have EX:"EX t>0. t < b - c & 
    f (b - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
      diff n (t + c) / real (fact n) * (b - c) ^ n" 
    by (rule Maclaurin)
  show ?thesis
  proof -
    from EX obtain x where 
      X: "0 < x & x < b - c & 
        f (b - c + c) = (∑m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
          diff n (x + c) / real (fact n) * (b - c) ^ n" ..
    let ?H = "x + c"
    from X have "c<?H & ?H<b ∧ f b = (∑m = 0..<n. diff m c / real (fact m) * (b - c) ^ m) +
      diff n ?H / real (fact n) * (b - c) ^ n"
      by fastsimp
    thus ?thesis by fastsimp
  qed
qed

lemma taylor_down:
  assumes INIT: "n>0" "diff 0 = f"
  and DERIV: "(∀ m t. m < n & a ≤ t & t ≤ b --> DERIV (diff m) t :> (diff (Suc m) t))"
  and INTERV: "a < c" "c ≤ b"
  shows "∃ t. a < t & t < c & 
    f a = setsum (% m. (diff m c / real (fact m)) * (a - c)^m) {0..<n} +
      (diff n t / real (fact n)) * (a - c)^n" 
proof -
  from INTERV have "a-c < 0" by arith
  moreover 
  from INIT have "n>0" "((λm x. diff m (x + c)) 0) = (λx. f (x + c))" by auto
  moreover
  have "ALL m t. m < n & a-c <= t & t <= 0 --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
  proof (rule allI impI)+
    fix m t
    assume "m < n & a-c <= t & t <= 0"
    with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto 
    moreover
    from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
    ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)" by (rule DERIV_chain2)
    thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
  qed
  ultimately 
  have EX: "EX t>a - c. t < 0 &
    f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
      diff n (t + c) / real (fact n) * (a - c) ^ n" 
    by (rule Maclaurin_minus)
  show ?thesis
  proof -
    from EX obtain x where X: "a - c < x & x < 0 &
      f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
        diff n (x + c) / real (fact n) * (a - c) ^ n" ..
    let ?H = "x + c"
    from X have "a<?H & ?H<c ∧ f a = (∑m = 0..<n. diff m c / real (fact m) * (a - c) ^ m) +
      diff n ?H / real (fact n) * (a - c) ^ n"
      by fastsimp
    thus ?thesis by fastsimp
  qed
qed

lemma taylor:
  assumes INIT: "n>0" "diff 0 = f"
  and DERIV: "(∀ m t. m < n & a ≤ t & t ≤ b --> DERIV (diff m) t :> (diff (Suc m) t))"
  and INTERV: "a ≤ c " "c ≤ b" "a ≤ x" "x ≤ b" "x ≠ c" 
  shows "∃ t. (if x<c then (x < t & t < c) else (c < t & t < x)) &
    f x = setsum (% m. (diff m c / real (fact m)) * (x - c)^m) {0..<n} +
      (diff n t / real (fact n)) * (x - c)^n" 
proof (cases "x<c")
  case True
  note INIT
  moreover from DERIV and INTERV
  have "∀m t. m < n ∧ x ≤ t ∧ t ≤ b --> DERIV (diff m) t :> diff (Suc m) t"
    by fastsimp
  moreover note True
  moreover from INTERV have "c ≤ b" by simp
  ultimately have EX: "∃t>x. t < c ∧ f x =
    (∑m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
      diff n t / real (fact n) * (x - c) ^ n"
    by (rule taylor_down)
  with True show ?thesis by simp
next
  case False
  note INIT
  moreover from DERIV and INTERV
  have "∀m t. m < n ∧ a ≤ t ∧ t ≤ x --> DERIV (diff m) t :> diff (Suc m) t"
    by fastsimp
  moreover from INTERV have "a ≤ c" by arith
  moreover from False and INTERV have "c < x" by arith
  ultimately have EX: "∃t>c. t < x ∧ f x =
    (∑m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
      diff n t / real (fact n) * (x - c) ^ n" 
    by (rule taylor_up)
  with False show ?thesis by simp
qed

end

lemma taylor_up:

  [| 0 < n; diff 0 = f;
     ∀m t. m < na  tt  b --> DERIV (diff m) t :> diff (Suc m) t; a  c;
     c < b |]
  ==> ∃t>c. t < bf b =
            (∑m = 0..<n. diff m c / real (fact m) * (b - c) ^ m) +
            diff n t / real (fact n) * (b - c) ^ n

lemma taylor_down:

  [| 0 < n; diff 0 = f;
     ∀m t. m < na  tt  b --> DERIV (diff m) t :> diff (Suc m) t; a < c;
     c  b |]
  ==> ∃t>a. t < cf a =
            (∑m = 0..<n. diff m c / real (fact m) * (a - c) ^ m) +
            diff n t / real (fact n) * (a - c) ^ n

lemma taylor:

  [| 0 < n; diff 0 = f;
     ∀m t. m < na  tt  b --> DERIV (diff m) t :> diff (Suc m) t; a  c;
     c  b; a  x; x  b; x  c |]
  ==> ∃t. (if x < c then x < tt < c else c < tt < x) ∧
          f x =
          (∑m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
          diff n t / real (fact n) * (x - c) ^ n