(* Title: HOL/Taylor.thy
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: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
and INTERV: "a \<le> c" "c < b"
shows "\<exists> 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" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. 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) = (\<Sum>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 \<and> f b = (\<Sum>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: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
and INTERV: "a < c" "c \<le> b"
shows "\<exists> 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" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. 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 \<and> f a = (\<Sum>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: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c"
shows "\<exists> 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 "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
by fastsimp
moreover note True
moreover from INTERV have "c \<le> b" by simp
ultimately have EX: "\<exists>t>x. t < c \<and> f x =
(\<Sum>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 "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
by fastsimp
moreover from INTERV have "a \<le> c" by arith
moreover from False and INTERV have "c < x" by arith
ultimately have EX: "\<exists>t>c. t < x \<and> f x =
(\<Sum>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