haftmann@28952
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(* Author : Jacques D. Fleuriot
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paulson@12224
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Copyright : 2001 University of Edinburgh
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paulson@15079
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3 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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bulwahn@41368
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Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005
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paulson@12224
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*)
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paulson@12224
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paulson@15944
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header{*MacLaurin Series*}
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paulson@15944
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nipkow@15131
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theory MacLaurin
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chaieb@29748
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imports Transcendental
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nipkow@15131
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11 |
begin
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obua@14738
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12 |
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paulson@15079
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subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
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paulson@15079
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14 |
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paulson@15079
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text{*This is a very long, messy proof even now that it's been broken down
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paulson@15079
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into lemmas.*}
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paulson@15079
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paulson@15079
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lemma Maclaurin_lemma:
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paulson@15079
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"0 < h ==>
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nipkow@15539
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\<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
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paulson@15079
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(B * ((h^n) / real(fact n)))"
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hoelzl@41412
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by (rule exI[where x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
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hoelzl@41412
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real(fact n) / (h^n)"]) simp
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paulson@15079
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paulson@15079
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lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
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paulson@15079
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by arith
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paulson@15079
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avigad@32031
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lemma fact_diff_Suc [rule_format]:
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avigad@32031
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"n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
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avigad@32031
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by (subst fact_reduce_nat, auto)
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avigad@32031
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paulson@15079
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lemma Maclaurin_lemma2:
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hoelzl@41412
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fixes B
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bulwahn@41368
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assumes DERIV : "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
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hoelzl@41412
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and INIT : "n = Suc k"
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hoelzl@41412
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defines "difg \<equiv> (\<lambda>m t. diff m t - ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
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hoelzl@41412
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B * (t ^ (n - m) / real (fact (n - m)))))" (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")
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bulwahn@41368
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shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
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hoelzl@41412
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proof (rule allI impI)+
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hoelzl@41412
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fix m t assume INIT2: "m < n & 0 \<le> t & t \<le> h"
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hoelzl@41412
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have "DERIV (difg m) t :> diff (Suc m) t -
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hoelzl@41412
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((\<Sum>x = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) +
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hoelzl@41412
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real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)))" unfolding difg_def
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hoelzl@41412
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by (auto intro!: DERIV_intros DERIV[rule_format, OF INIT2])
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bulwahn@41368
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moreover
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hoelzl@41412
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from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m"
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hoelzl@41412
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unfolding atLeast0LessThan[symmetric] by auto
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hoelzl@41412
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have "(\<Sum>x = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) =
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hoelzl@41412
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(\<Sum>x = 0..<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)))"
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hoelzl@41412
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unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex)
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hoelzl@41412
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moreover
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hoelzl@41412
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have fact_neq_0: "\<And>x::nat. real (fact x) + real x * real (fact x) \<noteq> 0"
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hoelzl@41412
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by (metis fact_gt_zero_nat not_add_less1 real_of_nat_add real_of_nat_mult real_of_nat_zero_iff)
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hoelzl@41412
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have "\<And>x. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)) =
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hoelzl@41412
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diff (Suc m + x) 0 * t^x / real (fact x)"
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hoelzl@41412
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by (auto simp: field_simps real_of_nat_Suc fact_neq_0 intro!: nonzero_divide_eq_eq[THEN iffD2])
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hoelzl@41412
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moreover
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hoelzl@41412
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have "real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)) =
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hoelzl@41412
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B * (t ^ (n - Suc m) / real (fact (n - Suc m)))"
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hoelzl@41412
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using `0 < n - m` by (simp add: fact_reduce_nat)
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hoelzl@41412
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ultimately show "DERIV (difg m) t :> difg (Suc m) t"
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hoelzl@41412
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unfolding difg_def by simp
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bulwahn@41368
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qed
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avigad@32031
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paulson@15079
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lemma Maclaurin:
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huffman@29187
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assumes h: "0 < h"
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huffman@29187
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assumes n: "0 < n"
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huffman@29187
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assumes diff_0: "diff 0 = f"
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huffman@29187
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assumes diff_Suc:
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huffman@29187
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"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
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huffman@29187
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shows
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huffman@29187
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"\<exists>t. 0 < t & t < h &
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paulson@15079
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f h =
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nipkow@15539
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setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
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hoelzl@41412
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(diff n t / real (fact n)) * h ^ n"
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huffman@29187
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proof -
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huffman@29187
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from n obtain m where m: "n = Suc m"
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hoelzl@41412
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by (cases n) (simp add: n)
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huffman@29187
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huffman@29187
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obtain B where f_h: "f h =
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huffman@29187
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(\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
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huffman@29187
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B * (h ^ n / real (fact n))"
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huffman@29187
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using Maclaurin_lemma [OF h] ..
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huffman@29187
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hoelzl@41412
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def g \<equiv> "(\<lambda>t. f t -
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hoelzl@41412
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(setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {0..<n}
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hoelzl@41412
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+ (B * (t^n / real(fact n)))))"
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huffman@29187
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huffman@29187
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have g2: "g 0 = 0 & g h = 0"
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huffman@29187
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apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
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huffman@30019
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apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
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huffman@29187
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apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
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huffman@29187
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done
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huffman@29187
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hoelzl@41412
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def difg \<equiv> "(%m t. diff m t -
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huffman@29187
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(setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
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hoelzl@41412
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+ (B * ((t ^ (n - m)) / real (fact (n - m))))))"
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huffman@29187
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98 |
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huffman@29187
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have difg_0: "difg 0 = g"
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huffman@29187
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unfolding difg_def g_def by (simp add: diff_0)
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huffman@29187
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101 |
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huffman@29187
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have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
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huffman@29187
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m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
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hoelzl@41412
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using diff_Suc m unfolding difg_def by (rule Maclaurin_lemma2)
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huffman@29187
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105 |
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huffman@29187
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have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
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huffman@29187
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apply clarify
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huffman@29187
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apply (simp add: m difg_def)
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huffman@29187
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apply (frule less_iff_Suc_add [THEN iffD1], clarify)
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huffman@29187
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apply (simp del: setsum_op_ivl_Suc)
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huffman@30019
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apply (insert sumr_offset4 [of "Suc 0"])
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avigad@32039
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apply (simp del: setsum_op_ivl_Suc fact_Suc)
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huffman@29187
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113 |
done
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huffman@29187
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114 |
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huffman@29187
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have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
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huffman@29187
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by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
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huffman@29187
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117 |
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huffman@29187
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have differentiable_difg:
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huffman@29187
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"\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
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huffman@29187
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by (rule differentiableI [OF difg_Suc [rule_format]]) simp
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huffman@29187
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121 |
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huffman@29187
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have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
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huffman@29187
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\<Longrightarrow> difg (Suc m) t = 0"
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huffman@29187
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by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
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huffman@29187
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125 |
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huffman@29187
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126 |
have "m < n" using m by simp
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huffman@29187
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127 |
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huffman@29187
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have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
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huffman@29187
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using `m < n`
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huffman@29187
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proof (induct m)
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hoelzl@41412
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131 |
case 0
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huffman@29187
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132 |
show ?case
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huffman@29187
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133 |
proof (rule Rolle)
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huffman@29187
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show "0 < h" by fact
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huffman@29187
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show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
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huffman@29187
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show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
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huffman@29187
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by (simp add: isCont_difg n)
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huffman@29187
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138 |
show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
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huffman@29187
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by (simp add: differentiable_difg n)
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huffman@29187
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140 |
qed
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huffman@29187
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141 |
next
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hoelzl@41412
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142 |
case (Suc m')
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huffman@29187
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hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
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huffman@29187
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144 |
then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
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huffman@29187
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have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
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huffman@29187
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146 |
proof (rule Rolle)
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huffman@29187
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147 |
show "0 < t" by fact
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huffman@29187
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148 |
show "difg (Suc m') 0 = difg (Suc m') t"
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huffman@29187
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using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
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huffman@29187
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150 |
show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
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huffman@29187
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using `t < h` `Suc m' < n` by (simp add: isCont_difg)
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huffman@29187
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152 |
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
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huffman@29187
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153 |
using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
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huffman@29187
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154 |
qed
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huffman@29187
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155 |
thus ?case
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huffman@29187
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156 |
using `t < h` by auto
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huffman@29187
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157 |
qed
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huffman@29187
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158 |
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huffman@29187
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159 |
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
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huffman@29187
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160 |
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huffman@29187
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161 |
hence "difg (Suc m) t = 0"
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huffman@29187
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162 |
using `m < n` by (simp add: difg_Suc_eq_0)
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huffman@29187
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163 |
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huffman@29187
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164 |
show ?thesis
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huffman@29187
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165 |
proof (intro exI conjI)
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huffman@29187
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166 |
show "0 < t" by fact
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huffman@29187
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167 |
show "t < h" by fact
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huffman@29187
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168 |
show "f h =
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huffman@29187
|
169 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
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huffman@29187
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170 |
diff n t / real (fact n) * h ^ n"
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huffman@29187
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171 |
using `difg (Suc m) t = 0`
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avigad@32039
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172 |
by (simp add: m f_h difg_def del: fact_Suc)
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huffman@29187
|
173 |
qed
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huffman@29187
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174 |
qed
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paulson@15079
|
175 |
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paulson@15079
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176 |
lemma Maclaurin_objl:
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nipkow@25162
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177 |
"0 < h & n>0 & diff 0 = f &
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nipkow@25134
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178 |
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
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nipkow@25134
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179 |
--> (\<exists>t. 0 < t & t < h &
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nipkow@25134
|
180 |
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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nipkow@25134
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181 |
diff n t / real (fact n) * h ^ n)"
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paulson@15079
|
182 |
by (blast intro: Maclaurin)
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paulson@15079
|
183 |
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paulson@15079
|
184 |
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paulson@15079
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185 |
lemma Maclaurin2:
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bulwahn@41368
|
186 |
assumes INIT1: "0 < h " and INIT2: "diff 0 = f"
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bulwahn@41368
|
187 |
and DERIV: "\<forall>m t.
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bulwahn@41368
|
188 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
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bulwahn@41368
|
189 |
shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h =
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bulwahn@41368
|
190 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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bulwahn@41368
|
191 |
diff n t / real (fact n) * h ^ n"
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bulwahn@41368
|
192 |
proof (cases "n")
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bulwahn@41368
|
193 |
case 0 with INIT1 INIT2 show ?thesis by fastsimp
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bulwahn@41368
|
194 |
next
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hoelzl@41412
|
195 |
case Suc
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bulwahn@41368
|
196 |
hence "n > 0" by simp
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bulwahn@41368
|
197 |
from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and>
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bulwahn@41368
|
198 |
f h =
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hoelzl@41412
|
199 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n"
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bulwahn@41368
|
200 |
by (rule Maclaurin)
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bulwahn@41368
|
201 |
thus ?thesis by fastsimp
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bulwahn@41368
|
202 |
qed
|
paulson@15079
|
203 |
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paulson@15079
|
204 |
lemma Maclaurin2_objl:
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paulson@15079
|
205 |
"0 < h & diff 0 = f &
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paulson@15079
|
206 |
(\<forall>m t.
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paulson@15079
|
207 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
|
paulson@15079
|
208 |
--> (\<exists>t. 0 < t &
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paulson@15079
|
209 |
t \<le> h &
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paulson@15079
|
210 |
f h =
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nipkow@15539
|
211 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
|
paulson@15079
|
212 |
diff n t / real (fact n) * h ^ n)"
|
paulson@15079
|
213 |
by (blast intro: Maclaurin2)
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paulson@15079
|
214 |
|
paulson@15079
|
215 |
lemma Maclaurin_minus:
|
hoelzl@41412
|
216 |
assumes "h < 0" "0 < n" "diff 0 = f"
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hoelzl@41412
|
217 |
and DERIV: "\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t"
|
bulwahn@41368
|
218 |
shows "\<exists>t. h < t & t < 0 &
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bulwahn@41368
|
219 |
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
|
bulwahn@41368
|
220 |
diff n t / real (fact n) * h ^ n"
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bulwahn@41368
|
221 |
proof -
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hoelzl@41412
|
222 |
txt "Transform @{text ABL'} into @{text DERIV_intros} format."
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hoelzl@41412
|
223 |
note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
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hoelzl@41412
|
224 |
from assms
|
hoelzl@41412
|
225 |
have "\<exists>t>0. t < - h \<and>
|
bulwahn@41368
|
226 |
f (- (- h)) =
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bulwahn@41368
|
227 |
(\<Sum>m = 0..<n.
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bulwahn@41368
|
228 |
(- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) +
|
hoelzl@41412
|
229 |
(- 1) ^ n * diff n (- t) / real (fact n) * (- h) ^ n"
|
hoelzl@41412
|
230 |
by (intro Maclaurin) (auto intro!: DERIV_intros DERIV')
|
hoelzl@41412
|
231 |
then guess t ..
|
bulwahn@41368
|
232 |
moreover
|
bulwahn@41368
|
233 |
have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)"
|
bulwahn@41368
|
234 |
by (auto simp add: power_mult_distrib[symmetric])
|
bulwahn@41368
|
235 |
moreover
|
bulwahn@41368
|
236 |
have "(SUM m = 0..<n. -1 ^ m * diff m 0 * (- h) ^ m / real (fact m)) = (SUM m = 0..<n. diff m 0 * h ^ m / real (fact m))"
|
bulwahn@41368
|
237 |
by (auto intro: setsum_cong simp add: power_mult_distrib[symmetric])
|
bulwahn@41368
|
238 |
ultimately have " h < - t \<and>
|
bulwahn@41368
|
239 |
- t < 0 \<and>
|
bulwahn@41368
|
240 |
f h =
|
bulwahn@41368
|
241 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n"
|
bulwahn@41368
|
242 |
by auto
|
bulwahn@41368
|
243 |
thus ?thesis ..
|
bulwahn@41368
|
244 |
qed
|
paulson@15079
|
245 |
|
paulson@15079
|
246 |
lemma Maclaurin_minus_objl:
|
nipkow@25162
|
247 |
"(h < 0 & n > 0 & diff 0 = f &
|
paulson@15079
|
248 |
(\<forall>m t.
|
paulson@15079
|
249 |
m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
|
paulson@15079
|
250 |
--> (\<exists>t. h < t &
|
paulson@15079
|
251 |
t < 0 &
|
paulson@15079
|
252 |
f h =
|
nipkow@15539
|
253 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
|
paulson@15079
|
254 |
diff n t / real (fact n) * h ^ n)"
|
paulson@15079
|
255 |
by (blast intro: Maclaurin_minus)
|
paulson@15079
|
256 |
|
paulson@15079
|
257 |
|
paulson@15079
|
258 |
subsection{*More Convenient "Bidirectional" Version.*}
|
paulson@15079
|
259 |
|
paulson@15079
|
260 |
(* not good for PVS sin_approx, cos_approx *)
|
paulson@15079
|
261 |
|
paulson@15079
|
262 |
lemma Maclaurin_bi_le_lemma [rule_format]:
|
nipkow@25162
|
263 |
"n>0 \<longrightarrow>
|
nipkow@25134
|
264 |
diff 0 0 =
|
nipkow@25134
|
265 |
(\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
|
nipkow@25134
|
266 |
diff n 0 * 0 ^ n / real (fact n)"
|
hoelzl@41412
|
267 |
by (induct "n") auto
|
paulson@15079
|
268 |
|
paulson@15079
|
269 |
lemma Maclaurin_bi_le:
|
hoelzl@41412
|
270 |
assumes "diff 0 = f"
|
bulwahn@41368
|
271 |
and DERIV : "\<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t"
|
bulwahn@41368
|
272 |
shows "\<exists>t. abs t \<le> abs x &
|
paulson@15079
|
273 |
f x =
|
nipkow@15539
|
274 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
|
hoelzl@41412
|
275 |
diff n t / real (fact n) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t")
|
hoelzl@41412
|
276 |
proof cases
|
hoelzl@41412
|
277 |
assume "n = 0" with `diff 0 = f` show ?thesis by force
|
bulwahn@41368
|
278 |
next
|
hoelzl@41412
|
279 |
assume "n \<noteq> 0"
|
hoelzl@41412
|
280 |
show ?thesis
|
hoelzl@41412
|
281 |
proof (cases rule: linorder_cases)
|
hoelzl@41412
|
282 |
assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV
|
hoelzl@41412
|
283 |
have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (force simp add: Maclaurin_bi_le_lemma)
|
hoelzl@41412
|
284 |
thus ?thesis ..
|
bulwahn@41368
|
285 |
next
|
hoelzl@41412
|
286 |
assume "x < 0"
|
hoelzl@41412
|
287 |
with `n \<noteq> 0` DERIV
|
hoelzl@41412
|
288 |
have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t" by (intro Maclaurin_minus) auto
|
hoelzl@41412
|
289 |
then guess t ..
|
hoelzl@41412
|
290 |
with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
|
hoelzl@41412
|
291 |
thus ?thesis ..
|
hoelzl@41412
|
292 |
next
|
hoelzl@41412
|
293 |
assume "x > 0"
|
hoelzl@41412
|
294 |
with `n \<noteq> 0` `diff 0 = f` DERIV
|
hoelzl@41412
|
295 |
have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto
|
hoelzl@41412
|
296 |
then guess t ..
|
hoelzl@41412
|
297 |
with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
|
hoelzl@41412
|
298 |
thus ?thesis ..
|
bulwahn@41368
|
299 |
qed
|
bulwahn@41368
|
300 |
qed
|
bulwahn@41368
|
301 |
|
paulson@15079
|
302 |
lemma Maclaurin_all_lt:
|
bulwahn@41368
|
303 |
assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0"
|
bulwahn@41368
|
304 |
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x"
|
hoelzl@41412
|
305 |
shows "\<exists>t. 0 < abs t & abs t < abs x & f x =
|
hoelzl@41412
|
306 |
(\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
hoelzl@41412
|
307 |
(diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t")
|
hoelzl@41412
|
308 |
proof (cases rule: linorder_cases)
|
hoelzl@41412
|
309 |
assume "x = 0" with INIT3 show "?thesis"..
|
hoelzl@41412
|
310 |
next
|
hoelzl@41412
|
311 |
assume "x < 0"
|
hoelzl@41412
|
312 |
with assms have "\<exists>t>x. t < 0 \<and> f x = ?f x t" by (intro Maclaurin_minus) auto
|
hoelzl@41412
|
313 |
then guess t ..
|
hoelzl@41412
|
314 |
with `x < 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp
|
hoelzl@41412
|
315 |
thus ?thesis ..
|
hoelzl@41412
|
316 |
next
|
hoelzl@41412
|
317 |
assume "x > 0"
|
hoelzl@41412
|
318 |
with assms have "\<exists>t>0. t < x \<and> f x = ?f x t " by (intro Maclaurin) auto
|
hoelzl@41412
|
319 |
then guess t ..
|
hoelzl@41412
|
320 |
with `x > 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp
|
hoelzl@41412
|
321 |
thus ?thesis ..
|
bulwahn@41368
|
322 |
qed
|
bulwahn@41368
|
323 |
|
paulson@15079
|
324 |
|
paulson@15079
|
325 |
lemma Maclaurin_all_lt_objl:
|
paulson@15079
|
326 |
"diff 0 = f &
|
paulson@15079
|
327 |
(\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
|
nipkow@25162
|
328 |
x ~= 0 & n > 0
|
paulson@15079
|
329 |
--> (\<exists>t. 0 < abs t & abs t < abs x &
|
nipkow@15539
|
330 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
331 |
(diff n t / real (fact n)) * x ^ n)"
|
paulson@15079
|
332 |
by (blast intro: Maclaurin_all_lt)
|
paulson@15079
|
333 |
|
paulson@15079
|
334 |
lemma Maclaurin_zero [rule_format]:
|
paulson@15079
|
335 |
"x = (0::real)
|
nipkow@25134
|
336 |
==> n \<noteq> 0 -->
|
nipkow@15539
|
337 |
(\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
|
paulson@15079
|
338 |
diff 0 0"
|
paulson@15079
|
339 |
by (induct n, auto)
|
paulson@15079
|
340 |
|
bulwahn@41368
|
341 |
|
bulwahn@41368
|
342 |
lemma Maclaurin_all_le:
|
bulwahn@41368
|
343 |
assumes INIT: "diff 0 = f"
|
bulwahn@41368
|
344 |
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff (Suc m) x"
|
hoelzl@41412
|
345 |
shows "\<exists>t. abs t \<le> abs x & f x =
|
hoelzl@41412
|
346 |
(\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
hoelzl@41412
|
347 |
(diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t")
|
hoelzl@41412
|
348 |
proof cases
|
hoelzl@41412
|
349 |
assume "n = 0" with INIT show ?thesis by force
|
bulwahn@41368
|
350 |
next
|
hoelzl@41412
|
351 |
assume "n \<noteq> 0"
|
hoelzl@41412
|
352 |
show ?thesis
|
hoelzl@41412
|
353 |
proof cases
|
hoelzl@41412
|
354 |
assume "x = 0"
|
hoelzl@41412
|
355 |
with `n \<noteq> 0` have "(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0"
|
hoelzl@41412
|
356 |
by (intro Maclaurin_zero) auto
|
hoelzl@41412
|
357 |
with INIT `x = 0` `n \<noteq> 0` have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by force
|
hoelzl@41412
|
358 |
thus ?thesis ..
|
hoelzl@41412
|
359 |
next
|
hoelzl@41412
|
360 |
assume "x \<noteq> 0"
|
hoelzl@41412
|
361 |
with INIT `n \<noteq> 0` DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
|
hoelzl@41412
|
362 |
by (intro Maclaurin_all_lt) auto
|
hoelzl@41412
|
363 |
then guess t ..
|
hoelzl@41412
|
364 |
hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
|
hoelzl@41412
|
365 |
thus ?thesis ..
|
bulwahn@41368
|
366 |
qed
|
bulwahn@41368
|
367 |
qed
|
bulwahn@41368
|
368 |
|
paulson@15079
|
369 |
lemma Maclaurin_all_le_objl: "diff 0 = f &
|
paulson@15079
|
370 |
(\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
|
paulson@15079
|
371 |
--> (\<exists>t. abs t \<le> abs x &
|
nipkow@15539
|
372 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
|
paulson@15079
|
373 |
(diff n t / real (fact n)) * x ^ n)"
|
paulson@15079
|
374 |
by (blast intro: Maclaurin_all_le)
|
paulson@15079
|
375 |
|
paulson@15079
|
376 |
|
paulson@15079
|
377 |
subsection{*Version for Exponential Function*}
|
paulson@15079
|
378 |
|
nipkow@25162
|
379 |
lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
|
paulson@15079
|
380 |
==> (\<exists>t. 0 < abs t &
|
paulson@15079
|
381 |
abs t < abs x &
|
nipkow@15539
|
382 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
|
paulson@15079
|
383 |
(exp t / real (fact n)) * x ^ n)"
|
paulson@15079
|
384 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
|
paulson@15079
|
385 |
|
paulson@15079
|
386 |
|
paulson@15079
|
387 |
lemma Maclaurin_exp_le:
|
paulson@15079
|
388 |
"\<exists>t. abs t \<le> abs x &
|
nipkow@15539
|
389 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
|
paulson@15079
|
390 |
(exp t / real (fact n)) * x ^ n"
|
paulson@15079
|
391 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
|
paulson@15079
|
392 |
|
paulson@15079
|
393 |
|
paulson@15079
|
394 |
subsection{*Version for Sine Function*}
|
paulson@15079
|
395 |
|
paulson@15079
|
396 |
lemma mod_exhaust_less_4:
|
nipkow@25134
|
397 |
"m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
|
webertj@20217
|
398 |
by auto
|
paulson@15079
|
399 |
|
paulson@15079
|
400 |
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
|
nipkow@25134
|
401 |
"n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
|
paulson@15251
|
402 |
by (induct "n", auto)
|
paulson@15079
|
403 |
|
paulson@15079
|
404 |
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
|
nipkow@25134
|
405 |
"n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
|
paulson@15251
|
406 |
by (induct "n", auto)
|
paulson@15079
|
407 |
|
paulson@15079
|
408 |
lemma Suc_mult_two_diff_one [rule_format, simp]:
|
nipkow@25134
|
409 |
"n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
|
paulson@15251
|
410 |
by (induct "n", auto)
|
paulson@15079
|
411 |
|
paulson@15234
|
412 |
|
paulson@15234
|
413 |
text{*It is unclear why so many variant results are needed.*}
|
paulson@15079
|
414 |
|
huffman@36974
|
415 |
lemma sin_expansion_lemma:
|
hoelzl@41412
|
416 |
"sin (x + real (Suc m) * pi / 2) =
|
huffman@36974
|
417 |
cos (x + real (m) * pi / 2)"
|
huffman@36974
|
418 |
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
|
huffman@36974
|
419 |
|
huffman@45166
|
420 |
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
|
huffman@45166
|
421 |
unfolding sin_coeff_def by simp (* TODO: move *)
|
huffman@45166
|
422 |
|
paulson@15079
|
423 |
lemma Maclaurin_sin_expansion2:
|
paulson@15079
|
424 |
"\<exists>t. abs t \<le> abs x &
|
paulson@15079
|
425 |
sin x =
|
huffman@45166
|
426 |
(\<Sum>m=0..<n. sin_coeff m * x ^ m)
|
paulson@15079
|
427 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
428 |
apply (cut_tac f = sin and n = n and x = x
|
paulson@15079
|
429 |
and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
|
paulson@15079
|
430 |
apply safe
|
paulson@15079
|
431 |
apply (simp (no_asm))
|
huffman@36974
|
432 |
apply (simp (no_asm) add: sin_expansion_lemma)
|
huffman@45166
|
433 |
apply (subst (asm) setsum_0', clarify, case_tac "a", simp, simp)
|
huffman@45166
|
434 |
apply (cases n, simp, simp)
|
paulson@15079
|
435 |
apply (rule ccontr, simp)
|
paulson@15079
|
436 |
apply (drule_tac x = x in spec, simp)
|
paulson@15079
|
437 |
apply (erule ssubst)
|
paulson@15079
|
438 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
439 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
440 |
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
441 |
done
|
paulson@15079
|
442 |
|
paulson@15234
|
443 |
lemma Maclaurin_sin_expansion:
|
paulson@15234
|
444 |
"\<exists>t. sin x =
|
huffman@45166
|
445 |
(\<Sum>m=0..<n. sin_coeff m * x ^ m)
|
paulson@15234
|
446 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
hoelzl@41412
|
447 |
apply (insert Maclaurin_sin_expansion2 [of x n])
|
hoelzl@41412
|
448 |
apply (blast intro: elim:)
|
paulson@15234
|
449 |
done
|
paulson@15234
|
450 |
|
paulson@15079
|
451 |
lemma Maclaurin_sin_expansion3:
|
nipkow@25162
|
452 |
"[| n > 0; 0 < x |] ==>
|
paulson@15079
|
453 |
\<exists>t. 0 < t & t < x &
|
paulson@15079
|
454 |
sin x =
|
huffman@45166
|
455 |
(\<Sum>m=0..<n. sin_coeff m * x ^ m)
|
paulson@15079
|
456 |
+ ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
457 |
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
|
paulson@15079
|
458 |
apply safe
|
paulson@15079
|
459 |
apply simp
|
huffman@36974
|
460 |
apply (simp (no_asm) add: sin_expansion_lemma)
|
paulson@15079
|
461 |
apply (erule ssubst)
|
paulson@15079
|
462 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
463 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
464 |
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
465 |
done
|
paulson@15079
|
466 |
|
paulson@15079
|
467 |
lemma Maclaurin_sin_expansion4:
|
paulson@15079
|
468 |
"0 < x ==>
|
paulson@15079
|
469 |
\<exists>t. 0 < t & t \<le> x &
|
paulson@15079
|
470 |
sin x =
|
huffman@45166
|
471 |
(\<Sum>m=0..<n. sin_coeff m * x ^ m)
|
paulson@15079
|
472 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
473 |
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
|
paulson@15079
|
474 |
apply safe
|
paulson@15079
|
475 |
apply simp
|
huffman@36974
|
476 |
apply (simp (no_asm) add: sin_expansion_lemma)
|
paulson@15079
|
477 |
apply (erule ssubst)
|
paulson@15079
|
478 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
479 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
480 |
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
|
paulson@15079
|
481 |
done
|
paulson@15079
|
482 |
|
paulson@15079
|
483 |
|
paulson@15079
|
484 |
subsection{*Maclaurin Expansion for Cosine Function*}
|
paulson@15079
|
485 |
|
huffman@45166
|
486 |
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
|
huffman@45166
|
487 |
unfolding cos_coeff_def by simp (* TODO: move *)
|
huffman@45166
|
488 |
|
paulson@15079
|
489 |
lemma sumr_cos_zero_one [simp]:
|
huffman@45166
|
490 |
"(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"
|
paulson@15251
|
491 |
by (induct "n", auto)
|
paulson@15079
|
492 |
|
huffman@36974
|
493 |
lemma cos_expansion_lemma:
|
huffman@36974
|
494 |
"cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
|
huffman@36974
|
495 |
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
|
huffman@36974
|
496 |
|
paulson@15079
|
497 |
lemma Maclaurin_cos_expansion:
|
paulson@15079
|
498 |
"\<exists>t. abs t \<le> abs x &
|
paulson@15079
|
499 |
cos x =
|
huffman@45166
|
500 |
(\<Sum>m=0..<n. cos_coeff m * x ^ m)
|
paulson@15079
|
501 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
502 |
apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
|
paulson@15079
|
503 |
apply safe
|
paulson@15079
|
504 |
apply (simp (no_asm))
|
huffman@36974
|
505 |
apply (simp (no_asm) add: cos_expansion_lemma)
|
paulson@15079
|
506 |
apply (case_tac "n", simp)
|
nipkow@15561
|
507 |
apply (simp del: setsum_op_ivl_Suc)
|
paulson@15079
|
508 |
apply (rule ccontr, simp)
|
paulson@15079
|
509 |
apply (drule_tac x = x in spec, simp)
|
paulson@15079
|
510 |
apply (erule ssubst)
|
paulson@15079
|
511 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
512 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
513 |
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
514 |
done
|
paulson@15079
|
515 |
|
paulson@15079
|
516 |
lemma Maclaurin_cos_expansion2:
|
nipkow@25162
|
517 |
"[| 0 < x; n > 0 |] ==>
|
paulson@15079
|
518 |
\<exists>t. 0 < t & t < x &
|
paulson@15079
|
519 |
cos x =
|
huffman@45166
|
520 |
(\<Sum>m=0..<n. cos_coeff m * x ^ m)
|
paulson@15079
|
521 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
522 |
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
|
paulson@15079
|
523 |
apply safe
|
paulson@15079
|
524 |
apply simp
|
huffman@36974
|
525 |
apply (simp (no_asm) add: cos_expansion_lemma)
|
paulson@15079
|
526 |
apply (erule ssubst)
|
paulson@15079
|
527 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
528 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
529 |
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
530 |
done
|
paulson@15079
|
531 |
|
paulson@15234
|
532 |
lemma Maclaurin_minus_cos_expansion:
|
nipkow@25162
|
533 |
"[| x < 0; n > 0 |] ==>
|
paulson@15079
|
534 |
\<exists>t. x < t & t < 0 &
|
paulson@15079
|
535 |
cos x =
|
huffman@45166
|
536 |
(\<Sum>m=0..<n. cos_coeff m * x ^ m)
|
paulson@15079
|
537 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
|
paulson@15079
|
538 |
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
|
paulson@15079
|
539 |
apply safe
|
paulson@15079
|
540 |
apply simp
|
huffman@36974
|
541 |
apply (simp (no_asm) add: cos_expansion_lemma)
|
paulson@15079
|
542 |
apply (erule ssubst)
|
paulson@15079
|
543 |
apply (rule_tac x = t in exI, simp)
|
nipkow@15536
|
544 |
apply (rule setsum_cong[OF refl])
|
huffman@45166
|
545 |
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
|
paulson@15079
|
546 |
done
|
paulson@15079
|
547 |
|
paulson@15079
|
548 |
(* ------------------------------------------------------------------------- *)
|
paulson@15079
|
549 |
(* Version for ln(1 +/- x). Where is it?? *)
|
paulson@15079
|
550 |
(* ------------------------------------------------------------------------- *)
|
paulson@15079
|
551 |
|
paulson@15079
|
552 |
lemma sin_bound_lemma:
|
paulson@15081
|
553 |
"[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
|
paulson@15079
|
554 |
by auto
|
paulson@15079
|
555 |
|
paulson@15079
|
556 |
lemma Maclaurin_sin_bound:
|
huffman@45166
|
557 |
"abs(sin x - (\<Sum>m=0..<n. sin_coeff m * x ^ m))
|
huffman@45166
|
558 |
\<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
|
obua@14738
|
559 |
proof -
|
paulson@15079
|
560 |
have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
|
obua@14738
|
561 |
by (rule_tac mult_right_mono,simp_all)
|
obua@14738
|
562 |
note est = this[simplified]
|
huffman@22985
|
563 |
let ?diff = "\<lambda>(n::nat) x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)"
|
huffman@22985
|
564 |
have diff_0: "?diff 0 = sin" by simp
|
huffman@22985
|
565 |
have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
|
huffman@22985
|
566 |
apply (clarify)
|
huffman@22985
|
567 |
apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
|
huffman@22985
|
568 |
apply (cut_tac m=m in mod_exhaust_less_4)
|
hoelzl@31880
|
569 |
apply (safe, auto intro!: DERIV_intros)
|
huffman@22985
|
570 |
done
|
huffman@22985
|
571 |
from Maclaurin_all_le [OF diff_0 DERIV_diff]
|
huffman@22985
|
572 |
obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
|
huffman@22985
|
573 |
t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
|
huffman@22985
|
574 |
?diff n t / real (fact n) * x ^ n" by fast
|
huffman@22985
|
575 |
have diff_m_0:
|
huffman@22985
|
576 |
"\<And>m. ?diff m 0 = (if even m then 0
|
huffman@23177
|
577 |
else -1 ^ ((m - Suc 0) div 2))"
|
huffman@22985
|
578 |
apply (subst even_even_mod_4_iff)
|
huffman@22985
|
579 |
apply (cut_tac m=m in mod_exhaust_less_4)
|
huffman@22985
|
580 |
apply (elim disjE, simp_all)
|
huffman@22985
|
581 |
apply (safe dest!: mod_eqD, simp_all)
|
huffman@22985
|
582 |
done
|
obua@14738
|
583 |
show ?thesis
|
huffman@45166
|
584 |
unfolding sin_coeff_def
|
huffman@22985
|
585 |
apply (subst t2)
|
paulson@15079
|
586 |
apply (rule sin_bound_lemma)
|
nipkow@15536
|
587 |
apply (rule setsum_cong[OF refl])
|
huffman@22985
|
588 |
apply (subst diff_m_0, simp)
|
paulson@15079
|
589 |
apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
|
hoelzl@41412
|
590 |
simp add: est mult_nonneg_nonneg mult_ac divide_inverse
|
paulson@16924
|
591 |
power_abs [symmetric] abs_mult)
|
obua@14738
|
592 |
done
|
obua@14738
|
593 |
qed
|
obua@14738
|
594 |
|
paulson@15079
|
595 |
end
|