src/HOL/MacLaurin.thy
author huffman
Fri, 19 Aug 2011 08:39:43 -0700
changeset 44306 33572a766836
parent 41166 4b2a457b17e8
child 44308 d2a6f9af02f4
permissions -rw-r--r--
fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Author      : Jacques D. Fleuriot
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    Copyright   : 2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005
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*)
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header{*MacLaurin Series*}
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theory MacLaurin
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imports Transcendental
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begin
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subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
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text{*This is a very long, messy proof even now that it's been broken down
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into lemmas.*}
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lemma Maclaurin_lemma:
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    "0 < h ==>
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     \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
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               (B * ((h^n) / real(fact n)))"
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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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                 real(fact n) / (h^n)"]) simp
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lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
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by arith
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lemma fact_diff_Suc [rule_format]:
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  "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
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  by (subst fact_reduce_nat, auto)
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lemma Maclaurin_lemma2:
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  fixes B
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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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    and INIT : "n = Suc k"
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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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    B * (t ^ (n - m) / real (fact (n - m)))))" (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")
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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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proof (rule allI impI)+
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  fix m t assume INIT2: "m < n & 0 \<le> t & t \<le> h"
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  have "DERIV (difg m) t :> diff (Suc m) t -
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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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     real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)))" unfolding difg_def
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    by (auto intro!: DERIV_intros DERIV[rule_format, OF INIT2])
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      moreover
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  from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m"
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    unfolding atLeast0LessThan[symmetric] by auto
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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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      (\<Sum>x = 0..<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)))"
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    unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex)
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  moreover
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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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    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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  have "\<And>x. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)) =
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      diff (Suc m + x) 0 * t^x / real (fact x)"
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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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  moreover
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  have "real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)) =
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      B * (t ^ (n - Suc m) / real (fact (n - Suc m)))"
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    using `0 < n - m` by (simp add: fact_reduce_nat)
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  ultimately show "DERIV (difg m) t :> difg (Suc m) t"
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    unfolding difg_def by simp
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qed
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lemma Maclaurin:
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  assumes h: "0 < h"
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  assumes n: "0 < n"
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  assumes diff_0: "diff 0 = f"
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  assumes diff_Suc:
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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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  shows
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    "\<exists>t. 0 < t & t < h &
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              f h =
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              setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
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              (diff n t / real (fact n)) * h ^ n"
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proof -
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  from n obtain m where m: "n = Suc m"
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    by (cases n) (simp add: n)
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  obtain B where f_h: "f h =
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        (\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
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        B * (h ^ n / real (fact n))"
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    using Maclaurin_lemma [OF h] ..
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  def g \<equiv> "(\<lambda>t. f t -
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    (setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {0..<n}
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      + (B * (t^n / real(fact n)))))"
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    88
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  have g2: "g 0 = 0 & g h = 0"
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    90
    apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
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    91
    apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
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    92
    apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
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    93
    done
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    95
  def difg \<equiv> "(%m t. diff m t -
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    96
    (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
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    97
      + (B * ((t ^ (n - m)) / real (fact (n - m))))))"
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    98
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    99
  have difg_0: "difg 0 = g"
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   100
    unfolding difg_def g_def by (simp add: diff_0)
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   101
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   102
  have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
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   103
        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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   104
    using diff_Suc m unfolding difg_def by (rule Maclaurin_lemma2)
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   105
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   106
  have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
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   107
    apply clarify
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   108
    apply (simp add: m difg_def)
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   109
    apply (frule less_iff_Suc_add [THEN iffD1], clarify)
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   110
    apply (simp del: setsum_op_ivl_Suc)
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   111
    apply (insert sumr_offset4 [of "Suc 0"])
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   112
    apply (simp del: setsum_op_ivl_Suc fact_Suc)
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   113
    done
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   114
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   115
  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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   116
    by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
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   117
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  have differentiable_difg:
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   119
    "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
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   120
    by (rule differentiableI [OF difg_Suc [rule_format]]) simp
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   121
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   122
  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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   123
        \<Longrightarrow> difg (Suc m) t = 0"
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   124
    by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
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   125
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   126
  have "m < n" using m by simp
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   127
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diff changeset
   128
  have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   129
  using `m < n`
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   130
  proof (induct m)
41166
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hoelzl
parents: 41120
diff changeset
   131
    case 0
29187
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huffman
parents: 29168
diff changeset
   132
    show ?case
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   133
    proof (rule Rolle)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   134
      show "0 < h" by fact
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   135
      show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   136
      show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   137
        by (simp add: isCont_difg n)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   138
      show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   139
        by (simp add: differentiable_difg n)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   140
    qed
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   141
  next
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   142
    case (Suc m')
29187
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   143
    hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   144
    then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   145
    have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   146
    proof (rule Rolle)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   147
      show "0 < t" by fact
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   148
      show "difg (Suc m') 0 = difg (Suc m') t"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   149
        using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   150
      show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   151
        using `t < h` `Suc m' < n` by (simp add: isCont_difg)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   152
      show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   153
        using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   154
    qed
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   155
    thus ?case
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   156
      using `t < h` by auto
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   157
  qed
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   158
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   159
  then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   160
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   161
  hence "difg (Suc m) t = 0"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   162
    using `m < n` by (simp add: difg_Suc_eq_0)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   163
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   164
  show ?thesis
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   165
  proof (intro exI conjI)
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   166
    show "0 < t" by fact
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   167
    show "t < h" by fact
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   168
    show "f h =
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   169
      (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   170
      diff n t / real (fact n) * h ^ n"
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   171
      using `difg (Suc m) t = 0`
32047
c141f139ce26 Changed fact_Suc_nat back to fact_Suc
avigad
parents: 32038
diff changeset
   172
      by (simp add: m f_h difg_def del: fact_Suc)
29187
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   173
  qed
7b09385234f9 clean up proofs of lemma Maclaurin
huffman
parents: 29168
diff changeset
   174
qed
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   175
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   176
lemma Maclaurin_objl:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   177
  "0 < h & n>0 & diff 0 = f &
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   178
  (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   179
   --> (\<exists>t. 0 < t & t < h &
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   180
            f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   181
                  diff n t / real (fact n) * h ^ n)"
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   182
by (blast intro: Maclaurin)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   183
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   184
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   185
lemma Maclaurin2:
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   186
  assumes INIT1: "0 < h " and INIT2: "diff 0 = f"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   187
  and DERIV: "\<forall>m t.
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   188
  m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   189
  shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h =
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   190
  (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   191
  diff n t / real (fact n) * h ^ n"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   192
proof (cases "n")
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   193
  case 0 with INIT1 INIT2 show ?thesis by fastsimp
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   194
next
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   195
  case Suc
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   196
  hence "n > 0" by simp
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   197
  from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and>
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   198
    f h =
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   199
    (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n"
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   200
    by (rule Maclaurin)
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   201
  thus ?thesis by fastsimp
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   202
qed
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   203
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   204
lemma Maclaurin2_objl:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   205
     "0 < h & diff 0 = f &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   206
       (\<forall>m t.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   207
          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   208
    --> (\<exists>t. 0 < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   209
              t \<le> h &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   210
              f h =
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   211
              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   212
              diff n t / real (fact n) * h ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   213
by (blast intro: Maclaurin2)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   214
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   215
lemma Maclaurin_minus:
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   216
  assumes "h < 0" "0 < n" "diff 0 = f"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   217
  and DERIV: "\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t"
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   218
  shows "\<exists>t. h < t & t < 0 &
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   219
         f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   220
         diff n t / real (fact n) * h ^ n"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   221
proof -
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   222
  txt "Transform @{text ABL'} into @{text DERIV_intros} format."
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   223
  note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   224
  from assms
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   225
  have "\<exists>t>0. t < - h \<and>
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   226
    f (- (- h)) =
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   227
    (\<Sum>m = 0..<n.
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   228
    (- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) +
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   229
    (- 1) ^ n * diff n (- t) / real (fact n) * (- h) ^ n"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   230
    by (intro Maclaurin) (auto intro!: DERIV_intros DERIV')
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   231
  then guess t ..
41120
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   232
  moreover
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   233
  have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   234
    by (auto simp add: power_mult_distrib[symmetric])
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   235
  moreover
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   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))"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   237
    by (auto intro: setsum_cong simp add: power_mult_distrib[symmetric])
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   238
  ultimately have " h < - t \<and>
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   239
    - t < 0 \<and>
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   240
    f h =
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   241
    (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n"
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   242
    by auto
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   243
  thus ?thesis ..
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents: 36974
diff changeset
   244
qed
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   245
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   246
lemma Maclaurin_minus_objl:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   247
     "(h < 0 & n > 0 & diff 0 = f &
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   248
       (\<forall>m t.
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   249
          m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   250
    --> (\<exists>t. h < t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   251
              t < 0 &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   252
              f h =
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   253
              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   254
              diff n t / real (fact n) * h ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   255
by (blast intro: Maclaurin_minus)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   256
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   257
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   258
subsection{*More Convenient "Bidirectional" Version.*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   259
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   260
(* not good for PVS sin_approx, cos_approx *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   261
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lemma Maclaurin_bi_le_lemma [rule_format]:
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  "n>0 \<longrightarrow>
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   diff 0 0 =
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   (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
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   diff n 0 * 0 ^ n / real (fact n)"
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by (induct "n") auto
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   268
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lemma Maclaurin_bi_le:
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   assumes "diff 0 = f"
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   and DERIV : "\<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t"
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   shows "\<exists>t. abs t \<le> abs x &
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              f x =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
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     diff n t / real (fact n) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t")
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proof cases
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  assume "n = 0" with `diff 0 = f` show ?thesis by force
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next
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  assume "n \<noteq> 0"
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  show ?thesis
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  proof (cases rule: linorder_cases)
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    assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV
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   283
    have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (force simp add: Maclaurin_bi_le_lemma)
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    thus ?thesis ..
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  next
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    assume "x < 0"
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   287
    with `n \<noteq> 0` DERIV
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   288
    have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t" by (intro Maclaurin_minus) auto
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   289
    then guess t ..
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   290
    with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
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    thus ?thesis ..
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   292
  next
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    assume "x > 0"
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   294
    with `n \<noteq> 0` `diff 0 = f` DERIV
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   295
    have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto
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   296
    then guess t ..
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   297
    with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
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   298
    thus ?thesis ..
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  qed
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qed
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
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   301
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lemma Maclaurin_all_lt:
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  assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0"
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  and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x"
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   305
  shows "\<exists>t. 0 < abs t & abs t < abs x & f x =
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   306
    (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   307
                (diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t")
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   308
proof (cases rule: linorder_cases)
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  assume "x = 0" with INIT3 show "?thesis"..
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   310
next
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   311
  assume "x < 0"
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  with assms have "\<exists>t>x. t < 0 \<and> f x = ?f x t" by (intro Maclaurin_minus) auto
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   313
  then guess t ..
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   314
  with `x < 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp
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   315
  thus ?thesis ..
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   316
next
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   317
  assume "x > 0"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   318
  with assms have "\<exists>t>0. t < x \<and> f x = ?f x t " by (intro Maclaurin) auto
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   319
  then guess t ..
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   320
  with `x > 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp
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diff changeset
   321
  thus ?thesis ..
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   322
qed
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diff changeset
   323
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   324
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   325
lemma Maclaurin_all_lt_objl:
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diff changeset
   326
     "diff 0 = f &
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   327
      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
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   328
      x ~= 0 & n > 0
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   329
      --> (\<exists>t. 0 < abs t & abs t < abs x &
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   330
               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
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diff changeset
   331
                     (diff n t / real (fact n)) * x ^ n)"
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   332
by (blast intro: Maclaurin_all_lt)
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diff changeset
   333
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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   334
lemma Maclaurin_zero [rule_format]:
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   335
     "x = (0::real)
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   336
      ==> n \<noteq> 0 -->
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   337
          (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
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diff changeset
   338
          diff 0 0"
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   339
by (induct n, auto)
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diff changeset
   340
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diff changeset
   341
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   342
lemma Maclaurin_all_le:
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   343
  assumes INIT: "diff 0 = f"
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   344
  and DERIV: "\<forall>m x. DERIV (diff m) x :> diff (Suc m) x"
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   345
  shows "\<exists>t. abs t \<le> abs x & f x =
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   346
    (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   347
    (diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t")
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   348
proof cases
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   349
  assume "n = 0" with INIT show ?thesis by force
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   350
  next
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   351
  assume "n \<noteq> 0"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   352
  show ?thesis
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   353
  proof cases
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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   354
    assume "x = 0"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   355
    with `n \<noteq> 0` have "(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   356
      by (intro Maclaurin_zero) auto
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   357
    with INIT `x = 0` `n \<noteq> 0` have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by force
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   358
    thus ?thesis ..
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   359
  next
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   360
    assume "x \<noteq> 0"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   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"
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   362
      by (intro Maclaurin_all_lt) auto
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   363
    then guess t ..
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
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diff changeset
   364
    hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
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diff changeset
   365
    thus ?thesis ..
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   366
  qed
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   367
qed
74e41b2d48ea adding an Isar version of the MacLaurin theorem from some students' work in 2005
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diff changeset
   368
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diff changeset
   369
lemma Maclaurin_all_le_objl: "diff 0 = f &
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diff changeset
   370
      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   371
      --> (\<exists>t. abs t \<le> abs x &
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diff changeset
   372
              f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   373
                    (diff n t / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
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diff changeset
   374
by (blast intro: Maclaurin_all_le)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
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diff changeset
   375
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
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diff changeset
   376
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   377
subsection{*Version for Exponential Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   378
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   379
lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
15079
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diff changeset
   380
      ==> (\<exists>t. 0 < abs t &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   381
                abs t < abs x &
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diff changeset
   382
                exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
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diff changeset
   383
                        (exp t / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   384
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   385
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   386
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   387
lemma Maclaurin_exp_le:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   388
     "\<exists>t. abs t \<le> abs x &
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   389
            exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   390
                       (exp t / real (fact n)) * x ^ n"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   391
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   392
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   393
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   394
subsection{*Version for Sine Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   395
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   396
lemma mod_exhaust_less_4:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   397
  "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   398
by auto
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   399
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   400
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   401
  "n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   402
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   403
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   404
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   405
  "n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   406
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   407
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   408
lemma Suc_mult_two_diff_one [rule_format, simp]:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   409
  "n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   410
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   411
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   412
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   413
text{*It is unclear why so many variant results are needed.*}
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   414
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   415
lemma sin_expansion_lemma:
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   416
     "sin (x + real (Suc m) * pi / 2) =
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   417
      cos (x + real (m) * pi / 2)"
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   418
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   419
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   420
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   421
  unfolding sin_coeff_def by simp (* TODO: move *)
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   422
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   423
lemma Maclaurin_sin_expansion2:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   424
     "\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   425
       sin x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   426
       (\<Sum>m=0..<n. sin_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   427
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   428
apply (cut_tac f = sin and n = n and x = x
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   429
        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   430
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   431
apply (simp (no_asm))
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   432
apply (simp (no_asm) add: sin_expansion_lemma)
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   433
apply (subst (asm) setsum_0', clarify, case_tac "a", simp, simp)
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   434
apply (cases n, simp, simp)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   435
apply (rule ccontr, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   436
apply (drule_tac x = x in spec, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   437
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   438
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   439
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   440
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   441
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   442
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   443
lemma Maclaurin_sin_expansion:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   444
     "\<exists>t. sin x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   445
       (\<Sum>m=0..<n. sin_coeff m * x ^ m)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   446
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   447
apply (insert Maclaurin_sin_expansion2 [of x n])
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   448
apply (blast intro: elim:)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   449
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   450
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   451
lemma Maclaurin_sin_expansion3:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   452
     "[| n > 0; 0 < x |] ==>
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   453
       \<exists>t. 0 < t & t < x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   454
       sin x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   455
       (\<Sum>m=0..<n. sin_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   456
      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   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)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   458
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   459
apply simp
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   460
apply (simp (no_asm) add: sin_expansion_lemma)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   461
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   462
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   463
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   464
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   465
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   466
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   467
lemma Maclaurin_sin_expansion4:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   468
     "0 < x ==>
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   469
       \<exists>t. 0 < t & t \<le> x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   470
       sin x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   471
       (\<Sum>m=0..<n. sin_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   472
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   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)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   474
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   475
apply simp
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   476
apply (simp (no_asm) add: sin_expansion_lemma)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   477
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   478
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   479
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   480
apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   481
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   482
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   483
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   484
subsection{*Maclaurin Expansion for Cosine Function*}
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   485
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   486
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   487
  unfolding cos_coeff_def by simp (* TODO: move *)
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   488
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   489
lemma sumr_cos_zero_one [simp]:
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   490
  "(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   491
by (induct "n", auto)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   492
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   493
lemma cos_expansion_lemma:
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   494
  "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   495
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   496
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   497
lemma Maclaurin_cos_expansion:
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   498
     "\<exists>t. abs t \<le> abs x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   499
       cos x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   500
       (\<Sum>m=0..<n. cos_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   501
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   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)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   503
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   504
apply (simp (no_asm))
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   505
apply (simp (no_asm) add: cos_expansion_lemma)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   506
apply (case_tac "n", simp)
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15539
diff changeset
   507
apply (simp del: setsum_op_ivl_Suc)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   508
apply (rule ccontr, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   509
apply (drule_tac x = x in spec, simp)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   510
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   511
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   512
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   513
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   514
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   515
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   516
lemma Maclaurin_cos_expansion2:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   517
     "[| 0 < x; n > 0 |] ==>
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   518
       \<exists>t. 0 < t & t < x &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   519
       cos x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   520
       (\<Sum>m=0..<n. cos_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   521
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   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)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   523
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   524
apply simp
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   525
apply (simp (no_asm) add: cos_expansion_lemma)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   526
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   527
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   528
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   529
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   530
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   531
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   532
lemma Maclaurin_minus_cos_expansion:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   533
     "[| x < 0; n > 0 |] ==>
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   534
       \<exists>t. x < t & t < 0 &
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   535
       cos x =
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   536
       (\<Sum>m=0..<n. cos_coeff m * x ^ m)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   537
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   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)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   539
apply safe
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   540
apply simp
36974
b877866b5b00 remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents: 32047
diff changeset
   541
apply (simp (no_asm) add: cos_expansion_lemma)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   542
apply (erule ssubst)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   543
apply (rule_tac x = t in exI, simp)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   544
apply (rule setsum_cong[OF refl])
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   545
apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   546
done
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   547
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   548
(* ------------------------------------------------------------------------- *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   549
(* Version for ln(1 +/- x). Where is it??                                    *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   550
(* ------------------------------------------------------------------------- *)
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   551
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   552
lemma sin_bound_lemma:
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   553
    "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   554
by auto
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   555
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   556
lemma Maclaurin_sin_bound:
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   557
  "abs(sin x - (\<Sum>m=0..<n. sin_coeff m * x ^ m))
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   558
  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   559
proof -
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   560
  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   561
    by (rule_tac mult_right_mono,simp_all)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   562
  note est = this[simplified]
22985
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   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)"
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   564
  have diff_0: "?diff 0 = sin" by simp
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   565
  have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   566
    apply (clarify)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   567
    apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   568
    apply (cut_tac m=m in mod_exhaust_less_4)
31881
eba74a5790d2 use DERIV_intros
hoelzl
parents: 31148
diff changeset
   569
    apply (safe, auto intro!: DERIV_intros)
22985
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   570
    done
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   571
  from Maclaurin_all_le [OF diff_0 DERIV_diff]
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   572
  obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   573
    t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   574
      ?diff n t / real (fact n) * x ^ n" by fast
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   575
  have diff_m_0:
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   576
    "\<And>m. ?diff m 0 = (if even m then 0
23177
3004310c95b1 replace (- 1) with -1
huffman
parents: 23069
diff changeset
   577
         else -1 ^ ((m - Suc 0) div 2))"
22985
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   578
    apply (subst even_even_mod_4_iff)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   579
    apply (cut_tac m=m in mod_exhaust_less_4)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   580
    apply (elim disjE, simp_all)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   581
    apply (safe dest!: mod_eqD, simp_all)
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   582
    done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   583
  show ?thesis
44306
33572a766836 fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents: 41166
diff changeset
   584
    unfolding sin_coeff_def
22985
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   585
    apply (subst t2)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   586
    apply (rule sin_bound_lemma)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   587
    apply (rule setsum_cong[OF refl])
22985
501e6dfe4e5a cleaned up proof of Maclaurin_sin_bound
huffman
parents: 22983
diff changeset
   588
    apply (subst diff_m_0, simp)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   589
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
41166
4b2a457b17e8 beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents: 41120
diff changeset
   590
                simp add: est mult_nonneg_nonneg mult_ac divide_inverse
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   591
                          power_abs [symmetric] abs_mult)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   592
    done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   593
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 12224
diff changeset
   594
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 14738
diff changeset
   595
end