src/HOL/MacLaurin.thy
author hoelzl
Tue Jun 30 18:24:00 2009 +0200 (2009-06-30)
changeset 31882 3578434d645d
parent 31881 eba74a5790d2
child 32038 4127b89f48ab
permissions -rw-r--r--
remove DERIV_tac and deriv_tac, neither is used in Isabelle/HOL or the AFP
     1 (*  Author      : Jacques D. Fleuriot
     2     Copyright   : 2001 University of Edinburgh
     3     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     4 *)
     5 
     6 header{*MacLaurin Series*}
     7 
     8 theory MacLaurin
     9 imports Transcendental
    10 begin
    11 
    12 subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
    13 
    14 text{*This is a very long, messy proof even now that it's been broken down
    15 into lemmas.*}
    16 
    17 lemma Maclaurin_lemma:
    18     "0 < h ==>
    19      \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
    20                (B * ((h^n) / real(fact n)))"
    21 apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
    22                  real(fact n) / (h^n)"
    23        in exI)
    24 apply (simp) 
    25 done
    26 
    27 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
    28 by arith
    29 
    30 lemma Maclaurin_lemma2:
    31   assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
    32   assumes n: "n = Suc k"
    33   assumes difg: "difg =
    34         (\<lambda>m t. diff m t -
    35                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
    36                 B * (t ^ (n - m) / real (fact (n - m)))))"
    37   shows
    38       "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
    39 unfolding difg
    40  apply clarify
    41  apply (rule DERIV_diff)
    42   apply (simp add: diff)
    43  apply (simp only: n)
    44  apply (rule DERIV_add)
    45   apply (rule_tac [2] DERIV_cmult)
    46   apply (rule_tac [2] lemma_DERIV_subst)
    47    apply (rule_tac [2] DERIV_quotient)
    48      apply (rule_tac [3] DERIV_const)
    49     apply (rule_tac [2] DERIV_pow)
    50    prefer 3 apply (simp add: fact_diff_Suc)
    51   prefer 2 apply simp
    52  apply (frule less_iff_Suc_add [THEN iffD1], clarify)
    53  apply (simp del: setsum_op_ivl_Suc)
    54  apply (insert sumr_offset4 [of "Suc 0"])
    55  apply (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
    56  apply (rule lemma_DERIV_subst)
    57   apply (rule DERIV_add)
    58    apply (rule_tac [2] DERIV_const)
    59   apply (rule DERIV_sumr, clarify)
    60   prefer 2 apply simp
    61  apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc power_Suc)
    62  apply (rule DERIV_cmult)
    63  apply (rule lemma_DERIV_subst)
    64   apply (best intro!: DERIV_intros)
    65  apply (subst fact_Suc)
    66  apply (subst real_of_nat_mult)
    67  apply (simp add: mult_ac)
    68 done
    69 
    70 lemma Maclaurin:
    71   assumes h: "0 < h"
    72   assumes n: "0 < n"
    73   assumes diff_0: "diff 0 = f"
    74   assumes diff_Suc:
    75     "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
    76   shows
    77     "\<exists>t. 0 < t & t < h &
    78               f h =
    79               setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
    80               (diff n t / real (fact n)) * h ^ n"
    81 proof -
    82   from n obtain m where m: "n = Suc m"
    83     by (cases n, simp add: n)
    84 
    85   obtain B where f_h: "f h =
    86         (\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
    87         B * (h ^ n / real (fact n))"
    88     using Maclaurin_lemma [OF h] ..
    89 
    90   obtain g where g_def: "g = (%t. f t -
    91     (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n}
    92       + (B * (t^n / real(fact n)))))" by blast
    93 
    94   have g2: "g 0 = 0 & g h = 0"
    95     apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
    96     apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
    97     apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
    98     done
    99 
   100   obtain difg where difg_def: "difg = (%m t. diff m t -
   101     (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
   102       + (B * ((t ^ (n - m)) / real (fact (n - m))))))" by blast
   103 
   104   have difg_0: "difg 0 = g"
   105     unfolding difg_def g_def by (simp add: diff_0)
   106 
   107   have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
   108         m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
   109     using diff_Suc m difg_def by (rule Maclaurin_lemma2)
   110 
   111   have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
   112     apply clarify
   113     apply (simp add: m difg_def)
   114     apply (frule less_iff_Suc_add [THEN iffD1], clarify)
   115     apply (simp del: setsum_op_ivl_Suc)
   116     apply (insert sumr_offset4 [of "Suc 0"])
   117     apply (simp del: setsum_op_ivl_Suc fact_Suc)
   118     done
   119 
   120   have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
   121     by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
   122 
   123   have differentiable_difg:
   124     "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
   125     by (rule differentiableI [OF difg_Suc [rule_format]]) simp
   126 
   127   have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
   128         \<Longrightarrow> difg (Suc m) t = 0"
   129     by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
   130 
   131   have "m < n" using m by simp
   132 
   133   have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
   134   using `m < n`
   135   proof (induct m)
   136   case 0
   137     show ?case
   138     proof (rule Rolle)
   139       show "0 < h" by fact
   140       show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
   141       show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
   142         by (simp add: isCont_difg n)
   143       show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
   144         by (simp add: differentiable_difg n)
   145     qed
   146   next
   147   case (Suc m')
   148     hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
   149     then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
   150     have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
   151     proof (rule Rolle)
   152       show "0 < t" by fact
   153       show "difg (Suc m') 0 = difg (Suc m') t"
   154         using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
   155       show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
   156         using `t < h` `Suc m' < n` by (simp add: isCont_difg)
   157       show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
   158         using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
   159     qed
   160     thus ?case
   161       using `t < h` by auto
   162   qed
   163 
   164   then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
   165 
   166   hence "difg (Suc m) t = 0"
   167     using `m < n` by (simp add: difg_Suc_eq_0)
   168 
   169   show ?thesis
   170   proof (intro exI conjI)
   171     show "0 < t" by fact
   172     show "t < h" by fact
   173     show "f h =
   174       (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
   175       diff n t / real (fact n) * h ^ n"
   176       using `difg (Suc m) t = 0`
   177       by (simp add: m f_h difg_def del: fact_Suc)
   178   qed
   179 
   180 qed
   181 
   182 lemma Maclaurin_objl:
   183   "0 < h & n>0 & diff 0 = f &
   184   (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   185    --> (\<exists>t. 0 < t & t < h &
   186             f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   187                   diff n t / real (fact n) * h ^ n)"
   188 by (blast intro: Maclaurin)
   189 
   190 
   191 lemma Maclaurin2:
   192    "[| 0 < h; diff 0 = f;
   193        \<forall>m t.
   194           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   195     ==> \<exists>t. 0 < t &
   196               t \<le> h &
   197               f h =
   198               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   199               diff n t / real (fact n) * h ^ n"
   200 apply (case_tac "n", auto)
   201 apply (drule Maclaurin, auto)
   202 done
   203 
   204 lemma Maclaurin2_objl:
   205      "0 < h & diff 0 = f &
   206        (\<forall>m t.
   207           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   208     --> (\<exists>t. 0 < t &
   209               t \<le> h &
   210               f h =
   211               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   212               diff n t / real (fact n) * h ^ n)"
   213 by (blast intro: Maclaurin2)
   214 
   215 lemma Maclaurin_minus:
   216    "[| h < 0; n > 0; diff 0 = f;
   217        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
   218     ==> \<exists>t. h < t &
   219               t < 0 &
   220               f h =
   221               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   222               diff n t / real (fact n) * h ^ n"
   223 apply (cut_tac f = "%x. f (-x)"
   224         and diff = "%n x. (-1 ^ n) * diff n (-x)"
   225         and h = "-h" and n = n in Maclaurin_objl)
   226 apply (simp)
   227 apply safe
   228 apply (subst minus_mult_right)
   229 apply (rule DERIV_cmult)
   230 apply (rule lemma_DERIV_subst)
   231 apply (rule DERIV_chain2 [where g=uminus])
   232 apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_ident)
   233 prefer 2 apply force
   234 apply force
   235 apply (rule_tac x = "-t" in exI, auto)
   236 apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
   237                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
   238 apply (rule_tac [2] setsum_cong[OF refl])
   239 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
   240 done
   241 
   242 lemma Maclaurin_minus_objl:
   243      "(h < 0 & n > 0 & diff 0 = f &
   244        (\<forall>m t.
   245           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
   246     --> (\<exists>t. h < t &
   247               t < 0 &
   248               f h =
   249               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   250               diff n t / real (fact n) * h ^ n)"
   251 by (blast intro: Maclaurin_minus)
   252 
   253 
   254 subsection{*More Convenient "Bidirectional" Version.*}
   255 
   256 (* not good for PVS sin_approx, cos_approx *)
   257 
   258 lemma Maclaurin_bi_le_lemma [rule_format]:
   259   "n>0 \<longrightarrow>
   260    diff 0 0 =
   261    (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
   262    diff n 0 * 0 ^ n / real (fact n)"
   263 by (induct "n", auto)
   264 
   265 lemma Maclaurin_bi_le:
   266    "[| diff 0 = f;
   267        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
   268     ==> \<exists>t. abs t \<le> abs x &
   269               f x =
   270               (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
   271               diff n t / real (fact n) * x ^ n"
   272 apply (case_tac "n = 0", force)
   273 apply (case_tac "x = 0")
   274  apply (rule_tac x = 0 in exI)
   275  apply (force simp add: Maclaurin_bi_le_lemma)
   276 apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
   277  txt{*Case 1, where @{term "x < 0"}*}
   278  apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
   279   apply (simp add: abs_if)
   280  apply (rule_tac x = t in exI)
   281  apply (simp add: abs_if)
   282 txt{*Case 2, where @{term "0 < x"}*}
   283 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
   284  apply (simp add: abs_if)
   285 apply (rule_tac x = t in exI)
   286 apply (simp add: abs_if)
   287 done
   288 
   289 lemma Maclaurin_all_lt:
   290      "[| diff 0 = f;
   291          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
   292         x ~= 0; n > 0
   293       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
   294                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   295                      (diff n t / real (fact n)) * x ^ n"
   296 apply (rule_tac x = x and y = 0 in linorder_cases)
   297 prefer 2 apply blast
   298 apply (drule_tac [2] diff=diff in Maclaurin)
   299 apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
   300 apply (rule_tac [!] x = t in exI, auto)
   301 done
   302 
   303 lemma Maclaurin_all_lt_objl:
   304      "diff 0 = f &
   305       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
   306       x ~= 0 & n > 0
   307       --> (\<exists>t. 0 < abs t & abs t < abs x &
   308                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   309                      (diff n t / real (fact n)) * x ^ n)"
   310 by (blast intro: Maclaurin_all_lt)
   311 
   312 lemma Maclaurin_zero [rule_format]:
   313      "x = (0::real)
   314       ==> n \<noteq> 0 -->
   315           (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
   316           diff 0 0"
   317 by (induct n, auto)
   318 
   319 lemma Maclaurin_all_le: "[| diff 0 = f;
   320         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
   321       |] ==> \<exists>t. abs t \<le> abs x &
   322               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   323                     (diff n t / real (fact n)) * x ^ n"
   324 apply(cases "n=0")
   325 apply (force)
   326 apply (case_tac "x = 0")
   327 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
   328 apply (drule not0_implies_Suc)
   329 apply (rule_tac x = 0 in exI, force)
   330 apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
   331 apply (rule_tac x = t in exI, auto)
   332 done
   333 
   334 lemma Maclaurin_all_le_objl: "diff 0 = f &
   335       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
   336       --> (\<exists>t. abs t \<le> abs x &
   337               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   338                     (diff n t / real (fact n)) * x ^ n)"
   339 by (blast intro: Maclaurin_all_le)
   340 
   341 
   342 subsection{*Version for Exponential Function*}
   343 
   344 lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
   345       ==> (\<exists>t. 0 < abs t &
   346                 abs t < abs x &
   347                 exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
   348                         (exp t / real (fact n)) * x ^ n)"
   349 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
   350 
   351 
   352 lemma Maclaurin_exp_le:
   353      "\<exists>t. abs t \<le> abs x &
   354             exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
   355                        (exp t / real (fact n)) * x ^ n"
   356 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
   357 
   358 
   359 subsection{*Version for Sine Function*}
   360 
   361 lemma mod_exhaust_less_4:
   362   "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
   363 by auto
   364 
   365 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
   366   "n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
   367 by (induct "n", auto)
   368 
   369 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
   370   "n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
   371 by (induct "n", auto)
   372 
   373 lemma Suc_mult_two_diff_one [rule_format, simp]:
   374   "n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
   375 by (induct "n", auto)
   376 
   377 
   378 text{*It is unclear why so many variant results are needed.*}
   379 
   380 lemma Maclaurin_sin_expansion2:
   381      "\<exists>t. abs t \<le> abs x &
   382        sin x =
   383        (\<Sum>m=0..<n. (if even m then 0
   384                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   385                        x ^ m)
   386       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   387 apply (cut_tac f = sin and n = n and x = x
   388         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
   389 apply safe
   390 apply (simp (no_asm))
   391 apply (simp (no_asm))
   392 apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
   393 apply (rule ccontr, simp)
   394 apply (drule_tac x = x in spec, simp)
   395 apply (erule ssubst)
   396 apply (rule_tac x = t in exI, simp)
   397 apply (rule setsum_cong[OF refl])
   398 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   399 done
   400 
   401 lemma Maclaurin_sin_expansion:
   402      "\<exists>t. sin x =
   403        (\<Sum>m=0..<n. (if even m then 0
   404                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   405                        x ^ m)
   406       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   407 apply (insert Maclaurin_sin_expansion2 [of x n]) 
   408 apply (blast intro: elim:); 
   409 done
   410 
   411 
   412 lemma Maclaurin_sin_expansion3:
   413      "[| n > 0; 0 < x |] ==>
   414        \<exists>t. 0 < t & t < x &
   415        sin x =
   416        (\<Sum>m=0..<n. (if even m then 0
   417                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   418                        x ^ m)
   419       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
   420 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)
   421 apply safe
   422 apply simp
   423 apply (simp (no_asm))
   424 apply (erule ssubst)
   425 apply (rule_tac x = t in exI, simp)
   426 apply (rule setsum_cong[OF refl])
   427 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   428 done
   429 
   430 lemma Maclaurin_sin_expansion4:
   431      "0 < x ==>
   432        \<exists>t. 0 < t & t \<le> x &
   433        sin x =
   434        (\<Sum>m=0..<n. (if even m then 0
   435                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   436                        x ^ m)
   437       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   438 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)
   439 apply safe
   440 apply simp
   441 apply (simp (no_asm))
   442 apply (erule ssubst)
   443 apply (rule_tac x = t in exI, simp)
   444 apply (rule setsum_cong[OF refl])
   445 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   446 done
   447 
   448 
   449 subsection{*Maclaurin Expansion for Cosine Function*}
   450 
   451 lemma sumr_cos_zero_one [simp]:
   452  "(\<Sum>m=0..<(Suc n).
   453      (if even m then -1 ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
   454 by (induct "n", auto)
   455 
   456 lemma Maclaurin_cos_expansion:
   457      "\<exists>t. abs t \<le> abs x &
   458        cos x =
   459        (\<Sum>m=0..<n. (if even m
   460                        then -1 ^ (m div 2)/(real (fact m))
   461                        else 0) *
   462                        x ^ m)
   463       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   464 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)
   465 apply safe
   466 apply (simp (no_asm))
   467 apply (simp (no_asm))
   468 apply (case_tac "n", simp)
   469 apply (simp del: setsum_op_ivl_Suc)
   470 apply (rule ccontr, simp)
   471 apply (drule_tac x = x in spec, simp)
   472 apply (erule ssubst)
   473 apply (rule_tac x = t in exI, simp)
   474 apply (rule setsum_cong[OF refl])
   475 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   476 done
   477 
   478 lemma Maclaurin_cos_expansion2:
   479      "[| 0 < x; n > 0 |] ==>
   480        \<exists>t. 0 < t & t < x &
   481        cos x =
   482        (\<Sum>m=0..<n. (if even m
   483                        then -1 ^ (m div 2)/(real (fact m))
   484                        else 0) *
   485                        x ^ m)
   486       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   487 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)
   488 apply safe
   489 apply simp
   490 apply (simp (no_asm))
   491 apply (erule ssubst)
   492 apply (rule_tac x = t in exI, simp)
   493 apply (rule setsum_cong[OF refl])
   494 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   495 done
   496 
   497 lemma Maclaurin_minus_cos_expansion:
   498      "[| x < 0; n > 0 |] ==>
   499        \<exists>t. x < t & t < 0 &
   500        cos x =
   501        (\<Sum>m=0..<n. (if even m
   502                        then -1 ^ (m div 2)/(real (fact m))
   503                        else 0) *
   504                        x ^ m)
   505       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   506 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)
   507 apply safe
   508 apply simp
   509 apply (simp (no_asm))
   510 apply (erule ssubst)
   511 apply (rule_tac x = t in exI, simp)
   512 apply (rule setsum_cong[OF refl])
   513 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   514 done
   515 
   516 (* ------------------------------------------------------------------------- *)
   517 (* Version for ln(1 +/- x). Where is it??                                    *)
   518 (* ------------------------------------------------------------------------- *)
   519 
   520 lemma sin_bound_lemma:
   521     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
   522 by auto
   523 
   524 lemma Maclaurin_sin_bound:
   525   "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   526   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
   527 proof -
   528   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
   529     by (rule_tac mult_right_mono,simp_all)
   530   note est = this[simplified]
   531   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)"
   532   have diff_0: "?diff 0 = sin" by simp
   533   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
   534     apply (clarify)
   535     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
   536     apply (cut_tac m=m in mod_exhaust_less_4)
   537     apply (safe, auto intro!: DERIV_intros)
   538     done
   539   from Maclaurin_all_le [OF diff_0 DERIV_diff]
   540   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
   541     t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
   542       ?diff n t / real (fact n) * x ^ n" by fast
   543   have diff_m_0:
   544     "\<And>m. ?diff m 0 = (if even m then 0
   545          else -1 ^ ((m - Suc 0) div 2))"
   546     apply (subst even_even_mod_4_iff)
   547     apply (cut_tac m=m in mod_exhaust_less_4)
   548     apply (elim disjE, simp_all)
   549     apply (safe dest!: mod_eqD, simp_all)
   550     done
   551   show ?thesis
   552     apply (subst t2)
   553     apply (rule sin_bound_lemma)
   554     apply (rule setsum_cong[OF refl])
   555     apply (subst diff_m_0, simp)
   556     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
   557                    simp add: est mult_nonneg_nonneg mult_ac divide_inverse
   558                           power_abs [symmetric] abs_mult)
   559     done
   560 qed
   561 
   562 end