src/HOL/MacLaurin.thy
author haftmann
Fri Nov 27 08:41:10 2009 +0100 (2009-11-27)
changeset 33963 977b94b64905
parent 32047 c141f139ce26
child 36974 b877866b5b00
permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     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 fact_diff_Suc [rule_format]:
    31   "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
    32   by (subst fact_reduce_nat, auto)
    33 
    34 lemma Maclaurin_lemma2:
    35   assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
    36   assumes n: "n = Suc k"
    37   assumes difg: "difg =
    38         (\<lambda>m t. diff m t -
    39                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
    40                 B * (t ^ (n - m) / real (fact (n - m)))))"
    41   shows
    42       "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
    43 unfolding difg
    44  apply clarify
    45  apply (rule DERIV_diff)
    46   apply (simp add: diff)
    47  apply (simp only: n)
    48  apply (rule DERIV_add)
    49   apply (rule_tac [2] DERIV_cmult)
    50   apply (rule_tac [2] lemma_DERIV_subst)
    51    apply (rule_tac [2] DERIV_quotient)
    52      apply (rule_tac [3] DERIV_const)
    53     apply (rule_tac [2] DERIV_pow)
    54    prefer 3 
    55 
    56 apply (simp add: fact_diff_Suc)
    57   prefer 2 apply simp
    58  apply (frule less_iff_Suc_add [THEN iffD1], clarify)
    59  apply (simp del: setsum_op_ivl_Suc)
    60  apply (insert sumr_offset4 [of "Suc 0"])
    61  apply (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
    62  apply (rule lemma_DERIV_subst)
    63   apply (rule DERIV_add)
    64    apply (rule_tac [2] DERIV_const)
    65   apply (rule DERIV_sumr, clarify)
    66   prefer 2 apply simp
    67  apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc power_Suc)
    68  apply (rule DERIV_cmult)
    69  apply (rule lemma_DERIV_subst)
    70   apply (best intro!: DERIV_intros)
    71  apply (subst fact_Suc)
    72  apply (subst real_of_nat_mult)
    73  apply (simp add: mult_ac)
    74 done
    75 
    76 lemma Maclaurin:
    77   assumes h: "0 < h"
    78   assumes n: "0 < n"
    79   assumes diff_0: "diff 0 = f"
    80   assumes diff_Suc:
    81     "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
    82   shows
    83     "\<exists>t. 0 < t & t < h &
    84               f h =
    85               setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
    86               (diff n t / real (fact n)) * h ^ n"
    87 proof -
    88   from n obtain m where m: "n = Suc m"
    89     by (cases n, simp add: n)
    90 
    91   obtain B where f_h: "f h =
    92         (\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
    93         B * (h ^ n / real (fact n))"
    94     using Maclaurin_lemma [OF h] ..
    95 
    96   obtain g where g_def: "g = (%t. f t -
    97     (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n}
    98       + (B * (t^n / real(fact n)))))" by blast
    99 
   100   have g2: "g 0 = 0 & g h = 0"
   101     apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
   102     apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
   103     apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
   104     done
   105 
   106   obtain difg where difg_def: "difg = (%m t. diff m t -
   107     (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
   108       + (B * ((t ^ (n - m)) / real (fact (n - m))))))" by blast
   109 
   110   have difg_0: "difg 0 = g"
   111     unfolding difg_def g_def by (simp add: diff_0)
   112 
   113   have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
   114         m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
   115     using diff_Suc m difg_def by (rule Maclaurin_lemma2)
   116 
   117   have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
   118     apply clarify
   119     apply (simp add: m difg_def)
   120     apply (frule less_iff_Suc_add [THEN iffD1], clarify)
   121     apply (simp del: setsum_op_ivl_Suc)
   122     apply (insert sumr_offset4 [of "Suc 0"])
   123     apply (simp del: setsum_op_ivl_Suc fact_Suc)
   124     done
   125 
   126   have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
   127     by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
   128 
   129   have differentiable_difg:
   130     "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
   131     by (rule differentiableI [OF difg_Suc [rule_format]]) simp
   132 
   133   have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
   134         \<Longrightarrow> difg (Suc m) t = 0"
   135     by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
   136 
   137   have "m < n" using m by simp
   138 
   139   have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
   140   using `m < n`
   141   proof (induct m)
   142   case 0
   143     show ?case
   144     proof (rule Rolle)
   145       show "0 < h" by fact
   146       show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
   147       show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
   148         by (simp add: isCont_difg n)
   149       show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
   150         by (simp add: differentiable_difg n)
   151     qed
   152   next
   153   case (Suc m')
   154     hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
   155     then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
   156     have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
   157     proof (rule Rolle)
   158       show "0 < t" by fact
   159       show "difg (Suc m') 0 = difg (Suc m') t"
   160         using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
   161       show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
   162         using `t < h` `Suc m' < n` by (simp add: isCont_difg)
   163       show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
   164         using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
   165     qed
   166     thus ?case
   167       using `t < h` by auto
   168   qed
   169 
   170   then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
   171 
   172   hence "difg (Suc m) t = 0"
   173     using `m < n` by (simp add: difg_Suc_eq_0)
   174 
   175   show ?thesis
   176   proof (intro exI conjI)
   177     show "0 < t" by fact
   178     show "t < h" by fact
   179     show "f h =
   180       (\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
   181       diff n t / real (fact n) * h ^ n"
   182       using `difg (Suc m) t = 0`
   183       by (simp add: m f_h difg_def del: fact_Suc)
   184   qed
   185 
   186 qed
   187 
   188 lemma Maclaurin_objl:
   189   "0 < h & n>0 & diff 0 = f &
   190   (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   191    --> (\<exists>t. 0 < t & t < h &
   192             f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   193                   diff n t / real (fact n) * h ^ n)"
   194 by (blast intro: Maclaurin)
   195 
   196 
   197 lemma Maclaurin2:
   198    "[| 0 < h; diff 0 = f;
   199        \<forall>m t.
   200           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   201     ==> \<exists>t. 0 < t &
   202               t \<le> h &
   203               f h =
   204               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   205               diff n t / real (fact n) * h ^ n"
   206 apply (case_tac "n", auto)
   207 apply (drule Maclaurin, auto)
   208 done
   209 
   210 lemma Maclaurin2_objl:
   211      "0 < h & diff 0 = f &
   212        (\<forall>m t.
   213           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   214     --> (\<exists>t. 0 < t &
   215               t \<le> h &
   216               f h =
   217               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   218               diff n t / real (fact n) * h ^ n)"
   219 by (blast intro: Maclaurin2)
   220 
   221 lemma Maclaurin_minus:
   222    "[| h < 0; n > 0; diff 0 = f;
   223        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
   224     ==> \<exists>t. h < t &
   225               t < 0 &
   226               f h =
   227               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   228               diff n t / real (fact n) * h ^ n"
   229 apply (cut_tac f = "%x. f (-x)"
   230         and diff = "%n x. (-1 ^ n) * diff n (-x)"
   231         and h = "-h" and n = n in Maclaurin_objl)
   232 apply (simp)
   233 apply safe
   234 apply (subst minus_mult_right)
   235 apply (rule DERIV_cmult)
   236 apply (rule lemma_DERIV_subst)
   237 apply (rule DERIV_chain2 [where g=uminus])
   238 apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_ident)
   239 prefer 2 apply force
   240 apply force
   241 apply (rule_tac x = "-t" in exI, auto)
   242 apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
   243                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
   244 apply (rule_tac [2] setsum_cong[OF refl])
   245 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
   246 done
   247 
   248 lemma Maclaurin_minus_objl:
   249      "(h < 0 & n > 0 & diff 0 = f &
   250        (\<forall>m t.
   251           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
   252     --> (\<exists>t. h < t &
   253               t < 0 &
   254               f h =
   255               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   256               diff n t / real (fact n) * h ^ n)"
   257 by (blast intro: Maclaurin_minus)
   258 
   259 
   260 subsection{*More Convenient "Bidirectional" Version.*}
   261 
   262 (* not good for PVS sin_approx, cos_approx *)
   263 
   264 lemma Maclaurin_bi_le_lemma [rule_format]:
   265   "n>0 \<longrightarrow>
   266    diff 0 0 =
   267    (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
   268    diff n 0 * 0 ^ n / real (fact n)"
   269 by (induct "n", auto)
   270 
   271 lemma Maclaurin_bi_le:
   272    "[| diff 0 = f;
   273        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
   274     ==> \<exists>t. abs t \<le> abs x &
   275               f x =
   276               (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
   277               diff n t / real (fact n) * x ^ n"
   278 apply (case_tac "n = 0", force)
   279 apply (case_tac "x = 0")
   280  apply (rule_tac x = 0 in exI)
   281  apply (force simp add: Maclaurin_bi_le_lemma)
   282 apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
   283  txt{*Case 1, where @{term "x < 0"}*}
   284  apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
   285   apply (simp add: abs_if)
   286  apply (rule_tac x = t in exI)
   287  apply (simp add: abs_if)
   288 txt{*Case 2, where @{term "0 < x"}*}
   289 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
   290  apply (simp add: abs_if)
   291 apply (rule_tac x = t in exI)
   292 apply (simp add: abs_if)
   293 done
   294 
   295 lemma Maclaurin_all_lt:
   296      "[| diff 0 = f;
   297          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
   298         x ~= 0; n > 0
   299       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
   300                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   301                      (diff n t / real (fact n)) * x ^ n"
   302 apply (rule_tac x = x and y = 0 in linorder_cases)
   303 prefer 2 apply blast
   304 apply (drule_tac [2] diff=diff in Maclaurin)
   305 apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
   306 apply (rule_tac [!] x = t in exI, auto)
   307 done
   308 
   309 lemma Maclaurin_all_lt_objl:
   310      "diff 0 = f &
   311       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
   312       x ~= 0 & n > 0
   313       --> (\<exists>t. 0 < abs t & abs t < abs x &
   314                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   315                      (diff n t / real (fact n)) * x ^ n)"
   316 by (blast intro: Maclaurin_all_lt)
   317 
   318 lemma Maclaurin_zero [rule_format]:
   319      "x = (0::real)
   320       ==> n \<noteq> 0 -->
   321           (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
   322           diff 0 0"
   323 by (induct n, auto)
   324 
   325 lemma Maclaurin_all_le: "[| diff 0 = f;
   326         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
   327       |] ==> \<exists>t. abs t \<le> abs x &
   328               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   329                     (diff n t / real (fact n)) * x ^ n"
   330 apply(cases "n=0")
   331 apply (force)
   332 apply (case_tac "x = 0")
   333 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
   334 apply (drule not0_implies_Suc)
   335 apply (rule_tac x = 0 in exI, force)
   336 apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
   337 apply (rule_tac x = t in exI, auto)
   338 done
   339 
   340 lemma Maclaurin_all_le_objl: "diff 0 = f &
   341       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
   342       --> (\<exists>t. abs t \<le> abs x &
   343               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   344                     (diff n t / real (fact n)) * x ^ n)"
   345 by (blast intro: Maclaurin_all_le)
   346 
   347 
   348 subsection{*Version for Exponential Function*}
   349 
   350 lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
   351       ==> (\<exists>t. 0 < abs t &
   352                 abs t < abs x &
   353                 exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
   354                         (exp t / real (fact n)) * x ^ n)"
   355 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
   356 
   357 
   358 lemma Maclaurin_exp_le:
   359      "\<exists>t. abs t \<le> abs x &
   360             exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
   361                        (exp t / real (fact n)) * x ^ n"
   362 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
   363 
   364 
   365 subsection{*Version for Sine Function*}
   366 
   367 lemma mod_exhaust_less_4:
   368   "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
   369 by auto
   370 
   371 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
   372   "n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
   373 by (induct "n", auto)
   374 
   375 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
   376   "n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
   377 by (induct "n", auto)
   378 
   379 lemma Suc_mult_two_diff_one [rule_format, simp]:
   380   "n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
   381 by (induct "n", auto)
   382 
   383 
   384 text{*It is unclear why so many variant results are needed.*}
   385 
   386 lemma Maclaurin_sin_expansion2:
   387      "\<exists>t. abs t \<le> abs x &
   388        sin x =
   389        (\<Sum>m=0..<n. (if even m then 0
   390                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   391                        x ^ m)
   392       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   393 apply (cut_tac f = sin and n = n and x = x
   394         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
   395 apply safe
   396 apply (simp (no_asm))
   397 apply (simp (no_asm))
   398 apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
   399 apply (rule ccontr, simp)
   400 apply (drule_tac x = x in spec, simp)
   401 apply (erule ssubst)
   402 apply (rule_tac x = t in exI, simp)
   403 apply (rule setsum_cong[OF refl])
   404 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   405 done
   406 
   407 lemma Maclaurin_sin_expansion:
   408      "\<exists>t. sin x =
   409        (\<Sum>m=0..<n. (if even m then 0
   410                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   411                        x ^ m)
   412       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   413 apply (insert Maclaurin_sin_expansion2 [of x n]) 
   414 apply (blast intro: elim:); 
   415 done
   416 
   417 
   418 lemma Maclaurin_sin_expansion3:
   419      "[| n > 0; 0 < x |] ==>
   420        \<exists>t. 0 < t & t < x &
   421        sin x =
   422        (\<Sum>m=0..<n. (if even m then 0
   423                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   424                        x ^ m)
   425       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
   426 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)
   427 apply safe
   428 apply simp
   429 apply (simp (no_asm))
   430 apply (erule ssubst)
   431 apply (rule_tac x = t in exI, simp)
   432 apply (rule setsum_cong[OF refl])
   433 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   434 done
   435 
   436 lemma Maclaurin_sin_expansion4:
   437      "0 < x ==>
   438        \<exists>t. 0 < t & t \<le> x &
   439        sin x =
   440        (\<Sum>m=0..<n. (if even m then 0
   441                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   442                        x ^ m)
   443       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   444 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)
   445 apply safe
   446 apply simp
   447 apply (simp (no_asm))
   448 apply (erule ssubst)
   449 apply (rule_tac x = t in exI, simp)
   450 apply (rule setsum_cong[OF refl])
   451 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   452 done
   453 
   454 
   455 subsection{*Maclaurin Expansion for Cosine Function*}
   456 
   457 lemma sumr_cos_zero_one [simp]:
   458  "(\<Sum>m=0..<(Suc n).
   459      (if even m then -1 ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
   460 by (induct "n", auto)
   461 
   462 lemma Maclaurin_cos_expansion:
   463      "\<exists>t. abs t \<le> abs x &
   464        cos x =
   465        (\<Sum>m=0..<n. (if even m
   466                        then -1 ^ (m div 2)/(real (fact m))
   467                        else 0) *
   468                        x ^ m)
   469       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   470 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)
   471 apply safe
   472 apply (simp (no_asm))
   473 apply (simp (no_asm))
   474 apply (case_tac "n", simp)
   475 apply (simp del: setsum_op_ivl_Suc)
   476 apply (rule ccontr, simp)
   477 apply (drule_tac x = x in spec, simp)
   478 apply (erule ssubst)
   479 apply (rule_tac x = t in exI, simp)
   480 apply (rule setsum_cong[OF refl])
   481 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   482 done
   483 
   484 lemma Maclaurin_cos_expansion2:
   485      "[| 0 < x; n > 0 |] ==>
   486        \<exists>t. 0 < t & t < x &
   487        cos x =
   488        (\<Sum>m=0..<n. (if even m
   489                        then -1 ^ (m div 2)/(real (fact m))
   490                        else 0) *
   491                        x ^ m)
   492       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   493 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)
   494 apply safe
   495 apply simp
   496 apply (simp (no_asm))
   497 apply (erule ssubst)
   498 apply (rule_tac x = t in exI, simp)
   499 apply (rule setsum_cong[OF refl])
   500 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   501 done
   502 
   503 lemma Maclaurin_minus_cos_expansion:
   504      "[| x < 0; n > 0 |] ==>
   505        \<exists>t. x < t & t < 0 &
   506        cos x =
   507        (\<Sum>m=0..<n. (if even m
   508                        then -1 ^ (m div 2)/(real (fact m))
   509                        else 0) *
   510                        x ^ m)
   511       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   512 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)
   513 apply safe
   514 apply simp
   515 apply (simp (no_asm))
   516 apply (erule ssubst)
   517 apply (rule_tac x = t in exI, simp)
   518 apply (rule setsum_cong[OF refl])
   519 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   520 done
   521 
   522 (* ------------------------------------------------------------------------- *)
   523 (* Version for ln(1 +/- x). Where is it??                                    *)
   524 (* ------------------------------------------------------------------------- *)
   525 
   526 lemma sin_bound_lemma:
   527     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
   528 by auto
   529 
   530 lemma Maclaurin_sin_bound:
   531   "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   532   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
   533 proof -
   534   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
   535     by (rule_tac mult_right_mono,simp_all)
   536   note est = this[simplified]
   537   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)"
   538   have diff_0: "?diff 0 = sin" by simp
   539   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
   540     apply (clarify)
   541     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
   542     apply (cut_tac m=m in mod_exhaust_less_4)
   543     apply (safe, auto intro!: DERIV_intros)
   544     done
   545   from Maclaurin_all_le [OF diff_0 DERIV_diff]
   546   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
   547     t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
   548       ?diff n t / real (fact n) * x ^ n" by fast
   549   have diff_m_0:
   550     "\<And>m. ?diff m 0 = (if even m then 0
   551          else -1 ^ ((m - Suc 0) div 2))"
   552     apply (subst even_even_mod_4_iff)
   553     apply (cut_tac m=m in mod_exhaust_less_4)
   554     apply (elim disjE, simp_all)
   555     apply (safe dest!: mod_eqD, simp_all)
   556     done
   557   show ?thesis
   558     apply (subst t2)
   559     apply (rule sin_bound_lemma)
   560     apply (rule setsum_cong[OF refl])
   561     apply (subst diff_m_0, simp)
   562     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
   563                    simp add: est mult_nonneg_nonneg mult_ac divide_inverse
   564                           power_abs [symmetric] abs_mult)
   565     done
   566 qed
   567 
   568 end