src/HOL/Hyperreal/MacLaurin.thy
author chaieb
Sat Oct 20 12:09:33 2007 +0200 (2007-10-20)
changeset 25112 98824cc791c0
parent 24998 a339b33adfaf
child 25134 3d4953e88449
permissions -rw-r--r--
fixed proofs
     1 (*  ID          : $Id$
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 2001 University of Edinburgh
     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     5 *)
     6 
     7 header{*MacLaurin Series*}
     8 
     9 theory MacLaurin
    10 imports Transcendental
    11 begin
    12 
    13 subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
    14 
    15 text{*This is a very long, messy proof even now that it's been broken down
    16 into lemmas.*}
    17 
    18 lemma Maclaurin_lemma:
    19     "0 < h ==>
    20      \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
    21                (B * ((h^n) / real(fact n)))"
    22 apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
    23                  real(fact n) / (h^n)"
    24        in exI)
    25 apply (simp) 
    26 done
    27 
    28 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
    29 by arith
    30 
    31 text{*A crude tactic to differentiate by proof.*}
    32 
    33 lemmas deriv_rulesI =
    34   DERIV_ident DERIV_const DERIV_cos DERIV_cmult
    35   DERIV_sin DERIV_exp DERIV_inverse DERIV_pow
    36   DERIV_add DERIV_diff DERIV_mult DERIV_minus
    37   DERIV_inverse_fun DERIV_quotient DERIV_fun_pow
    38   DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos
    39   DERIV_ident DERIV_const DERIV_cos
    40 
    41 ML
    42 {*
    43 local
    44 exception DERIV_name;
    45 fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
    46 |   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
    47 |   get_fun_name _ = raise DERIV_name;
    48 
    49 in
    50 
    51 val deriv_tac =
    52   SUBGOAL (fn (prem,i) =>
    53    (resolve_tac @{thms deriv_rulesI} i) ORELSE
    54     ((rtac (read_instantiate [("f",get_fun_name prem)]
    55                      @{thm DERIV_chain2}) i) handle DERIV_name => no_tac));;
    56 
    57 val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
    58 
    59 end
    60 *}
    61 
    62 lemma Maclaurin_lemma2:
    63       "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
    64           n = Suc k;
    65         difg =
    66         (\<lambda>m t. diff m t -
    67                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
    68                 B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
    69         \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
    70                     DERIV (difg m) t :> difg (Suc m) t"
    71 apply clarify
    72 apply (rule DERIV_diff)
    73 apply (simp (no_asm_simp))
    74 apply (tactic DERIV_tac)
    75 apply (tactic DERIV_tac)
    76 apply (rule_tac [2] lemma_DERIV_subst)
    77 apply (rule_tac [2] DERIV_quotient)
    78 apply (rule_tac [3] DERIV_const)
    79 apply (rule_tac [2] DERIV_pow)
    80   prefer 3 apply (simp add: fact_diff_Suc)
    81  prefer 2 apply simp
    82 apply (frule_tac m = m in less_add_one, clarify)
    83 apply (simp del: setsum_op_ivl_Suc)
    84 apply (insert sumr_offset4 [of 1])
    85 apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
    86 apply (rule lemma_DERIV_subst)
    87 apply (rule DERIV_add)
    88 apply (rule_tac [2] DERIV_const)
    89 apply (rule DERIV_sumr, clarify)
    90  prefer 2 apply simp
    91 apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
    92 apply (rule DERIV_cmult)
    93 apply (rule lemma_DERIV_subst)
    94 apply (best intro: DERIV_chain2 intro!: DERIV_intros)
    95 apply (subst fact_Suc)
    96 apply (subst real_of_nat_mult)
    97 apply (simp add: mult_ac)
    98 done
    99 
   100 
   101 lemma Maclaurin_lemma3:
   102   fixes difg :: "nat => real => real" shows
   103      "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
   104         \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
   105         t < h|]
   106      ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
   107 apply (rule Rolle, assumption, simp)
   108 apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
   109 apply (rule DERIV_unique)
   110 prefer 2 apply assumption
   111 apply force
   112 apply (metis DERIV_isCont dlo_simps(4) dlo_simps(9) less_trans_Suc nat_less_le not_less_eq real_le_trans)
   113 apply (metis Suc_less_eq differentiableI dlo_simps(7) dlo_simps(8) dlo_simps(9)   real_le_trans xt1(8))
   114 done
   115 
   116 lemma Maclaurin:
   117    "[| 0 < h; 0 < n; diff 0 = f;
   118        \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   119     ==> \<exists>t. 0 < t &
   120               t < h &
   121               f h =
   122               setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
   123               (diff n t / real (fact n)) * h ^ n"
   124 apply (case_tac "n = 0", force)
   125 apply (drule not0_implies_Suc)
   126 apply (erule exE)
   127 apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
   128 apply (erule exE)
   129 apply (subgoal_tac "\<exists>g.
   130      g = (%t. f t - (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n} + (B * (t^n / real(fact n)))))")
   131  prefer 2 apply blast
   132 apply (erule exE)
   133 apply (subgoal_tac "g 0 = 0 & g h =0")
   134  prefer 2
   135  apply (simp del: setsum_op_ivl_Suc)
   136  apply (cut_tac n = m and k = 1 in sumr_offset2)
   137  apply (simp add: eq_diff_eq' del: setsum_op_ivl_Suc)
   138 apply (subgoal_tac "\<exists>difg. difg = (%m t. diff m t - (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m} + (B * ((t ^ (n - m)) / real (fact (n - m))))))")
   139  prefer 2 apply blast
   140 apply (erule exE)
   141 apply (subgoal_tac "difg 0 = g")
   142  prefer 2 apply simp
   143 apply (frule Maclaurin_lemma2, assumption+)
   144 apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
   145  apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
   146  apply (erule impE)
   147   apply (simp (no_asm_simp))
   148  apply (erule exE)
   149  apply (rule_tac x = t in exI)
   150  apply (simp del: realpow_Suc fact_Suc)
   151 apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
   152  prefer 2
   153  apply clarify
   154  apply simp
   155  apply (frule_tac m = ma in less_add_one, clarify)
   156  apply (simp del: setsum_op_ivl_Suc)
   157 apply (insert sumr_offset4 [of 1])
   158 apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
   159 apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
   160 apply (rule allI, rule impI)
   161 apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
   162 apply (erule impE, assumption)
   163 apply (erule exE)
   164 apply (rule_tac x = t in exI)
   165 (* do some tidying up *)
   166 apply (erule_tac [!] V= "difg = (%m t. diff m t - (setsum (%p. diff (m + p) 0 / real (fact p) * t ^ p) {0..<n-m} + B * (t ^ (n - m) / real (fact (n - m)))))"
   167        in thin_rl)
   168 apply (erule_tac [!] V="g = (%t. f t - (setsum (%m. diff m 0 / real (fact m) * t ^ m) {0..<n} + B * (t ^ n / real (fact n))))"
   169        in thin_rl)
   170 apply (erule_tac [!] V="f h = setsum (%m. diff m 0 / real (fact m) * h ^ m) {0..<n} + B * (h ^ n / real (fact n))"
   171        in thin_rl)
   172 (* back to business *)
   173 apply (simp (no_asm_simp))
   174 apply (rule DERIV_unique)
   175 prefer 2 apply blast
   176 apply force
   177 apply (rule allI, induct_tac "ma")
   178 apply (rule impI, rule Rolle, assumption, simp, simp)
   179 apply (metis DERIV_isCont zero_less_Suc)
   180 apply (metis One_nat_def differentiableI dlo_simps(7))
   181 apply safe
   182 apply force
   183 apply (frule Maclaurin_lemma3, assumption+, safe)
   184 apply (rule_tac x = ta in exI, force)
   185 done
   186 
   187 lemma Maclaurin_objl:
   188      "0 < h & 0 < n & diff 0 = f &
   189        (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   190     --> (\<exists>t. 0 < t &
   191               t < h &
   192               f h =
   193               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   194               diff n t / real (fact n) * h ^ n)"
   195 by (blast intro: Maclaurin)
   196 
   197 
   198 lemma Maclaurin2:
   199    "[| 0 < h; diff 0 = f;
   200        \<forall>m t.
   201           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   202     ==> \<exists>t. 0 < t &
   203               t \<le> h &
   204               f h =
   205               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   206               diff n t / real (fact n) * h ^ n"
   207 apply (case_tac "n", auto)
   208 apply (drule Maclaurin, auto)
   209 done
   210 
   211 lemma Maclaurin2_objl:
   212      "0 < h & diff 0 = f &
   213        (\<forall>m t.
   214           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   215     --> (\<exists>t. 0 < t &
   216               t \<le> h &
   217               f h =
   218               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   219               diff n t / real (fact n) * h ^ n)"
   220 by (blast intro: Maclaurin2)
   221 
   222 lemma Maclaurin_minus:
   223    "[| h < 0; 0 < n; diff 0 = f;
   224        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
   225     ==> \<exists>t. h < t &
   226               t < 0 &
   227               f h =
   228               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   229               diff n t / real (fact n) * h ^ n"
   230 apply (cut_tac f = "%x. f (-x)"
   231         and diff = "%n x. (-1 ^ n) * diff n (-x)"
   232         and h = "-h" and n = n in Maclaurin_objl)
   233 apply (simp)
   234 apply safe
   235 apply (subst minus_mult_right)
   236 apply (rule DERIV_cmult)
   237 apply (rule lemma_DERIV_subst)
   238 apply (rule DERIV_chain2 [where g=uminus])
   239 apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_ident)
   240 prefer 2 apply force
   241 apply force
   242 apply (rule_tac x = "-t" in exI, auto)
   243 apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
   244                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
   245 apply (rule_tac [2] setsum_cong[OF refl])
   246 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
   247 done
   248 
   249 lemma Maclaurin_minus_objl:
   250      "(h < 0 & 0 < n & diff 0 = f &
   251        (\<forall>m t.
   252           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
   253     --> (\<exists>t. h < t &
   254               t < 0 &
   255               f h =
   256               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
   257               diff n t / real (fact n) * h ^ n)"
   258 by (blast intro: Maclaurin_minus)
   259 
   260 
   261 subsection{*More Convenient "Bidirectional" Version.*}
   262 
   263 (* not good for PVS sin_approx, cos_approx *)
   264 
   265 lemma Maclaurin_bi_le_lemma [rule_format]:
   266      "0 < n \<longrightarrow>
   267        diff 0 0 =
   268        (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
   269        diff n 0 * 0 ^ n / real (fact n)"
   270 by (induct "n", auto)
   271 
   272 lemma Maclaurin_bi_le:
   273    "[| diff 0 = f;
   274        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
   275     ==> \<exists>t. abs t \<le> abs x &
   276               f x =
   277               (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
   278               diff n t / real (fact n) * x ^ n"
   279 apply (case_tac "n = 0", force)
   280 apply (case_tac "x = 0")
   281 apply (rule_tac x = 0 in exI)
   282 apply (force simp add: neq0_conv Maclaurin_bi_le_lemma)
   283 apply (cut_tac x = x and y = 0 in linorder_less_linear, auto simp add: neq0_conv)
   284 txt{*Case 1, where @{term "x < 0"}*}
   285 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
   286 apply (simp add: abs_if neq0_conv)
   287 apply (rule_tac x = t in exI)
   288 apply (simp add: abs_if)
   289 txt{*Case 2, where @{term "0 < x"}*}
   290 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
   291 apply (simp add: abs_if)
   292 apply (rule_tac x = t in exI)
   293 apply (simp add: abs_if)
   294 done
   295 
   296 lemma Maclaurin_all_lt:
   297      "[| diff 0 = f;
   298          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
   299         x ~= 0; 0 < n
   300       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
   301                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   302                      (diff n t / real (fact n)) * x ^ n"
   303 apply (rule_tac x = x and y = 0 in linorder_cases)
   304 prefer 2 apply blast
   305 apply (drule_tac [2] diff=diff in Maclaurin)
   306 apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
   307 apply (rule_tac [!] x = t in exI, auto)
   308 done
   309 
   310 lemma Maclaurin_all_lt_objl:
   311      "diff 0 = f &
   312       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
   313       x ~= 0 & 0 < n
   314       --> (\<exists>t. 0 < abs t & abs t < abs x &
   315                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   316                      (diff n t / real (fact n)) * x ^ n)"
   317 by (blast intro: Maclaurin_all_lt)
   318 
   319 lemma Maclaurin_zero [rule_format]:
   320      "x = (0::real)
   321       ==> 0 < n -->
   322           (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
   323           diff 0 0"
   324 by (induct n, auto)
   325 
   326 lemma Maclaurin_all_le: "[| diff 0 = f;
   327         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
   328       |] ==> \<exists>t. abs t \<le> abs x &
   329               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   330                     (diff n t / real (fact n)) * x ^ n"
   331 apply (insert linorder_le_less_linear [of n 0])
   332 apply (erule disjE, force)
   333 apply (case_tac "x = 0")
   334 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
   335 apply (drule gr_implies_not0 [THEN not0_implies_Suc])
   336 apply (rule_tac x = 0 in exI, force)
   337 apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
   338 apply (rule_tac x = t in exI, auto)
   339 done
   340 
   341 lemma Maclaurin_all_le_objl: "diff 0 = f &
   342       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
   343       --> (\<exists>t. abs t \<le> abs x &
   344               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
   345                     (diff n t / real (fact n)) * x ^ n)"
   346 by (blast intro: Maclaurin_all_le)
   347 
   348 
   349 subsection{*Version for Exponential Function*}
   350 
   351 lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
   352       ==> (\<exists>t. 0 < abs t &
   353                 abs t < 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_lt_objl, auto)
   357 
   358 
   359 lemma Maclaurin_exp_le:
   360      "\<exists>t. abs t \<le> abs x &
   361             exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
   362                        (exp t / real (fact n)) * x ^ n"
   363 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
   364 
   365 
   366 subsection{*Version for Sine Function*}
   367 
   368 lemma MVT2:
   369      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
   370       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
   371 apply (drule MVT)
   372 apply (blast intro: DERIV_isCont)
   373 apply (force dest: order_less_imp_le simp add: differentiable_def)
   374 apply (blast dest: DERIV_unique order_less_imp_le)
   375 done
   376 
   377 lemma mod_exhaust_less_4:
   378      "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
   379 by auto
   380 
   381 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
   382      "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
   383 by (induct "n", auto)
   384 
   385 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
   386      "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
   387 by (induct "n", auto)
   388 
   389 lemma Suc_mult_two_diff_one [rule_format, simp]:
   390       "0 < n --> Suc (2 * n - 1) = 2*n"
   391 by (induct "n", auto)
   392 
   393 
   394 text{*It is unclear why so many variant results are needed.*}
   395 
   396 lemma Maclaurin_sin_expansion2:
   397      "\<exists>t. abs t \<le> abs x &
   398        sin x =
   399        (\<Sum>m=0..<n. (if even m then 0
   400                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   401                        x ^ m)
   402       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   403 apply (cut_tac f = sin and n = n and x = x
   404         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
   405 apply safe
   406 apply (simp (no_asm))
   407 apply (simp (no_asm))
   408 apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
   409 apply (rule ccontr, simp)
   410 apply (drule_tac x = x in spec, simp)
   411 apply (erule ssubst)
   412 apply (rule_tac x = t in exI, simp)
   413 apply (rule setsum_cong[OF refl])
   414 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   415 done
   416 
   417 lemma Maclaurin_sin_expansion:
   418      "\<exists>t. sin x =
   419        (\<Sum>m=0..<n. (if even m then 0
   420                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   421                        x ^ m)
   422       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   423 apply (insert Maclaurin_sin_expansion2 [of x n]) 
   424 apply (blast intro: elim:); 
   425 done
   426 
   427 
   428 
   429 lemma Maclaurin_sin_expansion3:
   430      "[| 0 < n; 0 < x |] ==>
   431        \<exists>t. 0 < t & t < x &
   432        sin x =
   433        (\<Sum>m=0..<n. (if even m then 0
   434                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   435                        x ^ m)
   436       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
   437 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)
   438 apply safe
   439 apply simp
   440 apply (simp (no_asm))
   441 apply (erule ssubst)
   442 apply (rule_tac x = t in exI, simp)
   443 apply (rule setsum_cong[OF refl])
   444 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   445 done
   446 
   447 lemma Maclaurin_sin_expansion4:
   448      "0 < x ==>
   449        \<exists>t. 0 < t & t \<le> x &
   450        sin x =
   451        (\<Sum>m=0..<n. (if even m then 0
   452                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   453                        x ^ m)
   454       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   455 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)
   456 apply safe
   457 apply simp
   458 apply (simp (no_asm))
   459 apply (erule ssubst)
   460 apply (rule_tac x = t in exI, simp)
   461 apply (rule setsum_cong[OF refl])
   462 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
   463 done
   464 
   465 
   466 subsection{*Maclaurin Expansion for Cosine Function*}
   467 
   468 lemma sumr_cos_zero_one [simp]:
   469  "(\<Sum>m=0..<(Suc n).
   470      (if even m then -1 ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
   471 by (induct "n", auto)
   472 
   473 lemma Maclaurin_cos_expansion:
   474      "\<exists>t. abs t \<le> abs x &
   475        cos x =
   476        (\<Sum>m=0..<n. (if even m
   477                        then -1 ^ (m div 2)/(real (fact m))
   478                        else 0) *
   479                        x ^ m)
   480       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   481 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)
   482 apply safe
   483 apply (simp (no_asm))
   484 apply (simp (no_asm))
   485 apply (case_tac "n", simp)
   486 apply (simp del: setsum_op_ivl_Suc)
   487 apply (rule ccontr, simp)
   488 apply (drule_tac x = x in spec, simp)
   489 apply (erule ssubst)
   490 apply (rule_tac x = t in exI, simp)
   491 apply (rule setsum_cong[OF refl])
   492 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   493 done
   494 
   495 lemma Maclaurin_cos_expansion2:
   496      "[| 0 < x; 0 < n |] ==>
   497        \<exists>t. 0 < t & t < x &
   498        cos x =
   499        (\<Sum>m=0..<n. (if even m
   500                        then -1 ^ (m div 2)/(real (fact m))
   501                        else 0) *
   502                        x ^ m)
   503       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   504 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)
   505 apply safe
   506 apply simp
   507 apply (simp (no_asm))
   508 apply (erule ssubst)
   509 apply (rule_tac x = t in exI, simp)
   510 apply (rule setsum_cong[OF refl])
   511 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   512 done
   513 
   514 lemma Maclaurin_minus_cos_expansion:
   515      "[| x < 0; 0 < n |] ==>
   516        \<exists>t. x < t & t < 0 &
   517        cos x =
   518        (\<Sum>m=0..<n. (if even m
   519                        then -1 ^ (m div 2)/(real (fact m))
   520                        else 0) *
   521                        x ^ m)
   522       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   523 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)
   524 apply safe
   525 apply simp
   526 apply (simp (no_asm))
   527 apply (erule ssubst)
   528 apply (rule_tac x = t in exI, simp)
   529 apply (rule setsum_cong[OF refl])
   530 apply (auto simp add: cos_zero_iff even_mult_two_ex)
   531 done
   532 
   533 (* ------------------------------------------------------------------------- *)
   534 (* Version for ln(1 +/- x). Where is it??                                    *)
   535 (* ------------------------------------------------------------------------- *)
   536 
   537 lemma sin_bound_lemma:
   538     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
   539 by auto
   540 
   541 lemma Maclaurin_sin_bound:
   542   "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
   543   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
   544 proof -
   545   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
   546     by (rule_tac mult_right_mono,simp_all)
   547   note est = this[simplified]
   548   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)"
   549   have diff_0: "?diff 0 = sin" by simp
   550   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
   551     apply (clarify)
   552     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
   553     apply (cut_tac m=m in mod_exhaust_less_4)
   554     apply (safe, simp_all)
   555     apply (rule DERIV_minus, simp)
   556     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
   557     done
   558   from Maclaurin_all_le [OF diff_0 DERIV_diff]
   559   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
   560     t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
   561       ?diff n t / real (fact n) * x ^ n" by fast
   562   have diff_m_0:
   563     "\<And>m. ?diff m 0 = (if even m then 0
   564          else -1 ^ ((m - Suc 0) div 2))"
   565     apply (subst even_even_mod_4_iff)
   566     apply (cut_tac m=m in mod_exhaust_less_4)
   567     apply (elim disjE, simp_all)
   568     apply (safe dest!: mod_eqD, simp_all)
   569     done
   570   show ?thesis
   571     apply (subst t2)
   572     apply (rule sin_bound_lemma)
   573     apply (rule setsum_cong[OF refl])
   574     apply (subst diff_m_0, simp)
   575     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
   576                    simp add: est mult_nonneg_nonneg mult_ac divide_inverse
   577                           power_abs [symmetric] abs_mult)
   578     done
   579 qed
   580 
   581 end