src/HOL/Hyperreal/MacLaurin.thy
author nipkow
Mon Aug 16 14:22:27 2004 +0200 (2004-08-16)
changeset 15131 c69542757a4d
parent 15081 32402f5624d1
child 15140 322485b816ac
permissions -rw-r--r--
New theory header syntax.
     1 (*  Title       : MacLaurin.thy
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 2001 University of Edinburgh
     4     Description : MacLaurin series
     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     6 *)
     7 
     8 theory MacLaurin
     9 import Log
    10 begin
    11 
    12 lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
    13 by (induct_tac "n", auto)
    14 
    15 lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
    16 by (induct_tac "n", auto)
    17 
    18 lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
    19 by (simp  add: sumr_offset)
    20 
    21 lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
    22 by (simp add: sumr_offset)
    23 
    24 lemma sumr_from_1_from_0: "0 < n ==>
    25       sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else
    26              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =
    27       sumr 0 (Suc n) (%n. (if even(n) then 0 else
    28              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"
    29 by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
    30 
    31 
    32 subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
    33 
    34 text{*This is a very long, messy proof even now that it's been broken down
    35 into lemmas.*}
    36 
    37 lemma Maclaurin_lemma:
    38     "0 < h ==>
    39      \<exists>B. f h = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +
    40                (B * ((h^n) / real(fact n)))"
    41 by (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *
    42                  real(fact n) / (h^n)"
    43        in exI, auto)
    44 
    45 
    46 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
    47 by arith
    48 
    49 text{*A crude tactic to differentiate by proof.*}
    50 ML
    51 {*
    52 exception DERIV_name;
    53 fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
    54 |   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
    55 |   get_fun_name _ = raise DERIV_name;
    56 
    57 val deriv_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,
    58                     DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,
    59                     DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,
    60                     DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,
    61                     DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,
    62                     DERIV_Id,DERIV_const,DERIV_cos];
    63 
    64 val deriv_tac =
    65   SUBGOAL (fn (prem,i) =>
    66    (resolve_tac deriv_rulesI i) ORELSE
    67     ((rtac (read_instantiate [("f",get_fun_name prem)]
    68                      DERIV_chain2) i) handle DERIV_name => no_tac));;
    69 
    70 val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
    71 *}
    72 
    73 lemma Maclaurin_lemma2:
    74       "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
    75           n = Suc k;
    76         difg =
    77         (\<lambda>m t. diff m t -
    78                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
    79                 B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
    80         \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
    81                     DERIV (difg m) t :> difg (Suc m) t"
    82 apply clarify
    83 apply (rule DERIV_diff)
    84 apply (simp (no_asm_simp))
    85 apply (tactic DERIV_tac)
    86 apply (tactic DERIV_tac)
    87 apply (rule_tac [2] lemma_DERIV_subst)
    88 apply (rule_tac [2] DERIV_quotient)
    89 apply (rule_tac [3] DERIV_const)
    90 apply (rule_tac [2] DERIV_pow)
    91   prefer 3 apply (simp add: fact_diff_Suc)
    92  prefer 2 apply simp
    93 apply (frule_tac m = m in less_add_one, clarify)
    94 apply (simp del: sumr_Suc)
    95 apply (insert sumr_offset4 [of 1])
    96 apply (simp del: sumr_Suc fact_Suc realpow_Suc)
    97 apply (rule lemma_DERIV_subst)
    98 apply (rule DERIV_add)
    99 apply (rule_tac [2] DERIV_const)
   100 apply (rule DERIV_sumr, clarify)
   101  prefer 2 apply simp
   102 apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
   103 apply (rule DERIV_cmult)
   104 apply (rule lemma_DERIV_subst)
   105 apply (best intro: DERIV_chain2 intro!: DERIV_intros)
   106 apply (subst fact_Suc)
   107 apply (subst real_of_nat_mult)
   108 apply (simp add: inverse_mult_distrib mult_ac)
   109 done
   110 
   111 
   112 lemma Maclaurin_lemma3:
   113      "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
   114         \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
   115         t < h|]
   116      ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
   117 apply (rule Rolle, assumption, simp)
   118 apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
   119 apply (rule DERIV_unique)
   120 prefer 2 apply assumption
   121 apply force
   122 apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")
   123 apply (simp add: differentiable_def)
   124 apply (blast dest!: DERIV_isCont)
   125 apply (simp add: differentiable_def, clarify)
   126 apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
   127 apply force
   128 apply (simp add: differentiable_def, clarify)
   129 apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
   130 apply force
   131 done
   132 
   133 lemma Maclaurin:
   134    "[| 0 < h; 0 < n; diff 0 = f;
   135        \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   136     ==> \<exists>t. 0 < t &
   137               t < h &
   138               f h =
   139               sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) +
   140               (diff n t / real (fact n)) * h ^ n"
   141 apply (case_tac "n = 0", force)
   142 apply (drule not0_implies_Suc)
   143 apply (erule exE)
   144 apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
   145 apply (erule exE)
   146 apply (subgoal_tac "\<exists>g.
   147      g = (%t. f t - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (B * (t^n / real(fact n)))))")
   148  prefer 2 apply blast
   149 apply (erule exE)
   150 apply (subgoal_tac "g 0 = 0 & g h =0")
   151  prefer 2
   152  apply (simp del: sumr_Suc)
   153  apply (cut_tac n = m and k = 1 in sumr_offset2)
   154  apply (simp add: eq_diff_eq' del: sumr_Suc)
   155 apply (subgoal_tac "\<exists>difg. difg = (%m t. diff m t - (sumr 0 (n - m) (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) + (B * ((t ^ (n - m)) / real (fact (n - m))))))")
   156  prefer 2 apply blast
   157 apply (erule exE)
   158 apply (subgoal_tac "difg 0 = g")
   159  prefer 2 apply simp
   160 apply (frule Maclaurin_lemma2, assumption+)
   161 apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
   162 apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
   163 apply (erule impE)
   164 apply (simp (no_asm_simp))
   165 apply (erule exE)
   166 apply (rule_tac x = t in exI)
   167 apply (simp del: realpow_Suc fact_Suc)
   168 apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
   169  prefer 2
   170  apply clarify
   171  apply simp
   172  apply (frule_tac m = ma in less_add_one, clarify)
   173  apply (simp del: sumr_Suc)
   174 apply (insert sumr_offset4 [of 1])
   175 apply (simp del: sumr_Suc fact_Suc realpow_Suc)
   176 apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
   177 apply (rule allI, rule impI)
   178 apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
   179 apply (erule impE, assumption)
   180 apply (erule exE)
   181 apply (rule_tac x = t in exI)
   182 (* do some tidying up *)
   183 apply (erule_tac [!] V= "difg = (%m t. diff m t - (sumr 0 (n - m) (%p. diff (m + p) 0 / real (fact p) * t ^ p) + B * (t ^ (n - m) / real (fact (n - m)))))"
   184        in thin_rl)
   185 apply (erule_tac [!] V="g = (%t. f t - (sumr 0 n (%m. diff m 0 / real (fact m) * t ^ m) + B * (t ^ n / real (fact n))))"
   186        in thin_rl)
   187 apply (erule_tac [!] V="f h = sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))"
   188        in thin_rl)
   189 (* back to business *)
   190 apply (simp (no_asm_simp))
   191 apply (rule DERIV_unique)
   192 prefer 2 apply blast
   193 apply force
   194 apply (rule allI, induct_tac "ma")
   195 apply (rule impI, rule Rolle, assumption, simp, simp)
   196 apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")
   197 apply (simp add: differentiable_def)
   198 apply (blast dest: DERIV_isCont)
   199 apply (simp add: differentiable_def, clarify)
   200 apply (rule_tac x = "difg (Suc 0) t" in exI)
   201 apply force
   202 apply (simp add: differentiable_def, clarify)
   203 apply (rule_tac x = "difg (Suc 0) x" in exI)
   204 apply force
   205 apply safe
   206 apply force
   207 apply (frule Maclaurin_lemma3, assumption+, safe)
   208 apply (rule_tac x = ta in exI, force)
   209 done
   210 
   211 lemma Maclaurin_objl:
   212      "0 < h & 0 < n & diff 0 = f &
   213        (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   214     --> (\<exists>t. 0 < t &
   215               t < h &
   216               f h =
   217               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
   218               diff n t / real (fact n) * h ^ n)"
   219 by (blast intro: Maclaurin)
   220 
   221 
   222 lemma Maclaurin2:
   223    "[| 0 < h; diff 0 = f;
   224        \<forall>m t.
   225           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
   226     ==> \<exists>t. 0 < t &
   227               t \<le> h &
   228               f h =
   229               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
   230               diff n t / real (fact n) * h ^ n"
   231 apply (case_tac "n", auto)
   232 apply (drule Maclaurin, auto)
   233 done
   234 
   235 lemma Maclaurin2_objl:
   236      "0 < h & diff 0 = f &
   237        (\<forall>m t.
   238           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
   239     --> (\<exists>t. 0 < t &
   240               t \<le> h &
   241               f h =
   242               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
   243               diff n t / real (fact n) * h ^ n)"
   244 by (blast intro: Maclaurin2)
   245 
   246 lemma Maclaurin_minus:
   247    "[| h < 0; 0 < n; diff 0 = f;
   248        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
   249     ==> \<exists>t. h < t &
   250               t < 0 &
   251               f h =
   252               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
   253               diff n t / real (fact n) * h ^ n"
   254 apply (cut_tac f = "%x. f (-x)"
   255         and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
   256         and h = "-h" and n = n in Maclaurin_objl)
   257 apply simp
   258 apply safe
   259 apply (subst minus_mult_right)
   260 apply (rule DERIV_cmult)
   261 apply (rule lemma_DERIV_subst)
   262 apply (rule DERIV_chain2 [where g=uminus])
   263 apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
   264 prefer 2 apply force
   265 apply force
   266 apply (rule_tac x = "-t" in exI, auto)
   267 apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
   268                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
   269 apply (rule_tac [2] sumr_fun_eq)
   270 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
   271 done
   272 
   273 lemma Maclaurin_minus_objl:
   274      "(h < 0 & 0 < n & diff 0 = f &
   275        (\<forall>m t.
   276           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
   277     --> (\<exists>t. h < t &
   278               t < 0 &
   279               f h =
   280               sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) +
   281               diff n t / real (fact n) * h ^ n)"
   282 by (blast intro: Maclaurin_minus)
   283 
   284 
   285 subsection{*More Convenient "Bidirectional" Version.*}
   286 
   287 (* not good for PVS sin_approx, cos_approx *)
   288 
   289 lemma Maclaurin_bi_le_lemma [rule_format]:
   290      "0 < n \<longrightarrow>
   291        diff 0 0 =
   292        (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
   293        diff n 0 * 0 ^ n / real (fact n)"
   294 by (induct_tac "n", auto)
   295 
   296 lemma Maclaurin_bi_le:
   297    "[| diff 0 = f;
   298        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
   299     ==> \<exists>t. abs t \<le> abs x &
   300               f x =
   301               sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) +
   302               diff n t / real (fact n) * x ^ n"
   303 apply (case_tac "n = 0", force)
   304 apply (case_tac "x = 0")
   305 apply (rule_tac x = 0 in exI)
   306 apply (force simp add: Maclaurin_bi_le_lemma)
   307 apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
   308 txt{*Case 1, where @{term "x < 0"}*}
   309 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
   310 apply (simp add: abs_if)
   311 apply (rule_tac x = t in exI)
   312 apply (simp add: abs_if)
   313 txt{*Case 2, where @{term "0 < x"}*}
   314 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
   315 apply (simp add: abs_if)
   316 apply (rule_tac x = t in exI)
   317 apply (simp add: abs_if)
   318 done
   319 
   320 lemma Maclaurin_all_lt:
   321      "[| diff 0 = f;
   322          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
   323         x ~= 0; 0 < n
   324       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
   325                f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
   326                      (diff n t / real (fact n)) * x ^ n"
   327 apply (rule_tac x = x and y = 0 in linorder_cases)
   328 prefer 2 apply blast
   329 apply (drule_tac [2] diff=diff in Maclaurin)
   330 apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
   331 apply (rule_tac [!] x = t in exI, auto, arith+)
   332 done
   333 
   334 lemma Maclaurin_all_lt_objl:
   335      "diff 0 = f &
   336       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
   337       x ~= 0 & 0 < n
   338       --> (\<exists>t. 0 < abs t & abs t < abs x &
   339                f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
   340                      (diff n t / real (fact n)) * x ^ n)"
   341 by (blast intro: Maclaurin_all_lt)
   342 
   343 lemma Maclaurin_zero [rule_format]:
   344      "x = (0::real)
   345       ==> 0 < n -->
   346           sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) =
   347           diff 0 0"
   348 by (induct n, auto)
   349 
   350 lemma Maclaurin_all_le: "[| diff 0 = f;
   351         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
   352       |] ==> \<exists>t. abs t \<le> abs x &
   353               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
   354                     (diff n t / real (fact n)) * x ^ n"
   355 apply (insert linorder_le_less_linear [of n 0])
   356 apply (erule disjE, force)
   357 apply (case_tac "x = 0")
   358 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
   359 apply (drule gr_implies_not0 [THEN not0_implies_Suc])
   360 apply (rule_tac x = 0 in exI, force)
   361 apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
   362 apply (rule_tac x = t in exI, auto)
   363 done
   364 
   365 lemma Maclaurin_all_le_objl: "diff 0 = f &
   366       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
   367       --> (\<exists>t. abs t \<le> abs x &
   368               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
   369                     (diff n t / real (fact n)) * x ^ n)"
   370 by (blast intro: Maclaurin_all_le)
   371 
   372 
   373 subsection{*Version for Exponential Function*}
   374 
   375 lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
   376       ==> (\<exists>t. 0 < abs t &
   377                 abs t < abs x &
   378                 exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
   379                         (exp t / real (fact n)) * x ^ n)"
   380 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
   381 
   382 
   383 lemma Maclaurin_exp_le:
   384      "\<exists>t. abs t \<le> abs x &
   385             exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
   386                        (exp t / real (fact n)) * x ^ n"
   387 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
   388 
   389 
   390 subsection{*Version for Sine Function*}
   391 
   392 lemma MVT2:
   393      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
   394       ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
   395 apply (drule MVT)
   396 apply (blast intro: DERIV_isCont)
   397 apply (force dest: order_less_imp_le simp add: differentiable_def)
   398 apply (blast dest: DERIV_unique order_less_imp_le)
   399 done
   400 
   401 lemma mod_exhaust_less_4:
   402      "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
   403 by (case_tac "m mod 4", auto, arith)
   404 
   405 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
   406      "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
   407 by (induct_tac "n", auto)
   408 
   409 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
   410      "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
   411 by (induct_tac "n", auto)
   412 
   413 lemma Suc_mult_two_diff_one [rule_format, simp]:
   414       "0 < n --> Suc (2 * n - 1) = 2*n"
   415 by (induct_tac "n", auto)
   416 
   417 lemma Maclaurin_sin_expansion:
   418      "\<exists>t. sin x =
   419        (sumr 0 n (%m. (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 (cut_tac f = sin and n = n and x = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
   424 apply safe
   425 apply (simp (no_asm))
   426 apply (simp (no_asm))
   427 apply (case_tac "n", clarify, simp)
   428 apply (drule_tac x = 0 in spec, simp, simp)
   429 apply (rule ccontr, simp)
   430 apply (drule_tac x = x in spec, simp)
   431 apply (erule ssubst)
   432 apply (rule_tac x = t in exI, simp)
   433 apply (rule sumr_fun_eq)
   434 apply (auto simp add: odd_Suc_mult_two_ex)
   435 apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
   436 (*Could sin_zero_iff help?*)
   437 done
   438 
   439 lemma Maclaurin_sin_expansion2:
   440      "\<exists>t. abs t \<le> abs x &
   441        sin x =
   442        (sumr 0 n (%m. (if even m then 0
   443                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
   444                        x ^ m))
   445       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   446 apply (cut_tac f = sin and n = n and x = x
   447         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
   448 apply safe
   449 apply (simp (no_asm))
   450 apply (simp (no_asm))
   451 apply (case_tac "n", clarify, simp, simp)
   452 apply (rule ccontr, simp)
   453 apply (drule_tac x = x in spec, simp)
   454 apply (erule ssubst)
   455 apply (rule_tac x = t in exI, simp)
   456 apply (rule sumr_fun_eq)
   457 apply (auto simp add: odd_Suc_mult_two_ex)
   458 apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
   459 done
   460 
   461 lemma Maclaurin_sin_expansion3:
   462      "[| 0 < n; 0 < x |] ==>
   463        \<exists>t. 0 < t & t < x &
   464        sin x =
   465        (sumr 0 n (%m. (if even m then 0
   466                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
   467                        x ^ m))
   468       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
   469 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)
   470 apply safe
   471 apply simp
   472 apply (simp (no_asm))
   473 apply (erule ssubst)
   474 apply (rule_tac x = t in exI, simp)
   475 apply (rule sumr_fun_eq)
   476 apply (auto simp add: odd_Suc_mult_two_ex)
   477 apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
   478 done
   479 
   480 lemma Maclaurin_sin_expansion4:
   481      "0 < x ==>
   482        \<exists>t. 0 < t & t \<le> x &
   483        sin x =
   484        (sumr 0 n (%m. (if even m then 0
   485                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
   486                        x ^ m))
   487       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   488 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)
   489 apply safe
   490 apply simp
   491 apply (simp (no_asm))
   492 apply (erule ssubst)
   493 apply (rule_tac x = t in exI, simp)
   494 apply (rule sumr_fun_eq)
   495 apply (auto simp add: odd_Suc_mult_two_ex)
   496 apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
   497 done
   498 
   499 
   500 subsection{*Maclaurin Expansion for Cosine Function*}
   501 
   502 lemma sumr_cos_zero_one [simp]:
   503      "sumr 0 (Suc n)
   504          (%m. (if even m
   505                then (- 1) ^ (m div 2)/(real  (fact m))
   506                else 0) *
   507               0 ^ m) = 1"
   508 by (induct_tac "n", auto)
   509 
   510 lemma Maclaurin_cos_expansion:
   511      "\<exists>t. abs t \<le> abs x &
   512        cos x =
   513        (sumr 0 n (%m. (if even m
   514                        then (- 1) ^ (m div 2)/(real (fact m))
   515                        else 0) *
   516                        x ^ m))
   517       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   518 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)
   519 apply safe
   520 apply (simp (no_asm))
   521 apply (simp (no_asm))
   522 apply (case_tac "n", simp)
   523 apply (simp del: sumr_Suc)
   524 apply (rule ccontr, simp)
   525 apply (drule_tac x = x in spec, simp)
   526 apply (erule ssubst)
   527 apply (rule_tac x = t in exI, simp)
   528 apply (rule sumr_fun_eq)
   529 apply (auto simp add: odd_Suc_mult_two_ex)
   530 apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
   531 apply (simp add: mult_commute [of _ pi])
   532 done
   533 
   534 lemma Maclaurin_cos_expansion2:
   535      "[| 0 < x; 0 < n |] ==>
   536        \<exists>t. 0 < t & t < x &
   537        cos x =
   538        (sumr 0 n (%m. (if even m
   539                        then (- 1) ^ (m div 2)/(real (fact m))
   540                        else 0) *
   541                        x ^ m))
   542       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   543 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)
   544 apply safe
   545 apply simp
   546 apply (simp (no_asm))
   547 apply (erule ssubst)
   548 apply (rule_tac x = t in exI, simp)
   549 apply (rule sumr_fun_eq)
   550 apply (auto simp add: odd_Suc_mult_two_ex)
   551 apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
   552 apply (simp add: mult_commute [of _ pi])
   553 done
   554 
   555 lemma Maclaurin_minus_cos_expansion: "[| x < 0; 0 < n |] ==>
   556        \<exists>t. x < t & t < 0 &
   557        cos x =
   558        (sumr 0 n (%m. (if even m
   559                        then (- 1) ^ (m div 2)/(real (fact m))
   560                        else 0) *
   561                        x ^ m))
   562       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
   563 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)
   564 apply safe
   565 apply simp
   566 apply (simp (no_asm))
   567 apply (erule ssubst)
   568 apply (rule_tac x = t in exI, simp)
   569 apply (rule sumr_fun_eq)
   570 apply (auto simp add: odd_Suc_mult_two_ex)
   571 apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
   572 apply (simp add: mult_commute [of _ pi])
   573 done
   574 
   575 (* ------------------------------------------------------------------------- *)
   576 (* Version for ln(1 +/- x). Where is it??                                    *)
   577 (* ------------------------------------------------------------------------- *)
   578 
   579 lemma sin_bound_lemma:
   580     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
   581 by auto
   582 
   583 lemma Maclaurin_sin_bound:
   584   "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
   585   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
   586 proof -
   587   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
   588     by (rule_tac mult_right_mono,simp_all)
   589   note est = this[simplified]
   590   show ?thesis
   591     apply (cut_tac f=sin and n=n and x=x and
   592       diff = "%n 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)"
   593       in Maclaurin_all_le_objl)
   594     apply safe
   595     apply simp
   596     apply (subst mod_Suc_eq_Suc_mod)
   597     apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
   598     apply (rule DERIV_minus, simp+)
   599     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
   600     apply (erule ssubst)
   601     apply (rule sin_bound_lemma)
   602     apply (rule sumr_fun_eq, safe)
   603     apply (rule_tac f = "%u. u * (x^r)" in arg_cong)
   604     apply (subst even_even_mod_4_iff)
   605     apply (cut_tac m=r in mod_exhaust_less_4, simp, safe)
   606     apply (simp_all add:even_num_iff)
   607     apply (drule lemma_even_mod_4_div_2[simplified])
   608     apply(simp add: numeral_2_eq_2 divide_inverse)
   609     apply (drule lemma_odd_mod_4_div_2)
   610     apply (simp add: numeral_2_eq_2 divide_inverse)
   611     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
   612                    simp add: est mult_pos_le mult_ac divide_inverse
   613                           power_abs [symmetric])
   614     done
   615 qed
   616 
   617 end