src/HOL/MacLaurin.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30273 ecd6f0ca62ea child 31148 7ba7c1f8bc22 permissions -rw-r--r--
simplified method setup;
```     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 text{*A crude tactic to differentiate by proof.*}
```
```    31
```
```    32 lemmas deriv_rulesI =
```
```    33   DERIV_ident DERIV_const DERIV_cos DERIV_cmult
```
```    34   DERIV_sin DERIV_exp DERIV_inverse DERIV_pow
```
```    35   DERIV_add DERIV_diff DERIV_mult DERIV_minus
```
```    36   DERIV_inverse_fun DERIV_quotient DERIV_fun_pow
```
```    37   DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos
```
```    38   DERIV_ident DERIV_const DERIV_cos
```
```    39
```
```    40 ML
```
```    41 {*
```
```    42 local
```
```    43 exception DERIV_name;
```
```    44 fun get_fun_name (_ \$ (Const ("Lim.deriv",_) \$ Abs(_,_, Const (f,_) \$ _) \$ _ \$ _)) = f
```
```    45 |   get_fun_name (_ \$ (_ \$ (Const ("Lim.deriv",_) \$ Abs(_,_, Const (f,_) \$ _) \$ _ \$ _))) = f
```
```    46 |   get_fun_name _ = raise DERIV_name;
```
```    47
```
```    48 in
```
```    49
```
```    50 fun deriv_tac ctxt = SUBGOAL (fn (prem, i) =>
```
```    51   resolve_tac @{thms deriv_rulesI} i ORELSE
```
```    52     ((rtac (read_instantiate ctxt [(("f", 0), get_fun_name prem)]
```
```    53                      @{thm DERIV_chain2}) i) handle DERIV_name => no_tac));
```
```    54
```
```    55 fun DERIV_tac ctxt = ALLGOALS (fn i => REPEAT (deriv_tac ctxt i));
```
```    56
```
```    57 end
```
```    58 *}
```
```    59
```
```    60 lemma Maclaurin_lemma2:
```
```    61   assumes diff: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
```
```    62   assumes n: "n = Suc k"
```
```    63   assumes difg: "difg =
```
```    64         (\<lambda>m t. diff m t -
```
```    65                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
```
```    66                 B * (t ^ (n - m) / real (fact (n - m)))))"
```
```    67   shows
```
```    68       "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
```
```    69 unfolding difg
```
```    70  apply clarify
```
```    71  apply (rule DERIV_diff)
```
```    72   apply (simp add: diff)
```
```    73  apply (simp only: n)
```
```    74  apply (rule DERIV_add)
```
```    75   apply (rule_tac [2] DERIV_cmult)
```
```    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 less_iff_Suc_add [THEN iffD1], clarify)
```
```    83  apply (simp del: setsum_op_ivl_Suc)
```
```    84  apply (insert sumr_offset4 [of "Suc 0"])
```
```    85  apply (simp del: setsum_op_ivl_Suc fact_Suc power_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 power_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:
```
```   102   assumes h: "0 < h"
```
```   103   assumes n: "0 < n"
```
```   104   assumes diff_0: "diff 0 = f"
```
```   105   assumes diff_Suc:
```
```   106     "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
```
```   107   shows
```
```   108     "\<exists>t. 0 < t & t < h &
```
```   109               f h =
```
```   110               setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
```
```   111               (diff n t / real (fact n)) * h ^ n"
```
```   112 proof -
```
```   113   from n obtain m where m: "n = Suc m"
```
```   114     by (cases n, simp add: n)
```
```   115
```
```   116   obtain B where f_h: "f h =
```
```   117         (\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
```
```   118         B * (h ^ n / real (fact n))"
```
```   119     using Maclaurin_lemma [OF h] ..
```
```   120
```
```   121   obtain g where g_def: "g = (%t. f t -
```
```   122     (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n}
```
```   123       + (B * (t^n / real(fact n)))))" by blast
```
```   124
```
```   125   have g2: "g 0 = 0 & g h = 0"
```
```   126     apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
```
```   127     apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
```
```   128     apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
```
```   129     done
```
```   130
```
```   131   obtain difg where difg_def: "difg = (%m t. diff m t -
```
```   132     (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
```
```   133       + (B * ((t ^ (n - m)) / real (fact (n - m))))))" by blast
```
```   134
```
```   135   have difg_0: "difg 0 = g"
```
```   136     unfolding difg_def g_def by (simp add: diff_0)
```
```   137
```
```   138   have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
```
```   139         m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
```
```   140     using diff_Suc m difg_def by (rule Maclaurin_lemma2)
```
```   141
```
```   142   have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
```
```   143     apply clarify
```
```   144     apply (simp add: m difg_def)
```
```   145     apply (frule less_iff_Suc_add [THEN iffD1], clarify)
```
```   146     apply (simp del: setsum_op_ivl_Suc)
```
```   147     apply (insert sumr_offset4 [of "Suc 0"])
```
```   148     apply (simp del: setsum_op_ivl_Suc fact_Suc)
```
```   149     done
```
```   150
```
```   151   have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
```
```   152     by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
```
```   153
```
```   154   have differentiable_difg:
```
```   155     "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"
```
```   156     by (rule differentiableI [OF difg_Suc [rule_format]]) simp
```
```   157
```
```   158   have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
```
```   159         \<Longrightarrow> difg (Suc m) t = 0"
```
```   160     by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
```
```   161
```
```   162   have "m < n" using m by simp
```
```   163
```
```   164   have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
```
```   165   using `m < n`
```
```   166   proof (induct m)
```
```   167   case 0
```
```   168     show ?case
```
```   169     proof (rule Rolle)
```
```   170       show "0 < h" by fact
```
```   171       show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)
```
```   172       show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
```
```   173         by (simp add: isCont_difg n)
```
```   174       show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"
```
```   175         by (simp add: differentiable_difg n)
```
```   176     qed
```
```   177   next
```
```   178   case (Suc m')
```
```   179     hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp
```
```   180     then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast
```
```   181     have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
```
```   182     proof (rule Rolle)
```
```   183       show "0 < t" by fact
```
```   184       show "difg (Suc m') 0 = difg (Suc m') t"
```
```   185         using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)
```
```   186       show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
```
```   187         using `t < h` `Suc m' < n` by (simp add: isCont_difg)
```
```   188       show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
```
```   189         using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
```
```   190     qed
```
```   191     thus ?case
```
```   192       using `t < h` by auto
```
```   193   qed
```
```   194
```
```   195   then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
```
```   196
```
```   197   hence "difg (Suc m) t = 0"
```
```   198     using `m < n` by (simp add: difg_Suc_eq_0)
```
```   199
```
```   200   show ?thesis
```
```   201   proof (intro exI conjI)
```
```   202     show "0 < t" by fact
```
```   203     show "t < h" by fact
```
```   204     show "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       using `difg (Suc m) t = 0`
```
```   208       by (simp add: m f_h difg_def del: fact_Suc)
```
```   209   qed
```
```   210
```
```   211 qed
```
```   212
```
```   213 lemma Maclaurin_objl:
```
```   214   "0 < h & n>0 & diff 0 = f &
```
```   215   (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
```
```   216    --> (\<exists>t. 0 < t & t < h &
```
```   217             f h = (\<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: 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               (\<Sum>m=0..<n. 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               (\<Sum>m=0..<n. 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; n > 0; 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               (\<Sum>m=0..<n. 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_ident)
```
```   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] setsum_cong[OF refl])
```
```   270 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
```
```   271 done
```
```   272
```
```   273 lemma Maclaurin_minus_objl:
```
```   274      "(h < 0 & n > 0 & 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               (\<Sum>m=0..<n. 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   "n>0 \<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 "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               (\<Sum>m=0..<n. 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; n > 0
```
```   324       |] ==> \<exists>t. 0 < abs t & abs t < abs x &
```
```   325                f x = (\<Sum>m=0..<n. (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)
```
```   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 & n > 0
```
```   338       --> (\<exists>t. 0 < abs t & abs t < abs x &
```
```   339                f x = (\<Sum>m=0..<n. (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       ==> n \<noteq> 0 -->
```
```   346           (\<Sum>m=0..<n. (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 = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
```
```   354                     (diff n t / real (fact n)) * x ^ n"
```
```   355 apply(cases "n=0")
```
```   356 apply (force)
```
```   357 apply (case_tac "x = 0")
```
```   358 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
```
```   359 apply (drule 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 = (\<Sum>m=0..<n. (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; n > 0 |]
```
```   376       ==> (\<exists>t. 0 < abs t &
```
```   377                 abs t < abs x &
```
```   378                 exp x = (\<Sum>m=0..<n. (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 = (\<Sum>m=0..<n. (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 mod_exhaust_less_4:
```
```   393   "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
```
```   394 by auto
```
```   395
```
```   396 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
```
```   397   "n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
```
```   398 by (induct "n", auto)
```
```   399
```
```   400 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
```
```   401   "n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
```
```   402 by (induct "n", auto)
```
```   403
```
```   404 lemma Suc_mult_two_diff_one [rule_format, simp]:
```
```   405   "n\<noteq>0 --> Suc (2 * n - 1) = 2*n"
```
```   406 by (induct "n", auto)
```
```   407
```
```   408
```
```   409 text{*It is unclear why so many variant results are needed.*}
```
```   410
```
```   411 lemma Maclaurin_sin_expansion2:
```
```   412      "\<exists>t. abs t \<le> abs x &
```
```   413        sin x =
```
```   414        (\<Sum>m=0..<n. (if even m then 0
```
```   415                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
```
```   416                        x ^ m)
```
```   417       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
```
```   418 apply (cut_tac f = sin and n = n and x = x
```
```   419         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
```
```   420 apply safe
```
```   421 apply (simp (no_asm))
```
```   422 apply (simp (no_asm))
```
```   423 apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
```
```   424 apply (rule ccontr, simp)
```
```   425 apply (drule_tac x = x in spec, simp)
```
```   426 apply (erule ssubst)
```
```   427 apply (rule_tac x = t in exI, simp)
```
```   428 apply (rule setsum_cong[OF refl])
```
```   429 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
```
```   430 done
```
```   431
```
```   432 lemma Maclaurin_sin_expansion:
```
```   433      "\<exists>t. 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 (insert Maclaurin_sin_expansion2 [of x n])
```
```   439 apply (blast intro: elim:);
```
```   440 done
```
```   441
```
```   442
```
```   443 lemma Maclaurin_sin_expansion3:
```
```   444      "[| n > 0; 0 < x |] ==>
```
```   445        \<exists>t. 0 < t & t < x &
```
```   446        sin x =
```
```   447        (\<Sum>m=0..<n. (if even m then 0
```
```   448                        else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
```
```   449                        x ^ m)
```
```   450       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
```
```   451 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)
```
```   452 apply safe
```
```   453 apply simp
```
```   454 apply (simp (no_asm))
```
```   455 apply (erule ssubst)
```
```   456 apply (rule_tac x = t in exI, simp)
```
```   457 apply (rule setsum_cong[OF refl])
```
```   458 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
```
```   459 done
```
```   460
```
```   461 lemma Maclaurin_sin_expansion4:
```
```   462      "0 < x ==>
```
```   463        \<exists>t. 0 < t & t \<le> x &
```
```   464        sin x =
```
```   465        (\<Sum>m=0..<n. (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 Maclaurin2_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 setsum_cong[OF refl])
```
```   476 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
```
```   477 done
```
```   478
```
```   479
```
```   480 subsection{*Maclaurin Expansion for Cosine Function*}
```
```   481
```
```   482 lemma sumr_cos_zero_one [simp]:
```
```   483  "(\<Sum>m=0..<(Suc n).
```
```   484      (if even m then -1 ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
```
```   485 by (induct "n", auto)
```
```   486
```
```   487 lemma Maclaurin_cos_expansion:
```
```   488      "\<exists>t. abs t \<le> abs x &
```
```   489        cos x =
```
```   490        (\<Sum>m=0..<n. (if even m
```
```   491                        then -1 ^ (m div 2)/(real (fact m))
```
```   492                        else 0) *
```
```   493                        x ^ m)
```
```   494       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
```
```   495 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)
```
```   496 apply safe
```
```   497 apply (simp (no_asm))
```
```   498 apply (simp (no_asm))
```
```   499 apply (case_tac "n", simp)
```
```   500 apply (simp del: setsum_op_ivl_Suc)
```
```   501 apply (rule ccontr, simp)
```
```   502 apply (drule_tac x = x in spec, simp)
```
```   503 apply (erule ssubst)
```
```   504 apply (rule_tac x = t in exI, simp)
```
```   505 apply (rule setsum_cong[OF refl])
```
```   506 apply (auto simp add: cos_zero_iff even_mult_two_ex)
```
```   507 done
```
```   508
```
```   509 lemma Maclaurin_cos_expansion2:
```
```   510      "[| 0 < x; n > 0 |] ==>
```
```   511        \<exists>t. 0 < t & t < x &
```
```   512        cos x =
```
```   513        (\<Sum>m=0..<n. (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 h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
```
```   519 apply safe
```
```   520 apply simp
```
```   521 apply (simp (no_asm))
```
```   522 apply (erule ssubst)
```
```   523 apply (rule_tac x = t in exI, simp)
```
```   524 apply (rule setsum_cong[OF refl])
```
```   525 apply (auto simp add: cos_zero_iff even_mult_two_ex)
```
```   526 done
```
```   527
```
```   528 lemma Maclaurin_minus_cos_expansion:
```
```   529      "[| x < 0; n > 0 |] ==>
```
```   530        \<exists>t. x < t & t < 0 &
```
```   531        cos x =
```
```   532        (\<Sum>m=0..<n. (if even m
```
```   533                        then -1 ^ (m div 2)/(real (fact m))
```
```   534                        else 0) *
```
```   535                        x ^ m)
```
```   536       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
```
```   537 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)
```
```   538 apply safe
```
```   539 apply simp
```
```   540 apply (simp (no_asm))
```
```   541 apply (erule ssubst)
```
```   542 apply (rule_tac x = t in exI, simp)
```
```   543 apply (rule setsum_cong[OF refl])
```
```   544 apply (auto simp add: cos_zero_iff even_mult_two_ex)
```
```   545 done
```
```   546
```
```   547 (* ------------------------------------------------------------------------- *)
```
```   548 (* Version for ln(1 +/- x). Where is it??                                    *)
```
```   549 (* ------------------------------------------------------------------------- *)
```
```   550
```
```   551 lemma sin_bound_lemma:
```
```   552     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
```
```   553 by auto
```
```   554
```
```   555 text {* TODO: move to Parity.thy *}
```
```   556 lemma nat_odd_1 [simp]: "odd (1::nat)"
```
```   557   unfolding even_nat_def by simp
```
```   558
```
```   559 lemma Maclaurin_sin_bound:
```
```   560   "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
```
```   561   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
```
```   562 proof -
```
```   563   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
```
```   564     by (rule_tac mult_right_mono,simp_all)
```
```   565   note est = this[simplified]
```
```   566   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)"
```
```   567   have diff_0: "?diff 0 = sin" by simp
```
```   568   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
```
```   569     apply (clarify)
```
```   570     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
```
```   571     apply (cut_tac m=m in mod_exhaust_less_4)
```
```   572     apply (safe, simp_all)
```
```   573     apply (rule DERIV_minus, simp)
```
```   574     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
```
```   575     done
```
```   576   from Maclaurin_all_le [OF diff_0 DERIV_diff]
```
```   577   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
```
```   578     t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
```
```   579       ?diff n t / real (fact n) * x ^ n" by fast
```
```   580   have diff_m_0:
```
```   581     "\<And>m. ?diff m 0 = (if even m then 0
```
```   582          else -1 ^ ((m - Suc 0) div 2))"
```
```   583     apply (subst even_even_mod_4_iff)
```
```   584     apply (cut_tac m=m in mod_exhaust_less_4)
```
```   585     apply (elim disjE, simp_all)
```
```   586     apply (safe dest!: mod_eqD, simp_all)
```
```   587     done
```
```   588   show ?thesis
```
```   589     apply (subst t2)
```
```   590     apply (rule sin_bound_lemma)
```
```   591     apply (rule setsum_cong[OF refl])
```
```   592     apply (subst diff_m_0, simp)
```
```   593     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
```
```   594                    simp add: est mult_nonneg_nonneg mult_ac divide_inverse
```
```   595                           power_abs [symmetric] abs_mult)
```
```   596     done
```
```   597 qed
```
```   598
```
```   599 end
```