remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
1 (* Author : Jacques D. Fleuriot
2 Copyright : 2001 University of Edinburgh
3 Conversion to Isar and new proofs by Lawrence C Paulson, 2004
6 header{*MacLaurin Series*}
12 subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
14 text{*This is a very long, messy proof even now that it's been broken down
17 lemma Maclaurin_lemma:
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)) *
27 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
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)
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"
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)))))"
42 "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
45 apply (rule DERIV_diff)
46 apply (simp add: diff)
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)
56 apply (simp add: fact_diff_Suc)
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)
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)
79 assumes diff_0: "diff 0 = f"
81 "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
83 "\<exists>t. 0 < t & t < h &
85 setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
86 (diff n t / real (fact n)) * h ^ n"
88 from n obtain m where m: "n = Suc m"
89 by (cases n, simp add: n)
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] ..
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
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)
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
110 have difg_0: "difg 0 = g"
111 unfolding difg_def g_def by (simp add: diff_0)
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)
117 have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"
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)
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
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
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
137 have "m < n" using m by simp
139 have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
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)
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"
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)
167 using `t < h` by auto
170 then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
172 hence "difg (Suc m) t = 0"
173 using `m < n` by (simp add: difg_Suc_eq_0)
176 proof (intro exI conjI)
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)
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)
198 "[| 0 < h; diff 0 = f;
200 m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
201 ==> \<exists>t. 0 < t &
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)
210 lemma Maclaurin2_objl:
211 "0 < h & diff 0 = f &
213 m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
214 --> (\<exists>t. 0 < t &
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)
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 &
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)
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)
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])
248 lemma Maclaurin_minus_objl:
249 "(h < 0 & n > 0 & diff 0 = f &
251 m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
252 --> (\<exists>t. h < t &
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)
260 subsection{*More Convenient "Bidirectional" Version.*}
262 (* not good for PVS sin_approx, cos_approx *)
264 lemma Maclaurin_bi_le_lemma [rule_format]:
265 "n>0 \<longrightarrow>
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)
271 lemma Maclaurin_bi_le:
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 &
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)
295 lemma Maclaurin_all_lt:
297 \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
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)
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)
309 lemma Maclaurin_all_lt_objl:
311 (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
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)
318 lemma Maclaurin_zero [rule_format]:
321 (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
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"
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)
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)
348 subsection{*Version for Exponential Function*}
350 lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
351 ==> (\<exists>t. 0 < abs t &
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)
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)
365 subsection{*Version for Sine Function*}
367 lemma mod_exhaust_less_4:
368 "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
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)
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)
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)
384 text{*It is unclear why so many variant results are needed.*}
386 lemma sin_expansion_lemma:
387 "sin (x + real (Suc m) * pi / 2) =
388 cos (x + real (m) * pi / 2)"
389 by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
391 lemma Maclaurin_sin_expansion2:
392 "\<exists>t. abs t \<le> abs x &
394 (\<Sum>m=0..<n. (if even m then 0
395 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
397 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
398 apply (cut_tac f = sin and n = n and x = x
399 and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
401 apply (simp (no_asm))
402 apply (simp (no_asm) add: sin_expansion_lemma)
403 apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
404 apply (rule ccontr, simp)
405 apply (drule_tac x = x in spec, simp)
407 apply (rule_tac x = t in exI, simp)
408 apply (rule setsum_cong[OF refl])
409 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
412 lemma Maclaurin_sin_expansion:
414 (\<Sum>m=0..<n. (if even m then 0
415 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
417 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
418 apply (insert Maclaurin_sin_expansion2 [of x n])
419 apply (blast intro: elim:);
422 lemma Maclaurin_sin_expansion3:
423 "[| n > 0; 0 < x |] ==>
424 \<exists>t. 0 < t & t < x &
426 (\<Sum>m=0..<n. (if even m then 0
427 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
429 + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
430 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)
433 apply (simp (no_asm) add: sin_expansion_lemma)
435 apply (rule_tac x = t in exI, simp)
436 apply (rule setsum_cong[OF refl])
437 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
440 lemma Maclaurin_sin_expansion4:
442 \<exists>t. 0 < t & t \<le> x &
444 (\<Sum>m=0..<n. (if even m then 0
445 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
447 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
448 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)
451 apply (simp (no_asm) add: sin_expansion_lemma)
453 apply (rule_tac x = t in exI, simp)
454 apply (rule setsum_cong[OF refl])
455 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
459 subsection{*Maclaurin Expansion for Cosine Function*}
461 lemma sumr_cos_zero_one [simp]:
462 "(\<Sum>m=0..<(Suc n).
463 (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 0 ^ m) = 1"
464 by (induct "n", auto)
466 lemma cos_expansion_lemma:
467 "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
468 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
470 lemma Maclaurin_cos_expansion:
471 "\<exists>t. abs t \<le> abs x &
473 (\<Sum>m=0..<n. (if even m
474 then -1 ^ (m div 2)/(real (fact m))
477 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
478 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)
480 apply (simp (no_asm))
481 apply (simp (no_asm) add: cos_expansion_lemma)
482 apply (case_tac "n", simp)
483 apply (simp del: setsum_op_ivl_Suc)
484 apply (rule ccontr, simp)
485 apply (drule_tac x = x in spec, simp)
487 apply (rule_tac x = t in exI, simp)
488 apply (rule setsum_cong[OF refl])
489 apply (auto simp add: cos_zero_iff even_mult_two_ex)
492 lemma Maclaurin_cos_expansion2:
493 "[| 0 < x; n > 0 |] ==>
494 \<exists>t. 0 < t & t < x &
496 (\<Sum>m=0..<n. (if even m
497 then -1 ^ (m div 2)/(real (fact m))
500 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
501 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)
504 apply (simp (no_asm) add: cos_expansion_lemma)
506 apply (rule_tac x = t in exI, simp)
507 apply (rule setsum_cong[OF refl])
508 apply (auto simp add: cos_zero_iff even_mult_two_ex)
511 lemma Maclaurin_minus_cos_expansion:
512 "[| x < 0; n > 0 |] ==>
513 \<exists>t. x < t & t < 0 &
515 (\<Sum>m=0..<n. (if even m
516 then -1 ^ (m div 2)/(real (fact m))
519 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
520 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)
523 apply (simp (no_asm) add: cos_expansion_lemma)
525 apply (rule_tac x = t in exI, simp)
526 apply (rule setsum_cong[OF refl])
527 apply (auto simp add: cos_zero_iff even_mult_two_ex)
530 (* ------------------------------------------------------------------------- *)
531 (* Version for ln(1 +/- x). Where is it?? *)
532 (* ------------------------------------------------------------------------- *)
534 lemma sin_bound_lemma:
535 "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
538 lemma Maclaurin_sin_bound:
539 "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
540 x ^ m)) \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
542 have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
543 by (rule_tac mult_right_mono,simp_all)
544 note est = this[simplified]
545 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)"
546 have diff_0: "?diff 0 = sin" by simp
547 have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
549 apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
550 apply (cut_tac m=m in mod_exhaust_less_4)
551 apply (safe, auto intro!: DERIV_intros)
553 from Maclaurin_all_le [OF diff_0 DERIV_diff]
554 obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
555 t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
556 ?diff n t / real (fact n) * x ^ n" by fast
558 "\<And>m. ?diff m 0 = (if even m then 0
559 else -1 ^ ((m - Suc 0) div 2))"
560 apply (subst even_even_mod_4_iff)
561 apply (cut_tac m=m in mod_exhaust_less_4)
562 apply (elim disjE, simp_all)
563 apply (safe dest!: mod_eqD, simp_all)
567 apply (rule sin_bound_lemma)
568 apply (rule setsum_cong[OF refl])
569 apply (subst diff_m_0, simp)
570 apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
571 simp add: est mult_nonneg_nonneg mult_ac divide_inverse
572 power_abs [symmetric] abs_mult)