src/HOL/MacLaurin.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65273 917ae0ba03a2 child 67091 1393c2340eec permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
1 (*  Title:      HOL/MacLaurin.thy
2     Author:     Jacques D. Fleuriot, 2001 University of Edinburgh
3     Author:     Lawrence C Paulson, 2004
4     Author:     Lukas Bulwahn and Bernhard Häupler, 2005
5 *)
7 section \<open>MacLaurin and Taylor Series\<close>
9 theory MacLaurin
10 imports Transcendental
11 begin
13 subsection \<open>Maclaurin's Theorem with Lagrange Form of Remainder\<close>
15 text \<open>This is a very long, messy proof even now that it's been broken down
16   into lemmas.\<close>
18 lemma Maclaurin_lemma:
19   "0 < h \<Longrightarrow>
20     \<exists>B::real. f h = (\<Sum>m<n. (j m / (fact m)) * (h^m)) + (B * ((h^n) /(fact n)))"
21   by (rule exI[where x = "(f h - (\<Sum>m<n. (j m / (fact m)) * h^m)) * (fact n) / (h^n)"]) simp
23 lemma eq_diff_eq': "x = y - z \<longleftrightarrow> y = x + z"
24   for x y z :: real
25   by arith
27 lemma fact_diff_Suc: "n < Suc m \<Longrightarrow> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
28   by (subst fact_reduce) auto
30 lemma Maclaurin_lemma2:
31   fixes B
32   assumes DERIV: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
33     and INIT: "n = Suc k"
34   defines "difg \<equiv>
35     (\<lambda>m t::real. diff m t -
36       ((\<Sum>p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m))))"
37     (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")
38   shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
39 proof (rule allI impI)+
40   fix m t
41   assume INIT2: "m < n \<and> 0 \<le> t \<and> t \<le> h"
42   have "DERIV (difg m) t :> diff (Suc m) t -
43     ((\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) +
44      real (n - m) * t ^ (n - Suc m) * B / fact (n - m))"
45     by (auto simp: difg_def intro!: derivative_eq_intros DERIV[rule_format, OF INIT2])
46   moreover
47   from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m"
48     unfolding atLeast0LessThan[symmetric] by auto
49   have "(\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) =
50       (\<Sum>x<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x))"
51     unfolding intvl by (subst sum.insert) (auto simp add: sum.reindex)
52   moreover
53   have fact_neq_0: "\<And>x. (fact x) + real x * (fact x) \<noteq> 0"
55         less_numeral_extra(3) mult_less_0_iff of_nat_less_0_iff)
56   have "\<And>x. (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x) = diff (Suc m + x) 0 * t^x / fact x"
57     by (rule nonzero_divide_eq_eq[THEN iffD2]) auto
58   moreover
59   have "(n - m) * t ^ (n - Suc m) * B / fact (n - m) = B * (t ^ (n - Suc m) / fact (n - Suc m))"
60     using \<open>0 < n - m\<close> by (simp add: divide_simps fact_reduce)
61   ultimately show "DERIV (difg m) t :> difg (Suc m) t"
62     unfolding difg_def  by (simp add: mult.commute)
63 qed
65 lemma Maclaurin:
66   assumes h: "0 < h"
67     and n: "0 < n"
68     and diff_0: "diff 0 = f"
69     and diff_Suc: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
70   shows
71     "\<exists>t::real. 0 < t \<and> t < h \<and>
72       f h = sum (\<lambda>m. (diff m 0 / fact m) * h ^ m) {..<n} + (diff n t / fact n) * h ^ n"
73 proof -
74   from n obtain m where m: "n = Suc m"
75     by (cases n) (simp add: n)
76   from m have "m < n" by simp
78   obtain B where f_h: "f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + B * (h ^ n / fact n)"
79     using Maclaurin_lemma [OF h] ..
81   define g where [abs_def]: "g t =
82     f t - (sum (\<lambda>m. (diff m 0 / fact m) * t^m) {..<n} + B * (t^n / fact n))" for t
83   have g2: "g 0 = 0" "g h = 0"
84     by (simp_all add: m f_h g_def lessThan_Suc_eq_insert_0 image_iff diff_0 sum.reindex)
86   define difg where [abs_def]: "difg m t =
87     diff m t - (sum (\<lambda>p. (diff (m + p) 0 / fact p) * (t ^ p)) {..<n-m} +
88       B * ((t ^ (n - m)) / fact (n - m)))" for m t
89   have difg_0: "difg 0 = g"
90     by (simp add: difg_def g_def diff_0)
91   have difg_Suc: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
92     using diff_Suc m unfolding difg_def [abs_def] by (rule Maclaurin_lemma2)
93   have difg_eq_0: "\<forall>m<n. difg m 0 = 0"
94     by (auto simp: difg_def m Suc_diff_le lessThan_Suc_eq_insert_0 image_iff sum.reindex)
95   have isCont_difg: "\<And>m x. m < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> h \<Longrightarrow> isCont (difg m) x"
96     by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
97   have differentiable_difg: "\<And>m x. m < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> h \<Longrightarrow> difg m differentiable (at x)"
98     by (rule differentiableI [OF difg_Suc [rule_format]]) simp
99   have difg_Suc_eq_0:
100     "\<And>m t. m < n \<Longrightarrow> 0 \<le> t \<Longrightarrow> t \<le> h \<Longrightarrow> DERIV (difg m) t :> 0 \<Longrightarrow> difg (Suc m) t = 0"
101     by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
103   have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
104   using \<open>m < n\<close>
105   proof (induct m)
106     case 0
107     show ?case
108     proof (rule Rolle)
109       show "0 < h" by fact
110       show "difg 0 0 = difg 0 h"
111         by (simp add: difg_0 g2)
112       show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0::nat)) x"
113         by (simp add: isCont_difg n)
114       show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0::nat) differentiable (at x)"
115         by (simp add: differentiable_difg n)
116     qed
117   next
118     case (Suc m')
119     then have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0"
120       by simp
121     then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0"
122       by fast
123     have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"
124     proof (rule Rolle)
125       show "0 < t" by fact
126       show "difg (Suc m') 0 = difg (Suc m') t"
127         using t \<open>Suc m' < n\<close> by (simp add: difg_Suc_eq_0 difg_eq_0)
128       show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
129         using \<open>t < h\<close> \<open>Suc m' < n\<close> by (simp add: isCont_difg)
130       show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable (at x)"
131         using \<open>t < h\<close> \<open>Suc m' < n\<close> by (simp add: differentiable_difg)
132     qed
133     with \<open>t < h\<close> show ?case
134       by auto
135   qed
136   then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0"
137     by fast
138   with \<open>m < n\<close> have "difg (Suc m) t = 0"
139     by (simp add: difg_Suc_eq_0)
140   show ?thesis
141   proof (intro exI conjI)
142     show "0 < t" by fact
143     show "t < h" by fact
144     show "f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n"
145       using \<open>difg (Suc m) t = 0\<close> by (simp add: m f_h difg_def)
146   qed
147 qed
149 lemma Maclaurin_objl:
150   "0 < h \<and> n > 0 \<and> diff 0 = f \<and>
151     (\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
152     (\<exists>t. 0 < t \<and> t < h \<and> f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / fact n * h ^ n)"
153   for n :: nat and h :: real
154   by (blast intro: Maclaurin)
156 lemma Maclaurin2:
157   fixes n :: nat
158     and h :: real
159   assumes INIT1: "0 < h"
160     and INIT2: "diff 0 = f"
161     and DERIV: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
162   shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / fact n * h ^ n"
163 proof (cases n)
164   case 0
165   with INIT1 INIT2 show ?thesis by fastforce
166 next
167   case Suc
168   then have "n > 0" by simp
169   from INIT1 this INIT2 DERIV
170   have "\<exists>t>0. t < h \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
171     by (rule Maclaurin)
172   then show ?thesis by fastforce
173 qed
175 lemma Maclaurin2_objl:
176   "0 < h \<and> diff 0 = f \<and>
177     (\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
178     (\<exists>t. 0 < t \<and> t \<le> h \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n)"
179   for n :: nat and h :: real
180   by (blast intro: Maclaurin2)
182 lemma Maclaurin_minus:
183   fixes n :: nat and h :: real
184   assumes "h < 0" "0 < n" "diff 0 = f"
185     and DERIV: "\<forall>m t. m < n \<and> h \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
186   shows "\<exists>t. h < t \<and> t < 0 \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
187 proof -
188   txt \<open>Transform \<open>ABL'\<close> into \<open>derivative_intros\<close> format.\<close>
189   note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
190   let ?sum = "\<lambda>t.
191     (\<Sum>m<n. (- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) +
192     (- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n"
193   from assms have "\<exists>t>0. t < - h \<and> f (- (- h)) = ?sum t"
194     by (intro Maclaurin) (auto intro!: derivative_eq_intros DERIV')
195   then obtain t where "0 < t" "t < - h" "f (- (- h)) = ?sum t"
196     by blast
197   moreover have "(- 1) ^ n * diff n (- t) * (- h) ^ n / fact n = diff n (- t) * h ^ n / fact n"
198     by (auto simp: power_mult_distrib[symmetric])
199   moreover
200     have "(\<Sum>m<n. (- 1) ^ m * diff m 0 * (- h) ^ m / fact m) = (\<Sum>m<n. diff m 0 * h ^ m / fact m)"
201     by (auto intro: sum.cong simp add: power_mult_distrib[symmetric])
202   ultimately have "h < - t \<and> - t < 0 \<and>
203     f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n (- t) / (fact n) * h ^ n"
204     by auto
205   then show ?thesis ..
206 qed
208 lemma Maclaurin_minus_objl:
209   fixes n :: nat and h :: real
210   shows
211     "h < 0 \<and> n > 0 \<and> diff 0 = f \<and>
212       (\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
213     (\<exists>t. h < t \<and> t < 0 \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n)"
214   by (blast intro: Maclaurin_minus)
217 subsection \<open>More Convenient "Bidirectional" Version.\<close>
219 (* not good for PVS sin_approx, cos_approx *)
221 lemma Maclaurin_bi_le_lemma:
222   "n > 0 \<Longrightarrow>
223     diff 0 0 = (\<Sum>m<n. diff m 0 * 0 ^ m / (fact m)) + diff n 0 * 0 ^ n / (fact n :: real)"
224   by (induct n) auto
226 lemma Maclaurin_bi_le:
227   fixes n :: nat and x :: real
228   assumes "diff 0 = f"
229     and DERIV : "\<forall>m t. m < n \<and> \<bar>t\<bar> \<le> \<bar>x\<bar> \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
230   shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / (fact m) * x ^ m) + diff n t / (fact n) * x ^ n"
231     (is "\<exists>t. _ \<and> f x = ?f x t")
232 proof (cases "n = 0")
233   case True
234   with \<open>diff 0 = f\<close> show ?thesis by force
235 next
236   case False
237   show ?thesis
238   proof (cases rule: linorder_cases)
239     assume "x = 0"
240     with \<open>n \<noteq> 0\<close> \<open>diff 0 = f\<close> DERIV have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0"
241       by (auto simp add: Maclaurin_bi_le_lemma)
242     then show ?thesis ..
243   next
244     assume "x < 0"
245     with \<open>n \<noteq> 0\<close> DERIV have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t"
246       by (intro Maclaurin_minus) auto
247     then obtain t where "x < t" "t < 0"
248       "diff 0 x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
249       by blast
250     with \<open>x < 0\<close> \<open>diff 0 = f\<close> have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t"
251       by simp
252     then show ?thesis ..
253   next
254     assume "x > 0"
255     with \<open>n \<noteq> 0\<close> \<open>diff 0 = f\<close> DERIV have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t"
256       by (intro Maclaurin) auto
257     then obtain t where "0 < t" "t < x"
258       "diff 0 x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
259       by blast
260     with \<open>x > 0\<close> \<open>diff 0 = f\<close> have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
261     then show ?thesis ..
262   qed
263 qed
265 lemma Maclaurin_all_lt:
266   fixes x :: real
267   assumes INIT1: "diff 0 = f"
268     and INIT2: "0 < n"
269     and INIT3: "x \<noteq> 0"
270     and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x"
271   shows "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x =
272       (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
273     (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t")
274 proof (cases rule: linorder_cases)
275   assume "x = 0"
276   with INIT3 show ?thesis ..
277 next
278   assume "x < 0"
279   with assms have "\<exists>t>x. t < 0 \<and> f x = ?f x t"
280     by (intro Maclaurin_minus) auto
281   then obtain t where "t > x" "t < 0" "f x = ?f x t"
282     by blast
283   with \<open>x < 0\<close> have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
284     by simp
285   then show ?thesis ..
286 next
287   assume "x > 0"
288   with assms have "\<exists>t>0. t < x \<and> f x = ?f x t"
289     by (intro Maclaurin) auto
290   then obtain t where "t > 0" "t < x" "f x = ?f x t"
291     by blast
292   with \<open>x > 0\<close> have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
293     by simp
294   then show ?thesis ..
295 qed
298 lemma Maclaurin_all_lt_objl:
299   fixes x :: real
300   shows
301     "diff 0 = f \<and> (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) \<and> x \<noteq> 0 \<and> n > 0 \<longrightarrow>
302     (\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and>
303       f x = (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n)"
304   by (blast intro: Maclaurin_all_lt)
306 lemma Maclaurin_zero: "x = 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) = diff 0 0"
307   for x :: real and n :: nat
308   by (induct n) auto
311 lemma Maclaurin_all_le:
312   fixes x :: real and n :: nat
313   assumes INIT: "diff 0 = f"
314     and DERIV: "\<forall>m x. DERIV (diff m) x :> diff (Suc m) x"
315   shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
316     (is "\<exists>t. _ \<and> f x = ?f x t")
317 proof (cases "n = 0")
318   case True
319   with INIT show ?thesis by force
320 next
321   case False
322   show ?thesis
323   proof (cases "x = 0")
324     case True
325     with \<open>n \<noteq> 0\<close> have "(\<Sum>m<n. diff m 0 / (fact m) * x ^ m) = diff 0 0"
326       by (intro Maclaurin_zero) auto
327     with INIT \<open>x = 0\<close> \<open>n \<noteq> 0\<close> have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0"
328       by force
329     then show ?thesis ..
330   next
331     case False
332     with INIT \<open>n \<noteq> 0\<close> DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
333       by (intro Maclaurin_all_lt) auto
334     then obtain t where "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" ..
335     then have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t"
336       by simp
337     then show ?thesis ..
338   qed
339 qed
341 lemma Maclaurin_all_le_objl:
342   "diff 0 = f \<and> (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) \<longrightarrow>
343     (\<exists>t::real. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n)"
344   for x :: real and n :: nat
345   by (blast intro: Maclaurin_all_le)
348 subsection \<open>Version for Exponential Function\<close>
350 lemma Maclaurin_exp_lt:
351   fixes x :: real and n :: nat
352   shows
353     "x \<noteq> 0 \<Longrightarrow> n > 0 \<Longrightarrow>
354       (\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> exp x = (\<Sum>m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n)"
355  using Maclaurin_all_lt_objl [where diff = "\<lambda>n. exp" and f = exp and x = x and n = n] by auto
357 lemma Maclaurin_exp_le:
358   fixes x :: real and n :: nat
359   shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> exp x = (\<Sum>m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n"
360   using Maclaurin_all_le_objl [where diff = "\<lambda>n. exp" and f = exp and x = x and n = n] by auto
362 corollary exp_lower_taylor_quadratic: "0 \<le> x \<Longrightarrow> 1 + x + x\<^sup>2 / 2 \<le> exp x"
363   for x :: real
364   using Maclaurin_exp_le [of x 3] by (auto simp: numeral_3_eq_3 power2_eq_square)
366 corollary ln_2_less_1: "ln 2 < (1::real)"
367 proof -
368   have "2 < 5/(2::real)" by simp
369   also have "5/2 \<le> exp (1::real)" using exp_lower_taylor_quadratic[of 1, simplified] by simp
370   finally have "exp (ln 2) < exp (1::real)" by simp
371   thus "ln 2 < (1::real)" by (subst (asm) exp_less_cancel_iff) simp
372 qed
374 subsection \<open>Version for Sine Function\<close>
376 lemma mod_exhaust_less_4: "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = 3"
377   for m :: nat
378   by auto
380 lemma Suc_Suc_mult_two_diff_two [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (Suc (2 * n - 2)) = 2 * n"
381   by (induct n) auto
383 lemma lemma_Suc_Suc_4n_diff_2 [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (Suc (4 * n - 2)) = 4 * n"
384   by (induct n) auto
386 lemma Suc_mult_two_diff_one [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (2 * n - 1) = 2 * n"
387   by (induct n) auto
390 text \<open>It is unclear why so many variant results are needed.\<close>
392 lemma sin_expansion_lemma: "sin (x + real (Suc m) * pi / 2) = cos (x + real m * pi / 2)"
395 lemma Maclaurin_sin_expansion2:
396   "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
397     sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
398   using Maclaurin_all_lt_objl
399     [where f = sin and n = n and x = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
400   apply safe
401       apply simp
402      apply (simp add: sin_expansion_lemma del: of_nat_Suc)
403      apply (force intro!: derivative_eq_intros)
404     apply (subst (asm) sum.neutral; auto)
405    apply (rule ccontr)
406    apply simp
407    apply (drule_tac x = x in spec)
408    apply simp
409   apply (erule ssubst)
410   apply (rule_tac x = t in exI)
411   apply simp
412   apply (rule sum.cong[OF refl])
413   apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
414   done
416 lemma Maclaurin_sin_expansion:
417   "\<exists>t. sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
418   using Maclaurin_sin_expansion2 [of x n] by blast
420 lemma Maclaurin_sin_expansion3:
421   "n > 0 \<Longrightarrow> 0 < x \<Longrightarrow>
422     \<exists>t. 0 < t \<and> t < x \<and>
423        sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
424   using Maclaurin_objl
425     [where f = sin and n = n and h = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
426   apply safe
427     apply simp
428    apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
429    apply (force intro!: derivative_eq_intros)
430   apply (erule ssubst)
431   apply (rule_tac x = t in exI)
432   apply simp
433   apply (rule sum.cong[OF refl])
434   apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
435   done
437 lemma Maclaurin_sin_expansion4:
438   "0 < x \<Longrightarrow>
439     \<exists>t. 0 < t \<and> t \<le> x \<and>
440       sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
441   using Maclaurin2_objl
442     [where f = sin and n = n and h = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
443   apply safe
444     apply simp
445    apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
446    apply (force intro!: derivative_eq_intros)
447   apply (erule ssubst)
448   apply (rule_tac x = t in exI)
449   apply simp
450   apply (rule sum.cong[OF refl])
451   apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
452   done
455 subsection \<open>Maclaurin Expansion for Cosine Function\<close>
457 lemma sumr_cos_zero_one [simp]: "(\<Sum>m<Suc n. cos_coeff m * 0 ^ m) = 1"
458   by (induct n) auto
460 lemma cos_expansion_lemma: "cos (x + real (Suc m) * pi / 2) = - sin (x + real m * pi / 2)"
463 lemma Maclaurin_cos_expansion:
464   "\<exists>t::real. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
465     cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + (cos(t + 1/2 * real n * pi) / fact n) * x ^ n"
466   using Maclaurin_all_lt_objl
467     [where f = cos and n = n and x = x and diff = "\<lambda>n x. cos (x + 1/2 * real n * pi)"]
468   apply safe
469       apply simp
470      apply (simp add: cos_expansion_lemma del: of_nat_Suc)
471     apply (cases n)
472      apply simp
473     apply (simp del: sum_lessThan_Suc)
474    apply (rule ccontr)
475    apply simp
476    apply (drule_tac x = x in spec)
477    apply simp
478   apply (erule ssubst)
479   apply (rule_tac x = t in exI)
480   apply simp
481   apply (rule sum.cong[OF refl])
482   apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE)
483   done
485 lemma Maclaurin_cos_expansion2:
486   "0 < x \<Longrightarrow> n > 0 \<Longrightarrow>
487     \<exists>t. 0 < t \<and> t < x \<and>
488       cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + (cos (t + 1/2 * real n * pi) / fact n) * x ^ n"
489   using Maclaurin_objl
490     [where f = cos and n = n and h = x and diff = "\<lambda>n x. cos (x + 1/2 * real n * pi)"]
491   apply safe
492     apply simp
493    apply (simp add: cos_expansion_lemma del: of_nat_Suc)
494   apply (erule ssubst)
495   apply (rule_tac x = t in exI)
496   apply simp
497   apply (rule sum.cong[OF refl])
498   apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
499   done
501 lemma Maclaurin_minus_cos_expansion:
502   "x < 0 \<Longrightarrow> n > 0 \<Longrightarrow>
503     \<exists>t. x < t \<and> t < 0 \<and>
504       cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + ((cos (t + 1/2 * real n * pi) / fact n) * x ^ n)"
505   using Maclaurin_minus_objl
506     [where f = cos and n = n and h = x and diff = "\<lambda>n x. cos (x + 1/2 * real n *pi)"]
507   apply safe
508     apply simp
509    apply (simp add: cos_expansion_lemma del: of_nat_Suc)
510   apply (erule ssubst)
511   apply (rule_tac x = t in exI)
512   apply simp
513   apply (rule sum.cong[OF refl])
514   apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
515   done
518 (* Version for ln(1 +/- x). Where is it?? *)
521 lemma sin_bound_lemma: "x = y \<Longrightarrow> \<bar>u\<bar> \<le> v \<Longrightarrow> \<bar>(x + u) - y\<bar> \<le> v"
522   for x y u v :: real
523   by auto
525 lemma Maclaurin_sin_bound: "\<bar>sin x - (\<Sum>m<n. sin_coeff m * x ^ m)\<bar> \<le> inverse (fact n) * \<bar>x\<bar> ^ n"
526 proof -
527   have est: "x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y" for x y :: real
528     by (rule mult_right_mono) simp_all
529   let ?diff = "\<lambda>(n::nat) x.
530     if n mod 4 = 0 then sin x
531     else if n mod 4 = 1 then cos x
532     else if n mod 4 = 2 then - sin x
533     else - cos x"
534   have diff_0: "?diff 0 = sin" by simp
535   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
536     apply clarify
537     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
538     apply (cut_tac m=m in mod_exhaust_less_4)
539     apply safe
540        apply (auto intro!: derivative_eq_intros)
541     done
542   from Maclaurin_all_le [OF diff_0 DERIV_diff]
543   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>"
544     and t2: "sin x = (\<Sum>m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / (fact n) * x ^ n"
545     by fast
546   have diff_m_0: "\<And>m. ?diff m 0 = (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2))"
547     apply (subst even_even_mod_4_iff)
548     apply (cut_tac m=m in mod_exhaust_less_4)
549     apply (elim disjE)
550        apply simp_all
551      apply (safe dest!: mod_eqD)
552      apply simp_all
553     done
554   show ?thesis
555     unfolding sin_coeff_def
556     apply (subst t2)
557     apply (rule sin_bound_lemma)
558      apply (rule sum.cong[OF refl])
559      apply (subst diff_m_0, simp)
560     using est
561     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
562         simp: ac_simps divide_inverse power_abs [symmetric] abs_mult)
563     done
564 qed
567 section \<open>Taylor series\<close>
569 text \<open>
570   We use MacLaurin and the translation of the expansion point \<open>c\<close> to \<open>0\<close>
571   to prove Taylor's theorem.
572 \<close>
574 lemma taylor_up:
575   assumes INIT: "n > 0" "diff 0 = f"
576     and DERIV: "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t)"
577     and INTERV: "a \<le> c" "c < b"
578   shows "\<exists>t::real. c < t \<and> t < b \<and>
579     f b = (\<Sum>m<n. (diff m c / fact m) * (b - c)^m) + (diff n t / fact n) * (b - c)^n"
580 proof -
581   from INTERV have "0 < b - c" by arith
582   moreover from INIT have "n > 0" "(\<lambda>m x. diff m (x + c)) 0 = (\<lambda>x. f (x + c))"
583     by auto
584   moreover
585   have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> b - c \<longrightarrow> DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
586   proof (intro strip)
587     fix m t
588     assume "m < n \<and> 0 \<le> t \<and> t \<le> b - c"
589     with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)"
590       by auto
591     moreover from DERIV_ident and DERIV_const have "DERIV (\<lambda>x. x + c) t :> 1 + 0"
592       by (rule DERIV_add)
593     ultimately have "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
594       by (rule DERIV_chain2)
595     then show "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
596       by simp
597   qed
598   ultimately obtain x where
599     "0 < x \<and> x < b - c \<and>
600       f (b - c + c) =
601         (\<Sum>m<n. diff m (0 + c) / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n"
602      by (rule Maclaurin [THEN exE])
603    then have "c < x + c \<and> x + c < b \<and> f b =
604      (\<Sum>m<n. diff m c / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n"
605     by fastforce
606   then show ?thesis by fastforce
607 qed
609 lemma taylor_down:
610   fixes a :: real and n :: nat
611   assumes INIT: "n > 0" "diff 0 = f"
612     and DERIV: "(\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t)"
613     and INTERV: "a < c" "c \<le> b"
614   shows "\<exists>t. a < t \<and> t < c \<and>
615     f a = (\<Sum>m<n. (diff m c / fact m) * (a - c)^m) + (diff n t / fact n) * (a - c)^n"
616 proof -
617   from INTERV have "a-c < 0" by arith
618   moreover from INIT have "n > 0" "(\<lambda>m x. diff m (x + c)) 0 = (\<lambda>x. f (x + c))"
619     by auto
620   moreover
621   have "\<forall>m t. m < n \<and> a - c \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
622   proof (rule allI impI)+
623     fix m t
624     assume "m < n \<and> a - c \<le> t \<and> t \<le> 0"
625     with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)"
626       by auto
627     moreover from DERIV_ident and DERIV_const have "DERIV (\<lambda>x. x + c) t :> 1 + 0"
628       by (rule DERIV_add)
629     ultimately have "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
630       by (rule DERIV_chain2)
631     then show "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
632       by simp
633   qed
634   ultimately obtain x where
635     "a - c < x \<and> x < 0 \<and>
636       f (a - c + c) =
637         (\<Sum>m<n. diff m (0 + c) / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n"
638     by (rule Maclaurin_minus [THEN exE])
639   then have "a < x + c \<and> x + c < c \<and>
640     f a = (\<Sum>m<n. diff m c / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n"
641     by fastforce
642   then show ?thesis by fastforce
643 qed
645 theorem taylor:
646   fixes a :: real and n :: nat
647   assumes INIT: "n > 0" "diff 0 = f"
648     and DERIV: "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
649     and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c"
650   shows "\<exists>t.
651     (if x < c then x < t \<and> t < c else c < t \<and> t < x) \<and>
652     f x = (\<Sum>m<n. (diff m c / fact m) * (x - c)^m) + (diff n t / fact n) * (x - c)^n"
653 proof (cases "x < c")
654   case True
655   note INIT
656   moreover have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
657     using DERIV and INTERV by fastforce
658   moreover note True
659   moreover from INTERV have "c \<le> b"
660     by simp
661   ultimately have "\<exists>t>x. t < c \<and> f x =
662     (\<Sum>m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n"
663     by (rule taylor_down)
664   with True show ?thesis by simp
665 next
666   case False
667   note INIT
668   moreover have "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
669     using DERIV and INTERV by fastforce
670   moreover from INTERV have "a \<le> c"
671     by arith
672   moreover from False and INTERV have "c < x"
673     by arith
674   ultimately have "\<exists>t>c. t < x \<and> f x =
675     (\<Sum>m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n"
676     by (rule taylor_up)
677   with False show ?thesis by simp
678 qed
680 end