src/HOL/Deriv.thy
 author blanchet Wed Mar 04 10:45:52 2009 +0100 (2009-03-04) changeset 30240 5b25fee0362c parent 29803 c56a5571f60a child 30242 aea5d7fa7ef5 permissions -rw-r--r--
Merge.
1 (*  Title       : Deriv.thy
2     ID          : \$Id\$
3     Author      : Jacques D. Fleuriot
4     Copyright   : 1998  University of Cambridge
5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
6     GMVT by Benjamin Porter, 2005
7 *)
11 theory Deriv
12 imports Lim
13 begin
15 text{*Standard Definitions*}
17 definition
18   deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
19     --{*Differentiation: D is derivative of function f at x*}
20           ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
21   "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"
23 consts
24   Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
25 primrec
26   "Bolzano_bisect P a b 0 = (a,b)"
27   "Bolzano_bisect P a b (Suc n) =
28       (let (x,y) = Bolzano_bisect P a b n
29        in if P(x, (x+y)/2) then ((x+y)/2, y)
30                             else (x, (x+y)/2))"
33 subsection {* Derivatives *}
35 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
38 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
41 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
44 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
45 by (simp add: deriv_def cong: LIM_cong)
48   fixes a b c d :: "'a::ab_group_add"
49   shows "(a + c) - (b + d) = (a - b) + (c - d)"
50 by simp
53   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
56 lemma DERIV_minus:
57   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
58 by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)
60 lemma DERIV_diff:
61   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
62 by (simp only: diff_def DERIV_add DERIV_minus)
65   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
66 by (simp only: DERIV_add DERIV_minus)
68 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
69 proof (unfold isCont_iff)
70   assume "DERIV f x :> D"
71   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
72     by (rule DERIV_D)
73   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
74     by (intro LIM_mult LIM_ident)
75   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
76     by simp
77   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
78     by (simp cong: LIM_cong)
79   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
81 qed
83 lemma DERIV_mult_lemma:
84   fixes a b c d :: "'a::real_field"
85   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
88 lemma DERIV_mult':
89   assumes f: "DERIV f x :> D"
90   assumes g: "DERIV g x :> E"
91   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
92 proof (unfold deriv_def)
93   from f have "isCont f x"
94     by (rule DERIV_isCont)
95   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
96     by (simp only: isCont_iff)
97   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
98               ((f(x+h) - f x) / h) * g x)
99           -- 0 --> f x * E + D * g x"
100     by (intro LIM_add LIM_mult LIM_const DERIV_D f g)
101   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
102          -- 0 --> f x * E + D * g x"
103     by (simp only: DERIV_mult_lemma)
104 qed
106 lemma DERIV_mult:
107      "[| DERIV f x :> Da; DERIV g x :> Db |]
108       ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
109 by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
111 lemma DERIV_unique:
112       "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
114 apply (blast intro: LIM_unique)
115 done
117 text{*Differentiation of finite sum*}
119 lemma DERIV_sumr [rule_format (no_asm)]:
120      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
121       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
122 apply (induct "n")
124 done
126 text{*Alternative definition for differentiability*}
128 lemma DERIV_LIM_iff:
129      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
130       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
131 apply (rule iffI)
132 apply (drule_tac k="- a" in LIM_offset)
134 apply (drule_tac k="a" in LIM_offset)
136 done
138 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
139 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
141 lemma inverse_diff_inverse:
142   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
143    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
146 lemma DERIV_inverse_lemma:
147   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
148    \<Longrightarrow> (inverse a - inverse b) / h
149      = - (inverse a * ((a - b) / h) * inverse b)"
152 lemma DERIV_inverse':
153   assumes der: "DERIV f x :> D"
154   assumes neq: "f x \<noteq> 0"
155   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
156     (is "DERIV _ _ :> ?E")
157 proof (unfold DERIV_iff2)
158   from der have lim_f: "f -- x --> f x"
159     by (rule DERIV_isCont [unfolded isCont_def])
161   from neq have "0 < norm (f x)" by simp
162   with LIM_D [OF lim_f] obtain s
163     where s: "0 < s"
164     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
165                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
166     by fast
168   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
169   proof (rule LIM_equal2 [OF s])
170     fix z
171     assume "z \<noteq> x" "norm (z - x) < s"
172     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
173     hence "f z \<noteq> 0" by auto
174     thus "(inverse (f z) - inverse (f x)) / (z - x) =
175           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
176       using neq by (rule DERIV_inverse_lemma)
177   next
178     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
179       by (unfold DERIV_iff2)
180     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
181           -- x --> ?E"
182       by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
183   qed
184 qed
186 lemma DERIV_divide:
187   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
188    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
189 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
190           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
191 apply (erule subst)
192 apply (unfold divide_inverse)
193 apply (erule DERIV_mult')
194 apply (erule (1) DERIV_inverse')
195 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
197 done
199 lemma DERIV_power_Suc:
200   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
201   assumes f: "DERIV f x :> D"
202   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
203 proof (induct n)
204 case 0
205   show ?case by (simp add: power_Suc f)
206 case (Suc k)
207   from DERIV_mult' [OF f Suc] show ?case
208     apply (simp only: of_nat_Suc ring_distribs mult_1_left)
209     apply (simp only: power_Suc algebra_simps)
210     done
211 qed
213 lemma DERIV_power:
214   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
215   assumes f: "DERIV f x :> D"
216   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
217 by (cases "n", simp, simp add: DERIV_power_Suc f)
220 text {* Caratheodory formulation of derivative at a point *}
222 lemma CARAT_DERIV:
223      "(DERIV f x :> l) =
224       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
225       (is "?lhs = ?rhs")
226 proof
227   assume der: "DERIV f x :> l"
228   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
229   proof (intro exI conjI)
230     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
231     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
232     show "isCont ?g x" using der
233       by (simp add: isCont_iff DERIV_iff diff_minus
234                cong: LIM_equal [rule_format])
235     show "?g x = l" by simp
236   qed
237 next
238   assume "?rhs"
239   then obtain g where
240     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
241   thus "(DERIV f x :> l)"
242      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
243 qed
245 lemma DERIV_chain':
246   assumes f: "DERIV f x :> D"
247   assumes g: "DERIV g (f x) :> E"
248   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
249 proof (unfold DERIV_iff2)
250   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
251     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
252     using CARAT_DERIV [THEN iffD1, OF g] by fast
253   from f have "f -- x --> f x"
254     by (rule DERIV_isCont [unfolded isCont_def])
255   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
256     by (rule isCont_LIM_compose)
257   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
258           -- x --> d (f x) * D"
259     by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
260   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
261     by (simp add: d dfx real_scaleR_def)
262 qed
264 (* let's do the standard proof though theorem *)
265 (* LIM_mult2 follows from a NS proof          *)
267 lemma DERIV_cmult:
268       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
269 by (drule DERIV_mult' [OF DERIV_const], simp)
271 (* standard version *)
272 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
273 by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
275 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
276 by (auto dest: DERIV_chain simp add: o_def)
278 (*derivative of linear multiplication*)
279 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
280 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
282 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
283 apply (cut_tac DERIV_power [OF DERIV_ident])
284 apply (simp add: real_scaleR_def real_of_nat_def)
285 done
287 text{*Power of -1*}
289 lemma DERIV_inverse:
290   fixes x :: "'a::{real_normed_field,recpower}"
291   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
292 by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)
294 text{*Derivative of inverse*}
295 lemma DERIV_inverse_fun:
296   fixes x :: "'a::{real_normed_field,recpower}"
297   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
298       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
299 by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
301 text{*Derivative of quotient*}
302 lemma DERIV_quotient:
303   fixes x :: "'a::{real_normed_field,recpower}"
304   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
305        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
306 by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
308 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
309 by auto
312 subsection {* Differentiability predicate *}
314 definition
315   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
316     (infixl "differentiable" 60) where
317   "f differentiable x = (\<exists>D. DERIV f x :> D)"
319 lemma differentiableE [elim?]:
320   assumes "f differentiable x"
321   obtains df where "DERIV f x :> df"
322   using prems unfolding differentiable_def ..
324 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
327 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
328 by (force simp add: differentiable_def)
330 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
331   by (rule DERIV_ident [THEN differentiableI])
333 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
334   by (rule DERIV_const [THEN differentiableI])
336 lemma differentiable_compose:
337   assumes f: "f differentiable (g x)"
338   assumes g: "g differentiable x"
339   shows "(\<lambda>x. f (g x)) differentiable x"
340 proof -
341   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
342   moreover
343   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
344   ultimately
345   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
346   thus ?thesis by (rule differentiableI)
347 qed
349 lemma differentiable_sum [simp]:
350   assumes "f differentiable x"
351   and "g differentiable x"
352   shows "(\<lambda>x. f x + g x) differentiable x"
353 proof -
354   from `f differentiable x` obtain df where "DERIV f x :> df" ..
355   moreover
356   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
357   ultimately
358   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
359   thus ?thesis by (rule differentiableI)
360 qed
362 lemma differentiable_minus [simp]:
363   assumes "f differentiable x"
364   shows "(\<lambda>x. - f x) differentiable x"
365 proof -
366   from `f differentiable x` obtain df where "DERIV f x :> df" ..
367   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
368   thus ?thesis by (rule differentiableI)
369 qed
371 lemma differentiable_diff [simp]:
372   assumes "f differentiable x"
373   assumes "g differentiable x"
374   shows "(\<lambda>x. f x - g x) differentiable x"
375   unfolding diff_minus using prems by simp
377 lemma differentiable_mult [simp]:
378   assumes "f differentiable x"
379   assumes "g differentiable x"
380   shows "(\<lambda>x. f x * g x) differentiable x"
381 proof -
382   from `f differentiable x` obtain df where "DERIV f x :> df" ..
383   moreover
384   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
385   ultimately
386   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
387   thus ?thesis by (rule differentiableI)
388 qed
390 lemma differentiable_inverse [simp]:
391   assumes "f differentiable x" and "f x \<noteq> 0"
392   shows "(\<lambda>x. inverse (f x)) differentiable x"
393 proof -
394   from `f differentiable x` obtain df where "DERIV f x :> df" ..
395   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
396     using `f x \<noteq> 0` by (rule DERIV_inverse')
397   thus ?thesis by (rule differentiableI)
398 qed
400 lemma differentiable_divide [simp]:
401   assumes "f differentiable x"
402   assumes "g differentiable x" and "g x \<noteq> 0"
403   shows "(\<lambda>x. f x / g x) differentiable x"
404   unfolding divide_inverse using prems by simp
406 lemma differentiable_power [simp]:
407   fixes f :: "'a::{recpower,real_normed_field} \<Rightarrow> 'a"
408   assumes "f differentiable x"
409   shows "(\<lambda>x. f x ^ n) differentiable x"
410   by (induct n, simp, simp add: power_Suc prems)
413 subsection {* Nested Intervals and Bisection *}
415 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
416      All considerably tidied by lcp.*}
418 lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
419 apply (induct "no")
420 apply (auto intro: order_trans)
421 done
423 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
424          \<forall>n. g(Suc n) \<le> g(n);
425          \<forall>n. f(n) \<le> g(n) |]
426       ==> Bseq (f :: nat \<Rightarrow> real)"
427 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
428 apply (induct_tac "n")
429 apply (auto intro: order_trans)
430 apply (rule_tac y = "g (Suc na)" in order_trans)
431 apply (induct_tac [2] "na")
432 apply (auto intro: order_trans)
433 done
435 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
436          \<forall>n. g(Suc n) \<le> g(n);
437          \<forall>n. f(n) \<le> g(n) |]
438       ==> Bseq (g :: nat \<Rightarrow> real)"
439 apply (subst Bseq_minus_iff [symmetric])
440 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
441 apply auto
442 done
444 lemma f_inc_imp_le_lim:
445   fixes f :: "nat \<Rightarrow> real"
446   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
447 apply (rule linorder_not_less [THEN iffD1])
448 apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
449 apply (drule real_less_sum_gt_zero)
450 apply (drule_tac x = "f n + - lim f" in spec, safe)
451 apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
452 apply (subgoal_tac "lim f \<le> f (no + n) ")
453 apply (drule_tac no=no and m=n in lemma_f_mono_add)
455 apply (induct_tac "no")
456 apply simp
457 apply (auto intro: order_trans simp add: diff_minus abs_if)
458 done
460 lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
461 apply (rule LIMSEQ_minus [THEN limI])
463 done
465 lemma g_dec_imp_lim_le:
466   fixes g :: "nat \<Rightarrow> real"
467   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
468 apply (subgoal_tac "- (g n) \<le> - (lim g) ")
469 apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
470 apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
471 done
473 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
474          \<forall>n. g(Suc n) \<le> g(n);
475          \<forall>n. f(n) \<le> g(n) |]
476       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
477                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
478 apply (subgoal_tac "monoseq f & monoseq g")
479 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
480 apply (subgoal_tac "Bseq f & Bseq g")
481 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
482 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
483 apply (rule_tac x = "lim f" in exI)
484 apply (rule_tac x = "lim g" in exI)
485 apply (auto intro: LIMSEQ_le)
486 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
487 done
489 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
490          \<forall>n. g(Suc n) \<le> g(n);
491          \<forall>n. f(n) \<le> g(n);
492          (%n. f(n) - g(n)) ----> 0 |]
493       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
494                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
495 apply (drule lemma_nest, auto)
496 apply (subgoal_tac "l = m")
497 apply (drule_tac [2] X = f in LIMSEQ_diff)
498 apply (auto intro: LIMSEQ_unique)
499 done
501 text{*The universal quantifiers below are required for the declaration
502   of @{text Bolzano_nest_unique} below.*}
504 lemma Bolzano_bisect_le:
505  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
506 apply (rule allI)
507 apply (induct_tac "n")
508 apply (auto simp add: Let_def split_def)
509 done
511 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
512    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
513 apply (rule allI)
514 apply (induct_tac "n")
515 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
516 done
518 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
519    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
520 apply (rule allI)
521 apply (induct_tac "n")
522 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
523 done
525 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
526 apply (auto)
527 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
528 apply (simp)
529 done
531 lemma Bolzano_bisect_diff:
532      "a \<le> b ==>
533       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
534       (b-a) / (2 ^ n)"
535 apply (induct "n")
537 done
539 lemmas Bolzano_nest_unique =
540     lemma_nest_unique
541     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
544 lemma not_P_Bolzano_bisect:
545   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
546       and notP: "~ P(a,b)"
547       and le:   "a \<le> b"
548   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
549 proof (induct n)
550   case 0 show ?case using notP by simp
551  next
552   case (Suc n)
553   thus ?case
554  by (auto simp del: surjective_pairing [symmetric]
555              simp add: Let_def split_def Bolzano_bisect_le [OF le]
556      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
557 qed
559 (*Now we re-package P_prem as a formula*)
560 lemma not_P_Bolzano_bisect':
561      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
562          ~ P(a,b);  a \<le> b |] ==>
563       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
564 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
568 lemma lemma_BOLZANO:
569      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
570          \<forall>x. \<exists>d::real. 0 < d &
571                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
572          a \<le> b |]
573       ==> P(a,b)"
574 apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
575 apply (rule LIMSEQ_minus_cancel)
576 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
577 apply (rule ccontr)
578 apply (drule not_P_Bolzano_bisect', assumption+)
579 apply (rename_tac "l")
580 apply (drule_tac x = l in spec, clarify)
582 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
583 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
584 apply (drule real_less_half_sum, auto)
585 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
586 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
587 apply safe
588 apply (simp_all (no_asm_simp))
589 apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
590 apply (simp (no_asm_simp) add: abs_if)
591 apply (rule real_sum_of_halves [THEN subst])
593 apply (simp_all add: diff_minus [symmetric])
594 done
597 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
598        (\<forall>x. \<exists>d::real. 0 < d &
599                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
600       --> (\<forall>a b. a \<le> b --> P(a,b))"
601 apply clarify
602 apply (blast intro: lemma_BOLZANO)
603 done
606 subsection {* Intermediate Value Theorem *}
608 text {*Prove Contrapositive by Bisection*}
610 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
611          a \<le> b;
612          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
613       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
614 apply (rule contrapos_pp, assumption)
615 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
616 apply safe
617 apply simp_all
618 apply (simp add: isCont_iff LIM_def)
619 apply (rule ccontr)
620 apply (subgoal_tac "a \<le> x & x \<le> b")
621  prefer 2
622  apply simp
623  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
624 apply (drule_tac x = x in spec)+
625 apply simp
626 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
627 apply safe
628 apply simp
629 apply (drule_tac x = s in spec, clarify)
630 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
631 apply (drule_tac x = "ba-x" in spec)
633 apply (drule_tac x = "aa-x" in spec)
634 apply (case_tac "x \<le> aa", simp_all)
635 done
637 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
638          a \<le> b;
639          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
640       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
641 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
642 apply (drule IVT [where f = "%x. - f x"], assumption)
643 apply (auto intro: isCont_minus)
644 done
646 (*HOL style here: object-level formulations*)
647 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
648       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
649       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
650 apply (blast intro: IVT)
651 done
653 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
654       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
655       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
656 apply (blast intro: IVT2)
657 done
660 subsection {* Boundedness of continuous functions *}
662 text{*By bisection, function continuous on closed interval is bounded above*}
664 lemma isCont_bounded:
665      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
666       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
667 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
668 apply safe
669 apply simp_all
670 apply (rename_tac x xa ya M Ma)
671 apply (cut_tac x = M and y = Ma in linorder_linear, safe)
672 apply (rule_tac x = Ma in exI, clarify)
673 apply (cut_tac x = xb and y = xa in linorder_linear, force)
674 apply (rule_tac x = M in exI, clarify)
675 apply (cut_tac x = xb and y = xa in linorder_linear, force)
676 apply (case_tac "a \<le> x & x \<le> b")
677 apply (rule_tac [2] x = 1 in exI)
678 prefer 2 apply force
679 apply (simp add: LIM_def isCont_iff)
680 apply (drule_tac x = x in spec, auto)
681 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
682 apply (drule_tac x = 1 in spec, auto)
683 apply (rule_tac x = s in exI, clarify)
684 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
685 apply (drule_tac x = "xa-x" in spec)
686 apply (auto simp add: abs_ge_self)
687 done
689 text{*Refine the above to existence of least upper bound*}
691 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
692       (\<exists>t. isLub UNIV S t)"
693 by (blast intro: reals_complete)
695 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
696          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
697                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
698 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
699         in lemma_reals_complete)
700 apply auto
701 apply (drule isCont_bounded, assumption)
702 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
703 apply (rule exI, auto)
704 apply (auto dest!: spec simp add: linorder_not_less)
705 done
707 text{*Now show that it attains its upper bound*}
709 lemma isCont_eq_Ub:
710   assumes le: "a \<le> b"
711       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
712   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
713              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
714 proof -
715   from isCont_has_Ub [OF le con]
716   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
717              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
718   show ?thesis
719   proof (intro exI, intro conjI)
720     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
721     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
722     proof (rule ccontr)
723       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
724       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
725         by (fastsimp simp add: linorder_not_le [symmetric])
726       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
727         by (auto simp add: isCont_inverse isCont_diff con)
728       from isCont_bounded [OF le this]
729       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
730       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
731         by (simp add: M3 algebra_simps)
732       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
733         by (auto intro: order_le_less_trans [of _ k])
734       with Minv
735       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
736         by (intro strip less_imp_inverse_less, simp_all)
737       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
738         by simp
739       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
740         by (simp, arith)
741       from M2 [OF this]
742       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
743       thus False using invlt [of x] by force
744     qed
745   qed
746 qed
749 text{*Same theorem for lower bound*}
751 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
752          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
753                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
754 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
755 prefer 2 apply (blast intro: isCont_minus)
756 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
757 apply safe
758 apply auto
759 done
762 text{*Another version.*}
764 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
765       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
766           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
767 apply (frule isCont_eq_Lb)
768 apply (frule_tac [2] isCont_eq_Ub)
769 apply (assumption+, safe)
770 apply (rule_tac x = "f x" in exI)
771 apply (rule_tac x = "f xa" in exI, simp, safe)
772 apply (cut_tac x = x and y = xa in linorder_linear, safe)
773 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
774 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
775 apply (rule_tac [2] x = xb in exI)
776 apply (rule_tac [4] x = xb in exI, simp_all)
777 done
780 subsection {* Local extrema *}
782 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
784 lemma DERIV_left_inc:
785   fixes f :: "real => real"
786   assumes der: "DERIV f x :> l"
787       and l:   "0 < l"
788   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
789 proof -
790   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
791   have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
793   then obtain s
794         where s:   "0 < s"
795           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
796     by auto
797   thus ?thesis
798   proof (intro exI conjI strip)
799     show "0<s" using s .
800     fix h::real
801     assume "0 < h" "h < s"
802     with all [of h] show "f x < f (x+h)"
803     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
805       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
806       with l
807       have "0 < (f (x+h) - f x) / h" by arith
808       thus "f x < f (x+h)"
809   by (simp add: pos_less_divide_eq h)
810     qed
811   qed
812 qed
814 lemma DERIV_left_dec:
815   fixes f :: "real => real"
816   assumes der: "DERIV f x :> l"
817       and l:   "l < 0"
818   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
819 proof -
820   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
821   have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
823   then obtain s
824         where s:   "0 < s"
825           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
826     by auto
827   thus ?thesis
828   proof (intro exI conjI strip)
829     show "0<s" using s .
830     fix h::real
831     assume "0 < h" "h < s"
832     with all [of "-h"] show "f x < f (x-h)"
833     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
835       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
836       with l
837       have "0 < (f (x-h) - f x) / h" by arith
838       thus "f x < f (x-h)"
839   by (simp add: pos_less_divide_eq h)
840     qed
841   qed
842 qed
844 lemma DERIV_local_max:
845   fixes f :: "real => real"
846   assumes der: "DERIV f x :> l"
847       and d:   "0 < d"
848       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
849   shows "l = 0"
850 proof (cases rule: linorder_cases [of l 0])
851   case equal thus ?thesis .
852 next
853   case less
854   from DERIV_left_dec [OF der less]
855   obtain d' where d': "0 < d'"
856              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
857   from real_lbound_gt_zero [OF d d']
858   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
859   with lt le [THEN spec [where x="x-e"]]
860   show ?thesis by (auto simp add: abs_if)
861 next
862   case greater
863   from DERIV_left_inc [OF der greater]
864   obtain d' where d': "0 < d'"
865              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
866   from real_lbound_gt_zero [OF d d']
867   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
868   with lt le [THEN spec [where x="x+e"]]
869   show ?thesis by (auto simp add: abs_if)
870 qed
873 text{*Similar theorem for a local minimum*}
874 lemma DERIV_local_min:
875   fixes f :: "real => real"
876   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
877 by (drule DERIV_minus [THEN DERIV_local_max], auto)
880 text{*In particular, if a function is locally flat*}
881 lemma DERIV_local_const:
882   fixes f :: "real => real"
883   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
884 by (auto dest!: DERIV_local_max)
887 subsection {* Rolle's Theorem *}
889 text{*Lemma about introducing open ball in open interval*}
890 lemma lemma_interval_lt:
891      "[| a < x;  x < b |]
892       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
895 apply (insert linorder_linear [of "x-a" "b-x"], safe)
896 apply (rule_tac x = "x-a" in exI)
897 apply (rule_tac [2] x = "b-x" in exI, auto)
898 done
900 lemma lemma_interval: "[| a < x;  x < b |] ==>
901         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
902 apply (drule lemma_interval_lt, auto)
903 apply (auto intro!: exI)
904 done
906 text{*Rolle's Theorem.
907    If @{term f} is defined and continuous on the closed interval
908    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
909    and @{term "f(a) = f(b)"},
910    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
911 theorem Rolle:
912   assumes lt: "a < b"
913       and eq: "f(a) = f(b)"
914       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
915       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
916   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
917 proof -
918   have le: "a \<le> b" using lt by simp
919   from isCont_eq_Ub [OF le con]
920   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
921              and alex: "a \<le> x" and xleb: "x \<le> b"
922     by blast
923   from isCont_eq_Lb [OF le con]
924   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
925               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
926     by blast
927   show ?thesis
928   proof cases
929     assume axb: "a < x & x < b"
930         --{*@{term f} attains its maximum within the interval*}
931     hence ax: "a<x" and xb: "x<b" by arith +
932     from lemma_interval [OF ax xb]
933     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
934       by blast
935     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
936       by blast
937     from differentiableD [OF dif [OF axb]]
938     obtain l where der: "DERIV f x :> l" ..
939     have "l=0" by (rule DERIV_local_max [OF der d bound'])
940         --{*the derivative at a local maximum is zero*}
941     thus ?thesis using ax xb der by auto
942   next
943     assume notaxb: "~ (a < x & x < b)"
944     hence xeqab: "x=a | x=b" using alex xleb by arith
945     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
946     show ?thesis
947     proof cases
948       assume ax'b: "a < x' & x' < b"
949         --{*@{term f} attains its minimum within the interval*}
950       hence ax': "a<x'" and x'b: "x'<b" by arith+
951       from lemma_interval [OF ax' x'b]
952       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
953   by blast
954       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
955   by blast
956       from differentiableD [OF dif [OF ax'b]]
957       obtain l where der: "DERIV f x' :> l" ..
958       have "l=0" by (rule DERIV_local_min [OF der d bound'])
959         --{*the derivative at a local minimum is zero*}
960       thus ?thesis using ax' x'b der by auto
961     next
962       assume notax'b: "~ (a < x' & x' < b)"
963         --{*@{term f} is constant througout the interval*}
964       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
965       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
966       from dense [OF lt]
967       obtain r where ar: "a < r" and rb: "r < b" by blast
968       from lemma_interval [OF ar rb]
969       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
970   by blast
971       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
972       proof (clarify)
973         fix z::real
974         assume az: "a \<le> z" and zb: "z \<le> b"
975         show "f z = f b"
976         proof (rule order_antisym)
977           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
978           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
979         qed
980       qed
981       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
982       proof (intro strip)
983         fix y::real
984         assume lt: "\<bar>r-y\<bar> < d"
985         hence "f y = f b" by (simp add: eq_fb bound)
986         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
987       qed
988       from differentiableD [OF dif [OF conjI [OF ar rb]]]
989       obtain l where der: "DERIV f r :> l" ..
990       have "l=0" by (rule DERIV_local_const [OF der d bound'])
991         --{*the derivative of a constant function is zero*}
992       thus ?thesis using ar rb der by auto
993     qed
994   qed
995 qed
998 subsection{*Mean Value Theorem*}
1000 lemma lemma_MVT:
1001      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
1002 proof cases
1003   assume "a=b" thus ?thesis by simp
1004 next
1005   assume "a\<noteq>b"
1006   hence ba: "b-a \<noteq> 0" by arith
1007   show ?thesis
1008     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
1011 qed
1013 theorem MVT:
1014   assumes lt:  "a < b"
1015       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
1016       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
1017   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
1018                    (f(b) - f(a) = (b-a) * l)"
1019 proof -
1020   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
1021   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
1022     by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
1023   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
1024   proof (clarify)
1025     fix x::real
1026     assume ax: "a < x" and xb: "x < b"
1027     from differentiableD [OF dif [OF conjI [OF ax xb]]]
1028     obtain l where der: "DERIV f x :> l" ..
1029     show "?F differentiable x"
1030       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
1031           blast intro: DERIV_diff DERIV_cmult_Id der)
1032   qed
1033   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
1034   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
1035     by blast
1036   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
1037     by (rule DERIV_cmult_Id)
1038   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
1039                    :> 0 + (f b - f a) / (b - a)"
1040     by (rule DERIV_add [OF der])
1041   show ?thesis
1042   proof (intro exI conjI)
1043     show "a < z" using az .
1044     show "z < b" using zb .
1045     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
1046     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
1047   qed
1048 qed
1050 lemma MVT2:
1051      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
1052       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
1053 apply (drule MVT)
1054 apply (blast intro: DERIV_isCont)
1055 apply (force dest: order_less_imp_le simp add: differentiable_def)
1056 apply (blast dest: DERIV_unique order_less_imp_le)
1057 done
1060 text{*A function is constant if its derivative is 0 over an interval.*}
1062 lemma DERIV_isconst_end:
1063   fixes f :: "real => real"
1064   shows "[| a < b;
1065          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
1066          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
1067         ==> f b = f a"
1068 apply (drule MVT, assumption)
1069 apply (blast intro: differentiableI)
1070 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
1071 done
1073 lemma DERIV_isconst1:
1074   fixes f :: "real => real"
1075   shows "[| a < b;
1076          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
1077          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
1078         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
1079 apply safe
1080 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
1081 apply (drule_tac b = x in DERIV_isconst_end, auto)
1082 done
1084 lemma DERIV_isconst2:
1085   fixes f :: "real => real"
1086   shows "[| a < b;
1087          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
1088          \<forall>x. a < x & x < b --> DERIV f x :> 0;
1089          a \<le> x; x \<le> b |]
1090         ==> f x = f a"
1091 apply (blast dest: DERIV_isconst1)
1092 done
1094 lemma DERIV_isconst3: fixes a b x y :: real
1095   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
1096   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
1097   shows "f x = f y"
1098 proof (cases "x = y")
1099   case False
1100   let ?a = "min x y"
1101   let ?b = "max x y"
1103   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
1104   proof (rule allI, rule impI)
1105     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
1106     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
1107     hence "z \<in> {a<..<b}" by auto
1108     thus "DERIV f z :> 0" by (rule derivable)
1109   qed
1110   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
1111     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
1113   have "?a < ?b" using `x \<noteq> y` by auto
1114   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
1115   show ?thesis by auto
1116 qed auto
1118 lemma DERIV_isconst_all:
1119   fixes f :: "real => real"
1120   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
1121 apply (rule linorder_cases [of x y])
1122 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
1123 done
1125 lemma DERIV_const_ratio_const:
1126   fixes f :: "real => real"
1127   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
1128 apply (rule linorder_cases [of a b], auto)
1129 apply (drule_tac [!] f = f in MVT)
1130 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
1131 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
1132 done
1134 lemma DERIV_const_ratio_const2:
1135   fixes f :: "real => real"
1136   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
1137 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
1138 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
1139 done
1141 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
1142 by (simp)
1144 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
1145 by (simp)
1147 text{*Gallileo's "trick": average velocity = av. of end velocities*}
1149 lemma DERIV_const_average:
1150   fixes v :: "real => real"
1151   assumes neq: "a \<noteq> (b::real)"
1152       and der: "\<forall>x. DERIV v x :> k"
1153   shows "v ((a + b)/2) = (v a + v b)/2"
1154 proof (cases rule: linorder_cases [of a b])
1155   case equal with neq show ?thesis by simp
1156 next
1157   case less
1158   have "(v b - v a) / (b - a) = k"
1159     by (rule DERIV_const_ratio_const2 [OF neq der])
1160   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
1161   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
1162     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
1163   ultimately show ?thesis using neq by force
1164 next
1165   case greater
1166   have "(v b - v a) / (b - a) = k"
1167     by (rule DERIV_const_ratio_const2 [OF neq der])
1168   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
1169   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
1170     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
1172 qed
1175 subsection {* Continuous injective functions *}
1177 text{*Dull lemma: an continuous injection on an interval must have a
1178 strict maximum at an end point, not in the middle.*}
1180 lemma lemma_isCont_inj:
1181   fixes f :: "real \<Rightarrow> real"
1182   assumes d: "0 < d"
1183       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
1184       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
1185   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
1186 proof (rule ccontr)
1187   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
1188   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
1189   show False
1190   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
1191     case le
1192     from d cont all [of "x+d"]
1193     have flef: "f(x+d) \<le> f x"
1194      and xlex: "x - d \<le> x"
1195      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
1196        by (auto simp add: abs_if)
1197     from IVT [OF le flef xlex cont']
1198     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
1199     moreover
1200     hence "g(f x') = g (f(x+d))" by simp
1201     ultimately show False using d inj [of x'] inj [of "x+d"]
1203   next
1204     case ge
1205     from d cont all [of "x-d"]
1206     have flef: "f(x-d) \<le> f x"
1207      and xlex: "x \<le> x+d"
1208      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
1209        by (auto simp add: abs_if)
1210     from IVT2 [OF ge flef xlex cont']
1211     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
1212     moreover
1213     hence "g(f x') = g (f(x-d))" by simp
1214     ultimately show False using d inj [of x'] inj [of "x-d"]
1216   qed
1217 qed
1220 text{*Similar version for lower bound.*}
1222 lemma lemma_isCont_inj2:
1223   fixes f g :: "real \<Rightarrow> real"
1224   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
1225         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
1226       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
1227 apply (insert lemma_isCont_inj
1228           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
1229 apply (simp add: isCont_minus linorder_not_le)
1230 done
1232 text{*Show there's an interval surrounding @{term "f(x)"} in
1233 @{text "f[[x - d, x + d]]"} .*}
1235 lemma isCont_inj_range:
1236   fixes f :: "real \<Rightarrow> real"
1237   assumes d: "0 < d"
1238       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
1239       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
1240   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
1241 proof -
1242   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
1243     by (auto simp add: abs_le_iff)
1244   from isCont_Lb_Ub [OF this]
1245   obtain L M
1246   where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
1247     and all2 [rule_format]:
1248            "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
1249     by auto
1250   with d have "L \<le> f x & f x \<le> M" by simp
1251   moreover have "L \<noteq> f x"
1252   proof -
1253     from lemma_isCont_inj2 [OF d inj cont]
1254     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
1255     thus ?thesis using all1 [of u] by arith
1256   qed
1257   moreover have "f x \<noteq> M"
1258   proof -
1259     from lemma_isCont_inj [OF d inj cont]
1260     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
1261     thus ?thesis using all1 [of u] by arith
1262   qed
1263   ultimately have "L < f x & f x < M" by arith
1264   hence "0 < f x - L" "0 < M - f x" by arith+
1265   from real_lbound_gt_zero [OF this]
1266   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
1267   thus ?thesis
1268   proof (intro exI conjI)
1269     show "0<e" using e(1) .
1270     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
1271     proof (intro strip)
1272       fix y::real
1273       assume "\<bar>y - f x\<bar> \<le> e"
1274       with e have "L \<le> y \<and> y \<le> M" by arith
1275       from all2 [OF this]
1276       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
1277       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
1278         by (force simp add: abs_le_iff)
1279     qed
1280   qed
1281 qed
1284 text{*Continuity of inverse function*}
1286 lemma isCont_inverse_function:
1287   fixes f g :: "real \<Rightarrow> real"
1288   assumes d: "0 < d"
1289       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
1290       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
1291   shows "isCont g (f x)"
1292 proof (simp add: isCont_iff LIM_eq)
1293   show "\<forall>r. 0 < r \<longrightarrow>
1294          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
1295   proof (intro strip)
1296     fix r::real
1297     assume r: "0<r"
1298     from real_lbound_gt_zero [OF r d]
1299     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
1300     with inj cont
1301     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
1302                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
1303     from isCont_inj_range [OF e this]
1304     obtain e' where e': "0 < e'"
1305         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
1306           by blast
1307     show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
1308     proof (intro exI conjI)
1309       show "0<e'" using e' .
1310       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
1311       proof (intro strip)
1312         fix z::real
1313         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
1314         with e e_lt e_simps all [rule_format, of "f x + z"]
1315         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
1316       qed
1317     qed
1318   qed
1319 qed
1321 text {* Derivative of inverse function *}
1323 lemma DERIV_inverse_function:
1324   fixes f g :: "real \<Rightarrow> real"
1325   assumes der: "DERIV f (g x) :> D"
1326   assumes neq: "D \<noteq> 0"
1327   assumes a: "a < x" and b: "x < b"
1328   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
1329   assumes cont: "isCont g x"
1330   shows "DERIV g x :> inverse D"
1331 unfolding DERIV_iff2
1332 proof (rule LIM_equal2)
1333   show "0 < min (x - a) (b - x)"
1334     using a b by arith
1335 next
1336   fix y
1337   assume "norm (y - x) < min (x - a) (b - x)"
1338   hence "a < y" and "y < b"
1340   thus "(g y - g x) / (y - x) =
1341         inverse ((f (g y) - x) / (g y - g x))"
1343 next
1344   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
1345     by (rule der [unfolded DERIV_iff2])
1346   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
1347     using inj a b by simp
1348   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
1349   proof (safe intro!: exI)
1350     show "0 < min (x - a) (b - x)"
1351       using a b by simp
1352   next
1353     fix y
1354     assume "norm (y - x) < min (x - a) (b - x)"
1355     hence y: "a < y" "y < b"
1357     assume "g y = g x"
1358     hence "f (g y) = f (g x)" by simp
1359     hence "y = x" using inj y a b by simp
1360     also assume "y \<noteq> x"
1361     finally show False by simp
1362   qed
1363   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
1364     using cont 1 2 by (rule isCont_LIM_compose2)
1365   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
1366         -- x --> inverse D"
1367     using neq by (rule LIM_inverse)
1368 qed
1371 subsection {* Generalized Mean Value Theorem *}
1373 theorem GMVT:
1374   fixes a b :: real
1375   assumes alb: "a < b"
1376   and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
1377   and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
1378   and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
1379   and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
1380   shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
1381 proof -
1382   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
1383   from prems have "a < b" by simp
1384   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
1385   proof -
1386     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
1387     with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
1388       by (auto intro: isCont_mult)
1389     moreover
1390     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
1391     with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
1392       by (auto intro: isCont_mult)
1393     ultimately show ?thesis
1394       by (fastsimp intro: isCont_diff)
1395   qed
1396   moreover
1397   have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
1398   proof -
1399     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
1400     with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
1401     moreover
1402     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
1403     with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
1404     ultimately show ?thesis by (simp add: differentiable_diff)
1405   qed
1406   ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
1407   then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
1408   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
1410   from cdef have cint: "a < c \<and> c < b" by auto
1411   with gd have "g differentiable c" by simp
1412   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
1413   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
1415   from cdef have "a < c \<and> c < b" by auto
1416   with fd have "f differentiable c" by simp
1417   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
1418   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
1420   from cdef have "DERIV ?h c :> l" by auto
1421   moreover
1422   {
1423     have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
1424       apply (insert DERIV_const [where k="f b - f a"])
1425       apply (drule meta_spec [of _ c])
1426       apply (drule DERIV_mult [OF _ g'cdef])
1427       by simp
1428     moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
1429       apply (insert DERIV_const [where k="g b - g a"])
1430       apply (drule meta_spec [of _ c])
1431       apply (drule DERIV_mult [OF _ f'cdef])
1432       by simp
1433     ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
1435   }
1436   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
1438   {
1439     from cdef have "?h b - ?h a = (b - a) * l" by auto
1440     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
1441     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
1442   }
1443   moreover
1444   {
1445     have "?h b - ?h a =
1446          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
1447           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
1449     hence "?h b - ?h a = 0" by auto
1450   }
1451   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
1452   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
1453   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
1454   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
1456   with g'cdef f'cdef cint show ?thesis by auto
1457 qed
1460 subsection {* Theorems about Limits *}
1462 (* need to rename second isCont_inverse *)
1464 lemma isCont_inv_fun:
1465   fixes f g :: "real \<Rightarrow> real"
1466   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
1467          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
1468       ==> isCont g (f x)"
1469 by (rule isCont_inverse_function)
1471 lemma isCont_inv_fun_inv:
1472   fixes f g :: "real \<Rightarrow> real"
1473   shows "[| 0 < d;
1474          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
1475          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
1476        ==> \<exists>e. 0 < e &
1477              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
1478 apply (drule isCont_inj_range)
1479 prefer 2 apply (assumption, assumption, auto)
1480 apply (rule_tac x = e in exI, auto)
1481 apply (rotate_tac 2)
1482 apply (drule_tac x = y in spec, auto)
1483 done
1486 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
1487 lemma LIM_fun_gt_zero:
1488      "[| f -- c --> (l::real); 0 < l |]
1489          ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
1490 apply (auto simp add: LIM_def)
1491 apply (drule_tac x = "l/2" in spec, safe, force)
1492 apply (rule_tac x = s in exI)
1493 apply (auto simp only: abs_less_iff)
1494 done
1496 lemma LIM_fun_less_zero:
1497      "[| f -- c --> (l::real); l < 0 |]
1498       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
1499 apply (auto simp add: LIM_def)
1500 apply (drule_tac x = "-l/2" in spec, safe, force)
1501 apply (rule_tac x = s in exI)
1502 apply (auto simp only: abs_less_iff)
1503 done
1506 lemma LIM_fun_not_zero:
1507      "[| f -- c --> (l::real); l \<noteq> 0 |]
1508       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
1509 apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
1510 apply (drule LIM_fun_less_zero)
1511 apply (drule_tac [3] LIM_fun_gt_zero)
1512 apply force+
1513 done
1515 end