src/HOL/Limits.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62369 acfc4ad7b76a child 62379 340738057c8c permissions -rw-r--r--
generalize more theorems to support enat and ennreal
1 (*  Title:      HOL/Limits.thy
2     Author:     Brian Huffman
3     Author:     Jacques D. Fleuriot, University of Cambridge
4     Author:     Lawrence C Paulson
5     Author:     Jeremy Avigad
6 *)
8 section \<open>Limits on Real Vector Spaces\<close>
10 theory Limits
11 imports Real_Vector_Spaces
12 begin
14 subsection \<open>Filter going to infinity norm\<close>
16 definition at_infinity :: "'a::real_normed_vector filter" where
17   "at_infinity = (INF r. principal {x. r \<le> norm x})"
19 lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
20   unfolding at_infinity_def
21   by (subst eventually_INF_base)
22      (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
24 lemma at_infinity_eq_at_top_bot:
25   "(at_infinity :: real filter) = sup at_top at_bot"
26   apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
27                    eventually_at_top_linorder eventually_at_bot_linorder)
28   apply safe
29   apply (rule_tac x="b" in exI, simp)
30   apply (rule_tac x="- b" in exI, simp)
31   apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
32   done
34 lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
35   unfolding at_infinity_eq_at_top_bot by simp
37 lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
38   unfolding at_infinity_eq_at_top_bot by simp
40 lemma filterlim_at_top_imp_at_infinity:
41   fixes f :: "_ \<Rightarrow> real"
42   shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
43   by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
45 lemma lim_infinity_imp_sequentially:
46   "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
47 by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
50 subsubsection \<open>Boundedness\<close>
52 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
53   Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
55 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
56   "Bseq X \<equiv> Bfun X sequentially"
58 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
60 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
61   unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
63 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
64   unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
66 lemma Bfun_def:
67   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
68   unfolding Bfun_metric_def norm_conv_dist
69 proof safe
70   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
71   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
72     by (intro always_eventually) (metis dist_commute dist_triangle)
73   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
74     by eventually_elim auto
75   with \<open>0 < K\<close> show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
76     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
77 qed auto
79 lemma BfunI:
80   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
81 unfolding Bfun_def
82 proof (intro exI conjI allI)
83   show "0 < max K 1" by simp
84 next
85   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
86     using K by (rule eventually_mono, simp)
87 qed
89 lemma BfunE:
90   assumes "Bfun f F"
91   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
92 using assms unfolding Bfun_def by blast
94 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
95   unfolding Cauchy_def Bfun_metric_def eventually_sequentially
96   apply (erule_tac x=1 in allE)
97   apply simp
98   apply safe
99   apply (rule_tac x="X M" in exI)
100   apply (rule_tac x=1 in exI)
101   apply (erule_tac x=M in allE)
102   apply simp
103   apply (rule_tac x=M in exI)
104   apply (auto simp: dist_commute)
105   done
108 subsubsection \<open>Bounded Sequences\<close>
110 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
111   by (intro BfunI) (auto simp: eventually_sequentially)
113 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
114   by (intro BfunI) (auto simp: eventually_sequentially)
116 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
117   unfolding Bfun_def eventually_sequentially
118 proof safe
119   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
120   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
121     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
122        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
123 qed auto
125 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
126 unfolding Bseq_def by auto
128 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
129 by (simp add: Bseq_def)
131 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
132 by (auto simp add: Bseq_def)
134 lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
135 proof (elim BseqE, intro bdd_aboveI2)
136   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
137     by (auto elim!: allE[of _ n])
138 qed
140 lemma Bseq_bdd_above':
141   "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
142 proof (elim BseqE, intro bdd_aboveI2)
143   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
144     by (auto elim!: allE[of _ n])
145 qed
147 lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
148 proof (elim BseqE, intro bdd_belowI2)
149   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
150     by (auto elim!: allE[of _ n])
151 qed
153 lemma Bseq_eventually_mono:
154   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
155   shows   "Bseq f"
156 proof -
157   from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
158     by (auto simp: eventually_at_top_linorder)
159   moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
160   ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
161     apply (cases "n < N")
162     apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
163     apply (rule max.coboundedI1, force intro: order.trans[OF N K])
164     done
165   thus ?thesis by (blast intro: BseqI')
166 qed
168 lemma lemma_NBseq_def:
169   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
170 proof safe
171   fix K :: real
172   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
173   then have "K \<le> real (Suc n)" by auto
174   moreover assume "\<forall>m. norm (X m) \<le> K"
175   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
176     by (blast intro: order_trans)
177   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
178 next
179   show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
180     using of_nat_0_less_iff by blast
181 qed
183 text\<open>alternative definition for Bseq\<close>
184 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
185 apply (simp add: Bseq_def)
186 apply (simp (no_asm) add: lemma_NBseq_def)
187 done
189 lemma lemma_NBseq_def2:
190      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
191 apply (subst lemma_NBseq_def, auto)
192 apply (rule_tac x = "Suc N" in exI)
193 apply (rule_tac  x = N in exI)
194 apply (auto simp add: of_nat_Suc)
195  prefer 2 apply (blast intro: order_less_imp_le)
196 apply (drule_tac x = n in spec, simp)
197 done
199 (* yet another definition for Bseq *)
200 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
201 by (simp add: Bseq_def lemma_NBseq_def2)
203 subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
205 text\<open>alternative formulation for boundedness\<close>
206 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
207 apply (unfold Bseq_def, safe)
208 apply (rule_tac  x = "k + norm x" in exI)
209 apply (rule_tac x = K in exI, simp)
210 apply (rule exI [where x = 0], auto)
211 apply (erule order_less_le_trans, simp)
212 apply (drule_tac x=n in spec)
213 apply (drule order_trans [OF norm_triangle_ineq2])
214 apply simp
215 done
217 text\<open>alternative formulation for boundedness\<close>
218 lemma Bseq_iff3:
219   "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
220 proof
221   assume ?P
222   then obtain K
223     where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
224   from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
225   from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
226     by (auto intro: order_trans norm_triangle_ineq4)
227   then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
228     by simp
229   with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
230 next
231   assume ?Q then show ?P by (auto simp add: Bseq_iff2)
232 qed
234 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
235 apply (simp add: Bseq_def)
236 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
237 apply (drule_tac x = n in spec, arith)
238 done
241 subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
243 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
244   by (simp add: Bseq_def)
247   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
248   shows   "Bseq (\<lambda>x. f x + c)"
249 proof -
250   from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
251   {
252     fix x :: nat
253     have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
254     also have "norm (f x) \<le> K" by (rule K)
255     finally have "norm (f x + c) \<le> K + norm c" by simp
256   }
257   thus ?thesis by (rule BseqI')
258 qed
260 lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
261   using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
263 lemma Bseq_mult:
264   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
265   assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
266   shows   "Bseq (\<lambda>x. f x * g x)"
267 proof -
268   from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0"
269     unfolding Bseq_def by blast
270   hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
271   thus ?thesis by (rule BseqI')
272 qed
274 lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
275   unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
277 lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
278 proof
279   assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
280   find_theorems "Bfun (\<lambda>_. ?c) _"
281   from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
282   with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
283 qed (intro Bseq_mult Bfun_const)
285 lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
286   unfolding Bseq_def by auto
288 lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
289   using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
291 lemma increasing_Bseq_subseq_iff:
292   assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
293   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
294 proof
295   assume "Bseq (\<lambda>x. f (g x))"
296   then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
297   {
298     fix x :: nat
299     from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
300       by (auto simp: filterlim_at_top eventually_at_top_linorder)
301     hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
302     also have "norm (f (g y)) \<le> K" by (rule K)
303     finally have "norm (f x) \<le> K" .
304   }
305   thus "Bseq f" by (rule BseqI')
306 qed (insert Bseq_subseq[of f g], simp_all)
308 lemma nonneg_incseq_Bseq_subseq_iff:
309   assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
310   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
311   using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
313 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
314   apply (simp add: subset_eq)
315   apply (rule BseqI'[where K="max (norm a) (norm b)"])
316   apply (erule_tac x=n in allE)
317   apply auto
318   done
320 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
321   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
323 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
324   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
326 subsection \<open>Bounded Monotonic Sequences\<close>
328 subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
330 (* TODO: delete *)
331 (* FIXME: one use in NSA/HSEQ.thy *)
332 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
333   apply (rule_tac x="X m" in exI)
334   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
335   unfolding eventually_sequentially
336   apply blast
337   done
339 subsection \<open>Convergence to Zero\<close>
341 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
342   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
344 lemma ZfunI:
345   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
346   unfolding Zfun_def by simp
348 lemma ZfunD:
349   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
350   unfolding Zfun_def by simp
352 lemma Zfun_ssubst:
353   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
354   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
356 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
357   unfolding Zfun_def by simp
359 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
360   unfolding Zfun_def by simp
362 lemma Zfun_imp_Zfun:
363   assumes f: "Zfun f F"
364   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
365   shows "Zfun (\<lambda>x. g x) F"
366 proof (cases)
367   assume K: "0 < K"
368   show ?thesis
369   proof (rule ZfunI)
370     fix r::real assume "0 < r"
371     hence "0 < r / K" using K by simp
372     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
373       using ZfunD [OF f] by blast
374     with g show "eventually (\<lambda>x. norm (g x) < r) F"
375     proof eventually_elim
376       case (elim x)
377       hence "norm (f x) * K < r"
378         by (simp add: pos_less_divide_eq K)
379       thus ?case
380         by (simp add: order_le_less_trans [OF elim(1)])
381     qed
382   qed
383 next
384   assume "\<not> 0 < K"
385   hence K: "K \<le> 0" by (simp only: not_less)
386   show ?thesis
387   proof (rule ZfunI)
388     fix r :: real
389     assume "0 < r"
390     from g show "eventually (\<lambda>x. norm (g x) < r) F"
391     proof eventually_elim
392       case (elim x)
393       also have "norm (f x) * K \<le> norm (f x) * 0"
394         using K norm_ge_zero by (rule mult_left_mono)
395       finally show ?case
396         using \<open>0 < r\<close> by simp
397     qed
398   qed
399 qed
401 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
402   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
405   assumes f: "Zfun f F" and g: "Zfun g F"
406   shows "Zfun (\<lambda>x. f x + g x) F"
407 proof (rule ZfunI)
408   fix r::real assume "0 < r"
409   hence r: "0 < r / 2" by simp
410   have "eventually (\<lambda>x. norm (f x) < r/2) F"
411     using f r by (rule ZfunD)
412   moreover
413   have "eventually (\<lambda>x. norm (g x) < r/2) F"
414     using g r by (rule ZfunD)
415   ultimately
416   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
417   proof eventually_elim
418     case (elim x)
419     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
420       by (rule norm_triangle_ineq)
421     also have "\<dots> < r/2 + r/2"
422       using elim by (rule add_strict_mono)
423     finally show ?case
424       by simp
425   qed
426 qed
428 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
429   unfolding Zfun_def by simp
431 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
432   using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
434 lemma (in bounded_linear) Zfun:
435   assumes g: "Zfun g F"
436   shows "Zfun (\<lambda>x. f (g x)) F"
437 proof -
438   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
439     using bounded by blast
440   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
441     by simp
442   with g show ?thesis
443     by (rule Zfun_imp_Zfun)
444 qed
446 lemma (in bounded_bilinear) Zfun:
447   assumes f: "Zfun f F"
448   assumes g: "Zfun g F"
449   shows "Zfun (\<lambda>x. f x ** g x) F"
450 proof (rule ZfunI)
451   fix r::real assume r: "0 < r"
452   obtain K where K: "0 < K"
453     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
454     using pos_bounded by blast
455   from K have K': "0 < inverse K"
456     by (rule positive_imp_inverse_positive)
457   have "eventually (\<lambda>x. norm (f x) < r) F"
458     using f r by (rule ZfunD)
459   moreover
460   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
461     using g K' by (rule ZfunD)
462   ultimately
463   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
464   proof eventually_elim
465     case (elim x)
466     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
467       by (rule norm_le)
468     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
469       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
470     also from K have "r * inverse K * K = r"
471       by simp
472     finally show ?case .
473   qed
474 qed
476 lemma (in bounded_bilinear) Zfun_left:
477   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
478   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
480 lemma (in bounded_bilinear) Zfun_right:
481   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
482   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
484 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
485 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
486 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
488 lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
489   by (simp only: tendsto_iff Zfun_def dist_norm)
491 lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
492                      \<Longrightarrow> (g \<longlongrightarrow> 0) F"
493   by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
495 subsubsection \<open>Distance and norms\<close>
497 lemma tendsto_dist [tendsto_intros]:
498   fixes l m :: "'a :: metric_space"
499   assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
500   shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
501 proof (rule tendstoI)
502   fix e :: real assume "0 < e"
503   hence e2: "0 < e/2" by simp
504   from tendstoD [OF f e2] tendstoD [OF g e2]
505   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
506   proof (eventually_elim)
507     case (elim x)
508     then show "dist (dist (f x) (g x)) (dist l m) < e"
509       unfolding dist_real_def
510       using dist_triangle2 [of "f x" "g x" "l"]
511       using dist_triangle2 [of "g x" "l" "m"]
512       using dist_triangle3 [of "l" "m" "f x"]
513       using dist_triangle [of "f x" "m" "g x"]
514       by arith
515   qed
516 qed
518 lemma continuous_dist[continuous_intros]:
519   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
520   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
521   unfolding continuous_def by (rule tendsto_dist)
523 lemma continuous_on_dist[continuous_intros]:
524   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
525   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
526   unfolding continuous_on_def by (auto intro: tendsto_dist)
528 lemma tendsto_norm [tendsto_intros]:
529   "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
530   unfolding norm_conv_dist by (intro tendsto_intros)
532 lemma continuous_norm [continuous_intros]:
533   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
534   unfolding continuous_def by (rule tendsto_norm)
536 lemma continuous_on_norm [continuous_intros]:
537   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
538   unfolding continuous_on_def by (auto intro: tendsto_norm)
540 lemma tendsto_norm_zero:
541   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
542   by (drule tendsto_norm, simp)
544 lemma tendsto_norm_zero_cancel:
545   "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
546   unfolding tendsto_iff dist_norm by simp
548 lemma tendsto_norm_zero_iff:
549   "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
550   unfolding tendsto_iff dist_norm by simp
552 lemma tendsto_rabs [tendsto_intros]:
553   "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
554   by (fold real_norm_def, rule tendsto_norm)
556 lemma continuous_rabs [continuous_intros]:
557   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
558   unfolding real_norm_def[symmetric] by (rule continuous_norm)
560 lemma continuous_on_rabs [continuous_intros]:
561   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
562   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
564 lemma tendsto_rabs_zero:
565   "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
566   by (fold real_norm_def, rule tendsto_norm_zero)
568 lemma tendsto_rabs_zero_cancel:
569   "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
570   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
572 lemma tendsto_rabs_zero_iff:
573   "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
574   by (fold real_norm_def, rule tendsto_norm_zero_iff)
576 subsection \<open>Topological Monoid\<close>
578 class topological_monoid_add = topological_space + monoid_add +
579   assumes tendsto_add_Pair: "LIM x (nhds a \<times>\<^sub>F nhds b). fst x + snd x :> nhds (a + b)"
583 lemma tendsto_add [tendsto_intros]:
584   fixes a b :: "'a::topological_monoid_add"
585   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
586   using filterlim_compose[OF tendsto_add_Pair, of "\<lambda>x. (f x, g x)" a b F]
587   by (simp add: nhds_prod[symmetric] tendsto_Pair)
589 lemma continuous_add [continuous_intros]:
590   fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
591   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
592   unfolding continuous_def by (rule tendsto_add)
594 lemma continuous_on_add [continuous_intros]:
595   fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
596   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
597   unfolding continuous_on_def by (auto intro: tendsto_add)
600   fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
601   shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
602   by (drule (1) tendsto_add, simp)
604 lemma tendsto_setsum [tendsto_intros]:
605   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
606   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
607   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
608 proof (cases "finite S")
609   assume "finite S" thus ?thesis using assms
610     by (induct, simp, simp add: tendsto_add)
611 qed simp
613 lemma continuous_setsum [continuous_intros]:
614   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
615   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
616   unfolding continuous_def by (rule tendsto_setsum)
618 lemma continuous_on_setsum [continuous_intros]:
619   fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::topological_comm_monoid_add"
620   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
621   unfolding continuous_on_def by (auto intro: tendsto_setsum)
623 instance nat :: topological_comm_monoid_add
624   proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
626 instance int :: topological_comm_monoid_add
627   proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
629 subsubsection \<open>Addition and subtraction\<close>
631 instance real_normed_vector < topological_comm_monoid_add
632 proof
633   fix a b :: 'a show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
635     using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
636     by (intro Zfun_add)
637        (auto simp add: tendsto_Zfun_iff[symmetric] nhds_prod[symmetric] intro!: tendsto_fst)
638 qed
640 lemma tendsto_minus [tendsto_intros]:
641   fixes a :: "'a::real_normed_vector"
642   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
643   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
645 lemma continuous_minus [continuous_intros]:
646   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
647   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
648   unfolding continuous_def by (rule tendsto_minus)
650 lemma continuous_on_minus [continuous_intros]:
651   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
652   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
653   unfolding continuous_on_def by (auto intro: tendsto_minus)
655 lemma tendsto_minus_cancel:
656   fixes a :: "'a::real_normed_vector"
657   shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
658   by (drule tendsto_minus, simp)
660 lemma tendsto_minus_cancel_left:
661     "(f \<longlongrightarrow> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
662   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
663   by auto
665 lemma tendsto_diff [tendsto_intros]:
666   fixes a b :: "'a::real_normed_vector"
667   shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
668   using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
670 lemma continuous_diff [continuous_intros]:
671   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
672   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
673   unfolding continuous_def by (rule tendsto_diff)
675 lemma continuous_on_diff [continuous_intros]:
676   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
677   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
678   unfolding continuous_on_def by (auto intro: tendsto_diff)
680 lemma continuous_on_op_minus: "continuous_on (s::'a::real_normed_vector set) (op - x)"
681   by (rule continuous_intros | simp)+
683 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
685 subsubsection \<open>Linear operators and multiplication\<close>
687 lemma linear_times:
688   fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
689   by (auto simp: linearI distrib_left)
691 lemma (in bounded_linear) tendsto:
692   "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
693   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
695 lemma (in bounded_linear) continuous:
696   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
697   using tendsto[of g _ F] by (auto simp: continuous_def)
699 lemma (in bounded_linear) continuous_on:
700   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
701   using tendsto[of g] by (auto simp: continuous_on_def)
703 lemma (in bounded_linear) tendsto_zero:
704   "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
705   by (drule tendsto, simp only: zero)
707 lemma (in bounded_bilinear) tendsto:
708   "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
709   by (simp only: tendsto_Zfun_iff prod_diff_prod
710                  Zfun_add Zfun Zfun_left Zfun_right)
712 lemma (in bounded_bilinear) continuous:
713   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
714   using tendsto[of f _ F g] by (auto simp: continuous_def)
716 lemma (in bounded_bilinear) continuous_on:
717   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
718   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
720 lemma (in bounded_bilinear) tendsto_zero:
721   assumes f: "(f \<longlongrightarrow> 0) F"
722   assumes g: "(g \<longlongrightarrow> 0) F"
723   shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
724   using tendsto [OF f g] by (simp add: zero_left)
726 lemma (in bounded_bilinear) tendsto_left_zero:
727   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
728   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
730 lemma (in bounded_bilinear) tendsto_right_zero:
731   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
732   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
734 lemmas tendsto_of_real [tendsto_intros] =
735   bounded_linear.tendsto [OF bounded_linear_of_real]
737 lemmas tendsto_scaleR [tendsto_intros] =
738   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
740 lemmas tendsto_mult [tendsto_intros] =
741   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
743 lemma tendsto_mult_left:
744   fixes c::"'a::real_normed_algebra"
745   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
746 by (rule tendsto_mult [OF tendsto_const])
748 lemma tendsto_mult_right:
749   fixes c::"'a::real_normed_algebra"
750   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
751 by (rule tendsto_mult [OF _ tendsto_const])
753 lemmas continuous_of_real [continuous_intros] =
754   bounded_linear.continuous [OF bounded_linear_of_real]
756 lemmas continuous_scaleR [continuous_intros] =
757   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
759 lemmas continuous_mult [continuous_intros] =
760   bounded_bilinear.continuous [OF bounded_bilinear_mult]
762 lemmas continuous_on_of_real [continuous_intros] =
763   bounded_linear.continuous_on [OF bounded_linear_of_real]
765 lemmas continuous_on_scaleR [continuous_intros] =
766   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
768 lemmas continuous_on_mult [continuous_intros] =
769   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
771 lemmas tendsto_mult_zero =
772   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
774 lemmas tendsto_mult_left_zero =
775   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
777 lemmas tendsto_mult_right_zero =
778   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
780 lemma tendsto_power [tendsto_intros]:
781   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
782   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
783   by (induct n) (simp_all add: tendsto_mult)
785 lemma continuous_power [continuous_intros]:
786   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
787   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
788   unfolding continuous_def by (rule tendsto_power)
790 lemma continuous_on_power [continuous_intros]:
791   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
792   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
793   unfolding continuous_on_def by (auto intro: tendsto_power)
795 lemma tendsto_setprod [tendsto_intros]:
796   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
797   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
798   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
799 proof (cases "finite S")
800   assume "finite S" thus ?thesis using assms
801     by (induct, simp, simp add: tendsto_mult)
802 qed simp
804 lemma continuous_setprod [continuous_intros]:
805   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
806   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
807   unfolding continuous_def by (rule tendsto_setprod)
809 lemma continuous_on_setprod [continuous_intros]:
810   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
811   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
812   unfolding continuous_on_def by (auto intro: tendsto_setprod)
814 lemma tendsto_of_real_iff:
815   "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
816   unfolding tendsto_iff by simp
819   "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
820   using tendsto_add[OF tendsto_const[of c], of f d]
821         tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
824 subsubsection \<open>Inverse and division\<close>
826 lemma (in bounded_bilinear) Zfun_prod_Bfun:
827   assumes f: "Zfun f F"
828   assumes g: "Bfun g F"
829   shows "Zfun (\<lambda>x. f x ** g x) F"
830 proof -
831   obtain K where K: "0 \<le> K"
832     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
833     using nonneg_bounded by blast
834   obtain B where B: "0 < B"
835     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
836     using g by (rule BfunE)
837   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
838   using norm_g proof eventually_elim
839     case (elim x)
840     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
841       by (rule norm_le)
842     also have "\<dots> \<le> norm (f x) * B * K"
843       by (intro mult_mono' order_refl norm_g norm_ge_zero
844                 mult_nonneg_nonneg K elim)
845     also have "\<dots> = norm (f x) * (B * K)"
846       by (rule mult.assoc)
847     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
848   qed
849   with f show ?thesis
850     by (rule Zfun_imp_Zfun)
851 qed
853 lemma (in bounded_bilinear) Bfun_prod_Zfun:
854   assumes f: "Bfun f F"
855   assumes g: "Zfun g F"
856   shows "Zfun (\<lambda>x. f x ** g x) F"
857   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
859 lemma Bfun_inverse_lemma:
860   fixes x :: "'a::real_normed_div_algebra"
861   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
862   apply (subst nonzero_norm_inverse, clarsimp)
863   apply (erule (1) le_imp_inverse_le)
864   done
866 lemma Bfun_inverse:
867   fixes a :: "'a::real_normed_div_algebra"
868   assumes f: "(f \<longlongrightarrow> a) F"
869   assumes a: "a \<noteq> 0"
870   shows "Bfun (\<lambda>x. inverse (f x)) F"
871 proof -
872   from a have "0 < norm a" by simp
873   hence "\<exists>r>0. r < norm a" by (rule dense)
874   then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
875   have "eventually (\<lambda>x. dist (f x) a < r) F"
876     using tendstoD [OF f r1] by blast
877   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
878   proof eventually_elim
879     case (elim x)
880     hence 1: "norm (f x - a) < r"
881       by (simp add: dist_norm)
882     hence 2: "f x \<noteq> 0" using r2 by auto
883     hence "norm (inverse (f x)) = inverse (norm (f x))"
884       by (rule nonzero_norm_inverse)
885     also have "\<dots> \<le> inverse (norm a - r)"
886     proof (rule le_imp_inverse_le)
887       show "0 < norm a - r" using r2 by simp
888     next
889       have "norm a - norm (f x) \<le> norm (a - f x)"
890         by (rule norm_triangle_ineq2)
891       also have "\<dots> = norm (f x - a)"
892         by (rule norm_minus_commute)
893       also have "\<dots> < r" using 1 .
894       finally show "norm a - r \<le> norm (f x)" by simp
895     qed
896     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
897   qed
898   thus ?thesis by (rule BfunI)
899 qed
901 lemma tendsto_inverse [tendsto_intros]:
902   fixes a :: "'a::real_normed_div_algebra"
903   assumes f: "(f \<longlongrightarrow> a) F"
904   assumes a: "a \<noteq> 0"
905   shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
906 proof -
907   from a have "0 < norm a" by simp
908   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
909     by (rule tendstoD)
910   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
911     unfolding dist_norm by (auto elim!: eventually_mono)
912   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
913     - (inverse (f x) * (f x - a) * inverse a)) F"
914     by (auto elim!: eventually_mono simp: inverse_diff_inverse)
915   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
916     by (intro Zfun_minus Zfun_mult_left
917       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
918       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
919   ultimately show ?thesis
920     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
921 qed
923 lemma continuous_inverse:
924   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
925   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
926   shows "continuous F (\<lambda>x. inverse (f x))"
927   using assms unfolding continuous_def by (rule tendsto_inverse)
929 lemma continuous_at_within_inverse[continuous_intros]:
930   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
931   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
932   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
933   using assms unfolding continuous_within by (rule tendsto_inverse)
935 lemma isCont_inverse[continuous_intros, simp]:
936   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
937   assumes "isCont f a" and "f a \<noteq> 0"
938   shows "isCont (\<lambda>x. inverse (f x)) a"
939   using assms unfolding continuous_at by (rule tendsto_inverse)
941 lemma continuous_on_inverse[continuous_intros]:
942   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
943   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
944   shows "continuous_on s (\<lambda>x. inverse (f x))"
945   using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
947 lemma tendsto_divide [tendsto_intros]:
948   fixes a b :: "'a::real_normed_field"
949   shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
950     \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
951   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
953 lemma continuous_divide:
954   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
955   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
956   shows "continuous F (\<lambda>x. (f x) / (g x))"
957   using assms unfolding continuous_def by (rule tendsto_divide)
959 lemma continuous_at_within_divide[continuous_intros]:
960   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
961   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
962   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
963   using assms unfolding continuous_within by (rule tendsto_divide)
965 lemma isCont_divide[continuous_intros, simp]:
966   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
967   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
968   shows "isCont (\<lambda>x. (f x) / g x) a"
969   using assms unfolding continuous_at by (rule tendsto_divide)
971 lemma continuous_on_divide[continuous_intros]:
972   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
973   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
974   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
975   using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
977 lemma tendsto_sgn [tendsto_intros]:
978   fixes l :: "'a::real_normed_vector"
979   shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
980   unfolding sgn_div_norm by (simp add: tendsto_intros)
982 lemma continuous_sgn:
983   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
984   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
985   shows "continuous F (\<lambda>x. sgn (f x))"
986   using assms unfolding continuous_def by (rule tendsto_sgn)
988 lemma continuous_at_within_sgn[continuous_intros]:
989   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
990   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
991   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
992   using assms unfolding continuous_within by (rule tendsto_sgn)
994 lemma isCont_sgn[continuous_intros]:
995   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
996   assumes "isCont f a" and "f a \<noteq> 0"
997   shows "isCont (\<lambda>x. sgn (f x)) a"
998   using assms unfolding continuous_at by (rule tendsto_sgn)
1000 lemma continuous_on_sgn[continuous_intros]:
1001   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1002   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
1003   shows "continuous_on s (\<lambda>x. sgn (f x))"
1004   using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
1006 lemma filterlim_at_infinity:
1007   fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
1008   assumes "0 \<le> c"
1009   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
1010   unfolding filterlim_iff eventually_at_infinity
1011 proof safe
1012   fix P :: "'a \<Rightarrow> bool" and b
1013   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
1014     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
1015   have "max b (c + 1) > c" by auto
1016   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
1017     by auto
1018   then show "eventually (\<lambda>x. P (f x)) F"
1019   proof eventually_elim
1020     fix x assume "max b (c + 1) \<le> norm (f x)"
1021     with P show "P (f x)" by auto
1022   qed
1023 qed force
1025 lemma not_tendsto_and_filterlim_at_infinity:
1026   assumes "F \<noteq> bot"
1027   assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
1028   assumes "filterlim f at_infinity F"
1029   shows   False
1030 proof -
1031   from tendstoD[OF assms(2), of "1/2"]
1032     have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
1033   moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
1034     have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
1035   ultimately have "eventually (\<lambda>x. False) F"
1036   proof eventually_elim
1037     fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
1038     note B
1039     also have "norm (f x) = dist (f x) 0" by (simp add: norm_conv_dist)
1040     also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
1041     also note A
1042     finally show False by (simp add: norm_conv_dist)
1043   qed
1044   with assms show False by simp
1045 qed
1047 lemma filterlim_at_infinity_imp_not_convergent:
1048   assumes "filterlim f at_infinity sequentially"
1049   shows   "\<not>convergent f"
1050   by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
1051      (simp_all add: convergent_LIMSEQ_iff)
1053 lemma filterlim_at_infinity_imp_eventually_ne:
1054   assumes "filterlim f at_infinity F"
1055   shows   "eventually (\<lambda>z. f z \<noteq> c) F"
1056 proof -
1057   have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
1058   with filterlim_at_infinity[OF order.refl, of f F] assms
1059     have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
1060   thus ?thesis by eventually_elim auto
1061 qed
1063 lemma tendsto_of_nat [tendsto_intros]:
1064   "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
1065 proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
1066   fix r :: real assume r: "r > 0"
1067   def n \<equiv> "nat \<lceil>r\<rceil>"
1068   from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
1069   from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
1070     by eventually_elim (insert n, simp_all)
1071 qed
1074 subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
1076 text \<open>
1078 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
1079 @{term "at_right x"} and also @{term "at_right 0"}.
1081 \<close>
1083 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
1085 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
1086   by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
1087      (auto intro!: tendsto_eq_intros filterlim_ident)
1089 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
1090   by (rule filtermap_fun_inverse[where g=uminus])
1091      (auto intro!: tendsto_eq_intros filterlim_ident)
1093 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
1094   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
1096 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
1097   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
1099 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
1100   using filtermap_at_right_shift[of "-a" 0] by simp
1102 lemma filterlim_at_right_to_0:
1103   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
1104   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
1106 lemma eventually_at_right_to_0:
1107   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
1108   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
1110 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
1111   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
1113 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
1114   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
1116 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
1117   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
1119 lemma filterlim_at_left_to_right:
1120   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
1121   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
1123 lemma eventually_at_left_to_right:
1124   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
1125   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
1127 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
1128   unfolding filterlim_at_top eventually_at_bot_dense
1129   by (metis leI minus_less_iff order_less_asym)
1131 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
1132   unfolding filterlim_at_bot eventually_at_top_dense
1133   by (metis leI less_minus_iff order_less_asym)
1135 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
1136   by (rule filtermap_fun_inverse[symmetric, of uminus])
1137      (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
1139 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
1140   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
1142 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
1143   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
1145 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
1146   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
1148 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
1149   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
1150   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
1151   by auto
1153 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
1154   unfolding filterlim_uminus_at_top by simp
1156 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
1157   unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
1158 proof safe
1159   fix Z :: real assume [arith]: "0 < Z"
1160   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
1161     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
1162   then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
1163     by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
1164 qed
1166 lemma tendsto_inverse_0:
1167   fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
1168   shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
1169   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
1170 proof safe
1171   fix r :: real assume "0 < r"
1172   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
1173   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
1174     fix x :: 'a
1175     from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
1176     also assume *: "inverse (r / 2) \<le> norm x"
1177     finally show "norm (inverse x) < r"
1178       using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
1179   qed
1180 qed
1183   assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
1184   assumes "filterlim g at_infinity F"
1185   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
1186 proof (subst filterlim_at_infinity[OF order_refl], safe)
1187   fix r :: real assume r: "r > 0"
1188   from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
1189   hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
1190   moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
1191   with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
1192     unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
1193   ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
1194   proof eventually_elim
1195     fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
1196     from A B have "r \<le> norm (g x) - norm (f x)" by simp
1197     also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
1198     finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
1199   qed
1200 qed
1203   assumes "filterlim f at_infinity F"
1204   assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
1205   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
1206   by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
1208 lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
1209   unfolding filterlim_at
1210   by (auto simp: eventually_at_top_dense)
1211      (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
1213 lemma filterlim_inverse_at_top:
1214   "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
1215   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
1216      (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
1218 lemma filterlim_inverse_at_bot_neg:
1219   "LIM x (at_left (0::real)). inverse x :> at_bot"
1220   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
1222 lemma filterlim_inverse_at_bot:
1223   "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
1224   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
1225   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
1227 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
1228   by (intro filtermap_fun_inverse[symmetric, where g=inverse])
1229      (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
1231 lemma eventually_at_right_to_top:
1232   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
1233   unfolding at_right_to_top eventually_filtermap ..
1235 lemma filterlim_at_right_to_top:
1236   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
1237   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
1239 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
1240   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
1242 lemma eventually_at_top_to_right:
1243   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
1244   unfolding at_top_to_right eventually_filtermap ..
1246 lemma filterlim_at_top_to_right:
1247   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
1248   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
1250 lemma filterlim_inverse_at_infinity:
1251   fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
1252   shows "filterlim inverse at_infinity (at (0::'a))"
1253   unfolding filterlim_at_infinity[OF order_refl]
1254 proof safe
1255   fix r :: real assume "0 < r"
1256   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
1257     unfolding eventually_at norm_inverse
1258     by (intro exI[of _ "inverse r"])
1259        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
1260 qed
1262 lemma filterlim_inverse_at_iff:
1263   fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
1264   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
1265   unfolding filterlim_def filtermap_filtermap[symmetric]
1266 proof
1267   assume "filtermap g F \<le> at_infinity"
1268   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
1269     by (rule filtermap_mono)
1270   also have "\<dots> \<le> at 0"
1271     using tendsto_inverse_0[where 'a='b]
1272     by (auto intro!: exI[of _ 1]
1273              simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
1274   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
1275 next
1276   assume "filtermap inverse (filtermap g F) \<le> at 0"
1277   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
1278     by (rule filtermap_mono)
1279   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
1280     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
1281 qed
1283 lemma tendsto_mult_filterlim_at_infinity:
1284   assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
1285   assumes "filterlim g at_infinity F"
1286   shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
1287 proof -
1288   have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
1289     by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
1290   hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
1291     unfolding filterlim_at using assms
1292     by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
1293   thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
1294 qed
1296 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
1297  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
1299 lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
1300   by (rule filterlim_subseq) (auto simp: subseq_def)
1302 lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
1303   by (rule filterlim_subseq) (auto simp: subseq_def)
1305 lemma at_to_infinity:
1306   fixes x :: "'a :: {real_normed_field,field}"
1307   shows "(at (0::'a)) = filtermap inverse at_infinity"
1308 proof (rule antisym)
1309   have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
1310     by (fact tendsto_inverse_0)
1311   then show "filtermap inverse at_infinity \<le> at (0::'a)"
1312     apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
1313     apply (rule_tac x="1" in exI, auto)
1314     done
1315 next
1316   have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
1317     using filterlim_inverse_at_infinity unfolding filterlim_def
1318     by (rule filtermap_mono)
1319   then show "at (0::'a) \<le> filtermap inverse at_infinity"
1320     by (simp add: filtermap_ident filtermap_filtermap)
1321 qed
1323 lemma lim_at_infinity_0:
1324   fixes l :: "'a :: {real_normed_field,field}"
1325   shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
1326 by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
1328 lemma lim_zero_infinity:
1329   fixes l :: "'a :: {real_normed_field,field}"
1330   shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
1331 by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
1334 text \<open>
1336 We only show rules for multiplication and addition when the functions are either against a real
1337 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
1339 \<close>
1341 lemma filterlim_tendsto_pos_mult_at_top:
1342   assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
1343   assumes g: "LIM x F. g x :> at_top"
1344   shows "LIM x F. (f x * g x :: real) :> at_top"
1345   unfolding filterlim_at_top_gt[where c=0]
1346 proof safe
1347   fix Z :: real assume "0 < Z"
1348   from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
1349     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
1350              simp: dist_real_def abs_real_def split: split_if_asm)
1351   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
1352     unfolding filterlim_at_top by auto
1353   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
1354   proof eventually_elim
1355     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
1356     with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
1357       by (intro mult_mono) (auto simp: zero_le_divide_iff)
1358     with \<open>0 < c\<close> show "Z \<le> f x * g x"
1359        by simp
1360   qed
1361 qed
1363 lemma filterlim_at_top_mult_at_top:
1364   assumes f: "LIM x F. f x :> at_top"
1365   assumes g: "LIM x F. g x :> at_top"
1366   shows "LIM x F. (f x * g x :: real) :> at_top"
1367   unfolding filterlim_at_top_gt[where c=0]
1368 proof safe
1369   fix Z :: real assume "0 < Z"
1370   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
1371     unfolding filterlim_at_top by auto
1372   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
1373     unfolding filterlim_at_top by auto
1374   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
1375   proof eventually_elim
1376     fix x assume "1 \<le> f x" "Z \<le> g x"
1377     with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
1378       by (intro mult_mono) (auto simp: zero_le_divide_iff)
1379     then show "Z \<le> f x * g x"
1380        by simp
1381   qed
1382 qed
1384 lemma filterlim_tendsto_pos_mult_at_bot:
1385   assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
1386   shows "LIM x F. f x * g x :> at_bot"
1387   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
1388   unfolding filterlim_uminus_at_bot by simp
1390 lemma filterlim_tendsto_neg_mult_at_bot:
1391   assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
1392   shows "LIM x F. f x * g x :> at_bot"
1393   using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
1394   unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
1396 lemma filterlim_pow_at_top:
1397   fixes f :: "real \<Rightarrow> real"
1398   assumes "0 < n" and f: "LIM x F. f x :> at_top"
1399   shows "LIM x F. (f x)^n :: real :> at_top"
1400 using \<open>0 < n\<close> proof (induct n)
1401   case (Suc n) with f show ?case
1402     by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
1403 qed simp
1405 lemma filterlim_pow_at_bot_even:
1406   fixes f :: "real \<Rightarrow> real"
1407   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
1408   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
1410 lemma filterlim_pow_at_bot_odd:
1411   fixes f :: "real \<Rightarrow> real"
1412   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
1413   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
1416   assumes f: "(f \<longlongrightarrow> c) F"
1417   assumes g: "LIM x F. g x :> at_top"
1418   shows "LIM x F. (f x + g x :: real) :> at_top"
1419   unfolding filterlim_at_top_gt[where c=0]
1420 proof safe
1421   fix Z :: real assume "0 < Z"
1422   from f have "eventually (\<lambda>x. c - 1 < f x) F"
1423     by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
1424   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
1425     unfolding filterlim_at_top by auto
1426   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
1427     by eventually_elim simp
1428 qed
1430 lemma LIM_at_top_divide:
1431   fixes f g :: "'a \<Rightarrow> real"
1432   assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
1433   assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
1434   shows "LIM x F. f x / g x :> at_top"
1435   unfolding divide_inverse
1436   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
1439   assumes f: "LIM x F. f x :> at_top"
1440   assumes g: "LIM x F. g x :> at_top"
1441   shows "LIM x F. (f x + g x :: real) :> at_top"
1442   unfolding filterlim_at_top_gt[where c=0]
1443 proof safe
1444   fix Z :: real assume "0 < Z"
1445   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
1446     unfolding filterlim_at_top by auto
1447   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
1448     unfolding filterlim_at_top by auto
1449   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
1450     by eventually_elim simp
1451 qed
1453 lemma tendsto_divide_0:
1454   fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
1455   assumes f: "(f \<longlongrightarrow> c) F"
1456   assumes g: "LIM x F. g x :> at_infinity"
1457   shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
1458   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
1460 lemma linear_plus_1_le_power:
1461   fixes x :: real
1462   assumes x: "0 \<le> x"
1463   shows "real n * x + 1 \<le> (x + 1) ^ n"
1464 proof (induct n)
1465   case (Suc n)
1466   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
1467     by (simp add: field_simps of_nat_Suc x)
1468   also have "\<dots> \<le> (x + 1)^Suc n"
1469     using Suc x by (simp add: mult_left_mono)
1470   finally show ?case .
1471 qed simp
1473 lemma filterlim_realpow_sequentially_gt1:
1474   fixes x :: "'a :: real_normed_div_algebra"
1475   assumes x[arith]: "1 < norm x"
1476   shows "LIM n sequentially. x ^ n :> at_infinity"
1477 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
1478   fix y :: real assume "0 < y"
1479   have "0 < norm x - 1" by simp
1480   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
1481   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
1482   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
1483   also have "\<dots> = norm x ^ N" by simp
1484   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
1485     by (metis order_less_le_trans power_increasing order_less_imp_le x)
1486   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
1487     unfolding eventually_sequentially
1488     by (auto simp: norm_power)
1489 qed simp
1492 subsection \<open>Limits of Sequences\<close>
1494 lemma [trans]: "X = Y \<Longrightarrow> Y \<longlonglongrightarrow> z \<Longrightarrow> X \<longlonglongrightarrow> z"
1495   by simp
1497 lemma LIMSEQ_iff:
1498   fixes L :: "'a::real_normed_vector"
1499   shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
1500 unfolding lim_sequentially dist_norm ..
1502 lemma LIMSEQ_I:
1503   fixes L :: "'a::real_normed_vector"
1504   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
1505 by (simp add: LIMSEQ_iff)
1507 lemma LIMSEQ_D:
1508   fixes L :: "'a::real_normed_vector"
1509   shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
1510 by (simp add: LIMSEQ_iff)
1512 lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
1513   unfolding tendsto_def eventually_sequentially
1514   by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
1516 lemma Bseq_inverse_lemma:
1517   fixes x :: "'a::real_normed_div_algebra"
1518   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
1519 apply (subst nonzero_norm_inverse, clarsimp)
1520 apply (erule (1) le_imp_inverse_le)
1521 done
1523 lemma Bseq_inverse:
1524   fixes a :: "'a::real_normed_div_algebra"
1525   shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
1526   by (rule Bfun_inverse)
1528 text\<open>Transformation of limit.\<close>
1530 lemma eventually_at2:
1531   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1532   unfolding eventually_at by auto
1534 lemma Lim_transform:
1535   fixes a b :: "'a::real_normed_vector"
1536   shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
1537   using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
1539 lemma Lim_transform2:
1540   fixes a b :: "'a::real_normed_vector"
1541   shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
1542   by (erule Lim_transform) (simp add: tendsto_minus_cancel)
1544 lemma Lim_transform_eventually:
1545   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
1546   apply (rule topological_tendstoI)
1547   apply (drule (2) topological_tendstoD)
1548   apply (erule (1) eventually_elim2, simp)
1549   done
1551 lemma Lim_transform_within:
1552   assumes "(f \<longlongrightarrow> l) (at x within S)"
1553     and "0 < d"
1554     and "\<And>x'. \<lbrakk>x'\<in>S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
1555   shows "(g \<longlongrightarrow> l) (at x within S)"
1556 proof (rule Lim_transform_eventually)
1557   show "eventually (\<lambda>x. f x = g x) (at x within S)"
1558     using assms by (auto simp: eventually_at)
1559   show "(f \<longlongrightarrow> l) (at x within S)" by fact
1560 qed
1562 text\<open>Common case assuming being away from some crucial point like 0.\<close>
1564 lemma Lim_transform_away_within:
1565   fixes a b :: "'a::t1_space"
1566   assumes "a \<noteq> b"
1567     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1568     and "(f \<longlongrightarrow> l) (at a within S)"
1569   shows "(g \<longlongrightarrow> l) (at a within S)"
1570 proof (rule Lim_transform_eventually)
1571   show "(f \<longlongrightarrow> l) (at a within S)" by fact
1572   show "eventually (\<lambda>x. f x = g x) (at a within S)"
1573     unfolding eventually_at_topological
1574     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1575 qed
1577 lemma Lim_transform_away_at:
1578   fixes a b :: "'a::t1_space"
1579   assumes ab: "a\<noteq>b"
1580     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1581     and fl: "(f \<longlongrightarrow> l) (at a)"
1582   shows "(g \<longlongrightarrow> l) (at a)"
1583   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
1585 text\<open>Alternatively, within an open set.\<close>
1587 lemma Lim_transform_within_open:
1588   assumes "(f \<longlongrightarrow> l) (at a within T)"
1589     and "open s" and "a \<in> s"
1590     and "\<And>x. \<lbrakk>x\<in>s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
1591   shows "(g \<longlongrightarrow> l) (at a within T)"
1592 proof (rule Lim_transform_eventually)
1593   show "eventually (\<lambda>x. f x = g x) (at a within T)"
1594     unfolding eventually_at_topological
1595     using assms by auto
1596   show "(f \<longlongrightarrow> l) (at a within T)" by fact
1597 qed
1599 text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
1601 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1603 lemma Lim_cong_within(*[cong add]*):
1604   assumes "a = b"
1605     and "x = y"
1606     and "S = T"
1607     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1608   shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
1609   unfolding tendsto_def eventually_at_topological
1610   using assms by simp
1612 lemma Lim_cong_at(*[cong add]*):
1613   assumes "a = b" "x = y"
1614     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1615   shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
1616   unfolding tendsto_def eventually_at_topological
1617   using assms by simp
1618 text\<open>An unbounded sequence's inverse tends to 0\<close>
1620 lemma LIMSEQ_inverse_zero:
1621   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
1622   apply (rule filterlim_compose[OF tendsto_inverse_0])
1623   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
1624   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
1625   done
1627 text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
1629 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
1630   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
1631             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
1633 text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
1634 infinity is now easily proved\<close>
1637      "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
1638   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
1641      "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
1642   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
1643   by auto
1646      "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
1647   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
1648   by auto
1650 lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
1651   using lim_1_over_n by (simp add: inverse_eq_divide)
1653 lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
1654 proof (rule Lim_transform_eventually)
1655   show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
1656     using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
1657   have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
1658     by (intro tendsto_add tendsto_const lim_inverse_n)
1659   thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
1660 qed
1662 lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
1663 proof (rule Lim_transform_eventually)
1664   show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
1665                         of_nat n / of_nat (Suc n)) sequentially"
1666     using eventually_gt_at_top[of "0::nat"]
1667     by eventually_elim (simp add: field_simps del: of_nat_Suc)
1668   have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
1669     by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
1670   thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
1671 qed
1673 subsection \<open>Convergence on sequences\<close>
1675 lemma convergent_cong:
1676   assumes "eventually (\<lambda>x. f x = g x) sequentially"
1677   shows   "convergent f \<longleftrightarrow> convergent g"
1678   unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
1680 lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
1681   by (auto simp: convergent_def LIMSEQ_Suc_iff)
1683 lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
1684 proof (induction m arbitrary: f)
1685   case (Suc m)
1686   have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
1687   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
1688   also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
1689   finally show ?case .
1690 qed simp_all
1693   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
1694   assumes "convergent (\<lambda>n. X n)"
1695   assumes "convergent (\<lambda>n. Y n)"
1696   shows "convergent (\<lambda>n. X n + Y n)"
1697   using assms unfolding convergent_def by (blast intro: tendsto_add)
1699 lemma convergent_setsum:
1700   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
1701   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
1702   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
1703 proof (cases "finite A")
1704   case True from this and assms show ?thesis
1705     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
1706 qed (simp add: convergent_const)
1708 lemma (in bounded_linear) convergent:
1709   assumes "convergent (\<lambda>n. X n)"
1710   shows "convergent (\<lambda>n. f (X n))"
1711   using assms unfolding convergent_def by (blast intro: tendsto)
1713 lemma (in bounded_bilinear) convergent:
1714   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
1715   shows "convergent (\<lambda>n. X n ** Y n)"
1716   using assms unfolding convergent_def by (blast intro: tendsto)
1718 lemma convergent_minus_iff:
1719   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1720   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
1721 apply (simp add: convergent_def)
1722 apply (auto dest: tendsto_minus)
1723 apply (drule tendsto_minus, auto)
1724 done
1726 lemma convergent_diff:
1727   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
1728   assumes "convergent (\<lambda>n. X n)"
1729   assumes "convergent (\<lambda>n. Y n)"
1730   shows "convergent (\<lambda>n. X n - Y n)"
1731   using assms unfolding convergent_def by (blast intro: tendsto_diff)
1733 lemma convergent_norm:
1734   assumes "convergent f"
1735   shows   "convergent (\<lambda>n. norm (f n))"
1736 proof -
1737   from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
1738   hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
1739   thus ?thesis by (auto simp: convergent_def)
1740 qed
1742 lemma convergent_of_real:
1743   "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
1744   unfolding convergent_def by (blast intro!: tendsto_of_real)
1747   "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
1748 proof
1749   assume "convergent (\<lambda>n. c + f n)"
1750   from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
1751 next
1752   assume "convergent f"
1753   from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
1754 qed
1757   "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
1760 lemma convergent_diff_const_right_iff:
1761   "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
1764 lemma convergent_mult:
1765   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
1766   assumes "convergent (\<lambda>n. X n)"
1767   assumes "convergent (\<lambda>n. Y n)"
1768   shows "convergent (\<lambda>n. X n * Y n)"
1769   using assms unfolding convergent_def by (blast intro: tendsto_mult)
1771 lemma convergent_mult_const_iff:
1772   assumes "c \<noteq> 0"
1773   shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
1774 proof
1775   assume "convergent (\<lambda>n. c * f n)"
1776   from assms convergent_mult[OF this convergent_const[of "inverse c"]]
1777     show "convergent f" by (simp add: field_simps)
1778 next
1779   assume "convergent f"
1780   from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
1781 qed
1783 lemma convergent_mult_const_right_iff:
1784   assumes "c \<noteq> 0"
1785   shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
1786   using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
1788 lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
1789   by (simp add: Cauchy_Bseq convergent_Cauchy)
1792 text \<open>A monotone sequence converges to its least upper bound.\<close>
1794 lemma LIMSEQ_incseq_SUP:
1795   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
1796   assumes u: "bdd_above (range X)"
1797   assumes X: "incseq X"
1798   shows "X \<longlonglongrightarrow> (SUP i. X i)"
1799   by (rule order_tendstoI)
1800      (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
1802 lemma LIMSEQ_decseq_INF:
1803   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
1804   assumes u: "bdd_below (range X)"
1805   assumes X: "decseq X"
1806   shows "X \<longlonglongrightarrow> (INF i. X i)"
1807   by (rule order_tendstoI)
1808      (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
1810 text\<open>Main monotonicity theorem\<close>
1812 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
1813   by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
1815 lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
1816   by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
1818 lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
1819   using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
1821 lemma Bseq_monoseq_convergent'_inc:
1822   "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
1823   by (subst convergent_ignore_initial_segment [symmetric, of _ M])
1824      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
1826 lemma Bseq_monoseq_convergent'_dec:
1827   "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
1828   by (subst convergent_ignore_initial_segment [symmetric, of _ M])
1829      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
1831 lemma Cauchy_iff:
1832   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1833   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
1834   unfolding Cauchy_def dist_norm ..
1836 lemma CauchyI:
1837   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1838   shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
1839 by (simp add: Cauchy_iff)
1841 lemma CauchyD:
1842   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
1843   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
1844 by (simp add: Cauchy_iff)
1846 lemma incseq_convergent:
1847   fixes X :: "nat \<Rightarrow> real"
1848   assumes "incseq X" and "\<forall>i. X i \<le> B"
1849   obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
1850 proof atomize_elim
1851   from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
1852   obtain L where "X \<longlonglongrightarrow> L"
1853     by (auto simp: convergent_def monoseq_def incseq_def)
1854   with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
1855     by (auto intro!: exI[of _ L] incseq_le)
1856 qed
1858 lemma decseq_convergent:
1859   fixes X :: "nat \<Rightarrow> real"
1860   assumes "decseq X" and "\<forall>i. B \<le> X i"
1861   obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
1862 proof atomize_elim
1863   from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
1864   obtain L where "X \<longlonglongrightarrow> L"
1865     by (auto simp: convergent_def monoseq_def decseq_def)
1866   with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
1867     by (auto intro!: exI[of _ L] decseq_le)
1868 qed
1870 subsubsection \<open>Cauchy Sequences are Bounded\<close>
1872 text\<open>A Cauchy sequence is bounded -- this is the standard
1873   proof mechanization rather than the nonstandard proof\<close>
1875 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
1876           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
1877 apply (clarify, drule spec, drule (1) mp)
1878 apply (simp only: norm_minus_commute)
1879 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
1880 apply simp
1881 done
1883 subsection \<open>Power Sequences\<close>
1885 text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
1886 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
1887   also fact that bounded and monotonic sequence converges.\<close>
1889 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
1890 apply (simp add: Bseq_def)
1891 apply (rule_tac x = 1 in exI)
1892 apply (simp add: power_abs)
1893 apply (auto dest: power_mono)
1894 done
1896 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
1897 apply (clarify intro!: mono_SucI2)
1898 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
1899 done
1901 lemma convergent_realpow:
1902   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
1903 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
1905 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
1906   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
1908 lemma LIMSEQ_realpow_zero:
1909   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
1910 proof cases
1911   assume "0 \<le> x" and "x \<noteq> 0"
1912   hence x0: "0 < x" by simp
1913   assume x1: "x < 1"
1914   from x0 x1 have "1 < inverse x"
1915     by (rule one_less_inverse)
1916   hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
1917     by (rule LIMSEQ_inverse_realpow_zero)
1918   thus ?thesis by (simp add: power_inverse)
1919 qed (rule LIMSEQ_imp_Suc, simp)
1921 lemma LIMSEQ_power_zero:
1922   fixes x :: "'a::{real_normed_algebra_1}"
1923   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
1924 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
1925 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
1926 apply (simp add: power_abs norm_power_ineq)
1927 done
1929 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
1930   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
1932 text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
1934 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
1935   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
1937 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
1938   by (rule LIMSEQ_power_zero) simp
1941 subsection \<open>Limits of Functions\<close>
1943 lemma LIM_eq:
1944   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1945   shows "f \<midarrow>a\<rightarrow> L =
1946      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
1947 by (simp add: LIM_def dist_norm)
1949 lemma LIM_I:
1950   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1951   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
1952       ==> f \<midarrow>a\<rightarrow> L"
1953 by (simp add: LIM_eq)
1955 lemma LIM_D:
1956   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
1957   shows "[| f \<midarrow>a\<rightarrow> L; 0<r |]
1958       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
1959 by (simp add: LIM_eq)
1961 lemma LIM_offset:
1962   fixes a :: "'a::real_normed_vector"
1963   shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
1964   unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
1966 lemma LIM_offset_zero:
1967   fixes a :: "'a::real_normed_vector"
1968   shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
1969 by (drule_tac k="a" in LIM_offset, simp add: add.commute)
1971 lemma LIM_offset_zero_cancel:
1972   fixes a :: "'a::real_normed_vector"
1973   shows "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
1974 by (drule_tac k="- a" in LIM_offset, simp)
1976 lemma LIM_offset_zero_iff:
1977   fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
1978   shows  "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
1979   using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
1981 lemma LIM_zero:
1982   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1983   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
1984 unfolding tendsto_iff dist_norm by simp
1986 lemma LIM_zero_cancel:
1987   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1988   shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
1989 unfolding tendsto_iff dist_norm by simp
1991 lemma LIM_zero_iff:
1992   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1993   shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
1994 unfolding tendsto_iff dist_norm by simp
1996 lemma LIM_imp_LIM:
1997   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
1998   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
1999   assumes f: "f \<midarrow>a\<rightarrow> l"
2000   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
2001   shows "g \<midarrow>a\<rightarrow> m"
2002   by (rule metric_LIM_imp_LIM [OF f],
2003     simp add: dist_norm le)
2005 lemma LIM_equal2:
2006   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
2007   assumes 1: "0 < R"
2008   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
2009   shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
2010 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
2012 lemma LIM_compose2:
2013   fixes a :: "'a::real_normed_vector"
2014   assumes f: "f \<midarrow>a\<rightarrow> b"
2015   assumes g: "g \<midarrow>b\<rightarrow> c"
2016   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
2017   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
2018 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
2020 lemma real_LIM_sandwich_zero:
2021   fixes f g :: "'a::topological_space \<Rightarrow> real"
2022   assumes f: "f \<midarrow>a\<rightarrow> 0"
2023   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
2024   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
2025   shows "g \<midarrow>a\<rightarrow> 0"
2026 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
2027   fix x assume x: "x \<noteq> a"
2028   have "norm (g x - 0) = g x" by (simp add: 1 x)
2029   also have "g x \<le> f x" by (rule 2 [OF x])
2030   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
2031   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
2032   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
2033 qed
2036 subsection \<open>Continuity\<close>
2038 lemma LIM_isCont_iff:
2039   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
2040   shows "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
2041 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
2043 lemma isCont_iff:
2044   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
2045   shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
2046 by (simp add: isCont_def LIM_isCont_iff)
2048 lemma isCont_LIM_compose2:
2049   fixes a :: "'a::real_normed_vector"
2050   assumes f [unfolded isCont_def]: "isCont f a"
2051   assumes g: "g \<midarrow>f a\<rightarrow> l"
2052   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
2053   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
2054 by (rule LIM_compose2 [OF f g inj])
2057 lemma isCont_norm [simp]:
2058   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
2059   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
2060   by (fact continuous_norm)
2062 lemma isCont_rabs [simp]:
2063   fixes f :: "'a::t2_space \<Rightarrow> real"
2064   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
2065   by (fact continuous_rabs)
2067 lemma isCont_add [simp]:
2068   fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
2069   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
2070   by (fact continuous_add)
2072 lemma isCont_minus [simp]:
2073   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
2074   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
2075   by (fact continuous_minus)
2077 lemma isCont_diff [simp]:
2078   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
2079   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
2080   by (fact continuous_diff)
2082 lemma isCont_mult [simp]:
2083   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
2084   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
2085   by (fact continuous_mult)
2087 lemma (in bounded_linear) isCont:
2088   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
2089   by (fact continuous)
2091 lemma (in bounded_bilinear) isCont:
2092   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
2093   by (fact continuous)
2095 lemmas isCont_scaleR [simp] =
2096   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
2098 lemmas isCont_of_real [simp] =
2099   bounded_linear.isCont [OF bounded_linear_of_real]
2101 lemma isCont_power [simp]:
2102   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
2103   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
2104   by (fact continuous_power)
2106 lemma isCont_setsum [simp]:
2107   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
2108   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
2109   by (auto intro: continuous_setsum)
2111 subsection \<open>Uniform Continuity\<close>
2113 definition
2114   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
2115   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
2117 lemma isUCont_isCont: "isUCont f ==> isCont f x"
2118 by (simp add: isUCont_def isCont_def LIM_def, force)
2120 lemma isUCont_Cauchy:
2121   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
2122 unfolding isUCont_def
2123 apply (rule metric_CauchyI)
2124 apply (drule_tac x=e in spec, safe)
2125 apply (drule_tac e=s in metric_CauchyD, safe)
2126 apply (rule_tac x=M in exI, simp)
2127 done
2129 lemma (in bounded_linear) isUCont: "isUCont f"
2130 unfolding isUCont_def dist_norm
2131 proof (intro allI impI)
2132   fix r::real assume r: "0 < r"
2133   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
2134     using pos_bounded by blast
2135   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
2136   proof (rule exI, safe)
2137     from r K show "0 < r / K" by simp
2138   next
2139     fix x y :: 'a
2140     assume xy: "norm (x - y) < r / K"
2141     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
2142     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
2143     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
2144     finally show "norm (f x - f y) < r" .
2145   qed
2146 qed
2148 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
2149 by (rule isUCont [THEN isUCont_Cauchy])
2151 lemma LIM_less_bound:
2152   fixes f :: "real \<Rightarrow> real"
2153   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
2154   shows "0 \<le> f x"
2155 proof (rule tendsto_le_const)
2156   show "(f \<longlongrightarrow> f x) (at_left x)"
2157     using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
2158   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
2159     using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
2160 qed simp
2163 subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
2165 lemma nested_sequence_unique:
2166   assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
2167   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
2168 proof -
2169   have "incseq f" unfolding incseq_Suc_iff by fact
2170   have "decseq g" unfolding decseq_Suc_iff by fact
2172   { fix n
2173     from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
2174     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
2175   then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
2176     using incseq_convergent[OF \<open>incseq f\<close>] by auto
2177   moreover
2178   { fix n
2179     from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
2180     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
2181   then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
2182     using decseq_convergent[OF \<open>decseq g\<close>] by auto
2183   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
2184   ultimately show ?thesis by auto
2185 qed
2187 lemma Bolzano[consumes 1, case_names trans local]:
2188   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
2189   assumes [arith]: "a \<le> b"
2190   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
2191   assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
2192   shows "P a b"
2193 proof -
2194   def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
2195   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
2196   have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
2197     and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
2198     by (simp_all add: l_def u_def bisect_def split: prod.split)
2200   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
2202   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
2203   proof (safe intro!: nested_sequence_unique)
2204     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
2205   next
2206     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
2207     then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
2208   qed fact
2209   then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
2210   obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
2211     using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
2213   show "P a b"
2214   proof (rule ccontr)
2215     assume "\<not> P a b"
2216     { fix n have "\<not> P (l n) (u n)"
2217       proof (induct n)
2218         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
2219       qed (simp add: \<open>\<not> P a b\<close>) }
2220     moreover
2221     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
2222         using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
2223       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
2224         using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
2225       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
2226       proof eventually_elim
2227         fix n assume "x - d / 2 < l n" "u n < x + d / 2"
2228         from add_strict_mono[OF this] have "u n - l n < d" by simp
2229         with x show "P (l n) (u n)" by (rule d)
2230       qed }
2231     ultimately show False by simp
2232   qed
2233 qed
2235 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
2236 proof (cases "a \<le> b", rule compactI)
2237   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
2238   def T == "{a .. b}"
2239   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
2240   proof (induct rule: Bolzano)
2241     case (trans a b c)
2242     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
2243     from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
2244       by (auto simp: *)
2245     with trans show ?case
2246       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
2247   next
2248     case (local x)
2249     then have "x \<in> \<Union>C" using C by auto
2250     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
2251     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
2252       by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
2253     with \<open>c \<in> C\<close> show ?case
2254       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
2255   qed
2256 qed simp
2259 lemma continuous_image_closed_interval:
2260   fixes a b and f :: "real \<Rightarrow> real"
2261   defines "S \<equiv> {a..b}"
2262   assumes "a \<le> b" and f: "continuous_on S f"
2263   shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
2264 proof -
2265   have S: "compact S" "S \<noteq> {}"
2266     using \<open>a \<le> b\<close> by (auto simp: S_def)
2267   obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
2268     using continuous_attains_sup[OF S f] by auto
2269   moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
2270     using continuous_attains_inf[OF S f] by auto
2271   moreover have "connected (f`S)"
2272     using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
2273   ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
2274     by (auto simp: connected_iff_interval)
2275   then show ?thesis
2276     by auto
2277 qed
2279 lemma open_Collect_positive:
2280  fixes f :: "'a::t2_space \<Rightarrow> real"
2281  assumes f: "continuous_on s f"
2282  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
2283  using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
2284  by (auto simp: Int_def field_simps)
2286 lemma open_Collect_less_Int:
2287  fixes f g :: "'a::t2_space \<Rightarrow> real"
2288  assumes f: "continuous_on s f" and g: "continuous_on s g"
2289  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
2290  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
2293 subsection \<open>Boundedness of continuous functions\<close>
2295 text\<open>By bisection, function continuous on closed interval is bounded above\<close>
2297 lemma isCont_eq_Ub:
2298   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2299   shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2300     \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
2301   using continuous_attains_sup[of "{a .. b}" f]
2302   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
2304 lemma isCont_eq_Lb:
2305   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2306   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2307     \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
2308   using continuous_attains_inf[of "{a .. b}" f]
2309   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
2311 lemma isCont_bounded:
2312   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2313   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
2314   using isCont_eq_Ub[of a b f] by auto
2316 lemma isCont_has_Ub:
2317   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
2318   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
2319     \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
2320   using isCont_eq_Ub[of a b f] by auto
2322 (*HOL style here: object-level formulations*)
2323 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
2324       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
2325       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
2326   by (blast intro: IVT)
2328 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
2329       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
2330       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
2331   by (blast intro: IVT2)
2333 lemma isCont_Lb_Ub:
2334   fixes f :: "real \<Rightarrow> real"
2335   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
2336   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
2337                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
2338 proof -
2339   obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
2340     using isCont_eq_Ub[OF assms] by auto
2341   obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
2342     using isCont_eq_Lb[OF assms] by auto
2343   show ?thesis
2344     using IVT[of f L _ M] IVT2[of f L _ M] M L assms
2345     apply (rule_tac x="f L" in exI)
2346     apply (rule_tac x="f M" in exI)
2347     apply (cases "L \<le> M")
2348     apply (simp, metis order_trans)
2349     apply (simp, metis order_trans)
2350     done
2351 qed
2354 text\<open>Continuity of inverse function\<close>
2356 lemma isCont_inverse_function:
2357   fixes f g :: "real \<Rightarrow> real"
2358   assumes d: "0 < d"
2359       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
2360       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
2361   shows "isCont g (f x)"
2362 proof -
2363   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
2365   have f: "continuous_on ?D f"
2366     using cont by (intro continuous_at_imp_continuous_on ballI) auto
2367   then have g: "continuous_on (f`?D) g"
2368     using inj by (intro continuous_on_inv) auto
2370   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
2371     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
2372   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
2373     by (rule continuous_on_subset)
2374   moreover
2375   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
2376     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
2377   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
2378     by auto
2379   ultimately
2380   show ?thesis
2381     by (simp add: continuous_on_eq_continuous_at)
2382 qed
2384 lemma isCont_inverse_function2:
2385   fixes f g :: "real \<Rightarrow> real" shows
2386   "\<lbrakk>a < x; x < b;
2387     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
2388     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
2389    \<Longrightarrow> isCont g (f x)"
2390 apply (rule isCont_inverse_function
2391        [where f=f and d="min (x - a) (b - x)"])
2392 apply (simp_all add: abs_le_iff)
2393 done
2395 (* need to rename second isCont_inverse *)
2397 lemma isCont_inv_fun:
2398   fixes f g :: "real \<Rightarrow> real"
2399   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
2400          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
2401       ==> isCont g (f x)"
2402 by (rule isCont_inverse_function)
2404 text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>
2405 lemma LIM_fun_gt_zero:
2406   fixes f :: "real \<Rightarrow> real"
2407   shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
2408 apply (drule (1) LIM_D, clarify)
2409 apply (rule_tac x = s in exI)
2410 apply (simp add: abs_less_iff)
2411 done
2413 lemma LIM_fun_less_zero:
2414   fixes f :: "real \<Rightarrow> real"
2415   shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
2416 apply (drule LIM_D [where r="-l"], simp, clarify)
2417 apply (rule_tac x = s in exI)
2418 apply (simp add: abs_less_iff)
2419 done
2421 lemma LIM_fun_not_zero:
2422   fixes f :: "real \<Rightarrow> real"
2423   shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
2424   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
2426 end