src/HOL/Library/Product_Vector.thy
 author bulwahn Tue Apr 05 09:38:28 2011 +0200 (2011-04-05) changeset 42231 bc1891226d00 parent 37678 0040bafffdef child 44066 d74182c93f04 permissions -rw-r--r--
removing bounded_forall code equation for characters when loading Code_Char
1 (*  Title:      HOL/Library/Product_Vector.thy
2     Author:     Brian Huffman
3 *)
5 header {* Cartesian Products as Vector Spaces *}
7 theory Product_Vector
8 imports Inner_Product Product_plus
9 begin
11 subsection {* Product is a real vector space *}
13 instantiation prod :: (real_vector, real_vector) real_vector
14 begin
16 definition scaleR_prod_def:
17   "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
19 lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
20   unfolding scaleR_prod_def by simp
22 lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
23   unfolding scaleR_prod_def by simp
25 lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
26   unfolding scaleR_prod_def by simp
28 instance proof
29   fix a b :: real and x y :: "'a \<times> 'b"
30   show "scaleR a (x + y) = scaleR a x + scaleR a y"
31     by (simp add: expand_prod_eq scaleR_right_distrib)
32   show "scaleR (a + b) x = scaleR a x + scaleR b x"
33     by (simp add: expand_prod_eq scaleR_left_distrib)
34   show "scaleR a (scaleR b x) = scaleR (a * b) x"
35     by (simp add: expand_prod_eq)
36   show "scaleR 1 x = x"
37     by (simp add: expand_prod_eq)
38 qed
40 end
42 subsection {* Product is a topological space *}
44 instantiation prod :: (topological_space, topological_space) topological_space
45 begin
47 definition open_prod_def:
48   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
49     (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
51 lemma open_prod_elim:
52   assumes "open S" and "x \<in> S"
53   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
54 using assms unfolding open_prod_def by fast
56 lemma open_prod_intro:
57   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
58   shows "open S"
59 using assms unfolding open_prod_def by fast
61 instance proof
62   show "open (UNIV :: ('a \<times> 'b) set)"
63     unfolding open_prod_def by auto
64 next
65   fix S T :: "('a \<times> 'b) set"
66   assume "open S" "open T"
67   show "open (S \<inter> T)"
68   proof (rule open_prod_intro)
69     fix x assume x: "x \<in> S \<inter> T"
70     from x have "x \<in> S" by simp
71     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
72       using `open S` and `x \<in> S` by (rule open_prod_elim)
73     from x have "x \<in> T" by simp
74     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
75       using `open T` and `x \<in> T` by (rule open_prod_elim)
76     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
77     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
78       using A B by (auto simp add: open_Int)
79     thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
80       by fast
81   qed
82 next
83   fix K :: "('a \<times> 'b) set set"
84   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
85     unfolding open_prod_def by fast
86 qed
88 end
90 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
91 unfolding open_prod_def by auto
93 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
94 by auto
96 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
97 by auto
99 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
100 by (simp add: fst_vimage_eq_Times open_Times)
102 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
103 by (simp add: snd_vimage_eq_Times open_Times)
105 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
106 unfolding closed_open vimage_Compl [symmetric]
107 by (rule open_vimage_fst)
109 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
110 unfolding closed_open vimage_Compl [symmetric]
111 by (rule open_vimage_snd)
113 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
114 proof -
115   have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
116   thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
117     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
118 qed
120 lemma openI: (* TODO: move *)
121   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
122   shows "open S"
123 proof -
124   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
125   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
126   ultimately show "open S" by simp
127 qed
129 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
130   unfolding image_def subset_eq by force
132 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
133   unfolding image_def subset_eq by force
135 lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
136 proof (rule openI)
137   fix x assume "x \<in> fst ` S"
138   then obtain y where "(x, y) \<in> S" by auto
139   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
140     using `open S` unfolding open_prod_def by auto
141   from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
142   with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
143   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
144 qed
146 lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
147 proof (rule openI)
148   fix y assume "y \<in> snd ` S"
149   then obtain x where "(x, y) \<in> S" by auto
150   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
151     using `open S` unfolding open_prod_def by auto
152   from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
153   with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
154   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
155 qed
157 subsection {* Product is a metric space *}
159 instantiation prod :: (metric_space, metric_space) metric_space
160 begin
162 definition dist_prod_def:
163   "dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
165 lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
166   unfolding dist_prod_def by simp
168 lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
169 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
171 lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
172 unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
174 instance proof
175   fix x y :: "'a \<times> 'b"
176   show "dist x y = 0 \<longleftrightarrow> x = y"
177     unfolding dist_prod_def expand_prod_eq by simp
178 next
179   fix x y z :: "'a \<times> 'b"
180   show "dist x y \<le> dist x z + dist y z"
181     unfolding dist_prod_def
182     by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
183         real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
184 next
185   (* FIXME: long proof! *)
186   (* Maybe it would be easier to define topological spaces *)
187   (* in terms of neighborhoods instead of open sets? *)
188   fix S :: "('a \<times> 'b) set"
189   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
190   proof
191     assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
192     proof
193       fix x assume "x \<in> S"
194       obtain A B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
195         using `open S` and `x \<in> S` by (rule open_prod_elim)
196       obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
197         using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
198       obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
199         using `open B` and `x \<in> A \<times> B` unfolding open_dist by auto
200       let ?e = "min r s"
201       have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> S)"
202       proof (intro allI impI conjI)
203         show "0 < min r s" by (simp add: r(1) s(1))
204       next
205         fix y assume "dist y x < min r s"
206         hence "dist y x < r" and "dist y x < s"
207           by simp_all
208         hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
209           by (auto intro: le_less_trans dist_fst_le dist_snd_le)
210         hence "fst y \<in> A" and "snd y \<in> B"
211           by (simp_all add: r(2) s(2))
212         hence "y \<in> A \<times> B" by (induct y, simp)
213         with `A \<times> B \<subseteq> S` show "y \<in> S" ..
214       qed
215       thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
216     qed
217   next
218     assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
219     unfolding open_prod_def open_dist
220     apply safe
221     apply (drule (1) bspec)
222     apply clarify
223     apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
224     apply clarify
225     apply (rule_tac x="{y. dist y a < r}" in exI)
226     apply (rule_tac x="{y. dist y b < s}" in exI)
227     apply (rule conjI)
228     apply clarify
229     apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
230     apply clarify
231     apply (simp add: less_diff_eq)
232     apply (erule le_less_trans [OF dist_triangle])
233     apply (rule conjI)
234     apply clarify
235     apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
236     apply clarify
237     apply (simp add: less_diff_eq)
238     apply (erule le_less_trans [OF dist_triangle])
239     apply (rule conjI)
240     apply simp
241     apply (clarify, rename_tac c d)
242     apply (drule spec, erule mp)
243     apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
244     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
245     apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
246     apply (simp add: power_divide)
247     done
248   qed
249 qed
251 end
253 subsection {* Continuity of operations *}
255 lemma tendsto_fst [tendsto_intros]:
256   assumes "(f ---> a) net"
257   shows "((\<lambda>x. fst (f x)) ---> fst a) net"
258 proof (rule topological_tendstoI)
259   fix S assume "open S" "fst a \<in> S"
260   then have "open (fst -` S)" "a \<in> fst -` S"
261     unfolding open_prod_def
262     apply simp_all
263     apply clarify
264     apply (rule exI, erule conjI)
265     apply (rule exI, rule conjI [OF open_UNIV])
266     apply auto
267     done
268   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
269     by (rule topological_tendstoD)
270   then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
271     by simp
272 qed
274 lemma tendsto_snd [tendsto_intros]:
275   assumes "(f ---> a) net"
276   shows "((\<lambda>x. snd (f x)) ---> snd a) net"
277 proof (rule topological_tendstoI)
278   fix S assume "open S" "snd a \<in> S"
279   then have "open (snd -` S)" "a \<in> snd -` S"
280     unfolding open_prod_def
281     apply simp_all
282     apply clarify
283     apply (rule exI, rule conjI [OF open_UNIV])
284     apply (rule exI, erule conjI)
285     apply auto
286     done
287   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
288     by (rule topological_tendstoD)
289   then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
290     by simp
291 qed
293 lemma tendsto_Pair [tendsto_intros]:
294   assumes "(f ---> a) net" and "(g ---> b) net"
295   shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
296 proof (rule topological_tendstoI)
297   fix S assume "open S" "(a, b) \<in> S"
298   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
299     unfolding open_prod_def by auto
300   have "eventually (\<lambda>x. f x \<in> A) net"
301     using `(f ---> a) net` `open A` `a \<in> A`
302     by (rule topological_tendstoD)
303   moreover
304   have "eventually (\<lambda>x. g x \<in> B) net"
305     using `(g ---> b) net` `open B` `b \<in> B`
306     by (rule topological_tendstoD)
307   ultimately
308   show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
309     by (rule eventually_elim2)
310        (simp add: subsetD [OF `A \<times> B \<subseteq> S`])
311 qed
313 lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
314 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
316 lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. snd (X n))"
317 unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])
319 lemma Cauchy_Pair:
320   assumes "Cauchy X" and "Cauchy Y"
321   shows "Cauchy (\<lambda>n. (X n, Y n))"
322 proof (rule metric_CauchyI)
323   fix r :: real assume "0 < r"
324   then have "0 < r / sqrt 2" (is "0 < ?s")
325     by (simp add: divide_pos_pos)
326   obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
327     using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
328   obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
329     using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
330   have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r"
331     using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
332   then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
333 qed
335 lemma isCont_Pair [simp]:
336   "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
337   unfolding isCont_def by (rule tendsto_Pair)
339 subsection {* Product is a complete metric space *}
341 instance prod :: (complete_space, complete_space) complete_space
342 proof
343   fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
344   have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
345     using Cauchy_fst [OF `Cauchy X`]
346     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
347   have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
348     using Cauchy_snd [OF `Cauchy X`]
349     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
350   have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
351     using tendsto_Pair [OF 1 2] by simp
352   then show "convergent X"
353     by (rule convergentI)
354 qed
356 subsection {* Product is a normed vector space *}
358 instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
359 begin
361 definition norm_prod_def:
362   "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
364 definition sgn_prod_def:
365   "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
367 lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
368   unfolding norm_prod_def by simp
370 instance proof
371   fix r :: real and x y :: "'a \<times> 'b"
372   show "0 \<le> norm x"
373     unfolding norm_prod_def by simp
374   show "norm x = 0 \<longleftrightarrow> x = 0"
375     unfolding norm_prod_def
376     by (simp add: expand_prod_eq)
377   show "norm (x + y) \<le> norm x + norm y"
378     unfolding norm_prod_def
379     apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
380     apply (simp add: add_mono power_mono norm_triangle_ineq)
381     done
382   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
383     unfolding norm_prod_def
384     apply (simp add: power_mult_distrib)
385     apply (simp add: right_distrib [symmetric])
386     apply (simp add: real_sqrt_mult_distrib)
387     done
388   show "sgn x = scaleR (inverse (norm x)) x"
389     by (rule sgn_prod_def)
390   show "dist x y = norm (x - y)"
391     unfolding dist_prod_def norm_prod_def
392     by (simp add: dist_norm)
393 qed
395 end
397 instance prod :: (banach, banach) banach ..
399 subsection {* Product is an inner product space *}
401 instantiation prod :: (real_inner, real_inner) real_inner
402 begin
404 definition inner_prod_def:
405   "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
407 lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
408   unfolding inner_prod_def by simp
410 instance proof
411   fix r :: real
412   fix x y z :: "'a::real_inner * 'b::real_inner"
413   show "inner x y = inner y x"
414     unfolding inner_prod_def
415     by (simp add: inner_commute)
416   show "inner (x + y) z = inner x z + inner y z"
417     unfolding inner_prod_def
419   show "inner (scaleR r x) y = r * inner x y"
420     unfolding inner_prod_def
421     by (simp add: right_distrib)
422   show "0 \<le> inner x x"
423     unfolding inner_prod_def
424     by (intro add_nonneg_nonneg inner_ge_zero)
425   show "inner x x = 0 \<longleftrightarrow> x = 0"
426     unfolding inner_prod_def expand_prod_eq
428   show "norm x = sqrt (inner x x)"
429     unfolding norm_prod_def inner_prod_def
430     by (simp add: power2_norm_eq_inner)
431 qed
433 end
435 subsection {* Pair operations are linear *}
437 interpretation fst: bounded_linear fst
438   apply (unfold_locales)
439   apply (rule fst_add)
440   apply (rule fst_scaleR)
441   apply (rule_tac x="1" in exI, simp add: norm_Pair)
442   done
444 interpretation snd: bounded_linear snd
445   apply (unfold_locales)
446   apply (rule snd_add)
447   apply (rule snd_scaleR)
448   apply (rule_tac x="1" in exI, simp add: norm_Pair)
449   done
451 text {* TODO: move to NthRoot *}
453   assumes x: "0 \<le> x" and y: "0 \<le> y"
454   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
455 apply (rule power2_le_imp_le)
456 apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
457 apply (simp add: mult_nonneg_nonneg x y)
458 apply (simp add: add_nonneg_nonneg x y)
459 done
461 lemma bounded_linear_Pair:
462   assumes f: "bounded_linear f"
463   assumes g: "bounded_linear g"
464   shows "bounded_linear (\<lambda>x. (f x, g x))"
465 proof
466   interpret f: bounded_linear f by fact
467   interpret g: bounded_linear g by fact
468   fix x y and r :: real
469   show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
471   show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
472     by (simp add: f.scaleR g.scaleR)
473   obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
474     using f.pos_bounded by fast
475   obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
476     using g.pos_bounded by fast
477   have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
478     apply (rule allI)
479     apply (simp add: norm_Pair)
480     apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
481     apply (simp add: right_distrib)
482     apply (rule add_mono [OF norm_f norm_g])
483     done
484   then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
485 qed
487 subsection {* Frechet derivatives involving pairs *}
489 lemma FDERIV_Pair:
490   assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
491   shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
492 apply (rule FDERIV_I)
493 apply (rule bounded_linear_Pair)
494 apply (rule FDERIV_bounded_linear [OF f])
495 apply (rule FDERIV_bounded_linear [OF g])
496 apply (simp add: norm_Pair)
497 apply (rule real_LIM_sandwich_zero)
498 apply (rule LIM_add_zero)
499 apply (rule FDERIV_D [OF f])
500 apply (rule FDERIV_D [OF g])
501 apply (rename_tac h)
502 apply (simp add: divide_nonneg_pos)
503 apply (rename_tac h)
504 apply (subst add_divide_distrib [symmetric])
505 apply (rule divide_right_mono [OF _ norm_ge_zero])